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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
{\color{red} 408 408 {\color{red}
This model is non-linear since we multiply some variable with another variable 409 409 This model is non-linear since we multiply some variable with another variable
and it is even non-quadratic, as $F$ does not have a known 410 410 and it is even non-quadratic, as $F$ does not have a known
linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations. 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). So $C_{ij}$ and $\pi_{ij}^C$ become constant and 414 414 functions from GNU Octave). So $C_{ij}$ and $\pi_{ij}^C$ become constant and
we defined $1 \leq j \leq p$ and the function $F$ can be estimate for each configurations 415 415 we defined $1 \leq j \leq p$ and the function $F$ can be estimate for each configurations
thanks our rejection criterion. We also defined binary 416 416 thanks our rejection criterion. We also defined binary
variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 417 417 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
and 0 otherwise. The new equations are as follows: 418 418 and 0 otherwise. The new equations are as follows:
} 419 419 }
420 420
\begin{align} 421 421 \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} \\ 422 422 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} \\ 423 423 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} \\ 424 424 \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} 425 425 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
\end{align} 426 426 \end{align}
427 427
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 428 428 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 429 429 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. 430 430 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
431 431
{\color{red} 432 432 {\color{red}
However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply 433 433 However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply
$\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can 434 434 $\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can
linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size 435 435 linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size
we define $0 < \pi_i^- \leq 128$ which is the maximal data size that we can process. 436 436 we define $0 < \pi_i^- \leq 128$ which is the maximal data size that we can process.
} 437 437 }
Moreover the Gurobi 438 438 Moreover the Gurobi
(\url{www.gurobi.com}) optimization software is used to solve this quadratic 439 439 (\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 440 440 model, and since Gurobi is able to linearize, the model is left as is. This model
has $O(np)$ variables and $O(n)$ constraints. 441 441 has $O(np)$ variables and $O(n)$ constraints.
442 442
% This model is non-linear and even non-quadratic, as $F$ does not have a known 443 443 % This model is non-linear and even non-quadratic, as $F$ does not have a known
% linear or quadratic expression. We introduce $p$ FIR configurations 444 444 % linear or quadratic expression. We introduce $p$ FIR configurations
% $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 445 445 % $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
% % r2.12 446 446 % % r2.12
% This variable must be defined by the user, it represent the number of different 447 447 % 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} 448 448 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
% functions from GNU Octave). 449 449 % functions from GNU Octave).
% We define binary 450 450 % We define binary
% variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 451 451 % variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
% and 0 otherwise. The new equations are as follows: 452 452 % and 0 otherwise. The new equations are as follows:
% 453 453 %
% \begin{align} 454 454 % \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} \\ 455 455 % 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} \\
% r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\ 456 456 % 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} \\ 457 457 % \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} \\
% \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} 458 458 % \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
% \end{align} 459 459 % \end{align}
% 460 460 %
% Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 461 461 % Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
% respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 462 462 % 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. 463 463 % Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
% 464 464 %
% % r2.13 465 465 % % r2.13
% This modified model is quadratic since we multiply two variables in the 466 466 % This modified model is quadratic since we multiply two variables in the
% equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary. 467 467 % equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
% The Gurobi 468 468 % The Gurobi
% (\url{www.gurobi.com}) optimization software is used to solve this quadratic 469 469 % (\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 470 470 % model, and since Gurobi is able to linearize, the model is left as is. This model
% has $O(np)$ variables and $O(n)$ constraints. 471 471 % has $O(np)$ variables and $O(n)$ constraints.
472 472
Two problems will be addressed using the workflow described in the next section: on the one 473 473 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 474 474 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 475 475 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 476 476 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
objective function is replaced with: 477 477 objective function is replaced with:
\begin{align} 478 478 \begin{align}
\text{Minimize } & \sum_{i=1}^n a_i \notag 479 479 \text{Minimize } & \sum_{i=1}^n a_i \notag
\end{align} 480 480 \end{align}
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 481 481 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 482 482 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
rejection required. 483 483 rejection required.
484 484
\begin{align} 485 485 \begin{align}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 486 486 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
\end{align} 487 487 \end{align}
488 488
\section{Design workflow} 489 489 \section{Design workflow}
\label{sec:workflow} 490 490 \label{sec:workflow}
491 491
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 492 492 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 493 493 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
in the computation of the results. 494 494 in the computation of the results.
