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% fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee 1 1 % fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee
% demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de 2 2 % demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de
% rejection par bit et perte si moins de bits que rejection/6 3 3 % rejection par bit et perte si moins de bits que rejection/6
% developper programme lineaire en incluant le decalage de bits 4 4 % developper programme lineaire en incluant le decalage de bits
% insister que avant on etait synthetisable mais pas implementable, alors que maintenant on 5 5 % insister que avant on etait synthetisable mais pas implementable, alors que maintenant on
% implemente et on demontre que ca tourne 6 6 % implemente et on demontre que ca tourne
% gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ? 7 7 % gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ?
% Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer 8 8 % Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer
% (zedboard ou redpit) 9 9 % (zedboard ou redpit)
10 10
% label schema : verifier que "argumenter de la cascade de FIR" est fait 11 11 % label schema : verifier que "argumenter de la cascade de FIR" est fait
12 12
\documentclass[a4paper,journal]{IEEEtran/IEEEtran} 13 13 \documentclass[a4paper,journal]{IEEEtran/IEEEtran}
\usepackage{graphicx,color,hyperref} 14 14 \usepackage{graphicx,color,hyperref}
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\usepackage[normalem]{ulem} 21 21 \usepackage[normalem]{ulem}
\usepackage{tikz} 22 22 \usepackage{tikz}
\usetikzlibrary{positioning,fit} 23 23 \usetikzlibrary{positioning,fit}
\usepackage{multirow} 24 24 \usepackage{multirow}
\usepackage{scalefnt} 25 25 \usepackage{scalefnt}
\usepackage{caption} 26 26 \usepackage{caption}
\usepackage{subcaption} 27 27 \usepackage{subcaption}
28 28
% correct bad hyphenation here 29 29 % correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor} 30 30 \hyphenation{op-tical net-works semi-conduc-tor}
\textheight=26cm 31 31 \textheight=26cm
\setlength{\footskip}{30pt} 32 32 \setlength{\footskip}{30pt}
\pagenumbering{gobble} 33 33 \pagenumbering{gobble}
\begin{document} 34 34 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 35 35 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 36 36 to oscillator metrology}
37 37
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 38 38 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 39 39 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\ 40 40 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\ 41 41 \IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\
\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\ 42 42 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 43 43 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 44 44 }
\maketitle 45 45 \maketitle
\thispagestyle{plain} 46 46 \thispagestyle{plain}
\pagestyle{plain} 47 47 \pagestyle{plain}
\newtheorem{definition}{Definition} 48 48 \newtheorem{definition}{Definition}
49 49
\begin{abstract} 50 50 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 51 51 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 52 52 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 53 53 of ultrastable clocks, stringent filtering requirements are defined by spurious signal or
noise rejection needs. Since real time radiofrequency processing must be performed in a 54 54 noise rejection needs. Since real time radiofrequency processing must be performed in a
Field Programmable Array to meet timing constraints, we investigate optimization strategies 55 55 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 56 56 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. The 57 57 required and keeping timing constraints within the targeted measurement bandwidths. The
presented technique is applicable to scheduling any sequence of processing blocks characterized 58 58 presented technique is applicable to scheduling any sequence of processing blocks characterized
by a throughput, resource occupation and performance tabulated as a function of configuration 59 59 by a throughput, resource occupation and performance tabulated as a function of configuration
characateristics, as is the case for filters with their coefficients and resolution yielding 60 60 characateristics, as is the case for filters with their coefficients and resolution yielding
rejection and number of multipliers. 61 61 rejection and number of multipliers.
\end{abstract} 62 62 \end{abstract}
63 63
\begin{IEEEkeywords} 64 64 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 65 65 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 66 66 \end{IEEEkeywords}
67 67
\section{Digital signal processing of ultrastable clock signals} 68 68 \section{Digital signal processing of ultrastable clock signals}
69 69
Analog oscillator phase noise characteristics are classically performed by downconverting 70 70 Analog oscillator phase noise characteristics are classically performed by downconverting
the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband, 71 71 the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,
followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In 72 72 followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In
a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by 73 73 a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by
multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}. 74 74 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
75 75
\begin{figure}[h!tb] 76 76 \begin{figure}[h!tb]
\begin{center} 77 77 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 78 78 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 79 79 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 80 80 \caption{Fully digital oscillator phase noise characterization: the Device Under Test
(DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and 81 81 (DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and
downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals 82 82 downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals
and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite 83 83 and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite
Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays 84 84 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 85 85 the spectral characteristics of the phase fluctuations.}
\label{schema} 86 86 \label{schema}
\end{figure} 87 87 \end{figure}
88 88
As with the analog mixer, 89 89 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 90 90 the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as
well as the generation of the frequency sum signal in addition to the frequency difference. 91 91 well as the generation of the frequency sum signal in addition to the frequency difference.
These unwanted spectral characteristics must be rejected before decimating the data stream 92 92 These unwanted spectral characteristics must be rejected before decimating the data stream
for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the 93 93 for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
downconverter 94 94 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 95 95 and the decimation processing blocks are core characteristics of an oscillator characterization
system, and must reject out-of-band signals below the targeted phase noise -- typically in the 96 96 system, and must reject out-of-band signals below the targeted phase noise -- typically in the
sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will 97 97 sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will
use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency 98 98 use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency
datastream: optimizing the performance of the filter while reducing the needed resources is 99 99 datastream: optimizing the performance of the filter while reducing the needed resources is
hence tackled in a systematic approach using optimization techniques. Most significantly, we 100 100 hence tackled in a systematic approach using optimization techniques. Most significantly, we
tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with 101 101 tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with
tunable number of coefficients and tunable number of bits representing the coefficients and the 102 102 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 103 103 data being processed.
104 104
\section{Finite impulse response filter} 105 105 \section{Finite impulse response filter}
106 106
We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined 107 107 We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined
by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the 108 108 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 109 109 outputs $y_k$
\begin{align} 110 110 \begin{align}
y_n=\sum_{k=0}^N b_k x_{n-k} 111 111 y_n=\sum_{k=0}^N b_k x_{n-k}
\label{eq:fir_equation} 112 112 \label{eq:fir_equation}
\end{align} 113 113 \end{align}
114 114
As opposed to an implementation on a general purpose processor in which word size is defined by the 115 115 As opposed to an implementation on a general purpose processor in which word size is defined by the
processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since 116 116 processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since
not only the coefficient values and number of taps must be defined, but also the number of bits 117 117 not only the coefficient values and number of taps must be defined, but also the number of bits
defining the coefficients and the sample size. For this reason, and because we consider pipeline 118 118 defining the coefficients and the sample size. For this reason, and because we consider pipeline
processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency 119 119 processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency
signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but 120 120 signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but
the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language 121 121 the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language
(VHDL) level. 122 122 (VHDL) level.
{\color{red}Since latency is not an issue in a openloop phase noise characterization instrument, 123 123 {\color{red}Since latency is not an issue in a openloop phase noise characterization instrument,
the large 124 124 the large
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 125 125 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
is not considered as an issue as would be in a closed loop system.} % r2.4 126 126 is not considered as an issue as would be in a closed loop system.} % r2.4
127 127
The coefficients are classically expressed as floating point values. However, this binary 128 128 The coefficients are classically expressed as floating point values. However, this binary
number representation is not efficient for fast arithmetic computation by an FPGA. Instead, 129 129 number representation is not efficient for fast arithmetic computation by an FPGA. Instead,
we select to quantify these floating point values into integer values. This quantization 130 130 we select to quantify these floating point values into integer values. This quantization
will result in some precision loss. 131 131 will result in some precision loss.
132 132
\begin{figure}[h!tb] 133 133 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/zero_values} 134 134 \includegraphics[width=\linewidth]{images/zero_values}
\caption{Impact of the quantization resolution of the coefficients: the quantization is 135 135 \caption{Impact of the quantization resolution of the coefficients: the quantization is
set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting 136 136 set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting
the 30~first and 30~last coefficients out of the initial 128~band-pass 137 137 the 30~first and 30~last coefficients out of the initial 128~band-pass
filter coefficients to 0 (red dots).} 138 138 filter coefficients to 0 (red dots).}
\label{float_vs_int} 139 139 \label{float_vs_int}
\end{figure} 140 140 \end{figure}
141 141
The tradeoff between quantization resolution and number of coefficients when considering 142 142 The tradeoff between quantization resolution and number of coefficients when considering
integer operations is not trivial. As an illustration of the issue related to the 143 143 integer operations is not trivial. As an illustration of the issue related to the
relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits 144 144 relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits
a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon 145 145 a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon
quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the 146 146 quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the
taps become null, {\color{red}making the large number of coefficients irrelevant: processing 147 147 taps become null, {\color{red}making the large number of coefficients irrelevant: processing
resources % r1.1 148 148 resources % r1.1
are hence saved by shrinking the filter length.} This tradeoff aimed at minimizing resources 149 149 are hence saved by shrinking the filter length.} This tradeoff aimed at minimizing resources
to reach a given rejection level, or maximizing out of band rejection for a given computational 150 150 to reach a given rejection level, or maximizing out of band rejection for a given computational
resource, will drive the investigation on cascading filters designed with varying tap resolution 151 151 resource, will drive the investigation on cascading filters designed with varying tap resolution
and tap length, as will be shown in the next section. Indeed, our development strategy closely 152 152 and tap length, as will be shown in the next section. Indeed, our development strategy closely
follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards} 153 153 follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
in which basic blocks are defined and characterized before being assembled \cite{hide} 154 154 in which basic blocks are defined and characterized before being assembled \cite{hide}
in a complete processing chain. In our case, assembling the filter blocks is a simpler block 155 155 in a complete processing chain. In our case, assembling the filter blocks is a simpler block
combination process since we assume a single value to be processed and a single value to be 156 156 combination process since we assume a single value to be processed and a single value to be
generated at each clock cycle. The FIR filters will not be considered to decimate in the 157 157 generated at each clock cycle. The FIR filters will not be considered to decimate in the
current implementation: the decimation is assumed to be located after the FIR cascade at the 158 158 current implementation: the decimation is assumed to be located after the FIR cascade at the
moment. 159 159 moment.
160 160
\section{Methodology description} 161 161 \section{Methodology description}
162 162
Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP) 163 163 Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP)
chain obtained by assembling basic processing blocks, with hardware and manufacturer independence. 164 164 chain obtained by assembling basic processing blocks, with hardware and manufacturer independence.
Achieving such a target requires defining an abstract model to represent some basic properties 165 165 Achieving such a target requires defining an abstract model to represent some basic properties
of DSP blocks such as perfomance (i.e. rejection or ripples in the bandpass for filters) and 166 166 of DSP blocks such as perfomance (i.e. rejection or ripples in the bandpass for filters) and
resource occupation. These abstract properties, not necessarily related to the detailed hardware 167 167 resource occupation. These abstract properties, not necessarily related to the detailed hardware
implementation of a given platform, will feed a scheduler solver aimed at assembling the optimum 168 168 implementation of a given platform, will feed a scheduler solver aimed at assembling the optimum
target, whether in terms of maximizing performance for a given arbitrary resource occupation, or 169 169 target, whether in terms of maximizing performance for a given arbitrary resource occupation, or
minimizing resource occupation for a given perfomance. In our approach, the solution of the 170 170 minimizing resource occupation for a given perfomance. In our approach, the solution of the
solver is then synthesized using the dedicated tool provided by each platform manufacturer 171 171 solver is then synthesized using the dedicated tool provided by each platform manufacturer
to assess the validity of our abstract resource occupation indicator, and the result of running 172 172 to assess the validity of our abstract resource occupation indicator, and the result of running
the DSP chain on the FPGA allows for assessing the performance of the scheduler. We emphasize 173 173 the DSP chain on the FPGA allows for assessing the performance of the scheduler. We emphasize
that all solutions found by the solver are synthesized and executed on hardware at the end 174 174 that all solutions found by the solver are synthesized and executed on hardware at the end
of the analysis. 175 175 of the analysis.
176 176
In this demonstration, we focus on only two operations: filtering and shifting the number of 177 177 In this demonstration, we focus on only two operations: filtering and shifting the number of
bits needed to represent the data along the processing chain. 178 178 bits needed to represent the data along the processing chain.
We have chosen these basic operations because shifting and the filtering have already been studied 179 179 We have chosen these basic operations because shifting and the filtering have already been studied
in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for 180 180 in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for
assessing our results. Furthermore, filtering is a core step in any radiofrequency frontend 181 181 assessing our results. Furthermore, filtering is a core step in any radiofrequency frontend
requiring pipelined processing at full bandwidth for the earliest steps, including for 182 182 requiring pipelined processing at full bandwidth for the earliest steps, including for
time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}. 183 183 time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}.
