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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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\usetikzlibrary{positioning,fit} 23 23 \usetikzlibrary{positioning,fit}
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% 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 % Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion
% yields unacceptable results since notches overestimate the rejection capability of the filter. Furthermore, the losses within 296 296 % yields unacceptable results since notches overestimate the rejection capability of the filter. Furthermore, the losses within
% the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients. 297 297 % 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 298 298 Our criterion to compute the filter rejection considers
% r2.8 et r2.2 r2.3 299 299 % r2.8 et r2.2 r2.3
the maximum magnitude within the stopband, to which the {\color{red}sum of the absolute values 300 300 the maximum magnitude within the stopband, to which the {\color{red}sum of the absolute values
within the passband is subtracted to avoid filters with excessive ripples}. With this 301 301 within the passband is subtracted to avoid filters with excessive ripples}. With this
criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}. 302 302 criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
303 303
% \begin{figure} 304 304 % \begin{figure}
% \centering 305 305 % \centering
% \includegraphics[width=\linewidth]{images/colored_mean_criterion} 306 306 % \includegraphics[width=\linewidth]{images/colored_mean_criterion}
% \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters} 307 307 % \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}
% \label{fig:mean_criterion} 308 308 % \label{fig:mean_criterion}
% \end{figure} 309 309 % \end{figure}
310 310
\begin{figure} 311 311 \begin{figure}
\centering 312 312 \centering
\includegraphics[width=\linewidth]{images/colored_custom_criterion} 313 313 \includegraphics[width=\linewidth]{images/colored_custom_criterion}
\caption{Custom criterion (maximum rejection in the stopband minus the mean of the absolute value of the passband rejection) 314 314 \caption{Custom criterion (maximum rejection in the stopband minus the mean of the absolute value of the passband rejection)
comparison between monolithic filter and cascaded filters} 315 315 comparison between monolithic filter and cascaded filters}
\label{fig:custom_criterion} 316 316 \label{fig:custom_criterion}
\end{figure} 317 317 \end{figure}
318 318
Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps 319 319 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 320 320 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. 321 321 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. 322 322 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. 323 323 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 324 324 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. 325 325 the rejection. Hence the best coefficient set are on the vertex of the pyramid.
326 326
\begin{figure} 327 327 \begin{figure}
\centering 328 328 \centering
\includegraphics[width=\linewidth]{images/rejection_pyramid} 329 329 \includegraphics[width=\linewidth]{images/rejection_pyramid}
\caption{Rejection as a function of number of coefficients and number of bits} 330 330 \caption{Rejection as a function of number of coefficients and number of bits}
\label{fig:rejection_pyramid} 331 331 \label{fig:rejection_pyramid}
\end{figure} 332 332 \end{figure}
333 333
Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps), 334 334 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. 335 335 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: 336 336 If the FIR filter coefficients are the same between the stages, we have:
$$F_{total} = F_1 + F_2$$ 337 337 $$F_{total} = F_1 + F_2$$
But selecting two different sets of coefficient will yield a more complex situation in which 338 338 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 339 339 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. 340 340 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 341 341 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. 342 342 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
% r2.9 343 343 % r2.9
Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection 344 344 Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection
criteria of each filter. However since the this sum underestimates the rejection capability of the cascade, 345 345 criteria of each filter. However since the this sum underestimates the rejection capability of the cascade,
% r2.10 346 346 % r2.10
this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability 347 347 this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
of the filter cascade to meet design criteria. 348 348 of the filter cascade to meet design criteria.
349 349
\begin{figure} 350 350 \begin{figure}
\centering 351 351 \centering
\includegraphics[width=\linewidth]{images/cascaded_criterion} 352 352 \includegraphics[width=\linewidth]{images/cascaded_criterion}
\caption{Rejection of two cascaded filters} 353 353 \caption{Rejection of two cascaded filters}
\label{fig:sum_rejection} 354 354 \label{fig:sum_rejection}
\end{figure} 355 355 \end{figure}
356 356
% r2.6 357 357 % r2.6
Finally in our case, we consider that the input signal are fully known. So the 358 358 Finally in our case, we consider that the input signal are fully known. So the
resolution of the data stream are fixed and still the same for all experiments 359 359 resolution of the data stream are fixed and still the same for all experiments
in this paper. 360 360 in this paper.
361 361
Based on this analysis, we address the estimate of resource consumption (called 362 362 Based on this analysis, we address the estimate of resource consumption (called
% r2.11 363 363 % r2.11
silicon area -- in the case of FPGAs this means processing cells) as a function of 364 364 silicon area -- in the case of FPGAs this means processing cells) as a function of
filter characteristics. As a reminder, we do not aim at matching actual hardware 365 365 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, 366 366 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 367 367 and will assess after synthesis the adequation of this arbitrary unit with actual
hardware resources provided by FPGA manufacturers. The sum of individual processing 368 368 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. 369 369 unit areas is constrained by a total silicon area representative of FPGA global resources.