495 495
\begin{figure} 496 496 \begin{figure}
\centering 497 497 \centering
\begin{tikzpicture}[node distance=0.75cm and 2cm] 498 498 \begin{tikzpicture}[node distance=0.75cm and 2cm]
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 499 499 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\node (Start) [left= 3cm of Solver] { } ; 500 500 \node (Start) [left= 3cm of Solver] { } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 501 501 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\node (Input) [above= of TCL] { } ; 502 502 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 503 503 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 504 504 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 505 505 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 506 506 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node (Results) [left= of Postproc] { } ; 507 507 \node (Results) [left= of Postproc] { } ;
508 508
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 509 509 \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) ; 510 510 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 511 511 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 512 512 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 513 513 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 514 514 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 515 515 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 516 516 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 517 517 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->] (Postproc) -- (Results) ; 518 518 \draw[->] (Postproc) -- (Results) ;
\end{tikzpicture} 519 519 \end{tikzpicture}
\caption{Design workflow from the input parameters to the results} 520 520 \caption{Design workflow from the input parameters to the results}
\label{fig:workflow} 521 521 \label{fig:workflow}
\end{figure} 522 522 \end{figure}
523 523
The filter solver is a C++ program that takes as input the maximum area 524 524 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 $\Pi^I$, 525 525 $\mathcal{A}$, 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 526 526 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. 527 527 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}) 528 528 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
and a deploy script ((1b) on figure~\ref{fig:workflow}). 529 529 and a deploy script ((1b) on figure~\ref{fig:workflow}).
530 530
The TCL script describes the whole digital processing chain from the beginning 531 531 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 532 532 (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 533 533 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) 534 534 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. 535 535 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 536 536 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 537 537 comes from a skeleton library. The generic FIR is highly configurable
with the number of coefficients and the size of the coefficients. The coefficients 538 538 with the number of coefficients and the size of the coefficients. The coefficients
themselves are not stored in the script. 539 539 themselves are not stored in the script.
As the signal is processed in real-time, the output signal is stored as 540 540 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 541 541 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 542 542 implemented FIR cascade transfer function with the design criteria and the expected
transfer function. 543 543 transfer function.
544 544
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 545 545 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 546 546 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 547 547 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 548 548 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
provide a broadband noise source. 549 549 provide a broadband noise source.
The board runs the Linux kernel and surrounding environment produced from the 550 550 The board runs the Linux kernel and surrounding environment produced from the
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 551 551 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 552 552 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
fetching the results is automated. 553 553 fetching the results is automated.
554 554
The deploy script uploads the bitstream to the board ((3) on 555 555 The deploy script uploads the bitstream to the board ((3) on
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 556 556 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
configures the coefficients of the FIR filters. It then waits for the results 557 557 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}). 558 558 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
559 559
Finally, an Octave post-processing script computes the final results thanks to 560 560 Finally, an Octave post-processing script computes the final results thanks to
the output data ((5) on figure~\ref{fig:workflow}). 561 561 the output data ((5) on figure~\ref{fig:workflow}).
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 562 562 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
and the different configurations can be compared. 563 563 and the different configurations can be compared.
564 564
\section{Maximizing the rejection at fixed silicon area} 565 565 \section{Maximizing the rejection at fixed silicon area}
\label{sec:fixed_area} 566 566 \label{sec:fixed_area}
This section presents the output of the filter solver {\em i.e.} the computed 567 567 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. 568 568 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. 569 569 Such results allow for understanding the choices made by the solver to compute its solutions.
570 570
The experimental setup is composed of three cases. The raw input is generated 571 571 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 $\Pi^I$. 572 572 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 573 573 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. 574 574 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 $\pi^C$ 575 575 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 576 576 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. 577 577 result up to five stages ($n = 5$) in the cascaded filter.
578 578
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 579 579 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. 580 580 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. 581 581 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
582 582
\renewcommand{\arraystretch}{1.4} 583 583 \renewcommand{\arraystretch}{1.4}
584 584
\begin{table} 585 585 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 586 586 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
\label{tbl:gurobi_max_500} 587 587 \label{tbl:gurobi_max_500}
\centering 588 588 \centering
{\scalefont{0.77} 589 589 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 590 590 \begin{tabular}{|c|ccccc|c|c|}
\hline 591 591 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 592 592 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 593 593 \hline
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 594 594 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ 595 595 2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\
3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ 596 596 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 \\ 597 597 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 \\ 598 598 5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\
\hline 599 599 \hline
\end{tabular} 600 600 \end{tabular}
} 601 601 }
\end{table} 602 602 \end{table}
603 603
\begin{table} 604 604 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 605 605 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
\label{tbl:gurobi_max_1000} 606 606 \label{tbl:gurobi_max_1000}
\centering 607 607 \centering
{\scalefont{0.77} 608 608 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 609 609 \begin{tabular}{|c|ccccc|c|c|}
\hline 610 610 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 611 611 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 612 612 \hline
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 613 613 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ 614 614 2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\
3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ 615 615 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 \\ 616 616 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 \\ 617 617 5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\
\hline 618 618 \hline
\end{tabular} 619 619 \end{tabular}
} 620 620 }
\end{table} 621 621 \end{table}
622 622
\begin{table} 623 623 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 624 624 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
\label{tbl:gurobi_max_1500} 625 625 \label{tbl:gurobi_max_1500}
\centering 626 626 \centering
{\scalefont{0.77} 627 627 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 628 628 \begin{tabular}{|c|ccccc|c|c|}
\hline 629 629 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 630 630 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 631 631 \hline
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 632 632 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ 633 633 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\
3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ 634 634 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 \\ 635 635 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 \\ 636 636 5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\
\hline 637 637 \hline
\end{tabular} 638 638 \end{tabular}
} 639 639 }
\end{table} 640 640 \end{table}
641 641
\renewcommand{\arraystretch}{1} 642 642 \renewcommand{\arraystretch}{1}
643 643
From these tables, we can first state that the more stages are used to define 644 644 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 645 645 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 646 646 been previously observed that many small filters are better than
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 647 647 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 648 648 being hardly used in practice due to the lack of tools for identifying individual filter
coefficients in the cascaded approach. 649 649 coefficients in the cascaded approach.