184 184
Addressing only two operations allows for demonstrating the methodology but should not be 185 185 Addressing only two operations allows for demonstrating the methodology but should not be
considered as a limitation of the framework which can be extended to assembling any number 186 186 considered as a limitation of the framework which can be extended to assembling any number
of skeleton blocks as long as perfomance and resource occupation can be determined. {\color{red} 187 187 of skeleton blocks as long as perfomance and resource occupation can be determined. {\color{red}
Hence, 188 188 Hence,
in this paper we will apply our methodology on simple DSP chains: a white noise input signal % r1.2 189 189 in this paper we will apply our methodology on simple DSP chains: a white noise input signal % r1.2
is generated using a Pseudo-Random Number (PRN) generator or by sampling a wideband (125~MS/s) 190 190 is generated using a Pseudo-Random Number (PRN) generator or by sampling a wideband (125~MS/s)
14-bit Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.} Once samples have been 191 191 14-bit Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.} Once samples have been
digitized at a rate of 125~MS/s, filtering is applied to qualify the processing block performance -- 192 192 digitized at a rate of 125~MS/s, filtering is applied to qualify the processing block performance --
practically meeting the radiofrequency frontend requirement of noise and bandwidth reduction 193 193 practically meeting the radiofrequency frontend requirement of noise and bandwidth reduction
by filtering and decimating. Finally, bursts of filtered samples are stored for post-processing, 194 194 by filtering and decimating. Finally, bursts of filtered samples are stored for post-processing,
allowing to assess either filter rejection for a given resource usage, or validating the rejection 195 195 allowing to assess either filter rejection for a given resource usage, or validating the rejection
when implementing a solution minimizing resource occupation. 196 196 when implementing a solution minimizing resource occupation.
197 197
{\color{red} 198 198 {\color{red}
The first step of our approach is to model the DSP chain. Since we aim at only optimizing % r1.3 199 199 The first step of our approach is to model the DSP chain. Since we aim at only optimizing % r1.3
the filtering part of the signal processing chain, we have not included the PRN generator or the 200 200 the filtering part of the signal processing chain, we have not included the PRN generator or the
ADC in the model: the input data size and rate are considered fixed and defined by the hardware. 201 201 ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
The filtering can be done in two ways, either by considering a single monolithic FIR filter 202 202 The filtering can be done in two ways, either by considering a single monolithic FIR filter
requiring many coefficients to reach the targeted noise rejection ratio, or by 203 203 requiring many coefficients to reach the targeted noise rejection ratio, or by
cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.} 204 204 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.}
205 205
After each filter we leave the possibility of shifting the filtered data to consume 206 206 After each filter we leave the possibility of shifting the filtered data to consume
less resources. Hence in the case of cascaded filter, we define a stage as a filter 207 207 less resources. Hence in the case of cascaded filter, we define a stage as a filter
and a shifter (the shift could be omitted if we do not need to divide the filtered data). 208 208 and a shifter (the shift could be omitted if we do not need to divide the filtered data).
209 209
\subsection{Model of a FIR filter} 210 210 \subsection{Model of a FIR filter}
211 211
A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$) 212 212 A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$)
the FIR has $C_i$ coefficients and each coefficient is an integer value with $\pi^C_i$ 213 213 the FIR has $C_i$ coefficients and each coefficient is an integer value with $\pi^C_i$
bits while the filtered data are shifted by $\pi^S_i$ bits. We define also $\pi^-_i$ as 214 214 bits while the filtered data are shifted by $\pi^S_i$ bits. We define also $\pi^-_i$ as
the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage} 215 215 the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage}
shows a filtering stage. 216 216 shows a filtering stage.
217 217
\begin{figure} 218 218 \begin{figure}
\centering 219 219 \centering
\begin{tikzpicture}[node distance=2cm] 220 220 \begin{tikzpicture}[node distance=2cm]
\node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ; 221 221 \node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;
\node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ; 222 222 \node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;
\node (Start) [left of=FIR] { } ; 223 223 \node (Start) [left of=FIR] { } ;
\node (End) [right of=Shift] { } ; 224 224 \node (End) [right of=Shift] { } ;
225 225
\node[draw,fit=(FIR) (Shift)] (Filter) { } ; 226 226 \node[draw,fit=(FIR) (Shift)] (Filter) { } ;
227 227
\draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ; 228 228 \draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;
\draw[->] (FIR) -- (Shift) ; 229 229 \draw[->] (FIR) -- (Shift) ;
\draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ; 230 230 \draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;
\end{tikzpicture} 231 231 \end{tikzpicture}
\caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)} 232 232 \caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}
\label{fig:fir_stage} 233 233 \label{fig:fir_stage}
\end{figure} 234 234 \end{figure}
235 235
FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB. 236 236 FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB.
This rejection has been computed using GNU Octave software FIR coefficient design functions 237 237 This rejection has been computed using GNU Octave software FIR coefficient design functions
(\texttt{firls} and \texttt{fir1}). 238 238 (\texttt{firls} and \texttt{fir1}).
For each configuration $(C_i, \pi_i^C)$, we first create a FIR with floating point coefficients and a given $C_i$ number of coefficients. 239 239 For each configuration $(C_i, \pi_i^C)$, we first create a FIR with floating point coefficients and a given $C_i$ number of coefficients.
Then, the floating point coefficients are discretized into integers. In order to ensure that the coefficients are coded on $\pi_i^C$~bits effectively, 240 240 Then, the floating point coefficients are discretized into integers. In order to ensure that the coefficients are coded on $\pi_i^C$~bits effectively,
the coefficients are normalized by their absolute maximum before being scaled to integer coefficients. 241 241 the coefficients are normalized by their absolute maximum before being scaled to integer coefficients.
At least one coefficient is coded on $\pi_i^C$~bits, and in practice only $b_{C_i/2}$ is coded on $\pi_i^C$~bits while the others are coded on much fewer bits. 242 242 At least one coefficient is coded on $\pi_i^C$~bits, and in practice only $b_{C_i/2}$ is coded on $\pi_i^C$~bits while the others are coded on much fewer bits.
243 243
With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter 244 244 With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter
transfer function. 245 245 transfer function.
Comparing the performance between FIRs requires however defining a unique criterion. As shown in figure~\ref{fig:fir_mag}, 246 246 Comparing the performance between FIRs requires however defining a unique criterion. As shown in figure~\ref{fig:fir_mag},
the FIR magnitude exhibits two parts: we focus here on the transitions width and the rejection rather than on the 247 247 the FIR magnitude exhibits two parts: we focus here on the transitions width and the rejection rather than on the
bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. {\color{red}Throughout this demonstration, 248 248 bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. {\color{red}Throughout this demonstration,
we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\% 249 249 we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%
of the Nyquist frequency to the end of the band, as would be typically selected to prevent 250 250 of the Nyquist frequency to the end of the band, as would be typically selected to prevent
aliasing before decimating the dataflow by 2. The method is however generalized to any filter 251 251 aliasing before decimating the dataflow by 2. The method is however generalized to any filter
shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid} 252 252 shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid}
as described below is indeed unique for each filter shape.} 253 253 as described below is indeed unique for each filter shape.}
254 254
\begin{figure} 255 255 \begin{figure}
\begin{center} 256 256 \begin{center}
\scalebox{0.8}{ 257 257 \scalebox{0.8}{
\centering 258 258 \centering
\begin{tikzpicture}[scale=0.3] 259 259 \begin{tikzpicture}[scale=0.3]
\draw[<->] (0,15) -- (0,0) -- (21,0) ; 260 260 \draw[<->] (0,15) -- (0,0) -- (21,0) ;
\draw[thick] (0,12) -- (8,12) -- (20,0) ; 261 261 \draw[thick] (0,12) -- (8,12) -- (20,0) ;
262 262
\draw (0,14) node [left] { $P$ } ; 263 263 \draw (0,14) node [left] { $P$ } ;
\draw (20,0) node [below] { $f$ } ; 264 264 \draw (20,0) node [below] { $f$ } ;
265 265
\draw[>=latex,<->] (0,14) -- (8,14) ; 266 266 \draw[>=latex,<->] (0,14) -- (8,14) ;
\draw (4,14) node [above] { passband } node [below] { $40\%$ } ; 267 267 \draw (4,14) node [above] { passband } node [below] { $40\%$ } ;
268 268
\draw[>=latex,<->] (8,14) -- (12,14) ; 269 269 \draw[>=latex,<->] (8,14) -- (12,14) ;
\draw (10,14) node [above] { transition } node [below] { $20\%$ } ; 270 270 \draw (10,14) node [above] { transition } node [below] { $20\%$ } ;
271 271
\draw[>=latex,<->] (12,14) -- (20,14) ; 272 272 \draw[>=latex,<->] (12,14) -- (20,14) ;
\draw (16,14) node [above] { stopband } node [below] { $40\%$ } ; 273 273 \draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;
274 274
\draw[>=latex,<->] (16,12) -- (16,8) ; 275 275 \draw[>=latex,<->] (16,12) -- (16,8) ;
\draw (16,10) node [right] { rejection } ; 276 276 \draw (16,10) node [right] { rejection } ;
277 277
\draw[dashed] (8,-1) -- (8,14) ; 278 278 \draw[dashed] (8,-1) -- (8,14) ;
\draw[dashed] (12,-1) -- (12,14) ; 279 279 \draw[dashed] (12,-1) -- (12,14) ;
280 280
\draw[dashed] (8,12) -- (16,12) ; 281 281 \draw[dashed] (8,12) -- (16,12) ;
\draw[dashed] (12,8) -- (16,8) ; 282 282 \draw[dashed] (12,8) -- (16,8) ;
283 283
\end{tikzpicture} 284 284 \end{tikzpicture}
} 285 285 }
\end{center} 286 286 \end{center}
\caption{Shape of the filter transmitted power $P$ as a function of frequency $f$: 287 287 \caption{Shape of the filter transmitted power $P$ as a function of frequency $f$:
the passband is considered to occupy the initial 40\% of the Nyquist frequency range, 288 288 the passband is considered to occupy the initial 40\% of the Nyquist frequency range,
the stopband the last 40\%, allowing 20\% transition width.} 289 289 the stopband the last 40\%, allowing 20\% transition width.}
\label{fig:fir_mag} 290 290 \label{fig:fir_mag}
\end{figure} 291 291 \end{figure}
292 292
In the transition band, the behavior of the filter is left free, we only {\color{red}define} the passband and the stopband characteristics. 293 293 In the transition band, the behavior of the filter is left free, we only {\color{red}define} the passband and the stopband characteristics.
% r2.7 294 294 % r2.7
% Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion 295 295 {\color{red}Initial considered criteria include the mean value of the stopband rejection which yields unacceptable results since notches
% yields unacceptable results since notches overestimate the rejection capability of the filter. Furthermore, the losses within 296 296 overestimate the rejection capability of the filter.}
297 % Furthermore, the losses within
% the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients. 297 298 % the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients.
Our criterion to compute the filter rejection considers 298 299 Our final criterion to compute the filter rejection considers
% r2.8 et r2.2 r2.3 299 300 % r2.8 et r2.2 r2.3
the {\color{red}minimal} rejection within the stopband, to which the {\color{red}sum of the absolute values 300 301 the {\color{red}minimal} rejection within the stopband, to which the {\color{red}sum of the absolute values
within the passband is subtracted to avoid filters with excessive ripples, normalized to the 301 302 within the passband is subtracted to avoid filters with excessive ripples, normalized to the
bin width to remain consistent with the passband criterion (dBc/Hz units in all cases)}. With this 302 303 bin width to remain consistent with the passband criterion (dBc/Hz units in all cases)}. With this
criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}. 303 304 criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
304 305
% \begin{figure} 305 306 % \begin{figure}
% \centering 306 307 % \centering
% \includegraphics[width=\linewidth]{images/colored_mean_criterion} 307 308 % \includegraphics[width=\linewidth]{images/colored_mean_criterion}
% \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters} 308 309 % \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}
% \label{fig:mean_criterion} 309 310 % \label{fig:mean_criterion}
% \end{figure} 310 311 % \end{figure}
311 312
\begin{figure} 312 313 \begin{figure}
\centering 313 314 \centering
\includegraphics[width=\linewidth]{images/colored_custom_criterion} 314 315 \includegraphics[width=\linewidth]{images/colored_custom_criterion}
\caption{Custom criterion (maximum rejection in the stopband minus the {\color{red} sum of the 315 316 \caption{Custom criterion (maximum rejection in the stopband minus the {\color{red} sum of the
absolute values of the passband rejection normalized to the bandwidth}) 316 317 absolute values of the passband rejection normalized to the bandwidth})
comparison between monolithic filter and cascaded filters} 317 318 comparison between monolithic filter and cascaded filters}
\label{fig:custom_criterion} 318 319 \label{fig:custom_criterion}
\end{figure} 319 320 \end{figure}
320 321
Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps 321 322 Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps
and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the 322 323 and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the
rejection as a function of the number of coefficients and the number of bits representing these coefficients. 323 324 rejection as a function of the number of coefficients and the number of bits representing these coefficients.