Formally, variable $a_i$ is the area taken by filter~$i$ 370 370 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). 371 371 (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: 372 372 Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
373 373
\begin{align} 374 374 \begin{align}
\text{Maximize } & \sum_{i=1}^n r_i \notag \\ 375 375 \text{Maximize } & \sum_{i=1}^n r_i \notag \\
\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\ 376 376 \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} \\ 377 377 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} \\ 378 378 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} \\ 379 379 \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} \\ 380 380 \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} \\ 381 381 \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} 382 382 \pi_1^- &= \Pi^I \label{eq:init}
\end{align} 383 383 \end{align}
384 384
Equation~\ref{eq:area} states that the total area taken by the filters must be 385 385 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 386 386 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 387 387 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 388 388 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 389 389 $\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the
input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the 390 390 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 391 391 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, 392 392 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 393 393 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 394 394 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 395 395 $\pi_i^C$ 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 396 396 $\pi_i^S$ 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. 397 397 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 398 398 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 399 399 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 400 400 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 401 401 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 402 402 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 403 403 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: 404 404 shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is:
$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. 405 405 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
Finally, equation~\ref{eq:init} gives the number of bits of the global input. 406 406 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
407 407
This model is non-linear and even non-quadratic, as $F$ does not have a known 408 408 {\color{red}
linear or quadratic expression. We introduce $p$ FIR configurations 409 409 This model is non-linear since we multiply some variable with another variable
$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 410 410 and it is even non-quadratic, as $F$ does not have a known
% r2.12 411 411 linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
This variable must be defined by the user, it represent the number of different 412 412 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} 413 413 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave). 414 414 functions from GNU Octave). So $C_{ij}$ and $\pi_{ij}^C$ become constant and
We define binary 415 415 we defined $1 \leq j \leq p$ and the function $F$ can be estimate for each configurations
416 thanks our rejection criterion. We also defined binary
variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 416 417 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
and 0 otherwise. The new equations are as follows: 417 418 and 0 otherwise. The new equations are as follows:
419 }
418 420
\begin{align} 419 421 \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} \\ 420 422 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} \\ 421 423 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} \\ 422 424 \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} 423 425 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
\end{align} 424 426 \end{align}
425 427
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 426 428 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 427 429 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. 428 430 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
429 431
% r2.13 430 432 {\color{red}
This modified model is quadratic since we multiply two variables in the 431 433 However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply
equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary. 432 434 $\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can
The Gurobi 433 435 linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size
436 we define $0 < \pi_i^- \leq 128$ which is the maximal data size that we can process.
437 }
438 Moreover the Gurobi
(\url{www.gurobi.com}) optimization software is used to solve this quadratic 434 439 (\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 435 440 model, and since Gurobi is able to linearize, the model is left as is. This model
has $O(np)$ variables and $O(n)$ constraints. 436 441 has $O(np)$ variables and $O(n)$ constraints.
442
443 % This model is non-linear and even non-quadratic, as $F$ does not have a known
444 % linear or quadratic expression. We introduce $p$ FIR configurations
445 % $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
446 % % r2.12
447 % This variable must be defined by the user, it represent the number of different
448 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
449 % functions from GNU Octave).
450 % We define binary
451 % variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
452 % and 0 otherwise. The new equations are as follows:
453 %
454 % \begin{align}
455 % 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} \\
456 % r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
457 % \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} \\
458 % \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
459 % \end{align}
460 %
461 % Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
462 % respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
463 % Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
464 %
465 % % r2.13
466 % This modified model is quadratic since we multiply two variables in the
467 % equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
468 % The Gurobi
469 % (\url{www.gurobi.com}) optimization software is used to solve this quadratic
470 % model, and since Gurobi is able to linearize, the model is left as is. This model
471 % has $O(np)$ variables and $O(n)$ constraints.
437 472
Two problems will be addressed using the workflow described in the next section: on the one 438 473 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 439 474 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 440 475 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 441 476 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
objective function is replaced with: 442 477 objective function is replaced with:
\begin{align} 443 478 \begin{align}
\text{Minimize } & \sum_{i=1}^n a_i \notag 444 479 \text{Minimize } & \sum_{i=1}^n a_i \notag
\end{align} 445 480 \end{align}
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 446 481 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 447 482 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
rejection required. 448 483 rejection required.
449 484
\begin{align} 450 485 \begin{align}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 451 486 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
\end{align} 452 487 \end{align}
453 488
\section{Design workflow} 454 489 \section{Design workflow}
\label{sec:workflow} 455 490 \label{sec:workflow}
456 491
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 457 492 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 458 493 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
in the computation of the results. 459 494 in the computation of the results.
460 495
\begin{figure} 461 496 \begin{figure}
\centering 462 497 \centering
\begin{tikzpicture}[node distance=0.75cm and 2cm] 463 498 \begin{tikzpicture}[node distance=0.75cm and 2cm]
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 464 499 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\node (Start) [left= 3cm of Solver] { } ; 465 500 \node (Start) [left= 3cm of Solver] { } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 466 501 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\node (Input) [above= of TCL] { } ; 467 502 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 468 503 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 469 504 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 470 505 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 471 506 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node (Results) [left= of Postproc] { } ; 472 507 \node (Results) [left= of Postproc] { } ;
473 508
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 474 509 \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) ; 475 510 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 476 511 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 477 512 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 478 513 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 479 514 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 480 515 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 481 516 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 482 517 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->] (Postproc) -- (Results) ; 483 518 \draw[->] (Postproc) -- (Results) ;
\end{tikzpicture} 484 519 \end{tikzpicture}
\caption{Design workflow from the input parameters to the results} 485 520 \caption{Design workflow from the input parameters to the results}
\label{fig:workflow} 486 521 \label{fig:workflow}
\end{figure} 487 522 \end{figure}
488 523
The filter solver is a C++ program that takes as input the maximum area 489 524 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 $\Pi^I$, 490 525 $\mathcal{A}$, 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 491 526 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. 492 527 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}) 493 528 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
and a deploy script ((1b) on figure~\ref{fig:workflow}). 494 529 and a deploy script ((1b) on figure~\ref{fig:workflow}).