650 650
Second, the larger the silicon area, the better the rejection. This was also an 651 651 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 652 652 expected result as more area means a filter of better quality with more coefficients
or more bits per coefficient. 653 653 or more bits per coefficient.
654 654
Then, we also observe that the first stage can have a larger shift than the other 655 655 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 656 656 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 657 657 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} 658 658 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
gives the relation between both values. 659 659 gives the relation between both values.
660 660
Finally, we note that the solver consumes all the given silicon area. 661 661 Finally, we note that the solver consumes all the given silicon area.
662 662
The following graphs present the rejection for real data on the FPGA. In all the following 663 663 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 664 664 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 665 665 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. 666 666 given by the quadratic solver. The configurations are those computed in the previous section.
667 667
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 668 668 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. 669 669 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. 670 670 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
671 671
% \begin{figure} 672 672 % \begin{figure}
% \centering 673 673 % \centering
% \includegraphics[width=\linewidth]{images/max_500} 674 674 % \includegraphics[width=\linewidth]{images/max_500}
% \caption{Signal spectrum for MAX/500} 675 675 % \caption{Signal spectrum for MAX/500}
% \label{fig:max_500_result} 676 676 % \label{fig:max_500_result}
% \end{figure} 677 677 % \end{figure}
% 678 678 %
% \begin{figure} 679 679 % \begin{figure}
% \centering 680 680 % \centering
% \includegraphics[width=\linewidth]{images/max_1000} 681 681 % \includegraphics[width=\linewidth]{images/max_1000}
% \caption{Signal spectrum for MAX/1000} 682 682 % \caption{Signal spectrum for MAX/1000}
% \label{fig:max_1000_result} 683 683 % \label{fig:max_1000_result}
% \end{figure} 684 684 % \end{figure}
% 685 685 %
% \begin{figure} 686 686 % \begin{figure}
% \centering 687 687 % \centering
% \includegraphics[width=\linewidth]{images/max_1500} 688 688 % \includegraphics[width=\linewidth]{images/max_1500}
% \caption{Signal spectrum for MAX/1500} 689 689 % \caption{Signal spectrum for MAX/1500}
% \label{fig:max_1500_result} 690 690 % \label{fig:max_1500_result}
% \end{figure} 691 691 % \end{figure}
692 692
% r2.14 et r2.15 et r2.16 693 693 % r2.14 et r2.15 et r2.16
\begin{figure} 694 694 \begin{figure}
\centering 695 695 \centering
\begin{subfigure}{\linewidth} 696 696 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_500} 697 697 \includegraphics[width=\linewidth]{images/max_500}
\caption{Signal spectrum for MAX/500} 698 698 \caption{Signal spectrum for MAX/500}
\label{fig:max_500_result} 699 699 \label{fig:max_500_result}
\end{subfigure} 700 700 \end{subfigure}
701 701
\begin{subfigure}{\linewidth} 702 702 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1000} 703 703 \includegraphics[width=\linewidth]{images/max_1000}
\caption{Signal spectrum for MAX/1000} 704 704 \caption{Signal spectrum for MAX/1000}
\label{fig:max_1000_result} 705 705 \label{fig:max_1000_result}
\end{subfigure} 706 706 \end{subfigure}
707 707
\begin{subfigure}{\linewidth} 708 708 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1500} 709 709 \includegraphics[width=\linewidth]{images/max_1500}
\caption{Signal spectrum for MAX/1500} 710 710 \caption{Signal spectrum for MAX/1500}
\label{fig:max_1500_result} 711 711 \label{fig:max_1500_result}
\end{subfigure} 712 712 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500} 713 713 \caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500}
\end{figure} 714 714 \end{figure}
715 715
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 716 716 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
717 717
We compare the actual silicon resources given by Vivado to the 718 718 We compare the actual silicon resources given by Vivado to the
resources in arbitrary units. 719 719 resources in arbitrary units.
The goal is to check that our arbitrary units of silicon area models well enough 720 720 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 721 721 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 722 722 number of arbitrary units, the actual silicon resources do not depend on the
number of stages $n$. Most significantly, our approach aims 723 723 number of stages $n$. Most significantly, our approach aims
at remaining far enough from the practical logic gate implementation used by 724 724 at remaining far enough from the practical logic gate implementation used by
various vendors to remain platform independent and be portable from one 725 725 various vendors to remain platform independent and be portable from one
architecture to another. 726 726 architecture to another.