The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet. 324 325 The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet.
Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection. 325 326 Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection.
Conversely when setting the a given number of bits, increasing the number of coefficients will not improve 326 327 Conversely when setting the a given number of bits, increasing the number of coefficients will not improve
the rejection. Hence the best coefficient set are on the vertex of the pyramid. 327 328 the rejection. Hence the best coefficient set are on the vertex of the pyramid.
328 329
\begin{figure} 329 330 \begin{figure}
\centering 330 331 \centering
\includegraphics[width=\linewidth]{images/rejection_pyramid} 331 332 \includegraphics[width=\linewidth]{images/rejection_pyramid}
\caption{{\color{red}{Filter}} rejection as a function of number of coefficients and number of bits 332 333 \caption{{\color{red}{Filter}} rejection as a function of number of coefficients and number of bits
{\color{red}: this lookup table will be used to identify which filter parameters -- number of bits 333 334 {\color{red}: this lookup table will be used to identify which filter parameters -- number of bits
representing coefficients and number of coefficients -- best match the targeted transfer function.}} 334 335 representing coefficients and number of coefficients -- best match the targeted transfer function.}}
\label{fig:rejection_pyramid} 335 336 \label{fig:rejection_pyramid}
\end{figure} 336 337 \end{figure}
337 338
Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps), 338 339 Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps),
we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria. 339 340 we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria.
If the FIR filter coefficients are the same between the stages, we have: 340 341 If the FIR filter coefficients are the same between the stages, we have:
$$F_{total} = F_1 + F_2$$ 341 342 $$F_{total} = F_1 + F_2$$
But selecting two different sets of coefficient will yield a more complex situation in which 342 343 But selecting two different sets of coefficient will yield a more complex situation in which
the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves 343 344 the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves
are two different filters with maximums and notches not located at the same frequency offsets. 344 345 are two different filters with maximums and notches not located at the same frequency offsets.
Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved 345 346 Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved
with respect to a basic sum of the rejection criteria shown as a the dotted yellow line. 346 347 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
% r2.9 347 348 % r2.9
Thus, estimating the rejection of filter cascades is more complex than {\color{red}taking} the sum of all the rejection 348 349 Thus, estimating the rejection of filter cascades is more complex than {\color{red}taking} the sum of all the rejection
criteria of each filter. However since the {\color{red}individual filter rejection} sum underestimates the rejection capability of the cascade, 349 350 criteria of each filter. However since the {\color{red}individual filter rejection} sum underestimates the rejection capability of the cascade,
% r2.10 350 351 % r2.10
this upper bound is considered as a {\color{red}conservative} and acceptable criterion for deciding on the suitability 351 352 this upper bound is considered as a {\color{red}conservative} and acceptable criterion for deciding on the suitability
of the filter cascade to meet design criteria. 352 353 of the filter cascade to meet design criteria.
353 354
\begin{figure} 354 355 \begin{figure}
\centering 355 356 \centering
\includegraphics[width=\linewidth]{images/cascaded_criterion} 356 357 \includegraphics[width=\linewidth]{images/cascaded_criterion}
\caption{{\color{red}Transfer function of individual filters and after cascading} the two filters, 357 358 \caption{{\color{red}Transfer function of individual filters and after cascading} the two filters,
{\color{red}demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal 358 359 {\color{red}demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal
lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop 359 360 lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop
maximum of each individual filter.} 360 361 maximum of each individual filter.}
} 361 362 }
\label{fig:sum_rejection} 362 363 \label{fig:sum_rejection}
\end{figure} 363 364 \end{figure}
364 365
% r2.6 365 366 % r2.6
{\color{red} 366 367 {\color{red}
Finally in our case, we consider that the input signal are fully known. The 367 368 Finally in our case, we consider that the input signal are fully known. The
resolution of the input data stream are fixed and still the same for all experiments 368 369 resolution of the input data stream are fixed and still the same for all experiments
in this paper.} 369 370 in this paper.}
370 371
Based on this analysis, we address the estimate of resource consumption (called 371 372 Based on this analysis, we address the estimate of resource consumption (called
% r2.11 372 373 % r2.11
silicon area -- in the case of FPGAs {\color{red}this means} processing cells) as a function of 373 374 silicon area -- in the case of FPGAs {\color{red}this means} processing cells) as a function of
filter characteristics. As a reminder, we do not aim at matching actual hardware 374 375 filter characteristics. As a reminder, we do not aim at matching actual hardware
configuration but consider an arbitrary silicon area occupied by each processing function, 375 376 configuration but consider an arbitrary silicon area occupied by each processing function,
and will assess after synthesis the adequation of this arbitrary unit with actual 376 377 and will assess after synthesis the adequation of this arbitrary unit with actual
hardware resources provided by FPGA manufacturers. The sum of individual processing 377 378 hardware resources provided by FPGA manufacturers. The sum of individual processing
unit areas is constrained by a total silicon area representative of FPGA global resources. 378 379 unit areas is constrained by a total silicon area representative of FPGA global resources.
Formally, variable $a_i$ is the area taken by filter~$i$ 379 380 Formally, variable $a_i$ is the area taken by filter~$i$
(in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB). 380 381 (in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB).
Constant $\mathcal{A}$ is the total available area. We model our problem as follows: 381 382 Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
382 383
\begin{align} 383 384 \begin{align}
\text{Maximize } & \sum_{i=1}^n r_i \notag \\ 384 385 \text{Maximize } & \sum_{i=1}^n r_i \notag \\
\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\ 385 386 \sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\
a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\ 386 387 a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\
r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\ 387 388 r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\
\pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\ 388 389 \pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\
\pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\ 389 390 \pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\
\pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\ 390 391 \pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\
\pi_1^- &= \Pi^I \label{eq:init} 391 392 \pi_1^- &= \Pi^I \label{eq:init}
\end{align} 392 393 \end{align}
393 394
Equation~\ref{eq:area} states that the total area taken by the filters must be 394 395 Equation~\ref{eq:area} states that the total area taken by the filters must be
less than the available area. Equation~\ref{eq:areadef} gives the definition of 395 396 less than the available area. Equation~\ref{eq:areadef} gives the definition of
the area used by a filter, considered as the area of the FIR since the Shifter is 396 397 the area used by a filter, considered as the area of the FIR since the Shifter is
assumed not to require significant resources. We consider that the FIR needs $C_i$ registers of size 397 398 assumed not to require significant resources. We consider that the FIR needs $C_i$ registers of size
$\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the 398 399 $\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the
input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the 399 400 input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the
definition of the rejection of the filter thanks to the tabulated function~$F$ that we defined 400 401 definition of the rejection of the filter thanks to the tabulated function~$F$ that we defined
previously. The Shifter does not introduce negative rejection as we will explain later, 401 402 previously. The Shifter does not introduce negative rejection as we will explain later,
so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the 402 403 so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the
relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add 403 404 relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add
$\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes 404 405 $\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes
$\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of 405 406 $\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of
a filter is the same as the input number of bits of the next filter. 406 407 a filter is the same as the input number of bits of the next filter.
Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative 407 408 Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative
rejection. Indeed, the results of the FIR can be right shifted without compromising 408 409 rejection. Indeed, the results of the FIR can be right shifted without compromising
the quality of the rejection until a threshold. Each bit of the output data 409 410 the quality of the rejection until a threshold. Each bit of the output data
increases the maximum rejection level by 6~dB. We add one to take the sign bit 410 411 increases the maximum rejection level by 6~dB. We add one to take the sign bit
into account. If equation~\ref{eq:maxshift} was not present, the Shifter could 411 412 into account. If equation~\ref{eq:maxshift} was not present, the Shifter could
shift too much and introduce some noise in the output data. Each supplementary 412 413 shift too much and introduce some noise in the output data. Each supplementary
shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is: 413 414 shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is:
$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. 414 415 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
Finally, equation~\ref{eq:init} gives the number of bits of the global input. 415 416 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
416 417
{\color{red} 417 418 {\color{red}
This model is non-linear since we multiply some variable with another variable 418 419 This model is non-linear since we multiply some variable with another variable
and it is even non-quadratic, as the cost function $F$ does not have a known 419 420 and it is even non-quadratic, as the cost function $F$ does not have a known
linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations. 420 421 linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
% AH: conflit merge 421 422 % AH: conflit merge
% This variable must be defined by the user, it represent the number of different 422 423 % This variable must be defined by the user, it represent the number of different
% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1} 423 424 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
% functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid} 424 425 % functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid}
% to restrict the number of configurations. Indeed, it is useless to have too many coefficients or 425 426 % to restrict the number of configurations. Indeed, it is useless to have too many coefficients or
% too many bits, hence we take the configurations close to edge of pyramid. Thank to theses 426 427 % too many bits, hence we take the configurations close to edge of pyramid. Thank to theses
% configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant 427 428 % configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant
% and the function $F$ can be estimate for each configurations 428 429 % and the function $F$ can be estimate for each configurations
% thanks our rejection criterion. We also defined binary 429 430 % thanks our rejection criterion. We also defined binary
This variable $p$ is defined by the user, and represents the number of different 430 431 This variable $p$ is defined by the user, and represents the number of different
set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1} 431 432 set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave) based on the targeted filter characteristics and implementation 432 433 functions from GNU Octave) based on the targeted filter characteristics and implementation
assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and 433 434 assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and
$\pi_{ij}^C$ become constants and 434 435 $\pi_{ij}^C$ become constants and
we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table) 435 436 we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table)
for each configurations thanks to the rejection criterion. We also define the binary 436 437 for each configurations thanks to the rejection criterion. We also define the binary
variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 437 438 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
and 0 otherwise. The new equations are as follows: 438 439 and 0 otherwise. The new equations are as follows:
} 439 440 }
440 441
\begin{align} 441 442 \begin{align}
a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\ 442 443 a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\
r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\ 443 444 r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
\pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\ 444 445 \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\
\sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} 445 446 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
\end{align} 446 447 \end{align}
447 448
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 448 449 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 449 450 respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most. 450 451 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
451 452
{\color{red} 452 453 {\color{red}
% JM: conflict merge 453 454 % JM: conflict merge
% However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2} 454 455 % However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
% we multiply 455 456 % we multiply
% $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can 456 457 % $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
% linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size, 457 458 % linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size,
% we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is 458 459 % we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is
% assumed on hardware characteristics. 459 460 % assumed on hardware characteristics.
% The Gurobi (\url{www.gurobi.com}) optimization software used to solve this quadratic 460 461 % The Gurobi (\url{www.gurobi.com}) optimization software used to solve this quadratic
% model is able to linearize the model provided as is. This model 461 462 % model is able to linearize the model provided as is. This model
% has $O(np)$ variables and $O(n)$ constraints.} 462 463 % has $O(np)$ variables and $O(n)$ constraints.}
However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2} 463 464 The problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
we multiply 464 465 we multiply
$\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can 465 466 $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
linearise linearize this multiplication. The following formula shows how to linearize 466 467 linearise linearize this multiplication. The following formula shows how to linearize
this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$): 467 468 this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):
\begin{equation*} 468 469 \begin{equation*}
m = x \times y \implies 469 470 m = x \times y \implies
\left \{ 470 471 \left \{
\begin{split} 471 472 \begin{split}
m & \geq 0 \\ 472 473 m & \geq 0 \\
m & \leq y \times X^{max} \\ 473 474 m & \leq y \times X^{max} \\
m & \leq x \\ 474 475 m & \leq x \\
m & \geq x - (1 - y) \times X^{max} \\ 475 476 m & \geq x - (1 - y) \times X^{max} \\
\end{split} 476 477 \end{split}
\right . 477 478 \right .