495 530
The TCL script describes the whole digital processing chain from the beginning 496 531 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 497 532 (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 498 533 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) 499 534 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. 500 535 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 501 536 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 502 537 comes from a skeleton library. The generic FIR is highly configurable
with the number of coefficients and the size of the coefficients. The coefficients 503 538 with the number of coefficients and the size of the coefficients. The coefficients
themselves are not stored in the script. 504 539 themselves are not stored in the script.
As the signal is processed in real-time, the output signal is stored as 505 540 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 506 541 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 507 542 implemented FIR cascade transfer function with the design criteria and the expected
transfer function. 508 543 transfer function.
509 544
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 510 545 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 511 546 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 512 547 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 513 548 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
provide a broadband noise source. 514 549 provide a broadband noise source.
The board runs the Linux kernel and surrounding environment produced from the 515 550 The board runs the Linux kernel and surrounding environment produced from the
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 516 551 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 517 552 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
fetching the results is automated. 518 553 fetching the results is automated.
519 554
The deploy script uploads the bitstream to the board ((3) on 520 555 The deploy script uploads the bitstream to the board ((3) on
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 521 556 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
configures the coefficients of the FIR filters. It then waits for the results 522 557 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}). 523 558 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
524 559
Finally, an Octave post-processing script computes the final results thanks to 525 560 Finally, an Octave post-processing script computes the final results thanks to
the output data ((5) on figure~\ref{fig:workflow}). 526 561 the output data ((5) on figure~\ref{fig:workflow}).
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 527 562 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
and the different configurations can be compared. 528 563 and the different configurations can be compared.
529 564
\section{Maximizing the rejection at fixed silicon area} 530 565 \section{Maximizing the rejection at fixed silicon area}
\label{sec:fixed_area} 531 566 \label{sec:fixed_area}
This section presents the output of the filter solver {\em i.e.} the computed 532 567 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. 533 568 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. 534 569 Such results allow for understanding the choices made by the solver to compute its solutions.
535 570
The experimental setup is composed of three cases. The raw input is generated 536 571 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 $\Pi^I$. 537 572 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 538 573 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. 539 574 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 $\pi^C$ 540 575 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 541 576 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. 542 577 result up to five stages ($n = 5$) in the cascaded filter.
543 578
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 544 579 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. 545 580 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. 546 581 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
547 582
\renewcommand{\arraystretch}{1.4} 548 583 \renewcommand{\arraystretch}{1.4}
549 584
\begin{table} 550 585 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 551 586 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
\label{tbl:gurobi_max_500} 552 587 \label{tbl:gurobi_max_500}
\centering 553 588 \centering
{\scalefont{0.77} 554 589 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 555 590 \begin{tabular}{|c|ccccc|c|c|}
\hline 556 591 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 557 592 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 558 593 \hline
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 559 594 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ 560 595 2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\
3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ 561 596 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 \\ 562 597 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 \\ 563 598 5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\
\hline 564 599 \hline
\end{tabular} 565 600 \end{tabular}
} 566 601 }
\end{table} 567 602 \end{table}
568 603
\begin{table} 569 604 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 570 605 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
\label{tbl:gurobi_max_1000} 571 606 \label{tbl:gurobi_max_1000}
\centering 572 607 \centering
{\scalefont{0.77} 573 608 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 574 609 \begin{tabular}{|c|ccccc|c|c|}
\hline 575 610 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 576 611 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 577 612 \hline
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 578 613 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ 579 614 2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\
3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ 580 615 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 \\ 581 616 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 \\ 582 617 5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\
\hline 583 618 \hline
\end{tabular} 584 619 \end{tabular}
} 585 620 }
\end{table} 586 621 \end{table}
587 622
\begin{table} 588 623 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 589 624 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
\label{tbl:gurobi_max_1500} 590 625 \label{tbl:gurobi_max_1500}
\centering 591 626 \centering
{\scalefont{0.77} 592 627 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 593 628 \begin{tabular}{|c|ccccc|c|c|}
\hline 594 629 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 595 630 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 596 631 \hline
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 597 632 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ 598 633 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\
3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ 599 634 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 \\ 600 635 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 \\ 601 636 5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\
\hline 602 637 \hline
\end{tabular} 603 638 \end{tabular}
} 604 639 }
\end{table} 605 640 \end{table}
606 641
\renewcommand{\arraystretch}{1} 607 642 \renewcommand{\arraystretch}{1}
608 643
From these tables, we can first state that the more stages are used to define 609 644 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 610 645 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 611 646 been previously observed that many small filters are better than
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 612 647 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 613 648 being hardly used in practice due to the lack of tools for identifying individual filter
coefficients in the cascaded approach. 614 649 coefficients in the cascaded approach.
615 650
Second, the larger the silicon area, the better the rejection. This was also an 616 651 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 617 652 expected result as more area means a filter of better quality with more coefficients
or more bits per coefficient. 618 653 or more bits per coefficient.
619 654
Then, we also observe that the first stage can have a larger shift than the other 620 655 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 621 656 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 622 657 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} 623 658 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
gives the relation between both values. 624 659 gives the relation between both values.
625 660
Finally, we note that the solver consumes all the given silicon area. 626 661 Finally, we note that the solver consumes all the given silicon area.
627 662
The following graphs present the rejection for real data on the FPGA. In all the following 628 663 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 629 664 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 630 665 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. 631 666 given by the quadratic solver. The configurations are those computed in the previous section.