727 727
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 728 728 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 729 729 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 730 730 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 731 731 the FIR filters and remove additional processing blocks including FIFO and Programmable
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 732 732 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
733 733
\begin{table}[h!tb] 734 734 \begin{table}[h!tb]
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 735 735 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage} 736 736 \label{tbl:resources_usage}
\centering 737 737 \centering
\begin{tabular}{|c|c|ccc|c|} 738 738 \begin{tabular}{|c|c|ccc|c|}
\hline 739 739 \hline
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 740 740 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 249 & 453 & 627 & \emph{17600} \\ 741 741 & LUT & 249 & 453 & 627 & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 742 742 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 743 743 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
& LUT & 2374 & 5494 & 691 & \emph{17600} \\ 744 744 & LUT & 2374 & 5494 & 691 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 745 745 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 70 & \emph{80} \\ \hline 746 746 & DSP & 0 & 0 & 70 & \emph{80} \\ \hline
& LUT & 2443 & 3304 & 3521 & \emph{17600} \\ 747 747 & LUT & 2443 & 3304 & 3521 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 748 748 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 19 & 35 & \emph{80} \\ \hline 749 749 & DSP & 0 & 19 & 35 & \emph{80} \\ \hline
& LUT & 2634 & 3753 & 2557 & \emph{17600} \\ 750 750 & LUT & 2634 & 3753 & 2557 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & \emph{120} \\ 751 751 4 & BRAM & 4 & 4 & 4 & \emph{120} \\
& DPS & 0 & 19 & 46 & \emph{80} \\ \hline 752 752 & DPS & 0 & 19 & 46 & \emph{80} \\ \hline
& LUT & 2423 & 3047 & 2847 & \emph{17600} \\ 753 753 & LUT & 2423 & 3047 & 2847 & \emph{17600} \\
5 & BRAM & 5 & 5 & 5 & \emph{120} \\ 754 754 5 & BRAM & 5 & 5 & 5 & \emph{120} \\
& DPS & 0 & 22 & 46 & \emph{80} \\ \hline 755 755 & DPS & 0 & 22 & 46 & \emph{80} \\ \hline
\end{tabular} 756 756 \end{tabular}
\end{table} 757 757 \end{table}
758 758
In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 759 759 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 760 760 when the filter coefficients are small enough, or when the input size is small
enough, Vivado optimizes resource consumption by selecting multiplexers to 761 761 enough, Vivado optimizes resource consumption by selecting multiplexers to
implement the multiplications instead of a DSP. In this case, it is quite difficult 762 762 implement the multiplications instead of a DSP. In this case, it is quite difficult
to compare the whole silicon budget. 763 763 to compare the whole silicon budget.
764 764
However, a rough estimation can be made with a simple equivalence: looking at 765 765 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 $n \geq 2$, 766 766 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 767 767 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, 768 768 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 769 769 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 770 770 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 771 771 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. 772 772 by the optimizations done by Vivado based on the detailed map of available processing resources.
773 773
We now present the computation time needed to solve the quadratic problem. 774 774 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 775 775 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 776 776 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 777 777 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. 778 778 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
779 779
\begin{table}[h!tb] 780 780 \begin{table}[h!tb]
\caption{Time needed to solve the quadratic program with Gurobi} 781 781 \caption{Time needed to solve the quadratic program with Gurobi}
\label{tbl:area_time} 782 782 \label{tbl:area_time}
\centering 783 783 \centering
\begin{tabular}{|c|c|c|c|}\hline 784 784 \begin{tabular}{|c|c|c|c|}\hline
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 785 785 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
1 & 0.1~s & 0.1~s & 0.3~s \\ 786 786 1 & 0.1~s & 0.1~s & 0.3~s \\
2 & 1.1~s & 2.2~s & 12~s \\ 787 787 2 & 1.1~s & 2.2~s & 12~s \\
3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ 788 788 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) \\ 789 789 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 790 790 5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline
\end{tabular} 791 791 \end{tabular}
\end{table} 792 792 \end{table}
793 793
As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? 794 794 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 795 795 When the area is limited, the design exploration space is more limited and the solver is able to
find an optimal solution faster. 796 796 find an optimal solution faster.
797 797
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 798 798 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
799 799
This section presents the results of the complementary quadratic program aimed at 800 800 This section presents the results of the complementary quadratic program aimed at
minimizing the area occupation for a targeted rejection level. 801 801 minimizing the area occupation for a targeted rejection level.
802 802
The experimental setup is composed of four cases. The raw input is the same 803 803 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 $\Pi^I$. 804 804 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. 805 805 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. 806 806 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. 807 807 The number of configurations $p$ is the same as previous section.