\end{equation*} 478 479 \end{equation*}
So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is 479 480 So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is
assumed on hardware characteristics, 480 481 assumed on hardware characteristics,
the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize 481 482 the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize
for us the quadratic problem so the model is left as is. This model 482 483 for us the quadratic problem so the model is left as is. This model
has $O(np)$ variables and $O(n)$ constraints.} 483 484 has $O(np)$ variables and $O(n)$ constraints.}
484 485
% This model is non-linear and even non-quadratic, as $F$ does not have a known 485 486 % This model is non-linear and even non-quadratic, as $F$ does not have a known
% linear or quadratic expression. We introduce $p$ FIR configurations 486 487 % linear or quadratic expression. We introduce $p$ FIR configurations
% $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 487 488 % $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
% % r2.12 488 489 % % r2.12
% This variable must be defined by the user, it represent the number of different 489 490 % This variable must be defined by the user, it represent the number of different
% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1} 490 491 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
% functions from GNU Octave). 491 492 % functions from GNU Octave).
% We define binary 492 493 % We define binary
% variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 493 494 % variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
% and 0 otherwise. The new equations are as follows: 494 495 % and 0 otherwise. The new equations are as follows:
% 495 496 %
% \begin{align} 496 497 % \begin{align}
% a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\ 497 498 % a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\
% r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\ 498 499 % r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
% \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\ 499 500 % \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\
% \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} 500 501 % \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
% \end{align} 501 502 % \end{align}
% 502 503 %
% Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 503 504 % Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
% respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 504 505 % respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
% Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most. 505 506 % Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
% 506 507 %
% % r2.13 507 508 % % r2.13
% This modified model is quadratic since we multiply two variables in the 508 509 % This modified model is quadratic since we multiply two variables in the
% equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary. 509 510 % equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
% The Gurobi 510 511 % The Gurobi
% (\url{www.gurobi.com}) optimization software is used to solve this quadratic 511 512 % (\url{www.gurobi.com}) optimization software is used to solve this quadratic
% model, and since Gurobi is able to linearize, the model is left as is. This model 512 513 % model, and since Gurobi is able to linearize, the model is left as is. This model
% has $O(np)$ variables and $O(n)$ constraints. 513 514 % has $O(np)$ variables and $O(n)$ constraints.
514 515
Two problems will be addressed using the workflow described in the next section: on the one 515 516 Two problems will be addressed using the workflow described in the next section: on the one
hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary 516 517 hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary
silcon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area 517 518 silcon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area
for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the 518 519 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
objective function is replaced with: 519 520 objective function is replaced with:
\begin{align} 520 521 \begin{align}
\text{Minimize } & \sum_{i=1}^n a_i \notag 521 522 \text{Minimize } & \sum_{i=1}^n a_i \notag
\end{align} 522 523 \end{align}
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 523 524 We adapt our constraints of quadratic program to replace equation \ref{eq:area}
with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal 524 525 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
rejection required. 525 526 rejection required.
526 527
\begin{align} 527 528 \begin{align}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 528 529 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
\end{align} 529 530 \end{align}
530 531
\section{Design workflow} 531 532 \section{Design workflow}
\label{sec:workflow} 532 533 \label{sec:workflow}
533 534
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 534 535 In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area}
and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved 535 536 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
in the computation of the results. 536 537 in the computation of the results.
537 538
\begin{figure} 538 539 \begin{figure}
\centering 539 540 \centering
\begin{tikzpicture}[node distance=0.75cm and 2cm] 540 541 \begin{tikzpicture}[node distance=0.75cm and 2cm]
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 541 542 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\node (Start) [left= 3cm of Solver] { } ; 542 543 \node (Start) [left= 3cm of Solver] { } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 543 544 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\node (Input) [above= of TCL] { } ; 544 545 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 545 546 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 546 547 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 547 548 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 548 549 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node (Results) [left= of Postproc] { } ; 549 550 \node (Results) [left= of Postproc] { } ;
550 551
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 551 552 \draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ;
\draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ; 552 553 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 553 554 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 554 555 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 555 556 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 556 557 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 557 558 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 558 559 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 559 560 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->] (Postproc) -- (Results) ; 560 561 \draw[->] (Postproc) -- (Results) ;
\end{tikzpicture} 561 562 \end{tikzpicture}
\caption{Design workflow from the input parameters to the results {\color{red} allowing for 562 563 \caption{Design workflow from the input parameters to the results {\color{red} allowing for
a fully automated optimal solution search.}} 563 564 a fully automated optimal solution search.}}
\label{fig:workflow} 564 565 \label{fig:workflow}
\end{figure} 565 566 \end{figure}
566 567
The filter solver is a C++ program that takes as input the maximum area 567 568 The filter solver is a C++ program that takes as input the maximum area
$\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$, 568 569 $\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$,
the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates 569 570 the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates
the quadratic programs and uses the Gurobi solver to estimate the optimal results. 570 571 the quadratic programs and uses the Gurobi solver to estimate the optimal results.
Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow}) 571 572 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
and a deploy script ((1b) on figure~\ref{fig:workflow}). 572 573 and a deploy script ((1b) on figure~\ref{fig:workflow}).
573 574
The TCL script describes the whole digital processing chain from the beginning 574 575 The TCL script describes the whole digital processing chain from the beginning
(the raw signal data) to the end (the filtered data) in a language compatible 575 576 (the raw signal data) to the end (the filtered data) in a language compatible
with proprietary synthesis software, namely Vivado for Xilinx and Quartus for 576 577 with proprietary synthesis software, namely Vivado for Xilinx and Quartus for
Intel/Altera. The raw input data generated from a 20-bit Pseudo Random Number (PRN) 577 578 Intel/Altera. The raw input data generated from a 20-bit Pseudo Random Number (PRN)
generator inside the FPGA and $\Pi^I$ is fixed at 16~bits. 578 579 generator inside the FPGA and $\Pi^I$ is fixed at 16~bits.
Then the script builds each stage of the chain with a generic FIR task that 579 580 Then the script builds each stage of the chain with a generic FIR task that
comes from a skeleton library. The generic FIR is highly configurable 580 581 comes from a skeleton library. The generic FIR is highly configurable
with the number of coefficients and the size of the coefficients. The coefficients 581 582 with the number of coefficients and the size of the coefficients. The coefficients
themselves are not stored in the script. 582 583 themselves are not stored in the script.
As the signal is processed in real-time, the output signal is stored as 583 584 As the signal is processed in real-time, the output signal is stored as
consecutive bursts of data for post-processing, mainly assessing the consistency of the 584 585 consecutive bursts of data for post-processing, mainly assessing the consistency of the
implemented FIR cascade transfer function with the design criteria and the expected 585 586 implemented FIR cascade transfer function with the design criteria and the expected
transfer function. 586 587 transfer function.
587 588
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 588 589 The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}).
We use the 2018.2 version of Xilinx Vivado and we execute the synthesized 589 590 We use the 2018.2 version of Xilinx Vivado and we execute the synthesized
bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series 590 591 bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series
FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to 591 592 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
provide a broadband noise source. 592 593 provide a broadband noise source.
The board runs the Linux kernel and surrounding environment produced from the 593 594 The board runs the Linux kernel and surrounding environment produced from the
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 594 595 Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring
the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and 595 596 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
fetching the results is automated. 596 597 fetching the results is automated.
597 598
The deploy script uploads the bitstream to the board ((3) on 598 599 The deploy script uploads the bitstream to the board ((3) on
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 599 600 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
configures the coefficients of the FIR filters. It then waits for the results 600 601 configures the coefficients of the FIR filters. It then waits for the results
and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}). 601 602 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
602 603
Finally, an Octave post-processing script computes the final results thanks to 603 604 Finally, an Octave post-processing script computes the final results thanks to
the output data ((5) on figure~\ref{fig:workflow}). 604 605 the output data ((5) on figure~\ref{fig:workflow}).
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 605 606 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
and the different configurations can be compared. 606 607 and the different configurations can be compared.
607 608
\section{Maximizing the rejection at fixed silicon area} 608 609 \section{Maximizing the rejection at fixed silicon area}
\label{sec:fixed_area} 609 610 \label{sec:fixed_area}
This section presents the output of the filter solver {\em i.e.} the computed 610 611 This section presents the output of the filter solver {\em i.e.} the computed
configurations for each stage, the computed rejection and the computed silicon area. 611 612 configurations for each stage, the computed rejection and the computed silicon area.
Such results allow for understanding the choices made by the solver to compute its solutions. 612 613 Such results allow for understanding the choices made by the solver to compute its solutions.
613 614
The experimental setup is composed of three cases. The raw input is generated 614 615 The experimental setup is composed of three cases. The raw input is generated
by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$. 615 616 by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$.
Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500 616 617 Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500
arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500. 617 618 arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500.
The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$ 618 619 The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$
ranging from 2 to 22. In each case, the quadratic program has been able to give a 619 620 ranging from 2 to 22. In each case, the quadratic program has been able to give a
result up to five stages ($n = 5$) in the cascaded filter. 620 621 result up to five stages ($n = 5$) in the cascaded filter.
621 622
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 622 623 Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500.
Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000. 623 624 Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000.
Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500. 624 625 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
625 626
\renewcommand{\arraystretch}{1.4} 626 627 \renewcommand{\arraystretch}{1.4}
627 628
\begin{table} 628 629 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 629 630 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
\label{tbl:gurobi_max_500} 630 631 \label{tbl:gurobi_max_500}
\centering 631 632 \centering
{\scalefont{0.77} 632 633 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 633 634 \begin{tabular}{|c|ccccc|c|c|}
\hline 634 635 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 635 636 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 636 637 \hline
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 637 638 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ 638 639 2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\
3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ 639 640 3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\
4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\ 640 641 4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\
5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\ 641 642 5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\
\hline 642 643 \hline
\end{tabular} 643 644 \end{tabular}
} 644 645 }
\end{table} 645 646 \end{table}
646 647
\begin{table} 647 648 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 648 649 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
\label{tbl:gurobi_max_1000} 649 650 \label{tbl:gurobi_max_1000}
\centering 650 651 \centering
{\scalefont{0.77} 651 652 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 652 653 \begin{tabular}{|c|ccccc|c|c|}
\hline 653 654 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 654 655 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 655 656 \hline
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 656 657 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ 657 658 2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\
3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ 658 659 3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\
4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\ 659 660 4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\
5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\ 660 661 5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\
\hline 661 662 \hline
\end{tabular} 662 663 \end{tabular}
} 663 664 }
\end{table} 664 665 \end{table}
665 666
\begin{table} 666 667 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 667 668 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
\label{tbl:gurobi_max_1500} 668 669 \label{tbl:gurobi_max_1500}
\centering 669 670 \centering
{\scalefont{0.77} 670 671 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 671 672 \begin{tabular}{|c|ccccc|c|c|}
\hline 672 673 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 673 674 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 674 675 \hline
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 675 676 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ 676 677 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\
3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ 677 678 3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\
4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\ 678 679 4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\
5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\ 679 680 5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\
\hline 680 681 \hline
\end{tabular} 681 682 \end{tabular}
} 682 683 }
\end{table} 683 684 \end{table}
684 685
\renewcommand{\arraystretch}{1} 685 686 \renewcommand{\arraystretch}{1}
686 687
From these tables, we can first state that the more stages are used to define 687 688 From these tables, we can first state that the more stages are used to define
the cascaded FIR filters, the better the rejection. It was an expected result as it has 688 689 the cascaded FIR filters, the better the rejection. It was an expected result as it has
been previously observed that many small filters are better than 689 690 been previously observed that many small filters are better than
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 690 691 a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions
being hardly used in practice due to the lack of tools for identifying individual filter 691 692 being hardly used in practice due to the lack of tools for identifying individual filter
coefficients in the cascaded approach. 692 693 coefficients in the cascaded approach.
693 694
Second, the larger the silicon area, the better the rejection. This was also an 694 695 Second, the larger the silicon area, the better the rejection. This was also an
expected result as more area means a filter of better quality with more coefficients 695 696 expected result as more area means a filter of better quality with more coefficients
or more bits per coefficient. 696 697 or more bits per coefficient.
697 698
Then, we also observe that the first stage can have a larger shift than the other 698 699 Then, we also observe that the first stage can have a larger shift than the other
stages. This is explained by the fact that the solver tries to use just enough 699 700 stages. This is explained by the fact that the solver tries to use just enough
bits for the computed rejection after each stage. In the first stage, a 700 701 bits for the computed rejection after each stage. In the first stage, a
balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift} 701 702 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
gives the relation between both values. 702 703 gives the relation between both values.
703 704
Finally, we note that the solver consumes all the given silicon area. 704 705 Finally, we note that the solver consumes all the given silicon area.
705 706
The following graphs present the rejection for real data on the FPGA. In all the following 706 707 The following graphs present the rejection for real data on the FPGA. In all the following
figures, the solid line represents the actual rejection of the filtered 707 708 figures, the solid line represents the actual rejection of the filtered
data on the FPGA as measured experimentally and the dashed line are the noise levels 708 709 data on the FPGA as measured experimentally and the dashed line are the noise levels
given by the quadratic solver. The configurations are those computed in the previous section. 709 710 given by the quadratic solver. The configurations are those computed in the previous section.