632 667
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 633 668 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. 634 669 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. 635 670 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
636 671
% \begin{figure} 637 672 % \begin{figure}
% \centering 638 673 % \centering
% \includegraphics[width=\linewidth]{images/max_500} 639 674 % \includegraphics[width=\linewidth]{images/max_500}
% \caption{Signal spectrum for MAX/500} 640 675 % \caption{Signal spectrum for MAX/500}
% \label{fig:max_500_result} 641 676 % \label{fig:max_500_result}
% \end{figure} 642 677 % \end{figure}
% 643 678 %
% \begin{figure} 644 679 % \begin{figure}
% \centering 645 680 % \centering
% \includegraphics[width=\linewidth]{images/max_1000} 646 681 % \includegraphics[width=\linewidth]{images/max_1000}
% \caption{Signal spectrum for MAX/1000} 647 682 % \caption{Signal spectrum for MAX/1000}
% \label{fig:max_1000_result} 648 683 % \label{fig:max_1000_result}
% \end{figure} 649 684 % \end{figure}
% 650 685 %
% \begin{figure} 651 686 % \begin{figure}
% \centering 652 687 % \centering
% \includegraphics[width=\linewidth]{images/max_1500} 653 688 % \includegraphics[width=\linewidth]{images/max_1500}
% \caption{Signal spectrum for MAX/1500} 654 689 % \caption{Signal spectrum for MAX/1500}
% \label{fig:max_1500_result} 655 690 % \label{fig:max_1500_result}
% \end{figure} 656 691 % \end{figure}
657 692
% r2.14 et r2.15 et r2.16 658 693 % r2.14 et r2.15 et r2.16
\begin{figure} 659 694 \begin{figure}
\centering 660 695 \centering
\begin{subfigure}{\linewidth} 661 696 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_500} 662 697 \includegraphics[width=\linewidth]{images/max_500}
\caption{Signal spectrum for MAX/500} 663 698 \caption{Signal spectrum for MAX/500}
\label{fig:max_500_result} 664 699 \label{fig:max_500_result}
\end{subfigure} 665 700 \end{subfigure}
666 701
\begin{subfigure}{\linewidth} 667 702 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1000} 668 703 \includegraphics[width=\linewidth]{images/max_1000}
\caption{Signal spectrum for MAX/1000} 669 704 \caption{Signal spectrum for MAX/1000}
\label{fig:max_1000_result} 670 705 \label{fig:max_1000_result}
\end{subfigure} 671 706 \end{subfigure}
672 707
\begin{subfigure}{\linewidth} 673 708 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1500} 674 709 \includegraphics[width=\linewidth]{images/max_1500}
\caption{Signal spectrum for MAX/1500} 675 710 \caption{Signal spectrum for MAX/1500}
\label{fig:max_1500_result} 676 711 \label{fig:max_1500_result}
\end{subfigure} 677 712 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500} 678 713 \caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500}
\end{figure} 679 714 \end{figure}
680 715
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 681 716 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
682 717
We compare the actual silicon resources given by Vivado to the 683 718 We compare the actual silicon resources given by Vivado to the
resources in arbitrary units. 684 719 resources in arbitrary units.
The goal is to check that our arbitrary units of silicon area models well enough 685 720 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 686 721 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 687 722 number of arbitrary units, the actual silicon resources do not depend on the
number of stages $n$. Most significantly, our approach aims 688 723 number of stages $n$. Most significantly, our approach aims
at remaining far enough from the practical logic gate implementation used by 689 724 at remaining far enough from the practical logic gate implementation used by
various vendors to remain platform independent and be portable from one 690 725 various vendors to remain platform independent and be portable from one
architecture to another. 691 726 architecture to another.
692 727
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 693 728 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 694 729 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 695 730 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 696 731 the FIR filters and remove additional processing blocks including FIFO and Programmable
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 697 732 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
698 733
\begin{table}[h!tb] 699 734 \begin{table}[h!tb]
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 700 735 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage} 701 736 \label{tbl:resources_usage}
\centering 702 737 \centering
\begin{tabular}{|c|c|ccc|c|} 703 738 \begin{tabular}{|c|c|ccc|c|}
\hline 704 739 \hline
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 705 740 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 249 & 453 & 627 & \emph{17600} \\ 706 741 & LUT & 249 & 453 & 627 & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 707 742 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 708 743 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
& LUT & 2374 & 5494 & 691 & \emph{17600} \\ 709 744 & LUT & 2374 & 5494 & 691 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 710 745 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 70 & \emph{80} \\ \hline 711 746 & DSP & 0 & 0 & 70 & \emph{80} \\ \hline
& LUT & 2443 & 3304 & 3521 & \emph{17600} \\ 712 747 & LUT & 2443 & 3304 & 3521 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 713 748 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 19 & 35 & \emph{80} \\ \hline 714 749 & DSP & 0 & 19 & 35 & \emph{80} \\ \hline
& LUT & 2634 & 3753 & 2557 & \emph{17600} \\ 715 750 & LUT & 2634 & 3753 & 2557 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & \emph{120} \\ 716 751 4 & BRAM & 4 & 4 & 4 & \emph{120} \\
& DPS & 0 & 19 & 46 & \emph{80} \\ \hline 717 752 & DPS & 0 & 19 & 46 & \emph{80} \\ \hline
& LUT & 2423 & 3047 & 2847 & \emph{17600} \\ 718 753 & LUT & 2423 & 3047 & 2847 & \emph{17600} \\
5 & BRAM & 5 & 5 & 5 & \emph{120} \\ 719 754 5 & BRAM & 5 & 5 & 5 & \emph{120} \\
& DPS & 0 & 22 & 46 & \emph{80} \\ \hline 720 755 & DPS & 0 & 22 & 46 & \emph{80} \\ \hline
\end{tabular} 721 756 \end{tabular}
\end{table} 722 757 \end{table}
723 758
In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 724 759 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 725 760 when the filter coefficients are small enough, or when the input size is small
enough, Vivado optimizes resource consumption by selecting multiplexers to 726 761 enough, Vivado optimizes resource consumption by selecting multiplexers to
implement the multiplications instead of a DSP. In this case, it is quite difficult 727 762 implement the multiplications instead of a DSP. In this case, it is quite difficult
to compare the whole silicon budget. 728 763 to compare the whole silicon budget.