808 808
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 809 809 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. 810 810 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. 811 811 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. 812 812 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
813 813
\renewcommand{\arraystretch}{1.4} 814 814 \renewcommand{\arraystretch}{1.4}
815 815
\begin{table}[h!tb] 816 816 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 817 817 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\label{tbl:gurobi_min_40} 818 818 \label{tbl:gurobi_min_40}
\centering 819 819 \centering
{\scalefont{0.77} 820 820 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 821 821 \begin{tabular}{|c|ccccc|c|c|}
\hline 822 822 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 823 823 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 824 824 \hline
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 825 825 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ 826 826 2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\
3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ 827 827 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 \\ 828 828 4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\
\hline 829 829 \hline
\end{tabular} 830 830 \end{tabular}
} 831 831 }
\end{table} 832 832 \end{table}
833 833
\begin{table}[h!tb] 834 834 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 835 835 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\label{tbl:gurobi_min_60} 836 836 \label{tbl:gurobi_min_60}
\centering 837 837 \centering
{\scalefont{0.77} 838 838 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 839 839 \begin{tabular}{|c|ccccc|c|c|}
\hline 840 840 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 841 841 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 842 842 \hline
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 843 843 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ 844 844 2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\
3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ 845 845 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 \\ 846 846 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 \\ 847 847 5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\
\hline 848 848 \hline
\end{tabular} 849 849 \end{tabular}
} 850 850 }
\end{table} 851 851 \end{table}
852 852
\begin{table}[h!tb] 853 853 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 854 854 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\label{tbl:gurobi_min_80} 855 855 \label{tbl:gurobi_min_80}
\centering 856 856 \centering
{\scalefont{0.77} 857 857 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 858 858 \begin{tabular}{|c|ccccc|c|c|}
\hline 859 859 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 860 860 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 861 861 \hline
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 862 862 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ 863 863 2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\
3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ 864 864 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 \\ 865 865 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 \\ 866 866 5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\
\hline 867 867 \hline
\end{tabular} 868 868 \end{tabular}
} 869 869 }
\end{table} 870 870 \end{table}
871 871
\begin{table}[h!tb] 872 872 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 873 873 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\label{tbl:gurobi_min_100} 874 874 \label{tbl:gurobi_min_100}
\centering 875 875 \centering
{\scalefont{0.77} 876 876 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 877 877 \begin{tabular}{|c|ccccc|c|c|}
\hline 878 878 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 879 879 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 880 880 \hline
1 & - & - & - & - & - & - & - \\ 881 881 1 & - & - & - & - & - & - & - \\
2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\ 882 882 2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\
3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\ 883 883 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 \\ 884 884 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 \\ 885 885 5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\
\hline 886 886 \hline
\end{tabular} 887 887 \end{tabular}
} 888 888 }
\end{table} 889 889 \end{table}
\renewcommand{\arraystretch}{1} 890 890 \renewcommand{\arraystretch}{1}
891 891
From these tables, we can first state that almost all configurations reach the targeted rejection 892 892 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 893 893 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 894 894 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. 895 895 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 896 896 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 897 897 (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. 898 898 respectively). More generally, the more filters are cascaded, the lower the occupied area.
899 899
Like in previous section, the solver chooses always a little filter as first 900 900 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 901 901 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 902 902 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 903 903 signal and in the second filter selecting a better filter to improve rejection without
having too many bits in the output data. 904 904 having too many bits in the output data.
905 905
For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 906 906 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 907 907 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 $n = 4$. 908 908 solution is equivalent to the result for $n = 4$.
909 909
The following graphs present the rejection for real data on the FPGA. In all the following 910 910 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 911 911 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 912 912 data on the FPGA as measured experimentally and the dashed line is the noise level
given by the quadratic solver. 913 913 given by the quadratic solver.
914 914
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 915 915 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. 916 916 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. 917 917 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. 918 918 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
919 919
% \begin{figure} 920 920 % \begin{figure}
% \centering 921 921 % \centering
% \includegraphics[width=\linewidth]{images/min_40} 922 922 % \includegraphics[width=\linewidth]{images/min_40}
% \caption{Signal spectrum for MIN/40} 923 923 % \caption{Signal spectrum for MIN/40}
% \label{fig:min_40} 924 924 % \label{fig:min_40}
% \end{figure} 925 925 % \end{figure}
% 926 926 %
% \begin{figure} 927 927 % \begin{figure}
% \centering 928 928 % \centering
% \includegraphics[width=\linewidth]{images/min_60} 929 929 % \includegraphics[width=\linewidth]{images/min_60}