710 711
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 711 712 Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500.
Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000. 712 713 Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000.
Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500. 713 714 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
714 715
% \begin{figure} 715 716 % \begin{figure}
% \centering 716 717 % \centering
% \includegraphics[width=\linewidth]{images/max_500} 717 718 % \includegraphics[width=\linewidth]{images/max_500}
% \caption{Signal spectrum for MAX/500} 718 719 % \caption{Signal spectrum for MAX/500}
% \label{fig:max_500_result} 719 720 % \label{fig:max_500_result}
% \end{figure} 720 721 % \end{figure}
% 721 722 %
% \begin{figure} 722 723 % \begin{figure}
% \centering 723 724 % \centering
% \includegraphics[width=\linewidth]{images/max_1000} 724 725 % \includegraphics[width=\linewidth]{images/max_1000}
% \caption{Signal spectrum for MAX/1000} 725 726 % \caption{Signal spectrum for MAX/1000}
% \label{fig:max_1000_result} 726 727 % \label{fig:max_1000_result}
% \end{figure} 727 728 % \end{figure}
% 728 729 %
% \begin{figure} 729 730 % \begin{figure}
% \centering 730 731 % \centering
% \includegraphics[width=\linewidth]{images/max_1500} 731 732 % \includegraphics[width=\linewidth]{images/max_1500}
% \caption{Signal spectrum for MAX/1500} 732 733 % \caption{Signal spectrum for MAX/1500}
% \label{fig:max_1500_result} 733 734 % \label{fig:max_1500_result}
% \end{figure} 734 735 % \end{figure}
735 736
% r2.14 et r2.15 et r2.16 736 737 % r2.14 et r2.15 et r2.16
\begin{figure} 737 738 \begin{figure}
\centering 738 739 \centering
\begin{subfigure}{\linewidth} 739 740 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_500} 740 741 \includegraphics[width=\linewidth]{images/max_500}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 741 742 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).} 742 743 the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).}
\label{fig:max_500_result} 743 744 \label{fig:max_500_result}
\end{subfigure} 744 745 \end{subfigure}
745 746
\begin{subfigure}{\linewidth} 746 747 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1000} 747 748 \includegraphics[width=\linewidth]{images/max_1000}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 748 749 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).} 749 750 the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).}
\label{fig:max_1000_result} 750 751 \label{fig:max_1000_result}
\end{subfigure} 751 752 \end{subfigure}
752 753
\begin{subfigure}{\linewidth} 753 754 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1500} 754 755 \includegraphics[width=\linewidth]{images/max_1500}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 755 756 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).} 756 757 the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).}
\label{fig:max_1500_result} 757 758 \label{fig:max_1500_result}
\end{subfigure} 758 759 \end{subfigure}
\caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing 759 760 \caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing
rejection for a given resource allocation. 760 761 rejection for a given resource allocation.
The filter shape constraint (bandpass and bandstop) is shown as thick 761 762 The filter shape constraint (bandpass and bandstop) is shown as thick
horizontal lines on each chart.} 762 763 horizontal lines on each chart.}
\end{figure} 763 764 \end{figure}
764 765
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 765 766 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
766 767
We compare the actual silicon resources given by Vivado to the 767 768 We compare the actual silicon resources given by Vivado to the
resources in arbitrary units. 768 769 resources in arbitrary units.
The goal is to check that our arbitrary units of silicon area models well enough 769 770 The goal is to check that our arbitrary units of silicon area models well enough
the real resources on the FPGA. Especially we want to verify that, for a given 770 771 the real resources on the FPGA. Especially we want to verify that, for a given
number of arbitrary units, the actual silicon resources do not depend on the 771 772 number of arbitrary units, the actual silicon resources do not depend on the
number of stages $n$. Most significantly, our approach aims 772 773 number of stages $n$. Most significantly, our approach aims
at remaining far enough from the practical logic gate implementation used by 773 774 at remaining far enough from the practical logic gate implementation used by
various vendors to remain platform independent and be portable from one 774 775 various vendors to remain platform independent and be portable from one
architecture to another. 775 776 architecture to another.
776 777
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 777 778 Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and
MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000 778 779 MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000
and 1500 arbitrary units. We have taken care to extract solely the resources used by 779 780 and 1500 arbitrary units. We have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and Programmable 780 781 the FIR filters and remove additional processing blocks including FIFO and Programmable
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 781 782 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
782 783
\begin{table}[h!tb] 783 784 \begin{table}[h!tb]
\caption{Resource occupation {\color{red}following synthesis of the solutions found for 784 785 \caption{Resource occupation {\color{red}following synthesis of the solutions found for
the problem of maximizing rejection for a given resource allocation}. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 785 786 the problem of maximizing rejection for a given resource allocation}. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage} 786 787 \label{tbl:resources_usage}
\centering 787 788 \centering
\begin{tabular}{|c|c|ccc|c|} 788 789 \begin{tabular}{|c|c|ccc|c|}
\hline 789 790 \hline
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 790 791 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 249 & 453 & 627 & \emph{17600} \\ 791 792 & LUT & 249 & 453 & 627 & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 792 793 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 793 794 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
& LUT & 2374 & 5494 & 691 & \emph{17600} \\ 794 795 & LUT & 2374 & 5494 & 691 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 795 796 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 70 & \emph{80} \\ \hline 796 797 & DSP & 0 & 0 & 70 & \emph{80} \\ \hline
& LUT & 2443 & 3304 & 3521 & \emph{17600} \\ 797 798 & LUT & 2443 & 3304 & 3521 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 798 799 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 19 & 35 & \emph{80} \\ \hline 799 800 & DSP & 0 & 19 & 35 & \emph{80} \\ \hline
& LUT & 2634 & 3753 & 2557 & \emph{17600} \\ 800 801 & LUT & 2634 & 3753 & 2557 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & \emph{120} \\ 801 802 4 & BRAM & 4 & 4 & 4 & \emph{120} \\
& DPS & 0 & 19 & 46 & \emph{80} \\ \hline 802 803 & DPS & 0 & 19 & 46 & \emph{80} \\ \hline
& LUT & 2423 & 3047 & 2847 & \emph{17600} \\ 803 804 & LUT & 2423 & 3047 & 2847 & \emph{17600} \\
5 & BRAM & 5 & 5 & 5 & \emph{120} \\ 804 805 5 & BRAM & 5 & 5 & 5 & \emph{120} \\
& DPS & 0 & 22 & 46 & \emph{80} \\ \hline 805 806 & DPS & 0 & 22 & 46 & \emph{80} \\ \hline
\end{tabular} 806 807 \end{tabular}
\end{table} 807 808 \end{table}
808 809
In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 809 810 In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that,
when the filter coefficients are small enough, or when the input size is small 810 811 when the filter coefficients are small enough, or when the input size is small
enough, Vivado optimizes resource consumption by selecting multiplexers to 811 812 enough, Vivado optimizes resource consumption by selecting multiplexers to
implement the multiplications instead of a DSP. In this case, it is quite difficult 812 813 implement the multiplications instead of a DSP. In this case, it is quite difficult
to compare the whole silicon budget. 813 814 to compare the whole silicon budget.
814 815
However, a rough estimation can be made with a simple equivalence: looking at 815 816 However, a rough estimation can be made with a simple equivalence: looking at
the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$, 816 817 the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$,
we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon 817 818 we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon
area use. With this equivalence, our 500 arbitraty units correspond to 2500 LUTs, 818 819 area use. With this equivalence, our 500 arbitraty units correspond to 2500 LUTs,
1000 arbitrary units correspond to 5000 LUTs and 1500 arbitrary units correspond 819 820 1000 arbitrary units correspond to 5000 LUTs and 1500 arbitrary units correspond
to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary 820 821 to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary
unit map well to actual hardware resources. The relatively small differences can probably be explained 821 822 unit map well to actual hardware resources. The relatively small differences can probably be explained
by the optimizations done by Vivado based on the detailed map of available processing resources. 822 823 by the optimizations done by Vivado based on the detailed map of available processing resources.
823 824
We now present the computation time needed to solve the quadratic problem. 824 825 We now present the computation time needed to solve the quadratic problem.
For each case, the filter solver software is executed on a Intel(R) Xeon(R) CPU E5606 825 826 For each case, the filter solver software is executed on a Intel(R) Xeon(R) CPU E5606
clocked at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve 826 827 clocked at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve
the quadratic problem. Table~\ref{tbl:area_time} shows the time needed to solve the quadratic 827 828 the quadratic problem. Table~\ref{tbl:area_time} shows the time needed to solve the quadratic
problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units. 828 829 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
829 830
\begin{table}[h!tb] 830 831 \begin{table}[h!tb]
\caption{Time needed to solve the quadratic program with Gurobi} 831 832 \caption{Time needed to solve the quadratic program with Gurobi}
\label{tbl:area_time} 832 833 \label{tbl:area_time}
\centering 833 834 \centering
\begin{tabular}{|c|c|c|c|}\hline 834 835 \begin{tabular}{|c|c|c|c|}\hline
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 835 836 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
1 & 0.1~s & 0.1~s & 0.3~s \\ 836 837 1 & 0.1~s & 0.1~s & 0.3~s \\
2 & 1.1~s & 2.2~s & 12~s \\ 837 838 2 & 1.1~s & 2.2~s & 12~s \\
3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ 838 839 3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\
4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\ 839 840 4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\
5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline 840 841 5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline
\end{tabular} 841 842 \end{tabular}
\end{table} 842 843 \end{table}
843 844
As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? 844 845 As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ?
When the area is limited, the design exploration space is more limited and the solver is able to 845 846 When the area is limited, the design exploration space is more limited and the solver is able to
find an optimal solution faster. 846 847 find an optimal solution faster.
847 848
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 848 849 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
849 850
This section presents the results of the complementary quadratic program aimed at 850 851 This section presents the results of the complementary quadratic program aimed at
minimizing the area occupation for a targeted rejection level. 851 852 minimizing the area occupation for a targeted rejection level.
852 853
The experimental setup is composed of four cases. The raw input is the same 853 854 The experimental setup is composed of four cases. The raw input is the same
as in the previous section, from a PRN generator, which fixes the input data size $\Pi^I$. 854 855 as in the previous section, from a PRN generator, which fixes the input data size $\Pi^I$.
Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB. 855 856 Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB.
Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100. 856 857 Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100.
The number of configurations $p$ is the same as previous section. 857 858 The number of configurations $p$ is the same as previous section.
858 859
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 859 860 Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40.
Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60. 860 861 Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60.
Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80. 861 862 Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80.
Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100. 862 863 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
863 864
\renewcommand{\arraystretch}{1.4} 864 865 \renewcommand{\arraystretch}{1.4}
865 866
\begin{table}[h!tb] 866 867 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 867 868 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\label{tbl:gurobi_min_40} 868 869 \label{tbl:gurobi_min_40}
\centering 869 870 \centering
{\scalefont{0.77} 870 871 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 871 872 \begin{tabular}{|c|ccccc|c|c|}
\hline 872 873 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 873 874 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 874 875 \hline
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 875 876 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ 876 877 2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\
3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ 877 878 3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\
4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\ 878 879 4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\
\hline 879 880 \hline
\end{tabular} 880 881 \end{tabular}
} 881 882 }
\end{table} 882 883 \end{table}
883 884
\begin{table}[h!tb] 884 885 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 885 886 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\label{tbl:gurobi_min_60} 886 887 \label{tbl:gurobi_min_60}
\centering 887 888 \centering
{\scalefont{0.77} 888 889 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 889 890 \begin{tabular}{|c|ccccc|c|c|}
\hline 890 891 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 891 892 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 892 893 \hline
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 893 894 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ 894 895 2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\
3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ 895 896 3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\
4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\ 896 897 4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\
5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\ 897 898 5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\
\hline 898 899 \hline
\end{tabular} 899 900 \end{tabular}
} 900 901 }
\end{table} 901 902 \end{table}
902 903
\begin{table}[h!tb] 903 904 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 904 905 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\label{tbl:gurobi_min_80} 905 906 \label{tbl:gurobi_min_80}
\centering 906 907 \centering
{\scalefont{0.77} 907 908 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 908 909 \begin{tabular}{|c|ccccc|c|c|}
\hline 909 910 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 910 911 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 911 912 \hline
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 912 913 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ 913 914 2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\
3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ 914 915 3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\
4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\ 915 916 4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\
5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\ 916 917 5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\
\hline 917 918 \hline
\end{tabular} 918 919 \end{tabular}
} 919 920 }
\end{table} 920 921 \end{table}
921 922
\begin{table}[h!tb] 922 923 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 923 924 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\label{tbl:gurobi_min_100} 924 925 \label{tbl:gurobi_min_100}
\centering 925 926 \centering
{\scalefont{0.77} 926 927 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 927 928 \begin{tabular}{|c|ccccc|c|c|}
\hline 928 929 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 929 930 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 930 931 \hline
1 & - & - & - & - & - & - & - \\ 931 932 1 & - & - & - & - & - & - & - \\
2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\ 932 933 2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\
3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\ 933 934 3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\
4 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 0) & - & 101~dB & 909 \\ 934 935 4 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 0) & - & 101~dB & 909 \\
5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\ 935 936 5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\
\hline 936 937 \hline
\end{tabular} 937 938 \end{tabular}
} 938 939 }
\end{table} 939 940 \end{table}
\renewcommand{\arraystretch}{1} 940 941 \renewcommand{\arraystretch}{1}
941 942
From these tables, we can first state that almost all configurations reach the targeted rejection 942 943 From these tables, we can first state that almost all configurations reach the targeted rejection
level or even better thanks to our underestimate of the cascade rejection as the sum of the 943 944 level or even better thanks to our underestimate of the cascade rejection as the sum of the
individual filter rejection. The only exception is for the monolithic case ($n = 1$) in 944 945 individual filter rejection. The only exception is for the monolithic case ($n = 1$) in
MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection. 945 946 MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection.