729 764
However, a rough estimation can be made with a simple equivalence: looking at 730 765 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 $n \geq 2$, 731 766 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 732 767 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, 733 768 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 734 769 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 735 770 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 736 771 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. 737 772 by the optimizations done by Vivado based on the detailed map of available processing resources.
738 773
We now present the computation time needed to solve the quadratic problem. 739 774 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 740 775 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 741 776 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 742 777 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. 743 778 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
744 779
\begin{table}[h!tb] 745 780 \begin{table}[h!tb]
\caption{Time needed to solve the quadratic program with Gurobi} 746 781 \caption{Time needed to solve the quadratic program with Gurobi}
\label{tbl:area_time} 747 782 \label{tbl:area_time}
\centering 748 783 \centering
\begin{tabular}{|c|c|c|c|}\hline 749 784 \begin{tabular}{|c|c|c|c|}\hline
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 750 785 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
1 & 0.1~s & 0.1~s & 0.3~s \\ 751 786 1 & 0.1~s & 0.1~s & 0.3~s \\
2 & 1.1~s & 2.2~s & 12~s \\ 752 787 2 & 1.1~s & 2.2~s & 12~s \\
3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ 753 788 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) \\ 754 789 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 755 790 5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline
\end{tabular} 756 791 \end{tabular}
\end{table} 757 792 \end{table}
758 793
As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? 759 794 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 760 795 When the area is limited, the design exploration space is more limited and the solver is able to
find an optimal solution faster. 761 796 find an optimal solution faster.
762 797
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 763 798 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
764 799
This section presents the results of the complementary quadratic program aimed at 765 800 This section presents the results of the complementary quadratic program aimed at
minimizing the area occupation for a targeted rejection level. 766 801 minimizing the area occupation for a targeted rejection level.
767 802
The experimental setup is composed of four cases. The raw input is the same 768 803 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 $\Pi^I$. 769 804 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. 770 805 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. 771 806 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. 772 807 The number of configurations $p$ is the same as previous section.
773 808
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 774 809 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. 775 810 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. 776 811 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. 777 812 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
778 813
\renewcommand{\arraystretch}{1.4} 779 814 \renewcommand{\arraystretch}{1.4}
780 815
\begin{table}[h!tb] 781 816 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 782 817 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\label{tbl:gurobi_min_40} 783 818 \label{tbl:gurobi_min_40}
\centering 784 819 \centering
{\scalefont{0.77} 785 820 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 786 821 \begin{tabular}{|c|ccccc|c|c|}
\hline 787 822 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 788 823 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 789 824 \hline
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 790 825 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ 791 826 2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\
3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ 792 827 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 \\ 793 828 4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\
\hline 794 829 \hline
\end{tabular} 795 830 \end{tabular}
} 796 831 }
\end{table} 797 832 \end{table}
798 833
\begin{table}[h!tb] 799 834 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 800 835 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\label{tbl:gurobi_min_60} 801 836 \label{tbl:gurobi_min_60}
\centering 802 837 \centering
{\scalefont{0.77} 803 838 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 804 839 \begin{tabular}{|c|ccccc|c|c|}
\hline 805 840 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 806 841 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 807 842 \hline
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 808 843 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ 809 844 2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\
3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ 810 845 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 \\ 811 846 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 \\ 812 847 5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\
\hline 813 848 \hline
\end{tabular} 814 849 \end{tabular}
} 815 850 }
\end{table} 816 851 \end{table}
817 852
\begin{table}[h!tb] 818 853 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 819 854 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\label{tbl:gurobi_min_80} 820 855 \label{tbl:gurobi_min_80}
\centering 821 856 \centering
{\scalefont{0.77} 822 857 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 823 858 \begin{tabular}{|c|ccccc|c|c|}
\hline 824 859 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 825 860 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 826 861 \hline
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 827 862 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ 828 863 2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\
3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ 829 864 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 \\ 830 865 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 \\ 831 866 5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\
\hline 832 867 \hline
\end{tabular} 833 868 \end{tabular}
} 834 869 }
\end{table} 835 870 \end{table}
836 871
\begin{table}[h!tb] 837 872 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 838 873 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\label{tbl:gurobi_min_100} 839 874 \label{tbl:gurobi_min_100}
\centering 840 875 \centering
{\scalefont{0.77} 841 876 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 842 877 \begin{tabular}{|c|ccccc|c|c|}
\hline 843 878 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 844 879 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 845 880 \hline
1 & - & - & - & - & - & - & - \\ 846 881 1 & - & - & - & - & - & - & - \\
2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\ 847 882 2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\
3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\ 848 883 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 \\ 849 884 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 \\ 850 885 5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\
\hline 851 886 \hline
\end{tabular} 852 887 \end{tabular}
} 853 888 }
\end{table} 854 889 \end{table}
\renewcommand{\arraystretch}{1} 855 890 \renewcommand{\arraystretch}{1}
856 891
From these tables, we can first state that almost all configurations reach the targeted rejection 857 892 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 858 893 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 859 894 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. 860 895 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 861 896 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 862 897 (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. 863 898 respectively). More generally, the more filters are cascaded, the lower the occupied area.