% \caption{Signal spectrum for MIN/60} 930 930 % \caption{Signal spectrum for MIN/60}
% \label{fig:min_60} 931 931 % \label{fig:min_60}
% \end{figure} 932 932 % \end{figure}
% 933 933 %
% \begin{figure} 934 934 % \begin{figure}
% \centering 935 935 % \centering
% \includegraphics[width=\linewidth]{images/min_80} 936 936 % \includegraphics[width=\linewidth]{images/min_80}
% \caption{Signal spectrum for MIN/80} 937 937 % \caption{Signal spectrum for MIN/80}
% \label{fig:min_80} 938 938 % \label{fig:min_80}
% \end{figure} 939 939 % \end{figure}
% 940 940 %
% \begin{figure} 941 941 % \begin{figure}
% \centering 942 942 % \centering
% \includegraphics[width=\linewidth]{images/min_100} 943 943 % \includegraphics[width=\linewidth]{images/min_100}
% \caption{Signal spectrum for MIN/100} 944 944 % \caption{Signal spectrum for MIN/100}
% \label{fig:min_100} 945 945 % \label{fig:min_100}
% \end{figure} 946 946 % \end{figure}
947 947
% r2.14 et r2.15 et r2.16 948 948 % r2.14 et r2.15 et r2.16
\begin{figure} 949 949 \begin{figure}
\centering 950 950 \centering
\begin{subfigure}{\linewidth} 951 951 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_40} 952 952 \includegraphics[width=\linewidth]{images/min_40}
\caption{Signal spectrum for MIN/40} 953 953 \caption{Signal spectrum for MIN/40}
\label{fig:min_40} 954 954 \label{fig:min_40}
\end{subfigure} 955 955 \end{subfigure}
956 956
\begin{subfigure}{\linewidth} 957 957 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_60} 958 958 \includegraphics[width=\linewidth]{images/min_60}
\caption{Signal spectrum for MIN/60} 959 959 \caption{Signal spectrum for MIN/60}
\label{fig:min_60} 960 960 \label{fig:min_60}
\end{subfigure} 961 961 \end{subfigure}
962 962
\begin{subfigure}{\linewidth} 963 963 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_80} 964 964 \includegraphics[width=\linewidth]{images/min_80}
\caption{Signal spectrum for MIN/80} 965 965 \caption{Signal spectrum for MIN/80}
\label{fig:min_80} 966 966 \label{fig:min_80}
\end{subfigure} 967 967 \end{subfigure}
968 968
\begin{subfigure}{\linewidth} 969 969 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_100} 970 970 \includegraphics[width=\linewidth]{images/min_100}
\caption{Signal spectrum for MIN/100} 971 971 \caption{Signal spectrum for MIN/100}
\label{fig:min_100} 972 972 \label{fig:min_100}
\end{subfigure} 973 973 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100} 974 974 \caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100}
\end{figure} 975 975 \end{figure}
976 976
We observe that all rejections given by the quadratic solver are close to the experimentally 977 977 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 978 978 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. 979 979 respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.
980 980
Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60; 981 981 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 982 982 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 983 983 have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and PL to 984 984 the FIR filters and remove additional processing blocks including FIFO and PL to
PS communication. 985 985 PS communication.
986 986
\renewcommand{\arraystretch}{1.2} 987 987 \renewcommand{\arraystretch}{1.2}
\begin{table} 988 988 \begin{table}
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 989 989 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage_comp} 990 990 \label{tbl:resources_usage_comp}
\centering 991 991 \centering
{\scalefont{0.90} 992 992 {\scalefont{0.90}
ifcs2018_journal_reponse.tex
Diff comments   View file @ 56f7c40
%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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%Giorgio Santarelli 69 69 %Giorgio Santarelli
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% 72 72 %
%**************************************************** 73 73 %****************************************************
%REVIEWERS' COMMENTS: 74 74 %REVIEWERS' COMMENTS:
75 75
\documentclass[a4paper]{article} 76 76 \documentclass[a4paper]{article}
\usepackage{fullpage,graphicx,amsmath} 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 - fait
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 To compare the performance of our FIR filters and the performance of device
fournis dans l'article (memes coefs et meme nombre de coefs) 123 123 manufacturers generic filter, we have added a paragraph and a table at the
124 end of experiments section. We compare the resources consumption with the same
125 FIR coefficients set.
124 126
{\bf 125 127 {\bf
Reviewer: 2 126 128 Reviewer: 2
} 127 129 }
128 130
%Comments to the Author 129 131 %Comments to the Author
%In the Manuscript, the Authors describe an optimization methodology for filter 130 132 %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 131 133 %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 132 134 %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, 133 135 %focus on filtering and shifting while the other aspects, in particular decimation,
%will be considered in a future work. The optimization problem is modelled 134 136 %will be considered in a future work. The optimization problem is modelled
%theoretically and then solved by means of a commercial software. The solutions 135 137 %theoretically and then solved by means of a commercial software. The solutions
%are tested experimentally on the Redpitaya platform with synthetic and real 136 138 %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: 137 139 %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 138 140 %maximum rejection given a fixed amount of resources and minimum resource
%utilization given a fixed amount of rejection. 139 141 %utilization given a fixed amount of rejection.
%The Authors find that filtering improves significantly when the number of 140 142 %The Authors find that filtering improves significantly when the number of
%filters increases. 141 143 %filters increases.
%A lot of work has been done in generalizing and automating the procedure so 142 144 %A lot of work has been done in generalizing and automating the procedure so
%that different approaches can be investigated quickly and efficiently. The 143 145 %that different approaches can be investigated quickly and efficiently. The
%results presented in the Manuscript seem to be just a case study based on 144 146 %results presented in the Manuscript seem to be just a case study based on
%the particular criterion chosen by the Authors. Different criteria, in 145 147 %the particular criterion chosen by the Authors. Different criteria, in
%general, could lead to different results and it is important to consider 146 148 %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 147 149 %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 148 150 %is adequate to compare the performance of filters and if multi-stage
%filters are really superior than monolithic filters. 149 151 %filters are really superior than monolithic filters.