Futhermore, the area of the monolithic filter is twice as big as the two cascaded filters 946 947 Futhermore, the area of the monolithic filter is twice as big as the two cascaded filters
(1131 and 1760 arbitrary units v.s 547 and 903 arbitrary units for 60 and 80~dB rejection 947 948 (1131 and 1760 arbitrary units v.s 547 and 903 arbitrary units for 60 and 80~dB rejection
respectively). More generally, the more filters are cascaded, the lower the occupied area. 948 949 respectively). More generally, the more filters are cascaded, the lower the occupied area.
949 950
Like in previous section, the solver chooses always a little filter as first 950 951 Like in previous section, the solver chooses always a little filter as first
filter stage and the second one is often the biggest filter. This choice can be explained 951 952 filter stage and the second one is often the biggest filter. This choice can be explained
as in the previous section, with the solver using just enough bits not to degrade the input 952 953 as in the previous section, with the solver using just enough bits not to degrade the input
signal and in the second filter selecting a better filter to improve rejection without 953 954 signal and in the second filter selecting a better filter to improve rejection without
having too many bits in the output data. 954 955 having too many bits in the output data.
955 956
For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 956 957 For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal
number of filters is 4 so it did not chose any configuration for the last filter. Hence this 957 958 number of filters is 4 so it did not chose any configuration for the last filter. Hence this
solution is equivalent to the result for $n = 4$. 958 959 solution is equivalent to the result for $n = 4$.
959 960
The following graphs present the rejection for real data on the FPGA. In all the following 960 961 The following graphs present the rejection for real data on the FPGA. In all the following
figures, the solid line represents the actual rejection of the filtered 961 962 figures, the solid line represents the actual rejection of the filtered
data on the FPGA as measured experimentally and the dashed line is the noise level 962 963 data on the FPGA as measured experimentally and the dashed line is the noise level
given by the quadratic solver. 963 964 given by the quadratic solver.
964 965
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 965 966 Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40.
Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60. 966 967 Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60.
Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80. 967 968 Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80.
Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100. 968 969 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
969 970
% \begin{figure} 970 971 % \begin{figure}
% \centering 971 972 % \centering
% \includegraphics[width=\linewidth]{images/min_40} 972 973 % \includegraphics[width=\linewidth]{images/min_40}
% \caption{Signal spectrum for MIN/40} 973 974 % \caption{Signal spectrum for MIN/40}
% \label{fig:min_40} 974 975 % \label{fig:min_40}
% \end{figure} 975 976 % \end{figure}
% 976 977 %
% \begin{figure} 977 978 % \begin{figure}
% \centering 978 979 % \centering
% \includegraphics[width=\linewidth]{images/min_60} 979 980 % \includegraphics[width=\linewidth]{images/min_60}
% \caption{Signal spectrum for MIN/60} 980 981 % \caption{Signal spectrum for MIN/60}
% \label{fig:min_60} 981 982 % \label{fig:min_60}
% \end{figure} 982 983 % \end{figure}
% 983 984 %
% \begin{figure} 984 985 % \begin{figure}
% \centering 985 986 % \centering
% \includegraphics[width=\linewidth]{images/min_80} 986 987 % \includegraphics[width=\linewidth]{images/min_80}
% \caption{Signal spectrum for MIN/80} 987 988 % \caption{Signal spectrum for MIN/80}
% \label{fig:min_80} 988 989 % \label{fig:min_80}
% \end{figure} 989 990 % \end{figure}
% 990 991 %
% \begin{figure} 991 992 % \begin{figure}
% \centering 992 993 % \centering
% \includegraphics[width=\linewidth]{images/min_100} 993 994 % \includegraphics[width=\linewidth]{images/min_100}
% \caption{Signal spectrum for MIN/100} 994 995 % \caption{Signal spectrum for MIN/100}
% \label{fig:min_100} 995 996 % \label{fig:min_100}
% \end{figure} 996 997 % \end{figure}
997 998
% r2.14 et r2.15 et r2.16 998 999 % r2.14 et r2.15 et r2.16
\begin{figure} 999 1000 \begin{figure}
\centering 1000 1001 \centering
\begin{subfigure}{\linewidth} 1001 1002 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_40} 1002 1003 \includegraphics[width=.91\linewidth]{images/min_40}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 1003 1004 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.} 1004 1005 the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.}
\label{fig:min_40} 1005 1006 \label{fig:min_40}
\end{subfigure} 1006 1007 \end{subfigure}
1007 1008
\begin{subfigure}{\linewidth} 1008 1009 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_60} 1009 1010 \includegraphics[width=.91\linewidth]{images/min_60}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 1010 1011 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.} 1011 1012 the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.}
\label{fig:min_60} 1012 1013 \label{fig:min_60}
\end{subfigure} 1013 1014 \end{subfigure}
1014 1015
\begin{subfigure}{\linewidth} 1015 1016 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_80} 1016 1017 \includegraphics[width=.91\linewidth]{images/min_80}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 1017 1018 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.} 1018 1019 the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.}
\label{fig:min_80} 1019 1020 \label{fig:min_80}
\end{subfigure} 1020 1021 \end{subfigure}
1021 1022
\begin{subfigure}{\linewidth} 1022 1023 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_100} 1023 1024 \includegraphics[width=.91\linewidth]{images/min_100}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 1024 1025 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.} 1025 1026 the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}
\label{fig:min_100} 1026 1027 \label{fig:min_100}
\end{subfigure} 1027 1028 \end{subfigure}
\caption{\color{red}Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a 1028 1029 \caption{\color{red}Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a
given rejection while minimizing resource allocation. The filter shape constraint (bandpass and 1029 1030 given rejection while minimizing resource allocation. The filter shape constraint (bandpass and
bandstop) is shown as thick 1030 1031 bandstop) is shown as thick
horizontal lines on each chart.} 1031 1032 horizontal lines on each chart.}
\end{figure} 1032 1033 \end{figure}
1033 1034
We observe that all rejections given by the quadratic solver are close to the experimentally 1034 1035 We observe that all rejections given by the quadratic solver are close to the experimentally
measured rejection. All curves prove that the constraint to reach the target rejection is 1035 1036 measured rejection. All curves prove that the constraint to reach the target rejection is
respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters. 1036 1037 respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.
1037 1038
Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60; 1038 1039 Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60;
MIN/80 and MIN/100 \emph{i.e.} when the target rejection is fixed to 40, 60, 80 and 100~dB. We 1039 1040 MIN/80 and MIN/100 \emph{i.e.} when the target rejection is fixed to 40, 60, 80 and 100~dB. We
have taken care to extract solely the resources used by 1040 1041 have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and PL to 1041 1042 the FIR filters and remove additional processing blocks including FIFO and PL to
PS communication. 1042 1043 PS communication.
1043 1044
\renewcommand{\arraystretch}{1.2} 1044 1045 \renewcommand{\arraystretch}{1.2}
\begin{table} 1045 1046 \begin{table}
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 1046 1047 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage_comp} 1047 1048 \label{tbl:resources_usage_comp}
\centering 1048 1049 \centering
{\scalefont{0.90} 1049 1050 {\scalefont{0.90}
\begin{tabular}{|c|c|cccc|c|} 1050 1051 \begin{tabular}{|c|c|cccc|c|}
\hline 1051 1052 \hline
$n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline 1052 1053 $n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 343 & 334 & 772 & - & \emph{17600} \\ 1053 1054 & LUT & 343 & 334 & 772 & - & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\ 1054 1055 1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\
ifcs2018_journal_reponse.tex
Diff comments   View file @ 4d90525
%Minor Revision - TUFFC-09469-2019 1 1 %Minor Revision - TUFFC-09469-2019
%Transactions on Ultrasonics, Ferroelectrics, and Frequency 2 2 %Transactions on Ultrasonics, Ferroelectrics, and Frequency
%Control (July 23, 2019 9:29 PM) 3 3 %Control (July 23, 2019 9:29 PM)
%To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr, 4 4 %To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr,
%gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr, 5 5 %gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr,
%jmfriedt@femto-st.fr 6 6 %jmfriedt@femto-st.fr
%CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu 7 7 %CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu
% 8 8 %
%Dear Mr. Arthur HUGEAT 9 9 %Dear Mr. Arthur HUGEAT
% 10 10 %
%Congratulations! Your manuscript 11 11 %Congratulations! Your manuscript
% 12 12 %
%MANUSCRIPT NO. TUFFC-09469-2019 13 13 %MANUSCRIPT NO. TUFFC-09469-2019
%MANUSCRIPT TYPE: Papers 14 14 %MANUSCRIPT TYPE: Papers
%TITLE: Filter optimization for real time digital processing of radiofrequency 15 15 %TITLE: Filter optimization for real time digital processing of radiofrequency
%signals: application to oscillator metrology 16 16 %signals: application to oscillator metrology
%AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, 17 17 %AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois,
%Pierre-Yves; Friedt, Jean-Michel 18 18 %Pierre-Yves; Friedt, Jean-Michel
% 19 19 %
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%**************************************************** 73 73 %****************************************************
%REVIEWERS' COMMENTS: 74 74 %REVIEWERS' COMMENTS:
75 75
\documentclass[a4paper]{article} 76 76 \documentclass[a4paper]{article}
\usepackage{fullpage,graphicx,amsmath, subcaption} 77 77 \usepackage{fullpage,graphicx,amsmath, subcaption}
\begin{document} 78 78 \begin{document}
{\bf Reviewer: 1} 79 79 \begin{center}
80 {\bf\Large
81 Rebuttal letter to the review of the manuscript entitled
80 82
83 ``Filter optimization for real time digital processing of radiofrequency
84 signals: application to oscillator metrology''
85 }
86
87 by A. Hugeat \& al.
88 \end{center}
89
90 \section*{Reviewer: 1}
91
%Comments to the Author 81 92 %Comments to the Author
%In general, the language/grammar is adequate. 82 93 %In general, the language/grammar is adequate.
83 94
{\bf 84 95 {\bf
On page 2, "...allowing to save processing resource..." could be improved. % r1.1 - fait 85 96 On page 2, "...allowing to save processing resource..." could be improved. % r1.1 - fait
} 86 97 }
87 98
The sentence was split and now reads ``number of coefficients irrelevant: processing 88 99 The sentence was split and now reads ``number of coefficients irrelevant: processing
resources are hence saved by shrinking the filter length.'' 89 100 resources are hence saved by shrinking the filter length.''
90 101
{\bf 91 102 {\bf
On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 - fait 92 103 On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 - fait
the author meant.} 93 104 the author meant.}
94 105
Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s) 95 106 Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s)
Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.'' 96 107 Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
97 108
{\bf 98 109 {\bf
On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 - fait 99 110 On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 - fait
could be improved. 100 111 could be improved.
} 101 112 }
102 113
Indeed this paragraph has be written again and now reads as\\ 103 114 Indeed this paragraph has be written again and now reads as\\
``The first step of our approach is to model the DSP chain. Since we aim at only optimizing 104 115 ``The first step of our approach is to model the DSP chain. Since we aim at only optimizing
the filtering part of the signal processing chain, we have not included the PRN generator or the 105 116 the filtering part of the signal processing chain, we have not included the PRN generator or the
ADC in the model: the input data size and rate are considered fixed and defined by the hardware. 106 117 ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
The filtering can be done in two ways, either by considering a single monolithic FIR filter 107 118 The filtering can be done in two ways, either by considering a single monolithic FIR filter
requiring many coefficients to reach the targeted noise rejection ratio, or by 108 119 requiring many coefficients to reach the targeted noise rejection ratio, or by
cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter. 109 120 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.