864 899
Like in previous section, the solver chooses always a little filter as first 865 900 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 866 901 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 867 902 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 868 903 signal and in the second filter selecting a better filter to improve rejection without
having too many bits in the output data. 869 904 having too many bits in the output data.
870 905
For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 871 906 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 872 907 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 $n = 4$. 873 908 solution is equivalent to the result for $n = 4$.
874 909
The following graphs present the rejection for real data on the FPGA. In all the following 875 910 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 876 911 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 877 912 data on the FPGA as measured experimentally and the dashed line is the noise level
given by the quadratic solver. 878 913 given by the quadratic solver.
879 914
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 880 915 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. 881 916 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. 882 917 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. 883 918 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
884 919
% \begin{figure} 885 920 % \begin{figure}
% \centering 886 921 % \centering
% \includegraphics[width=\linewidth]{images/min_40} 887 922 % \includegraphics[width=\linewidth]{images/min_40}
% \caption{Signal spectrum for MIN/40} 888 923 % \caption{Signal spectrum for MIN/40}
% \label{fig:min_40} 889 924 % \label{fig:min_40}
% \end{figure} 890 925 % \end{figure}
% 891 926 %
% \begin{figure} 892 927 % \begin{figure}
% \centering 893 928 % \centering
% \includegraphics[width=\linewidth]{images/min_60} 894 929 % \includegraphics[width=\linewidth]{images/min_60}
% \caption{Signal spectrum for MIN/60} 895 930 % \caption{Signal spectrum for MIN/60}
% \label{fig:min_60} 896 931 % \label{fig:min_60}
% \end{figure} 897 932 % \end{figure}
% 898 933 %
% \begin{figure} 899 934 % \begin{figure}
% \centering 900 935 % \centering
% \includegraphics[width=\linewidth]{images/min_80} 901 936 % \includegraphics[width=\linewidth]{images/min_80}
% \caption{Signal spectrum for MIN/80} 902 937 % \caption{Signal spectrum for MIN/80}
% \label{fig:min_80} 903 938 % \label{fig:min_80}
% \end{figure} 904 939 % \end{figure}
% 905 940 %
% \begin{figure} 906 941 % \begin{figure}
% \centering 907 942 % \centering
% \includegraphics[width=\linewidth]{images/min_100} 908 943 % \includegraphics[width=\linewidth]{images/min_100}
% \caption{Signal spectrum for MIN/100} 909 944 % \caption{Signal spectrum for MIN/100}
% \label{fig:min_100} 910 945 % \label{fig:min_100}
% \end{figure} 911 946 % \end{figure}
912 947
% r2.14 et r2.15 et r2.16 913 948 % r2.14 et r2.15 et r2.16
\begin{figure} 914 949 \begin{figure}
\centering 915 950 \centering
\begin{subfigure}{\linewidth} 916 951 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_40} 917 952 \includegraphics[width=\linewidth]{images/min_40}
\caption{Signal spectrum for MIN/40} 918 953 \caption{Signal spectrum for MIN/40}
\label{fig:min_40} 919 954 \label{fig:min_40}
\end{subfigure} 920 955 \end{subfigure}
921 956
\begin{subfigure}{\linewidth} 922 957 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_60} 923 958 \includegraphics[width=\linewidth]{images/min_60}
\caption{Signal spectrum for MIN/60} 924 959 \caption{Signal spectrum for MIN/60}
\label{fig:min_60} 925 960 \label{fig:min_60}
\end{subfigure} 926 961 \end{subfigure}
927 962
\begin{subfigure}{\linewidth} 928 963 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_80} 929 964 \includegraphics[width=\linewidth]{images/min_80}
\caption{Signal spectrum for MIN/80} 930 965 \caption{Signal spectrum for MIN/80}
\label{fig:min_80} 931 966 \label{fig:min_80}
\end{subfigure} 932 967 \end{subfigure}
933 968
\begin{subfigure}{\linewidth} 934 969 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_100} 935 970 \includegraphics[width=\linewidth]{images/min_100}
\caption{Signal spectrum for MIN/100} 936 971 \caption{Signal spectrum for MIN/100}
\label{fig:min_100} 937 972 \label{fig:min_100}
\end{subfigure} 938 973 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100} 939 974 \caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100}
\end{figure} 940 975 \end{figure}
941 976
We observe that all rejections given by the quadratic solver are close to the experimentally 942 977 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 943 978 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. 944 979 respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.
945 980
Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60; 946 981 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 947 982 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 948 983 have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and PL to 949 984 the FIR filters and remove additional processing blocks including FIFO and PL to
PS communication. 950 985 PS communication.
951 986
\renewcommand{\arraystretch}{1.2} 952 987 \renewcommand{\arraystretch}{1.2}
\begin{table} 953 988 \begin{table}
ifcs2018_journal_reponse.tex
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%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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%Sincerely, 67 67 %Sincerely,
% 68 68 %
%Giorgio Santarelli 69 69 %Giorgio Santarelli
%Associate Editor in Chief 70 70 %Associate Editor in Chief
%Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 71 71 %Transactions on Ultrasonics, Ferroelectrics, and Frequency Control
% 72 72 %
%**************************************************** 73 73 %****************************************************
%REVIEWERS' COMMENTS: 74 74 %REVIEWERS' COMMENTS:
75 75
\documentclass[a4paper]{article} 76 76 \documentclass[a4paper]{article}
\usepackage{fullpage,graphicx} 77 77 \usepackage{fullpage,graphicx,amsmath}
\begin{document} 78 78 \begin{document}
{\bf Reviewer: 1} 79 79 {\bf Reviewer: 1}
80 80
%Comments to the Author 81 81 %Comments to the Author
%In general, the language/grammar is adequate. 82 82 %In general, the language/grammar is adequate.