150 152
{\bf 151 153 {\bf
By observing the results presented in fig. 10-16, it is clear that the % r2.1 - fait 152 154 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 153 155 performances of multi-stage filters are obtained at the expense of their
selectivity and, in this sense, the filters presented in these figures 154 156 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, 155 157 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 156 158 the attenuation is almost 15 dB for n = 5, while it is not noticeable for
n = 1. 157 159 n = 1.
} 158 160 }
159 161
We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined 160 162 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. 161 163 the bandpass and the bandstop of the filter.
162 164
%Peut etre refaire une serie de simulation dans lesquelles on impose une coupure 163 165 We are aware of this non equivalence but we think that difference is not due to
%non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte 164 166 the cascaded filters but due to the definition of rejection criterion on the passband.
%au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque 165 167 Indeed, in this article we have choose to take the summation of absolute values divide
%a la cascade de filtres. 166 168 by the bandwidth but this criterion is maybe too permissive and when we cascade
%AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de 167 169 some filters this impact is more important.
%relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai 168
169 170
JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait 170 171 However if we change the passband
moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB 171 172 criterion by the summation of absolute value in passband, weighting given to the
quand on cascade 5 filtres ! 172 173 passband ripples are too strong and the solver are too restricted to provide
174 any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in
175 passband, the rejection evaluation are too close form the original criterion and
176 the result will not be improved.
173 177
178 In this article, we will highlight the methodology instead of the filter conception.
179 Even if our rejection criterion is not the best, our methodology was not impacted
180 by this. So to improve the results, we can choose another criterion to be more
181 selective in passband but it is not the main objective of our article.
182
183 % %Peut etre refaire une serie de simulation dans lesquelles on impose une coupure
184 % %non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte
185 % %au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque
186 % %a la cascade de filtres.
187 % %AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de
188 % %relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai
189 %
190 % densité spectrale de la bande passante
191 % sum des valeurs absolues / largeur de la bande passante (1/N) vs max dans la bande de coupure
192 %
193 % JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait
194 % moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB
195 % quand on cascade 5 filtres !
196 %
197 % Dire que la chute n'est pas du à la casacade mais à notre critère de rejection
198
{\bf 174 199 {\bf
The reason is in the criterion that considers the average attenuation in % r2.2 - fait 175 200 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 176 201 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 177 202 in this region, which is a very important parameter for specifying a filter
and for evaluating its performance. For example, with this criterion, a 178 203 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 179 204 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 180 205 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. 181 206 and in the results that are obtained and has to be reconsidered.
} 182 207 }
183 208
The manuscript erroneously stated that we considered the mean of the absolute 184 209 See above: If we choose the maximum absolute value in passband, we penalize the
value within the bandpass: the manuscript has now been corrected to properly state 185 210 case with 10 dB of ripple.
the selected criterion, namely the {\em sum} of the absolute value, so that any 186
ripple in the bandpass will reduce the chances of a given filter set from being 187
selected. The manuscript now states ``Our criterion to compute the filter rejection considers 188
% r2.8 et r2.2 r2.3 189
the maximum magnitude within the stopband, to which the {sum of the absolute values 190
within the passband is subtracted to avoid filters with excessive ripples}.'' 191
192 211
212 % The manuscript erroneously stated that we considered the mean of the absolute
213 % value within the bandpass: the manuscript has now been corrected to properly state
214 % the selected criterion, namely the {\em sum} of the absolute value, so that any
215 % ripple in the bandpass will reduce the chances of a given filter set from being
216 % selected. The manuscript now states ``Our criterion to compute the filter rejection considers
217 % % r2.8 et r2.2 r2.3
218 % the maximum magnitude within the stopband, to which the {sum of the absolute values
219 % within the passband is subtracted to avoid filters with excessive ripples}.''
220
{\bf 193 221 {\bf
I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait 194 222 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 195 223 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 196 224 fixing its value to a typical one, as it has been done for the transition
bandwidth. 197 225 bandwidth.
} 198 226 }
199 227
See above: the absolute value within the passband will reject filters with 200 228 See above: the absolute value within the passband will reject filters with
excessive ripples, including excessive attenuation, within the passband. 201 229 excessive ripples, including excessive attenuation, within the passband.
202 230
231 % TODO: test max(stopband) - max(abs(passband))
232
{\bf 203 233 {\bf
In addition, I suggest to address the following points: % r2.4 204 234 In addition, I suggest to address the following points: % r2.4
- Page 1, line 50: the Authors state that IIR have shorter impulse response 205 235 - 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. 206 236 than FIR. This is not true in general. The sentence should be reconsidered.
} 207 237 }
208 238
We have not stated that the IIR has a shorter impulse response but a shorter lag. 209 239 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 210 240 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 211 241 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 212 242 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 213 243 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 214 244 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 215 245 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, 216 246 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 217 247 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.'' 218 248 minimized to avoid oscillation conditions.''
219 249
{\bf 220 250 {\bf
- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 221 251 - 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 222 252 this transition bandwidth and if it is a typical requirement for phase-noise
metrology. 223 253 metrology.
} 224 254 }
225 255
The purpose of the paper is to demonstrate how a given filter shape can be achieved by 226 256 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 227 257 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 228 258 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 229 259 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 230 260 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, 231 261 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\% 232 262 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 233 263 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 234 264 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} 235 265 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.'' 236 266 as described below is indeed unique for each filter shape.''