'' 110 121 ''
111 122
{\bf 112 123 {\bf
I appreciate that the authors attempted and document two optimizations: that % r1.4 - fait 113 124 I appreciate that the authors attempted and document two optimizations: that % r1.4 - fait
of maximum rejection ratio at fixed silicon area, as well as minimum silicon 114 125 of maximum rejection ratio at fixed silicon area, as well as minimum silicon
area for a fixed minimum rejection ratio. For non-experts, it might be very 115 126 area for a fixed minimum rejection ratio. For non-experts, it might be very
useful to compare the results of both optimization paths to the performance and 116 127 useful to compare the results of both optimization paths to the performance and
resource-utilization of generic low-pass filter gateware offered by device 117 128 resource-utilization of generic low-pass filter gateware offered by device
manufacturers. I appreciate also that the authors have presented source code 118 129 manufacturers. I appreciate also that the authors have presented source code
for examination online. 119 130 for examination online.
} 120 131 }
121 132
To compare the performance of our FIR filters and the performance of device 122 133 We have compared the performance of our FIR filters and the performance of device
manufacturers generic filter, we have added a paragraph and a table at the 123 134 manufacturers generic filter: we have added a paragraph and a table at the
end of experiments section. We compare the resources consumption with the same 124 135 end of experiments section. We compare the resources consumption with the same
FIR coefficients set. 125 136 FIR coefficients set, demonstrating that the transfer functions match and that
137 resources are somewhat similar despite different assumptions (Xilinx sets coefficients
138 upon synthesis while we wish to be able to update taps without synthesis).
126 139
{\bf 127 140 \noindent
Reviewer: 2 128 141 \section*{Reviewer: 2}
} 129
130 142
%Comments to the Author 131 143 %Comments to the Author
%In the Manuscript, the Authors describe an optimization methodology for filter 132 144 %In the Manuscript, the Authors describe an optimization methodology for filter
%design to be used in phase noise metrology. The methodology is general and can 133 145 %design to be used in phase noise metrology. The methodology is general and can
%be used for many aspects of the processing chain. In the Manuscript, the Authors 134 146 %be used for many aspects of the processing chain. In the Manuscript, the Authors
%focus on filtering and shifting while the other aspects, in particular decimation, 135 147 %focus on filtering and shifting while the other aspects, in particular decimation,
%will be considered in a future work. The optimization problem is modelled 136 148 %will be considered in a future work. The optimization problem is modelled
%theoretically and then solved by means of a commercial software. The solutions 137 149 %theoretically and then solved by means of a commercial software. The solutions
%are tested experimentally on the Redpitaya platform with synthetic and real 138 150 %are tested experimentally on the Redpitaya platform with synthetic and real
%white noises. Two cases are considered as a function of the number of filters: 139 151 %white noises. Two cases are considered as a function of the number of filters:
%maximum rejection given a fixed amount of resources and minimum resource 140 152 %maximum rejection given a fixed amount of resources and minimum resource
%utilization given a fixed amount of rejection. 141 153 %utilization given a fixed amount of rejection.
%The Authors find that filtering improves significantly when the number of 142 154 %The Authors find that filtering improves significantly when the number of
%filters increases. 143 155 %filters increases.
%A lot of work has been done in generalizing and automating the procedure so 144 156 %A lot of work has been done in generalizing and automating the procedure so
%that different approaches can be investigated quickly and efficiently. The 145 157 %that different approaches can be investigated quickly and efficiently. The
%results presented in the Manuscript seem to be just a case study based on 146 158 %results presented in the Manuscript seem to be just a case study based on
%the particular criterion chosen by the Authors. Different criteria, in 147 159 %the particular criterion chosen by the Authors. Different criteria, in
%general, could lead to different results and it is important to consider 148 160 %general, could lead to different results and it is important to consider
%carefully the criterion adopted by the Authors, in order to check if it 149 161 %carefully the criterion adopted by the Authors, in order to check if it
%is adequate to compare the performance of filters and if multi-stage 150 162 %is adequate to compare the performance of filters and if multi-stage
%filters are really superior than monolithic filters. 151 163 %filters are really superior than monolithic filters.
152 164
{\bf 153 165 {\bf
By observing the results presented in fig. 10-16, it is clear that the % r2.1 154 166 By observing the results presented in fig. 10-16, it is clear that the % r2.1
performances of multi-stage filters are obtained at the expense of their 155 167 performances of multi-stage filters are obtained at the expense of their
selectivity and, in this sense, the filters presented in these figures 156 168 selectivity and, in this sense, the filters presented in these figures
are not equivalent. For example, in Fig. 14, at the limit of the pass band, 157 169 are not equivalent. For example, in Fig. 14, at the limit of the pass band,
the attenuation is almost 15 dB for n = 5, while it is not noticeable for 158 170 the attenuation is almost 15 dB for n = 5, while it is not noticeable for
n = 1. 159 171 n = 1.
} 160 172 }
161 173
We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined 162 174 We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to define
the bandpass and the bandstop of the filter. 163 175 the bandpass and the bandstop of the filter.
164 176
% We are aware of this non equivalence but we think that difference is not due to 165 177 % We are aware of this non equivalence but we think that difference is not due to
% the cascaded filters but due to the definition of rejection criterion on the passband. 166 178 % the cascaded filters but due to the definition of rejection criterion on the passband.
% Indeed, in this article we have choose to take the summation of absolute values divide 167 179 % Indeed, in this article we have choose to take the summation of absolute values divide
% by the bandwidth but this criterion is maybe too permissive and when we cascade 168 180 % by the bandwidth but this criterion is maybe too permissive and when we cascade
% some filters this impact is more important. 169 181 % some filters this impact is more important.
% 170 182 %
% However if we change the passband 171 183 % However if we change the passband
% criterion by the summation of absolute value in passband, weighting given to the 172 184 % criterion by the summation of absolute value in passband, weighting given to the
% passband ripples are too strong and the solver are too restricted to provide 173 185 % passband ripples are too strong and the solver are too restricted to provide
% any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in 174 186 % any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in
% passband, the rejection evaluation are too close form the original criterion and 175 187 % passband, the rejection evaluation are too close form the original criterion and
% the result will not be improved. 176 188 % the result will not be improved.
% 177 189 %
% In this article, we will highlight the methodology instead of the filter conception. 178 190 % In this article, we will highlight the methodology instead of the filter conception.
% Even if our rejection criterion is not the best, our methodology was not impacted 179 191 % Even if our rejection criterion is not the best, our methodology was not impacted
% by this. So to improve the results, we can choose another criterion to be more 180 192 % by this. So to improve the results, we can choose another criterion to be more
% selective in passband but it is not the main objective of our article. 181 193 % selective in passband but it is not the main objective of our article.
182 194
We are aware of this equivalence but to limit this ripples in passband we need to 183 195 We are aware of this difference between the cascaded and monolithic filters but
enforce the criterion in passband. If we takes a strong constraint like the sum of 184 196 we consider that limiting the ripples in passband is more a matter of enforcing some
absolute values in passband. This criterion si too selective because it considers 185 197 selection criteria rather than being intrinsic to cascading filters. Selecting
all bin on passband while on stopband we consider only the bin with the minimal 186 198 a strong constraint such as the sum of absolute values in the passband is too selective
rejection. The figure~\ref{fig:letter_sum_criterion} exhibits the results with this 187 199 because it considers all frequency bins in the passband while the stopband criterion is
criterion for the case MAX/1000. With this criterion, the solver find an optimal 188 200 limited to a single bin at which rejection is poorest. Fig.~\ref{fig:letter_sum_criterion}
solution with only two filters in expend of the resource consumption. 189 201 exhibits the results with this
202 criterion for the case MAX/1000. With this criterion, the solver finds an optimal
203 solution with only two filters. % in expend of the resource consumption.
190 204
191 205 Relaxing the criterion in the passband by considering only the maximum absolute
192 206 value, we penalize the ripple peak in the passband. Fig.~\ref{fig:letter_max_criterion}
If we relax a little the criterion on passband with taking only the maximum absolute 193
value, we will penalize the ripple peak on passband. The figure~\ref{fig:letter_max_criterion} 194
shows the results for the case MAX/1000. There as almost no difference with the 195 207 shows the results for the case MAX/1000. There as almost no difference with the
article results. Indeed the only little change are on the case $i = 4$ and $i = 5$ 196 208 article results. Indeed the only little change are on the case $i = 4$ and $i = 5$
which they have some minor differences on coefficients choices. 197 209 which exhibit some minor differences on coefficients choices.
198 210
\begin{figure}[h!tb] 199 211 \begin{figure}[h!tb]
\centering 200 212 \centering
\begin{subfigure}{0.48\linewidth} 201 213 \begin{subfigure}{0.48\linewidth}
\includegraphics[width=\linewidth]{images/letter_sum_criterion} 202 214 \includegraphics[width=\linewidth]{images/letter_sum_criterion}
\caption{Results for the case MAX/1000 with as criterion on passband the sum absolute values} 203 215 \caption{Results for the case MAX/1000 with as criterion on passband the sum absolute values}
\label{fig:letter_sum_criterion} 204 216 \label{fig:letter_sum_criterion}
\end{subfigure} 205 217 \end{subfigure}
\begin{subfigure}{0.48\linewidth} 206 218 \begin{subfigure}{0.48\linewidth}
\includegraphics[width=\linewidth]{images/letter_max_criterion} 207 219 \includegraphics[width=\linewidth]{images/letter_max_criterion}
\caption{Results for the case MAX/1000 with as criterion on passband the maximum absolute value} 208 220 \caption{Results for the case MAX/1000 with as criterion on passband the maximum absolute value}
\label{fig:letter_max_criterion} 209 221 \label{fig:letter_max_criterion}
\end{subfigure} 210 222 \end{subfigure}
\end{figure} 211 223 \end{figure}
212 224
Finally, if we ponder the maximum absolute on passband, we should improve the result. 213 225 Finally, if we weight the maximum absolute value in the passband, we might improve the result.
We have arbitrary pondered by 5 the maximum. Even with this weighting, the solver 214 226 We have arbitrary weighted by a factor of 5 the maximum of the absolute value in the passband.
choose the same coefficient set. 215 227 Even with this weighting, the solver chooses the same coefficient set.
216 228
To conclude, find a better criterion to avoid the ripples on the passband is difficult. 217 229 To conclude, finding a better criterion to avoid the ripples in the passband is challenging.
In this article we are focused on the methodology so even if our criterion could 218 230 In this article we focus on the methodology, so even if our criterion could
be improved, our methodology still the same and it works independently of rejection criterion. 219 231 be improved, our methodology still remains and works independently of rejection criterion.
232 The averaging of the absolute values is the passband is a matter of having consistent units
233 between the bandstop and banspass criteria: the bandstop criterion is the bin with poorest
234 rejection so in units of dB/Hz. Using a bandpass criterion of the sum of absolute values
235 in all bins would be a unit of dB: normalizing by the number of bins, equivalent to averaging
236 by dividing by the number of bins, brings back a criterion in dB/Hz consistent with the
237 former value.
220 238
% %Peut etre refaire une serie de simulation dans lesquelles on impose une coupure 221 239 % %Peut etre refaire une serie de simulation dans lesquelles on impose une coupure
% %non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte 222 240 % %non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte
% %au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque 223 241 % %au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque
% %a la cascade de filtres. 224 242 % %a la cascade de filtres.
% %AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de 225 243 % %AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de
% %relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai 226 244 % %relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai
% 227 245 %
% densité spectrale de la bande passante 228 246 % densité spectrale de la bande passante
% sum des valeurs absolues / largeur de la bande passante (1/N) vs max dans la bande de coupure 229 247 % sum des valeurs absolues / largeur de la bande passante (1/N) vs max dans la bande de coupure
% 230 248 %
% JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait 231 249 % JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait
% moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB 232 250 % moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB
% quand on cascade 5 filtres ! 233 251 % quand on cascade 5 filtres !
% 234 252 %
% Dire que la chute n'est pas du à la casacade mais à notre critère de rejection 235 253 % Dire que la chute n'est pas du à la casacade mais à notre critère de rejection
236 254
{\bf 237 255 {\bf
The reason is in the criterion that considers the average attenuation in % r2.2 238 256 The reason is in the criterion that considers the average attenuation in % r2.2
the pass band. This criterion does not take into account the maximum attenuation 239 257 the pass band. This criterion does not take into account the maximum attenuation
in this region, which is a very important parameter for specifying a filter 240 258 in this region, which is a very important parameter for specifying a filter
and for evaluating its performance. For example, with this criterion, a 241 259 and for evaluating its performance. For example, with this criterion, a
filter with 0.1 dB of ripple is considered equivalent to a filter with 242 260 filter with 0.1 dB of ripple is considered equivalent to a filter with
10 dB of ripple. This point has a strong impact in the optimization process 243 261 10 dB of ripple. This point has a strong impact in the optimization process
and in the results that are obtained and has to be reconsidered. 244 262 and in the results that are obtained and has to be reconsidered.
} 245 263 }
246 264
See above: Choose a criterion is difficult and depending on the context. The main 247 265 See above: choosing a criterion is challenging and dependent on the context. The main
contribution on this paper is the methodology not the criterion to quantify the 248 266 contribution on this paper is the methodology rather than the criterion to quantify the
rejection. 249 267 rejection.
250 268
% The manuscript erroneously stated that we considered the mean of the absolute 251 269 % The manuscript erroneously stated that we considered the mean of the absolute
% value within the bandpass: the manuscript has now been corrected to properly state 252 270 % value within the bandpass: the manuscript has now been corrected to properly state
% the selected criterion, namely the {\em sum} of the absolute value, so that any 253 271 % the selected criterion, namely the {\em sum} of the absolute value, so that any
% ripple in the bandpass will reduce the chances of a given filter set from being 254 272 % ripple in the bandpass will reduce the chances of a given filter set from being
% selected. The manuscript now states ``Our criterion to compute the filter rejection considers 255 273 % selected. The manuscript now states ``Our criterion to compute the filter rejection considers
% % r2.8 et r2.2 r2.3 256 274 % % r2.8 et r2.2 r2.3
% the maximum magnitude within the stopband, to which the {sum of the absolute values 257 275 % the maximum magnitude within the stopband, to which the {sum of the absolute values
% within the passband is subtracted to avoid filters with excessive ripples}.'' 258 276 % within the passband is subtracted to avoid filters with excessive ripples}.''
259 277
{\bf 260 278 {\bf
I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait 261 279 I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait
into account the maximum allowed attenuation in pass band, for example by 262 280 into account the maximum allowed attenuation in pass band, for example by
fixing its value to a typical one, as it has been done for the transition 263 281 fixing its value to a typical one, as it has been done for the transition
bandwidth. 264 282 bandwidth.
} 265 283 }
266 284
See above: the absolute value within the passband will reject filters with 267 285 See above: the absolute value within the passband will reject filters with
excessive ripples, including excessive attenuation, within the passband. 268 286 excessive ripples, including excessive attenuation, within the passband.
269 287
% TODO: test max(stopband) - max(abs(passband)) 270 288 % TODO: test max(stopband) - max(abs(passband))
271 289
{\bf 272 290 {\bf
In addition, I suggest to address the following points: % r2.4 - fait 273 291 In addition, I suggest to address the following points: % r2.4 - fait
- Page 1, line 50: the Authors state that IIR have shorter impulse response 274 292 - Page 1, line 50: the Authors state that IIR have shorter impulse response
than FIR. This is not true in general. The sentence should be reconsidered. 275 293 than FIR. This is not true in general. The sentence should be reconsidered.
} 276 294 }
277 295
We have not stated that the IIR has a shorter impulse response but a shorter lag. 278 296 We have not stated that the IIR has a shorter impulse response but a shorter lag.
Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filters 279 297 Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filters
have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing 280 298 have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing
the first output, an IIR will start providing outputs only 5 time steps after the initial 281 299 the first output, an IIR will start providing outputs only 5 time steps after the initial
input starts feeding the IIR. Hence, the issue we address here is lag and not impulse 282 300 input starts feeding the IIR. Hence, the issue we address here is lag and not impulse
response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue 283 301 response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue
in a openloop phase noise characterization instrument, the large 284 302 in a openloop phase noise characterization instrument, the large
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 285 303 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
is not considered as an issue as would be in a closed loop system in which lag aims at being 286 304 is not considered as an issue as would be in a closed loop system in which lag aims at being
minimized to avoid oscillation conditions.'' 287 305 minimized to avoid oscillation conditions.''
288 306
{\bf 289 307 {\bf
- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 - fait 290 308 - Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 - fait
this transition bandwidth and if it is a typical requirement for phase-noise 291 309 this transition bandwidth and if it is a typical requirement for phase-noise
metrology. 292 310 metrology.
} 293 311 }
294 312
The purpose of the paper is to demonstrate how a given filter shape can be achieved by 295 313 The purpose of the paper is to demonstrate how a given filter shape can be achieved by
minimizing varous resource criteria. Indeed the stopband and bandpass boundaries can 296 314 minimizing varous resource criteria. Indeed the stopband and bandpass boundaries can
be questioned: we have selected this filter shape as a typical anti-aliasing filter considering 297 315 be questioned: we have selected this filter shape as a typical anti-aliasing filter considering
the the dataflow is to be halved. Hence, selecting a cutoff frequency of 40\% the initial 298 316 the the dataflow is to be halved. Hence, selecting a cutoff frequency of 40\% the initial
Nyquist frequency prevents noise from reaching baseband after decimating the dataflow by a 299 317 Nyquist frequency prevents noise from reaching baseband after decimating the dataflow by a
factor of 2. Such ideas are now stated explicitly in the text as ``Throughout this demonstration, 300 318 factor of 2. Such ideas are now stated explicitly in the text as ``Throughout this demonstration,
we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\% 301 319 we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%
of the Nyquist frequency to the end of the band, as would be typically selected to prevent 302 320 of the Nyquist frequency to the end of the band, as would be typically selected to prevent
aliasing before decimating the dataflow by 2. The method is however generalized to any filter 303 321 aliasing before decimating the dataflow by 2. The method is however generalized to any filter
shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid} 304 322 shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid}
as described below is indeed unique for each filter shape.'' 305 323 as described below is indeed unique for each filter shape.''
306 324
{\bf 307 325 {\bf
- The impact of the coefficient resolution is discussed. What about the % r2.6 - fait 308 326 - The impact of the coefficient resolution is discussed. What about the % r2.6 - fait
resolution of the data stream? Is it fixed? If so, which value has been 309 327 resolution of the data stream? Is it fixed? If so, which value has been
used in the analysis? If not, how is it changed with respect to the 310 328 used in the analysis? If not, how is it changed with respect to the
coefficient resolution? 311 329 coefficient resolution?
} 312 330 }
313 331
We have now stated in the beginning of the document that ``we have not included the PRN generator 314 332 We have now stated in the beginning of the document that ``we have not included the PRN generator
or the ADC in the model: the input data size and rate are considered fixed and defined by the 315 333 or the ADC in the model: the input data size and rate are considered fixed and defined by the
hardware.'' so indeed the input datastream resolution is considered as a given. 316 334 hardware.'' so indeed the input datastream resolution is considered as a given.
317 335
{\bf 318 336 {\bf
- Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait 319 337 - Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait
Fig. 5 can be removed. 320 338 Fig. 5 can be removed.
} 321 339 }
322 340
Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats 323 341 We have kept a sentence stating our initial line of thought to avoid readers from performing
342 the same mistake, but have removed the associated figure as requested.
324 343
{\bf 325 344 {\bf
- Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait 326 345 - Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait
It should be ``minimum'' 327 346 It should be ``minimum''
- Page e, line 55, second column: ``takin'' % r2.9 - fait 328 347 - Page e, line 55, second column: ``takin'' % r2.9 - fait
- Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait 329 348 - Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait
- Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait 330 349 - Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait
} 331 350 }
332 351
All typos and grammatical errors have been corrected. 333 352 All typos and grammatical errors have been corrected.
334 353
{\bf 335 354 {\bf
- Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait 336 355 - Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait
these particular configurations? Are they chosen automatically? 337 356 these particular configurations? Are they chosen automatically?
} 338 357 }
% C'est le nombre de coefficients et un taille raisonnable 339 358 % C'est le nombre de coefficients et un taille raisonnable
% Troncature de la pyramide 340 359 % Troncature de la pyramide
341 360
See below: we have added a better description of $p$ during the transformation explanation. 342 361 See below: we have added a better description of $p$ during the transformation explanation.
``This variable $p$ is defined by the user, and represents the number of different 343 362 ``This variable $p$ is defined by the user, and represents the number of different
set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1} 344 363 set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave) based on the targeted filter characteristics and implementation 345 364 functions from GNU Octave) based on the targeted filter characteristics and implementation
assumptions (estimated number of bits defining the coefficients)'' 346 365 assumptions (estimated number of bits defining the coefficients)''
347 366
{\bf 348 367 {\bf
- Page 4, line 31: how does the delta function transform model from non-linear % r2.13 349 368 - Page 4, line 31: how does the delta function transform model from non-linear % r2.13
and non-quadratic to a quadratic?} 350 369 and non-quadratic to a quadratic?}
351 370
The first model is non-quadratic but when we introduce the $p$ configurations, 352 371 The first model is non-quadratic but when we introduce the $p$ configurations,
we can estimate the function $F$ by computing 353 372 we can estimate the function $F$ by computing
the rejection for each configuration, so the model become quadratic because we have 354 373 the rejection for each configuration, so the model becomes quadratic because we have
some multiplication between variables. With the definition of $\delta_{ij}$ we can 355 374 some multiplication between variables. With the definition of $\delta_{ij}$ we can
replace the multiplication between variables by multiplication with binary variable and 356 375 replace the multiplication between variables by multiplication with binary variables which
this one can be linearise as follow:\\ 357 376 can be linearised as follows:\\
$y$ is a binary variable \\ 358 377 $y$ is a binary variable \\
$x$ is a real variable bounded by $X^{max}$ \\ 359 378 $x$ is a real variable bounded by $X^{max}$ \\
\begin{equation*} 360 379 \begin{equation*}
m = x \times y \implies 361 380 m = x \times y \implies
\left \{ 362 381 \left \{
\begin{split} 363 382 \begin{split}
m & \geq 0 \\ 364 383 m & \geq 0 \\
m & \leq y \times X^{max} \\ 365 384 m & \leq y \times X^{max} \\
m & \leq x \\ 366 385 m & \leq x \\
m & \geq x - (1 - y) \times X^{max} \\ 367 386 m & \geq x - (1 - y) \times X^{max} \\
\end{split} 368 387 \end{split}
\right . 369 388 \right .
\end{equation*} 370 389 \end{equation*}
Gurobi does the linearization so we don't explain this step to keep the model more 371 390 as explained now in the manuscript. Gurobi does the linearization so we do not explain
simple. However, to improve the transformation explanation we have rewrote the 372 391 this step to keep the model more
392 simple. However, to improve the transformation explanation we have rewritten the
paragraph ``This model is non-linear and even non-quadratic...''. 373 393 paragraph ``This model is non-linear and even non-quadratic...''.
374 394
% JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article 375 395 % JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
% je ne comprends pas comment ca repond a la question 376 396 % je ne comprends pas comment ca repond a la question
% 377 397 %
% AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux. 378 398 % AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
% 379 399 %
% Le problème n'est pas linéaire car nous multiplions des variables 380 400 % Le problème n'est pas linéaire car nous multiplions des variables
% entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent 381 401 % entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
% des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique 382 402 % des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
% quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore 383 403 % quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
% quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre 384 404 % quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
% $\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible 385 405 % $\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
% de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre 386 406 % de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
% cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée, 387 407 % cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
% nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur 388 408 % nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
% qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la 389 409 % qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
% linéarisation pour nous. 390 410 % linéarisation pour nous.
391 411
392 412
{\bf 393 413 {\bf
- Captions of figure and tables are too minimal. % r2.14 394 414 - Captions of figure and tables are too minimal. % r2.14
} 395 415 }
We have change the captions of tables and figures. 396
397 416
417 Captions of figures were expanded to make the description easier to grasp by the reader
418
{\bf 398 419 {\bf
- Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait 399 420 - Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait
of a single figure. Same for fig. 13-16. 400 421 of a single figure. Same for fig. 13-16.
} 401 422 }
We add two sub figure to group the fig.10-12 and fig. 13-16 402 423
424 We have grouped figures 10--12 and 13--16 as two sets of sub-figures.
403 425
{\bf 404 426 {\bf
- Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait 405 427 - Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait
of the curves is about 20 dBpk-pk and it doesn’t allow to appreciate the 406 428 of the curves is about 20 dBpk-pk and it doesn’t allow to appreciate the
differences among the curves. I suggest to reduce the noise below 1 dBpk-pk. 407 429 differences among the curves. I suggest to reduce the noise below 1 dBpk-pk.
} 408 430 }
409 431