83 83
{\bf 84 84 {\bf
On page 2, "...allowing to save processing resource..." could be improved. % r1.1 85 85 On page 2, "...allowing to save processing resource..." could be improved. % r1.1
} 86 86 }
87 87
The sentence was split and now reads ``number of coefficients irrelevant: processing 88 88 The sentence was split and now reads ``number of coefficients irrelevant: processing
resources are hence saved by shrinking the filter length.'' 89 89 resources are hence saved by shrinking the filter length.''
90 90
{\bf 91 91 {\bf
On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 92 92 On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2
the author meant.} 93 93 the author meant.}
94 94
Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s) 95 95 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 96 Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
97 97
{\bf 98 98 {\bf
On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 99 99 On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3
could be improved. 100 100 could be improved.
} 101 101 }
102 102
Indeed this paragraph has be written again and now reads as\\ 103 103 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 104 104 ``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 105 105 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. 106 106 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 107 107 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 108 108 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. 109 109 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.
'' 110 110 ''
111 111
{\bf 112 112 {\bf
I appreciate that the authors attempted and document two optimizations: that % r1.4 - en attente des résultats 113 113 I appreciate that the authors attempted and document two optimizations: that % r1.4 - en attente des résultats
of maximum rejection ratio at fixed silicon area, as well as minimum silicon 114 114 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 115 115 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 116 116 useful to compare the results of both optimization paths to the performance and
resource-utilization of generic low-pass filter gateware offered by device 117 117 resource-utilization of generic low-pass filter gateware offered by device
manufacturers. I appreciate also that the authors have presented source code 118 118 manufacturers. I appreciate also that the authors have presented source code
for examination online. 119 119 for examination online.
} 120 120 }
121 121
TODO : FIR Compiler et regarder les ressources pour un FIR comparable a ceux monolithiques 122 122 TODO : FIR Compiler et regarder les ressources pour un FIR comparable a ceux monolithiques
fournis dans l'article (memes coefs et meme nombre de coefs) 123 123 fournis dans l'article (memes coefs et meme nombre de coefs)
124 124
{\bf 125 125 {\bf
Reviewer: 2 126 126 Reviewer: 2
} 127 127 }
128 128
%Comments to the Author 129 129 %Comments to the Author
%In the Manuscript, the Authors describe an optimization methodology for filter 130 130 %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 131 131 %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 132 132 %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, 133 133 %focus on filtering and shifting while the other aspects, in particular decimation,
%will be considered in a future work. The optimization problem is modelled 134 134 %will be considered in a future work. The optimization problem is modelled
%theoretically and then solved by means of a commercial software. The solutions 135 135 %theoretically and then solved by means of a commercial software. The solutions
%are tested experimentally on the Redpitaya platform with synthetic and real 136 136 %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: 137 137 %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 138 138 %maximum rejection given a fixed amount of resources and minimum resource
%utilization given a fixed amount of rejection. 139 139 %utilization given a fixed amount of rejection.
%The Authors find that filtering improves significantly when the number of 140 140 %The Authors find that filtering improves significantly when the number of
%filters increases. 141 141 %filters increases.
%A lot of work has been done in generalizing and automating the procedure so 142 142 %A lot of work has been done in generalizing and automating the procedure so
%that different approaches can be investigated quickly and efficiently. The 143 143 %that different approaches can be investigated quickly and efficiently. The
%results presented in the Manuscript seem to be just a case study based on 144 144 %results presented in the Manuscript seem to be just a case study based on
%the particular criterion chosen by the Authors. Different criteria, in 145 145 %the particular criterion chosen by the Authors. Different criteria, in
%general, could lead to different results and it is important to consider 146 146 %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 147 147 %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 148 148 %is adequate to compare the performance of filters and if multi-stage
%filters are really superior than monolithic filters. 149 149 %filters are really superior than monolithic filters.
150 150
{\bf 151 151 {\bf
By observing the results presented in fig. 10-16, it is clear that the % r2.1 - fait 152 152 By observing the results presented in fig. 10-16, it is clear that the % r2.1 - fait
performances of multi-stage filters are obtained at the expense of their 153 153 performances of multi-stage filters are obtained at the expense of their
selectivity and, in this sense, the filters presented in these figures 154 154 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, 155 155 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 156 156 the attenuation is almost 15 dB for n = 5, while it is not noticeable for
n = 1. 157 157 n = 1.
} 158 158 }
159 159
We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined 160 160 We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined
the bandpass and the bandstop of the filter. 161 161 the bandpass and the bandstop of the filter.
162 162
%Peut etre refaire une serie de simulation dans lesquelles on impose une coupure 163 163 %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 164 164 %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 165 165 %au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque
%a la cascade de filtres. 166 166 %a la cascade de filtres.
%AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de 167 167 %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 168 168 %relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai
169 169
JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait 170 170 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 171 171 moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB
quand on cascade 5 filtres ! 172 172 quand on cascade 5 filtres !
173 173
{\bf 174 174 {\bf
The reason is in the criterion that considers the average attenuation in % r2.2 - fait 175 175 The reason is in the criterion that considers the average attenuation in % r2.2 - fait
the pass band. This criterion does not take into account the maximum attenuation 176 176 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 177 177 in this region, which is a very important parameter for specifying a filter
and for evaluating its performance. For example, with this criterion, a 178 178 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 179 179 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 180 180 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. 181 181 and in the results that are obtained and has to be reconsidered.
} 182 182 }
183 183
The manuscript erroneously stated that we considered the mean of the absolute 184 184 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 185 185 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 186 186 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 187 187 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 188 188 selected. The manuscript now states ``Our criterion to compute the filter rejection considers
% r2.8 et r2.2 r2.3 189 189 % r2.8 et r2.2 r2.3
the maximum magnitude within the stopband, to which the {sum of the absolute values 190 190 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}.'' 191 191 within the passband is subtracted to avoid filters with excessive ripples}.''
192 192
{\bf 193 193 {\bf
I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait 194 194 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 195 195 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 196 196 fixing its value to a typical one, as it has been done for the transition
bandwidth. 197 197 bandwidth.
} 198 198 }
199 199
See above: the absolute value within the passband will reject filters with 200 200 See above: the absolute value within the passband will reject filters with
excessive ripples, including excessive attenuation, within the passband. 201 201 excessive ripples, including excessive attenuation, within the passband.
202 202
{\bf 203 203 {\bf
In addition, I suggest to address the following points: % r2.4 204 204 In addition, I suggest to address the following points: % r2.4
- Page 1, line 50: the Authors state that IIR have shorter impulse response 205 205 - 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. 206 206 than FIR. This is not true in general. The sentence should be reconsidered.
} 207 207 }
208 208
We have not stated that the IIR has a shorter impulse response but a shorter lag. 209 209 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 210 210 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 211 211 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 212 212 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 213 213 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 214 214 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 215 215 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, 216 216 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 217 217 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. 218 218 minimized to avoid oscillation conditions.''
'' 219
220 219
{\bf 221 220 {\bf
- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 222 221 - Fig. 4: the Author should motivate in the text why it has been chosen % r2.5
this transition bandwidth and if it is a typical requirement for phase-noise 223 222 this transition bandwidth and if it is a typical requirement for phase-noise
metrology. 224 223 metrology.
} 225 224 }
226 225
The purpose of the paper is to demonstrate how a given filter shape can be achieved by 227 226 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 228 227 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 229 228 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 230 229 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 231 230 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, 232 231 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\% 233 232 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 234 233 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 235 234 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} 236 235 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.'' 237 236 as described below is indeed unique for each filter shape.''
238 237
{\bf 239 238 {\bf
- The impact of the coefficient resolution is discussed. What about the % r2.6 - fait 240 239 - 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 241 240 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 242 241 used in the analysis? If not, how is it changed with respect to the
coefficient resolution? 243 242 coefficient resolution?
} 244 243 }
245 244
We have now stated in the beginning of the document that ``we have not included the PRN generator 246 245 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 247 246 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. 248 247 hardware.'' so indeed the input datastream resolution is considered as a given.
249 248
{\bf 250 249 {\bf
- Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait 251 250 - Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait
Fig. 5 can be removed. 252 251 Fig. 5 can be removed.
- Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait 253 252 - Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait
It should be ``minimum'' 254 253 It should be ``minimum''
} 255 254 }
AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max 256 255 AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max
de la magnitude. J'ai corrigé en ce sens. 257 256 de la magnitude. J'ai corrigé en ce sens.
258 257
{\bf 259 258 {\bf
- Page e, line 55, second column: ``takin'' % r2.9 - fait 260 259 - Page e, line 55, second column: ``takin'' % r2.9 - fait
- Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait 261 260 - Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait
- Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait 262 261 - Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait
} 263 262 }
264 263
All typos and grammatical errors have been corrected. 265 264 All typos and grammatical errors have been corrected.
266 265
{\bf 267 266 {\bf
- Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait 268 267 - 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? 269 268 these particular configurations? Are they chosen automatically?
} 270 269 }
270 See below: we have added a better description of $p$ during the transformation explanation.
271 ``we introduce $p$ FIR configurations.
272 This variable must be defined by the user, it represent the number of different
273 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
274 functions from GNU Octave)''
271 275
JMF : repondre 272
273
{\bf 274 276 {\bf
- Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait 275 277 - Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait
and non-quadratic to a quadratic?} 276 278 and non-quadratic to a quadratic?}
277 279
280 The first model is non-quadratic but when we introduce the $p$ configurations,
281 we can estimate the function $F$ by computing
282 the rejection for each configuration, so the model become quadratic because we have
283 some multiplication between variables. With the definition of $\delta_{ij}$ we can
284 replace the multiplication between variables by multiplication with binary variable and
285 this one can be linearise as follow:\\
286 $y$ is a binary variable \\
287 $x$ is a real variable bounded by $X^{max}$ \\
288 \begin{equation*}
289 m = x \times y \implies
290 \left \{
291 \begin{split}
292 m & \geq 0 \\
293 m & \leq y \times X^{max} \\
294 m & \leq x \\
295 m & \geq x - (1 - y) \times X^{max} \\
296 \end{split}
297 \right .
298 \end{equation*}
299 Gurobi does the linearization so we don't explain this step to keep the model more
300 simple. However, to improve the transformation explanation we have rewrote the
301 paragraph ``This model is non-linear and even non-quadratic...''.
302
JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article 278 303 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 279 304 je ne comprends pas comment ca repond a la question
280 305
306 AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
307
308 Le problème n'est pas linéaire car nous multiplions des variables
309 entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
310 des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
311 quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
312 quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
313 $\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
314 de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
315 cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
316 nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
317 qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
318 linéarisation pour nous.
319
320
{\bf 281 321 {\bf
- Captions of figure and tables are too minimal. % r2.14 282 322 - Captions of figure and tables are too minimal. % r2.14
323 }
324 We have change the captions of fig 10-16.
325
326 {\bf
- Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait 283 327 - 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. 284 328 of a single figure. Same for fig. 13-16.
} 285 329 }