237 267
{\bf 238 268 {\bf
- The impact of the coefficient resolution is discussed. What about the % r2.6 - fait 239 269 - 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 240 270 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 241 271 used in the analysis? If not, how is it changed with respect to the
coefficient resolution? 242 272 coefficient resolution?
} 243 273 }
244 274
We have now stated in the beginning of the document that ``we have not included the PRN generator 245 275 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 246 276 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. 247 277 hardware.'' so indeed the input datastream resolution is considered as a given.
248 278
{\bf 249 279 {\bf
- Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait 250 280 - Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait
Fig. 5 can be removed. 251 281 Fig. 5 can be removed.
- Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait 252 282 - Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait
It should be ``minimum'' 253 283 It should be ``minimum''
} 254 284 }
AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max 255 285 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. 256 286 de la magnitude. J'ai corrigé en ce sens.
257 287
288 Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats
289
{\bf 258 290 {\bf
- Page e, line 55, second column: ``takin'' % r2.9 - fait 259 291 - Page e, line 55, second column: ``takin'' % r2.9 - fait
- Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait 260 292 - Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait
- Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait 261 293 - Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait
} 262 294 }
263 295
All typos and grammatical errors have been corrected. 264 296 All typos and grammatical errors have been corrected.
265 297
{\bf 266 298 {\bf
- Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait 267 299 - 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? 268 300 these particular configurations? Are they chosen automatically?
} 269 301 }
302 C'est le nombre de coefficients et un taille raisonnable
303 Troncature de la pyramide
304
See below: we have added a better description of $p$ during the transformation explanation. 270 305 See below: we have added a better description of $p$ during the transformation explanation.
``we introduce $p$ FIR configurations. 271 306 ``we introduce $p$ FIR configurations.
This variable must be defined by the user, it represent the number of different 272 307 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} 273 308 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave)'' 274 309 functions from GNU Octave)''
275 310
{\bf 276 311 {\bf
- Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait 277 312 - Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait
and non-quadratic to a quadratic?} 278 313 and non-quadratic to a quadratic?}
279 314
The first model is non-quadratic but when we introduce the $p$ configurations, 280 315 The first model is non-quadratic but when we introduce the $p$ configurations,
we can estimate the function $F$ by computing 281 316 we can estimate the function $F$ by computing
the rejection for each configuration, so the model become quadratic because we have 282 317 the rejection for each configuration, so the model become quadratic because we have
some multiplication between variables. With the definition of $\delta_{ij}$ we can 283 318 some multiplication between variables. With the definition of $\delta_{ij}$ we can
replace the multiplication between variables by multiplication with binary variable and 284 319 replace the multiplication between variables by multiplication with binary variable and
this one can be linearise as follow:\\ 285 320 this one can be linearise as follow:\\
$y$ is a binary variable \\ 286 321 $y$ is a binary variable \\
$x$ is a real variable bounded by $X^{max}$ \\ 287 322 $x$ is a real variable bounded by $X^{max}$ \\
\begin{equation*} 288 323 \begin{equation*}
m = x \times y \implies 289 324 m = x \times y \implies
\left \{ 290 325 \left \{
\begin{split} 291 326 \begin{split}
m & \geq 0 \\ 292 327 m & \geq 0 \\
m & \leq y \times X^{max} \\ 293 328 m & \leq y \times X^{max} \\
m & \leq x \\ 294 329 m & \leq x \\
m & \geq x - (1 - y) \times X^{max} \\ 295 330 m & \geq x - (1 - y) \times X^{max} \\
\end{split} 296 331 \end{split}
\right . 297 332 \right .
\end{equation*} 298 333 \end{equation*}
Gurobi does the linearization so we don't explain this step to keep the model more 299 334 Gurobi does the linearization so we don't explain this step to keep the model more
simple. However, to improve the transformation explanation we have rewrote the 300 335 simple. However, to improve the transformation explanation we have rewrote the
paragraph ``This model is non-linear and even non-quadratic...''. 301 336 paragraph ``This model is non-linear and even non-quadratic...''.
302 337
JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article 303 338 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 304 339 je ne comprends pas comment ca repond a la question
305 340
AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux. 306 341 AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
307 342
Le problème n'est pas linéaire car nous multiplions des variables 308 343 Le problème n'est pas linéaire car nous multiplions des variables
entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent 309 344 entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique 310 345 des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore 311 346 quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre 312 347 quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
$\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible 313 348 $\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre 314 349 de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée, 315 350 cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur 316 351 nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la 317 352 qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
linéarisation pour nous. 318 353 linéarisation pour nous.
319 354
320 355
{\bf 321 356 {\bf
- Captions of figure and tables are too minimal. % r2.14 322 357 - Captions of figure and tables are too minimal. % r2.14
} 323 358 }
We have change the captions of fig 10-16. 324 359 We have change the captions of fig 10-16.
325 360
{\bf 326 361 {\bf
- Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait 327 362 - Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait