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# source: https://tex.stackexchange.com/questions/40738/how-to-properly-make-a-latex-project 1 1 # source: https://tex.stackexchange.com/questions/40738/how-to-properly-make-a-latex-project
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% fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee 1 1 % fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee
% demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de 2 2 % demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de
% rejection par bit et perte si moins de bits que rejection/6 3 3 % rejection par bit et perte si moins de bits que rejection/6
% developper programme lineaire en incluant le decalage de bits 4 4 % developper programme lineaire en incluant le decalage de bits
% insister que avant on etait synthetisable mais pas implementable, alors que maintenant on 5 5 % insister que avant on etait synthetisable mais pas implementable, alors que maintenant on
% implemente et on demontre que ca tourne 6 6 % implemente et on demontre que ca tourne
% gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ? 7 7 % gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ?
% Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer 8 8 % Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer
% (zedboard ou redpit) 9 9 % (zedboard ou redpit)
10 10
% label schema : verifier que "argumenter de la cascade de FIR" est fait 11 11 % label schema : verifier que "argumenter de la cascade de FIR" est fait
12 12
\documentclass[a4paper,journal]{IEEEtran/IEEEtran} 13 13 \documentclass[a4paper,journal]{IEEEtran/IEEEtran}
\usepackage{graphicx,color,hyperref} 14 14 \usepackage{graphicx,color,hyperref}
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\usepackage{amsthm} 16 16 \usepackage{amsthm}
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\usepackage{amsmath} 18 18 \usepackage{amsmath}
\usepackage{algorithm2e} 19 19 \usepackage{algorithm2e}
\usepackage{url,balance} 20 20 \usepackage{url,balance}
\usepackage[normalem]{ulem} 21 21 \usepackage[normalem]{ulem}
\usepackage{tikz} 22 22 \usepackage{tikz}
\usetikzlibrary{positioning,fit} 23 23 \usetikzlibrary{positioning,fit}
\usepackage{multirow} 24 24 \usepackage{multirow}
\usepackage{scalefnt} 25 25 \usepackage{scalefnt}
26 \usepackage{caption}
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% correct bad hyphenation here 27 29 % correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor} 28 30 \hyphenation{op-tical net-works semi-conduc-tor}
\textheight=26cm 29 31 \textheight=26cm
\setlength{\footskip}{30pt} 30 32 \setlength{\footskip}{30pt}
\pagenumbering{gobble} 31 33 \pagenumbering{gobble}
\begin{document} 32 34 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 33 35 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 34 36 to oscillator metrology}
35 37
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 36 38 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 37 39 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}} 38 40 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France } 39 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 \\ 40 42 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 41 43 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 42 44 }
\maketitle 43 45 \maketitle
\thispagestyle{plain} 44 46 \thispagestyle{plain}
\pagestyle{plain} 45 47 \pagestyle{plain}
\newtheorem{definition}{Definition} 46 48 \newtheorem{definition}{Definition}
47 49
\begin{abstract} 48 50 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 49 51 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 50 52 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 51 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 52 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 53 55 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 54 56 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. The 55 57 required and keeping timing constraints within the targeted measurement bandwidths. The
presented technique is applicable to scheduling any sequence of processing blocks characterized 56 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 57 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 58 60 characateristics, as is the case for filters with their coefficients and resolution yielding
rejection and number of multipliers. 59 61 rejection and number of multipliers.
\end{abstract} 60 62 \end{abstract}
61 63
\begin{IEEEkeywords} 62 64 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 63 65 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 64 66 \end{IEEEkeywords}
65 67
\section{Digital signal processing of ultrastable clock signals} 66 68 \section{Digital signal processing of ultrastable clock signals}
67 69
Analog oscillator phase noise characteristics are classically performed by downconverting 68 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, 69 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 70 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 71 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}. 72 74 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
73 75
\begin{figure}[h!tb] 74 76 \begin{figure}[h!tb]
\begin{center} 75 77 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 76 78 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 77 79 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 78 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 79 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 80 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 81 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 82 84 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 83 85 the spectral characteristics of the phase fluctuations.}
\label{schema} 84 86 \label{schema}
\end{figure} 85 87 \end{figure}
86 88
As with the analog mixer, 87 89 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 88 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. 89 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 90 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 91 93 for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
downconverter 92 94 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 93 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 94 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 95 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 96 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 97 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 98 100 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 99 101 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 100 102 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 101 103 data being processed.
102 104
\section{Finite impulse response filter} 103 105 \section{Finite impulse response filter}
104 106
We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined 105 107 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 106 108 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 107 109 outputs $y_k$
\begin{align} 108 110 \begin{align}
y_n=\sum_{k=0}^N b_k x_{n-k} 109 111 y_n=\sum_{k=0}^N b_k x_{n-k}
\label{eq:fir_equation} 110 112 \label{eq:fir_equation}
\end{align} 111 113 \end{align}
112 114
As opposed to an implementation on a general purpose processor in which word size is defined by the 113 115 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 114 116 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 115 117 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 116 118 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 117 119 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 118 120 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 119 121 the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language
(VHDL) level. 120 122 (VHDL) level.
Since latency is not an issue in a openloop phase noise characterization instrument, the large 121 123 Since latency is not an issue 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, 122 124 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. 123 125 is not considered as an issue as would be in a closed loop system.
124 126
The coefficients are classically expressed as floating point values. However, this binary 125 127 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, 126 128 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 127 129 we select to quantify these floating point values into integer values. This quantization
will result in some precision loss. 128 130 will result in some precision loss.
129 131
\begin{figure}[h!tb] 130 132 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/zero_values} 131 133 \includegraphics[width=\linewidth]{images/zero_values}
\caption{Impact of the quantization resolution of the coefficients: the quantization is 132 134 \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 133 135 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 134 136 the 30~first and 30~last coefficients out of the initial 128~band-pass
filter coefficients to 0 (red dots).} 135 137 filter coefficients to 0 (red dots).}
\label{float_vs_int} 136 138 \label{float_vs_int}
\end{figure} 137 139 \end{figure}
138 140
The tradeoff between quantization resolution and number of coefficients when considering 139 141 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 140 142 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 141 143 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 142 144 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 143 145 quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the
taps become null, making the large number of coefficients irrelevant and allowing to save 144 146 taps become null, making the large number of coefficients irrelevant and allowing to save
processing resource by shrinking the filter length. This tradeoff aimed at minimizing resources 145 147 processing resource 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 146 148 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 147 149 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 148 150 and tap length, as will be shown in the next section. Indeed, our development strategy closely
follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards} 149 151 follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
in which basic blocks are defined and characterized before being assembled \cite{hide} 150 152 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 151 153 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 152 154 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 153 155 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 154 156 current implementation: the decimation is assumed to be located after the FIR cascade at the
moment. 155 157 moment.
156 158
\section{Methodology description} 157 159 \section{Methodology description}
158 160
Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP) 159 161 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. 160 162 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 161 163 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 162 164 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 163 165 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 164 166 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 165 167 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 166 168 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 167 169 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 168 170 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 169 171 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 170 172 that all solutions found by the solver are synthesized and executed on hardware at the end
of the analysis. 171 173 of the analysis.
172 174
In this demonstration , we focus on only two operations: filtering and shifting the number of 173 175 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. 174 176 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 175 177 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 176 178 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 177 179 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 178 180 requiring pipelined processing at full bandwidth for the earliest steps, including for
time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}. 179 181 time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}.
180 182
Addressing only two operations allows for demonstrating the methodology but should not be 181 183 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 182 184 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. Hence, 183 185 of skeleton blocks as long as perfomance and resource occupation can be determined. Hence,
in this paper we will apply our methodology on simple DSP chains: a white noise input signal 184 186 in this paper we will apply our methodology on simple DSP chains: a white noise input signal
is generated using a Pseudo-Random Number (PRN) generator or thanks at a radiofrequency-grade 185 187 is generated using a Pseudo-Random Number (PRN) generator or thanks at a radiofrequency-grade
Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor. Once samples have been 186 188 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 -- 187 189 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 188 190 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, 189 191 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 190 192 allowing to assess either filter rejection for a given resource usage, or validating the rejection
when implementing a solution minimizing resource occupation. 191 193 when implementing a solution minimizing resource occupation.
192 194
The first step of our approach is to model the DSP chain and since we just optimize 193 195 The first step of our approach is to model the DSP chain and since we just optimize
the filtering, we have not modeling the PRN generator or the ADC. The filtering can be 194 196 the filtering, we have not modeling the PRN generator or the ADC. The filtering can be
done by two ways. The first one we use only one FIR filter with lot of coefficients 195 197 done by two ways. The first one we use only one FIR filter with lot of coefficients
to rejection the noise, we called this approach a monolithic approach. And the second one 196 198 to rejection the noise, we called this approach a monolithic approach. And the second one
we select different FIR filters with less coefficients the monolithic filter and we cascaded 197 199 we select different FIR filters with less coefficients the monolithic filter and we cascaded
it to filtering the signal. 198 200 it to filtering the signal.
199 201
After each filter we leave the possibility of shifting the filtered data to consume 200 202 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 201 203 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). 202 204 and a shifter (the shift could be omitted if we do not need to divide the filtered data).
203 205
\subsection{Model of a FIR filter} 204 206 \subsection{Model of a FIR filter}
205 207
A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$) 206 208 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 $\pi^C_i$ 207 209 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 208 210 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} 209 211 the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage}
shows a filtering stage. 210 212 shows a filtering stage.
211 213
\begin{figure} 212 214 \begin{figure}
\centering 213 215 \centering
\begin{tikzpicture}[node distance=2cm] 214 216 \begin{tikzpicture}[node distance=2cm]
\node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ; 215 217 \node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;
\node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ; 216 218 \node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;
\node (Start) [left of=FIR] { } ; 217 219 \node (Start) [left of=FIR] { } ;
\node (End) [right of=Shift] { } ; 218 220 \node (End) [right of=Shift] { } ;
219 221
\node[draw,fit=(FIR) (Shift)] (Filter) { } ; 220 222 \node[draw,fit=(FIR) (Shift)] (Filter) { } ;
221 223
\draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ; 222 224 \draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;
\draw[->] (FIR) -- (Shift) ; 223 225 \draw[->] (FIR) -- (Shift) ;
\draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ; 224 226 \draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;
\end{tikzpicture} 225 227 \end{tikzpicture}
\caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)} 226 228 \caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}
\label{fig:fir_stage} 227 229 \label{fig:fir_stage}
\end{figure} 228 230 \end{figure}
229 231
FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB. 230 232 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 231 233 This rejection has been computed using GNU Octave software FIR coefficient design functions
(\texttt{firls} and \texttt{fir1}). 232 234 (\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. 233 235 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, 234 236 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. 235 237 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. 236 238 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.
237 239
With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter 238 240 With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter
transfer function. 239 241 transfer function.
Comparing the performance between FIRs requires however defining a unique criterion. As shown in figure~\ref{fig:fir_mag}, 240 242 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 241 243 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}. 242 244 bandpass ripples as emphasized in \cite{lim_1988,lim_1996}.
243 245
\begin{figure} 244 246 \begin{figure}
\begin{center} 245 247 \begin{center}
\scalebox{0.8}{ 246 248 \scalebox{0.8}{
\centering 247 249 \centering
\begin{tikzpicture}[scale=0.3] 248 250 \begin{tikzpicture}[scale=0.3]
\draw[<->] (0,15) -- (0,0) -- (21,0) ; 249 251 \draw[<->] (0,15) -- (0,0) -- (21,0) ;
\draw[thick] (0,12) -- (8,12) -- (20,0) ; 250 252 \draw[thick] (0,12) -- (8,12) -- (20,0) ;
251 253
\draw (0,14) node [left] { $P$ } ; 252 254 \draw (0,14) node [left] { $P$ } ;
\draw (20,0) node [below] { $f$ } ; 253 255 \draw (20,0) node [below] { $f$ } ;
254 256
\draw[>=latex,<->] (0,14) -- (8,14) ; 255 257 \draw[>=latex,<->] (0,14) -- (8,14) ;
\draw (4,14) node [above] { passband } node [below] { $40\%$ } ; 256 258 \draw (4,14) node [above] { passband } node [below] { $40\%$ } ;
257 259
\draw[>=latex,<->] (8,14) -- (12,14) ; 258 260 \draw[>=latex,<->] (8,14) -- (12,14) ;
\draw (10,14) node [above] { transition } node [below] { $20\%$ } ; 259 261 \draw (10,14) node [above] { transition } node [below] { $20\%$ } ;
260 262
\draw[>=latex,<->] (12,14) -- (20,14) ; 261 263 \draw[>=latex,<->] (12,14) -- (20,14) ;
\draw (16,14) node [above] { stopband } node [below] { $40\%$ } ; 262 264 \draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;
263 265
\draw[>=latex,<->] (16,12) -- (16,8) ; 264 266 \draw[>=latex,<->] (16,12) -- (16,8) ;
\draw (16,10) node [right] { rejection } ; 265 267 \draw (16,10) node [right] { rejection } ;
266 268
\draw[dashed] (8,-1) -- (8,14) ; 267 269 \draw[dashed] (8,-1) -- (8,14) ;
\draw[dashed] (12,-1) -- (12,14) ; 268 270 \draw[dashed] (12,-1) -- (12,14) ;
269 271
\draw[dashed] (8,12) -- (16,12) ; 270 272 \draw[dashed] (8,12) -- (16,12) ;
\draw[dashed] (12,8) -- (16,8) ; 271 273 \draw[dashed] (12,8) -- (16,8) ;
272 274
\end{tikzpicture} 273 275 \end{tikzpicture}
} 274 276 }
\end{center} 275 277 \end{center}
\caption{Shape of the filter transmitted power $P$ as a function of frequency $f$: 276 278 \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, 277 279 the passband is considered to occupy the initial 40\% of the Nyquist frequency range,
the stopband the last 40\%, allowing 20\% transition width.} 278 280 the stopband the last 40\%, allowing 20\% transition width.}
\label{fig:fir_mag} 279 281 \label{fig:fir_mag}
\end{figure} 280 282 \end{figure}
281 283
In the transition band, the behavior of the filter is left free, we only care about the passband and the stopband characteristics. 282 284 In the transition band, the behavior of the filter is left free, we only care about the passband and the stopband characteristics.
Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion 283 285 % r2.7
yields unacceptable results since notches overestimate the rejection capability of the filter. Furthermore, the losses within 284 286 % Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion
the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients. 285 287 % yields unacceptable results since notches overestimate the rejection capability of the filter. Furthermore, the losses within
Such biases are compensated for by the second considered criterion which is based on computing the maximum rejection within the stopband minus the mean of the absolute value of passband rejection. With this criterion, the results are significantly improved as shown in figure~\ref{fig:custom_criterion} and meet the expected rejection capability of low pass filters. 286 288 % the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients.
289 Our criterion to compute the filter rejection takes
290 % r2.8 et r2.2 r2.3
291 the maximum magnitude within the stopband minus the sum of the absolute value of passband rejection. With this criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
287 292
\begin{figure} 288 293 % \begin{figure}
\centering 289 294 % \centering
\includegraphics[width=\linewidth]{images/colored_mean_criterion} 290 295 % \includegraphics[width=\linewidth]{images/colored_mean_criterion}
\caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters} 291 296 % \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}
\label{fig:mean_criterion} 292 297 % \label{fig:mean_criterion}
\end{figure} 293 298 % \end{figure}
294 299
\begin{figure} 295 300 \begin{figure}
\centering 296 301 \centering
\includegraphics[width=\linewidth]{images/colored_custom_criterion} 297 302 \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) 298 303 \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} 299 304 comparison between monolithic filter and cascaded filters}
\label{fig:custom_criterion} 300 305 \label{fig:custom_criterion}
\end{figure} 301 306 \end{figure}
302 307
Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps 303 308 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 304 309 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. 305 310 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. 306 311 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. 307 312 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 308 313 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. 309 314 the rejection. Hence the best coefficient set are on the vertex of the pyramid.
310 315
\begin{figure} 311 316 \begin{figure}
\centering 312 317 \centering
\includegraphics[width=\linewidth]{images/rejection_pyramid} 313 318 \includegraphics[width=\linewidth]{images/rejection_pyramid}
\caption{Rejection as a function of number of coefficients and number of bits} 314 319 \caption{Rejection as a function of number of coefficients and number of bits}
\label{fig:rejection_pyramid} 315 320 \label{fig:rejection_pyramid}
\end{figure} 316 321 \end{figure}
317 322
Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps), 318 323 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. 319 324 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: 320 325 If the FIR filter coefficients are the same between the stages, we have:
$$F_{total} = F_1 + F_2$$ 321 326 $$F_{total} = F_1 + F_2$$
But selecting two different sets of coefficient will yield a more complex situation in which 322 327 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 323 328 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. 324 329 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 325 330 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. 326 331 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
Thus, estimating the rejection of filter cascades is more complex than takin the sum of all the rejection 327 332 % r2.9
333 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, 328 334 criteria of each filter. However since the this sum underestimates the rejection capability of the cascade,
this upper bound is considered as a pessimistic and acceptable criterion for deciding on the suitability 329 335 % r2.10
336 this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
of the filter cascade to meet design criteria. 330 337 of the filter cascade to meet design criteria.
331 338
\begin{figure} 332 339 \begin{figure}
\centering 333 340 \centering
\includegraphics[width=\linewidth]{images/cascaded_criterion} 334 341 \includegraphics[width=\linewidth]{images/cascaded_criterion}
\caption{Rejection of two cascaded filters} 335 342 \caption{Rejection of two cascaded filters}
\label{fig:sum_rejection} 336 343 \label{fig:sum_rejection}
\end{figure} 337 344 \end{figure}
338 345
346 % r2.6
347 Finally in our case, we consider that the input signal are fully known. So the
348 resolution of the data stream are fixed and still the same for all experiments
349 in this paper.
350
Based on this analysis, we address the estimate of resource consumption (called 339 351 Based on this analysis, we address the estimate of resource consumption (called
silicon area -- in the case of FPGAs meaning processing cells) as a function of 340 352 % r2.11
353 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 341 354 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, 342 355 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 343 356 and will assess after synthesis the adequation of this arbitrary unit with actual
hardware resources provided by FPGA manufacturers. The sum of individual processing 344 357 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. 345 358 unit areas is constrained by a total silicon area representative of FPGA global resources.
Formally, variable $a_i$ is the area taken by filter~$i$ 346 359 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). 347 360 (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: 348 361 Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
349 362
\begin{align} 350 363 \begin{align}
\text{Maximize } & \sum_{i=1}^n r_i \notag \\ 351 364 \text{Maximize } & \sum_{i=1}^n r_i \notag \\
\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\ 352 365 \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} \\ 353 366 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} \\ 354 367 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} \\ 355 368 \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} \\ 356 369 \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} \\ 357 370 \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} 358 371 \pi_1^- &= \Pi^I \label{eq:init}
\end{align} 359 372 \end{align}
360 373
Equation~\ref{eq:area} states that the total area taken by the filters must be 361 374 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 362 375 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 363 376 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 364 377 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 365 378 $\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the
input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the 366 379 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 367 380 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, 368 381 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 369 382 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 370 383 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 371 384 $\pi_i^C$ 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 372 385 $\pi_i^S$ 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. 373 386 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 374 387 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 375 388 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 376 389 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 377 390 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 378 391 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 379 392 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: 380 393 shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is:
$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. 381 394 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
Finally, equation~\ref{eq:init} gives the number of bits of the global input. 382 395 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
383 396
This model is non-linear and even non-quadratic, as $F$ does not have a known 384 397 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 385 398 linear or quadratic expression. We introduce $p$ FIR configurations
$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. We define binary 386 399 $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
400 % r2.12
401 This variable must be defined by the user, it represent the number of different
402 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
403 functions from GNU Octave).
404 We define binary
variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 387 405 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
and 0 otherwise. The new equations are as follows: 388 406 and 0 otherwise. The new equations are as follows:
389 407
\begin{align} 390 408 \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} \\ 391 409 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} \\ 392 410 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} \\ 393 411 \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} 394 412 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
\end{align} 395 413 \end{align}
396 414
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 397 415 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 398 416 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. 399 417 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
400 418
This modified model is quadratic, and it can be linearised if necessary. The Gurobi 401 419 % r2.13
420 This modified model is quadratic since we multiply two variables in the
421 equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
422 The Gurobi
(\url{www.gurobi.com}) optimization software is used to solve this quadratic 402 423 (\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 403 424 model, and since Gurobi is able to linearize, the model is left as is. This model
has $O(np)$ variables and $O(n)$ constraints. 404 425 has $O(np)$ variables and $O(n)$ constraints.
405 426
Two problems will be addressed using the workflow described in the next section: on the one 406 427 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 407 428 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 408 429 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 409 430 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
objective function is replaced with: 410 431 objective function is replaced with:
\begin{align} 411 432 \begin{align}
\text{Minimize } & \sum_{i=1}^n a_i \notag 412 433 \text{Minimize } & \sum_{i=1}^n a_i \notag
\end{align} 413 434 \end{align}
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 414 435 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 415 436 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
rejection required. 416 437 rejection required.
417 438
\begin{align} 418 439 \begin{align}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 419 440 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
\end{align} 420 441 \end{align}
421 442
\section{Design workflow} 422 443 \section{Design workflow}
\label{sec:workflow} 423 444 \label{sec:workflow}
424 445
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 425 446 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 426 447 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
in the computation of the results. 427 448 in the computation of the results.
428 449
\begin{figure} 429 450 \begin{figure}
\centering 430 451 \centering
\begin{tikzpicture}[node distance=0.75cm and 2cm] 431 452 \begin{tikzpicture}[node distance=0.75cm and 2cm]
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 432 453 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\node (Start) [left= 3cm of Solver] { } ; 433 454 \node (Start) [left= 3cm of Solver] { } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 434 455 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\node (Input) [above= of TCL] { } ; 435 456 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 436 457 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 437 458 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 438 459 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 439 460 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node (Results) [left= of Postproc] { } ; 440 461 \node (Results) [left= of Postproc] { } ;
441 462
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 442 463 \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) ; 443 464 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 444 465 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 445 466 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 446 467 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 447 468 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 448 469 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 449 470 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 450 471 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->] (Postproc) -- (Results) ; 451 472 \draw[->] (Postproc) -- (Results) ;
\end{tikzpicture} 452 473 \end{tikzpicture}
\caption{Design workflow from the input parameters to the results} 453 474 \caption{Design workflow from the input parameters to the results}
\label{fig:workflow} 454 475 \label{fig:workflow}
\end{figure} 455 476 \end{figure}
456 477
The filter solver is a C++ program that takes as input the maximum area 457 478 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 $\Pi^I$, 458 479 $\mathcal{A}$, 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 459 480 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. 460 481 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}) 461 482 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
and a deploy script ((1b) on figure~\ref{fig:workflow}). 462 483 and a deploy script ((1b) on figure~\ref{fig:workflow}).
463 484
The TCL script describes the whole digital processing chain from the beginning 464 485 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 465 486 (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 466 487 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) 467 488 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. 468 489 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 469 490 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 470 491 comes from a skeleton library. The generic FIR is highly configurable
with the number of coefficients and the size of the coefficients. The coefficients 471 492 with the number of coefficients and the size of the coefficients. The coefficients
themselves are not stored in the script. 472 493 themselves are not stored in the script.
As the signal is processed in real-time, the output signal is stored as 473 494 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 474 495 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 475 496 implemented FIR cascade transfer function with the design criteria and the expected
transfer function. 476 497 transfer function.
477 498
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 478 499 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 479 500 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 480 501 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 481 502 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
provide a broadband noise source. 482 503 provide a broadband noise source.
The board runs the Linux kernel and surrounding environment produced from the 483 504 The board runs the Linux kernel and surrounding environment produced from the
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 484 505 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 485 506 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
fetching the results is automated. 486 507 fetching the results is automated.
487 508
The deploy script uploads the bitstream to the board ((3) on 488 509 The deploy script uploads the bitstream to the board ((3) on
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 489 510 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
configures the coefficients of the FIR filters. It then waits for the results 490 511 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}). 491 512 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
492 513
Finally, an Octave post-processing script computes the final results thanks to 493 514 Finally, an Octave post-processing script computes the final results thanks to
the output data ((5) on figure~\ref{fig:workflow}). 494 515 the output data ((5) on figure~\ref{fig:workflow}).
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 495 516 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
and the different configurations can be compared. 496 517 and the different configurations can be compared.
497 518
\section{Maximizing the rejection at fixed silicon area} 498 519 \section{Maximizing the rejection at fixed silicon area}
\label{sec:fixed_area} 499 520 \label{sec:fixed_area}
This section presents the output of the filter solver {\em i.e.} the computed 500 521 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. 501 522 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. 502 523 Such results allow for understanding the choices made by the solver to compute its solutions.
503 524
The experimental setup is composed of three cases. The raw input is generated 504 525 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 $\Pi^I$. 505 526 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 506 527 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. 507 528 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 $\pi^C$ 508 529 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 509 530 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. 510 531 result up to five stages ($n = 5$) in the cascaded filter.
511 532
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 512 533 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. 513 534 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. 514 535 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
515 536
\renewcommand{\arraystretch}{1.4} 516 537 \renewcommand{\arraystretch}{1.4}
517 538
\begin{table} 518 539 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 519 540 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
\label{tbl:gurobi_max_500} 520 541 \label{tbl:gurobi_max_500}
\centering 521 542 \centering
{\scalefont{0.77} 522 543 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 523 544 \begin{tabular}{|c|ccccc|c|c|}
\hline 524 545 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 525 546 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 526 547 \hline
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 527 548 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ 528 549 2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\
3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ 529 550 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 \\ 530 551 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 \\ 531 552 5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\
\hline 532 553 \hline
\end{tabular} 533 554 \end{tabular}
} 534 555 }
\end{table} 535 556 \end{table}
536 557
\begin{table} 537 558 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 538 559 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
\label{tbl:gurobi_max_1000} 539 560 \label{tbl:gurobi_max_1000}
\centering 540 561 \centering
{\scalefont{0.77} 541 562 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 542 563 \begin{tabular}{|c|ccccc|c|c|}
\hline 543 564 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 544 565 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 545 566 \hline
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 546 567 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ 547 568 2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\
3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ 548 569 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 \\ 549 570 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 \\ 550 571 5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\
\hline 551 572 \hline
\end{tabular} 552 573 \end{tabular}
} 553 574 }
\end{table} 554 575 \end{table}
555 576
\begin{table} 556 577 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 557 578 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
\label{tbl:gurobi_max_1500} 558 579 \label{tbl:gurobi_max_1500}
\centering 559 580 \centering
{\scalefont{0.77} 560 581 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 561 582 \begin{tabular}{|c|ccccc|c|c|}
\hline 562 583 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 563 584 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 564 585 \hline
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 565 586 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ 566 587 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\
3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ 567 588 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 \\ 568 589 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 \\ 569 590 5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\
\hline 570 591 \hline
\end{tabular} 571 592 \end{tabular}
} 572 593 }
\end{table} 573 594 \end{table}
574 595
\renewcommand{\arraystretch}{1} 575 596 \renewcommand{\arraystretch}{1}
576 597
From these tables, we can first state that the more stages are used to define 577 598 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 578 599 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 579 600 been previously observed that many small filters are better than
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 580 601 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 581 602 being hardly used in practice due to the lack of tools for identifying individual filter
coefficients in the cascaded approach. 582 603 coefficients in the cascaded approach.
583 604
Second, the larger the silicon area, the better the rejection. This was also an 584 605 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 585 606 expected result as more area means a filter of better quality with more coefficients
or more bits per coefficient. 586 607 or more bits per coefficient.
587 608
Then, we also observe that the first stage can have a larger shift than the other 588 609 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 589 610 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 590 611 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} 591 612 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
gives the relation between both values. 592 613 gives the relation between both values.
593 614
Finally, we note that the solver consumes all the given silicon area. 594 615 Finally, we note that the solver consumes all the given silicon area.
595 616
The following graphs present the rejection for real data on the FPGA. In all the following 596 617 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 597 618 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 598 619 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. 599 620 given by the quadratic solver. The configurations are those computed in the previous section.
600 621
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 601 622 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. 602 623 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. 603 624 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
604 625
\begin{figure} 605 626 % \begin{figure}
\centering 606 627 % \centering
\includegraphics[width=\linewidth]{images/max_500} 607 628 % \includegraphics[width=\linewidth]{images/max_500}
\caption{Signal spectrum for MAX/500} 608 629 % \caption{Signal spectrum for MAX/500}
\label{fig:max_500_result} 609 630 % \label{fig:max_500_result}
\end{figure} 610 631 % \end{figure}
632 %
633 % \begin{figure}
634 % \centering
635 % \includegraphics[width=\linewidth]{images/max_1000}
636 % \caption{Signal spectrum for MAX/1000}
637 % \label{fig:max_1000_result}
638 % \end{figure}
639 %
640 % \begin{figure}
641 % \centering
642 % \includegraphics[width=\linewidth]{images/max_1500}
643 % \caption{Signal spectrum for MAX/1500}
644 % \label{fig:max_1500_result}
645 % \end{figure}
611 646
647 % r2.14 et r2.15 et r2.16
\begin{figure} 612 648 \begin{figure}
\centering 613 649 \centering
\includegraphics[width=\linewidth]{images/max_1000} 614 650 \begin{subfigure}{\linewidth}
\caption{Signal spectrum for MAX/1000} 615 651 \includegraphics[width=\linewidth]{images/max_500}
\label{fig:max_1000_result} 616 652 \caption{Signal spectrum for MAX/500}
\end{figure} 617 653 \label{fig:max_500_result}
654 \end{subfigure}
618 655
\begin{figure} 619 656 \begin{subfigure}{\linewidth}
\centering 620 657 \includegraphics[width=\linewidth]{images/max_1000}
\includegraphics[width=\linewidth]{images/max_1500} 621 658 \caption{Signal spectrum for MAX/1000}
\caption{Signal spectrum for MAX/1500} 622 659 \label{fig:max_1000_result}
\label{fig:max_1500_result} 623 660 \end{subfigure}
661
662 \begin{subfigure}{\linewidth}
663 \includegraphics[width=\linewidth]{images/max_1500}
664 \caption{Signal spectrum for MAX/1500}
665 \label{fig:max_1500_result}
666 \end{subfigure}
667 \caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500}
\end{figure} 624 668 \end{figure}
625 669
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 626 670 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
627 671
We compare the actual silicon resources given by Vivado to the 628 672 We compare the actual silicon resources given by Vivado to the
resources in arbitrary units. 629 673 resources in arbitrary units.
The goal is to check that our arbitrary units of silicon area models well enough 630 674 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 631 675 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 632 676 number of arbitrary units, the actual silicon resources do not depend on the
number of stages $n$. Most significantly, our approach aims 633 677 number of stages $n$. Most significantly, our approach aims
at remaining far enough from the practical logic gate implementation used by 634 678 at remaining far enough from the practical logic gate implementation used by
various vendors to remain platform independent and be portable from one 635 679 various vendors to remain platform independent and be portable from one
architecture to another. 636 680 architecture to another.
637 681
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 638 682 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 639 683 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 640 684 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 641 685 the FIR filters and remove additional processing blocks including FIFO and Programmable
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 642 686 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
643 687
\begin{table}[h!tb] 644 688 \begin{table}[h!tb]
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 645 689 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage} 646 690 \label{tbl:resources_usage}
\centering 647 691 \centering
\begin{tabular}{|c|c|ccc|c|} 648 692 \begin{tabular}{|c|c|ccc|c|}
\hline 649 693 \hline
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 650 694 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 249 & 453 & 627 & \emph{17600} \\ 651 695 & LUT & 249 & 453 & 627 & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 652 696 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 653 697 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
& LUT & 2374 & 5494 & 691 & \emph{17600} \\ 654 698 & LUT & 2374 & 5494 & 691 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 655 699 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 70 & \emph{80} \\ \hline 656 700 & DSP & 0 & 0 & 70 & \emph{80} \\ \hline
& LUT & 2443 & 3304 & 3521 & \emph{17600} \\ 657 701 & LUT & 2443 & 3304 & 3521 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 658 702 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 19 & 35 & \emph{80} \\ \hline 659 703 & DSP & 0 & 19 & 35 & \emph{80} \\ \hline
& LUT & 2634 & 3753 & 2557 & \emph{17600} \\ 660 704 & LUT & 2634 & 3753 & 2557 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & \emph{120} \\ 661 705 4 & BRAM & 4 & 4 & 4 & \emph{120} \\
& DPS & 0 & 19 & 46 & \emph{80} \\ \hline 662 706 & DPS & 0 & 19 & 46 & \emph{80} \\ \hline
& LUT & 2423 & 3047 & 2847 & \emph{17600} \\ 663 707 & LUT & 2423 & 3047 & 2847 & \emph{17600} \\
5 & BRAM & 5 & 5 & 5 & \emph{120} \\ 664 708 5 & BRAM & 5 & 5 & 5 & \emph{120} \\
& DPS & 0 & 22 & 46 & \emph{80} \\ \hline 665 709 & DPS & 0 & 22 & 46 & \emph{80} \\ \hline
\end{tabular} 666 710 \end{tabular}
\end{table} 667 711 \end{table}
668 712
In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 669 713 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 670 714 when the filter coefficients are small enough, or when the input size is small
enough, Vivado optimizes resource consumption by selecting multiplexers to 671 715 enough, Vivado optimizes resource consumption by selecting multiplexers to
implement the multiplications instead of a DSP. In this case, it is quite difficult 672 716 implement the multiplications instead of a DSP. In this case, it is quite difficult
to compare the whole silicon budget. 673 717 to compare the whole silicon budget.
674 718
However, a rough estimation can be made with a simple equivalence: looking at 675 719 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 $n \geq 2$, 676 720 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 677 721 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, 678 722 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 679 723 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 680 724 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 681 725 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. 682 726 by the optimizations done by Vivado based on the detailed map of available processing resources.
683 727
We now present the computation time needed to solve the quadratic problem. 684 728 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 685 729 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 686 730 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 687 731 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. 688 732 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
689 733
\begin{table}[h!tb] 690 734 \begin{table}[h!tb]
\caption{Time needed to solve the quadratic program with Gurobi} 691 735 \caption{Time needed to solve the quadratic program with Gurobi}
\label{tbl:area_time} 692 736 \label{tbl:area_time}
\centering 693 737 \centering
\begin{tabular}{|c|c|c|c|}\hline 694 738 \begin{tabular}{|c|c|c|c|}\hline
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 695 739 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
1 & 0.1~s & 0.1~s & 0.3~s \\ 696 740 1 & 0.1~s & 0.1~s & 0.3~s \\
2 & 1.1~s & 2.2~s & 12~s \\ 697 741 2 & 1.1~s & 2.2~s & 12~s \\
3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ 698 742 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) \\ 699 743 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 700 744 5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline
\end{tabular} 701 745 \end{tabular}
\end{table} 702 746 \end{table}
703 747
As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? 704 748 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 705 749 When the area is limited, the design exploration space is more limited and the solver is able to
find an optimal solution faster. 706 750 find an optimal solution faster.
707 751
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 708 752 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
709 753
This section presents the results of the complementary quadratic program aimed at 710 754 This section presents the results of the complementary quadratic program aimed at
minimizing the area occupation for a targeted rejection level. 711 755 minimizing the area occupation for a targeted rejection level.
712 756
The experimental setup is composed of four cases. The raw input is the same 713 757 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 $\Pi^I$. 714 758 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. 715 759 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. 716 760 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. 717 761 The number of configurations $p$ is the same as previous section.
718 762
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 719 763 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. 720 764 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. 721 765 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. 722 766 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
723 767
\renewcommand{\arraystretch}{1.4} 724 768 \renewcommand{\arraystretch}{1.4}
725 769
\begin{table}[h!tb] 726 770 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 727 771 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\label{tbl:gurobi_min_40} 728 772 \label{tbl:gurobi_min_40}
\centering 729 773 \centering
{\scalefont{0.77} 730 774 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 731 775 \begin{tabular}{|c|ccccc|c|c|}
\hline 732 776 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 733 777 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 734 778 \hline
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 735 779 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ 736 780 2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\
3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ 737 781 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 \\ 738 782 4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\
\hline 739 783 \hline
\end{tabular} 740 784 \end{tabular}
} 741 785 }
\end{table} 742 786 \end{table}
743 787
\begin{table}[h!tb] 744 788 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 745 789 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\label{tbl:gurobi_min_60} 746 790 \label{tbl:gurobi_min_60}
\centering 747 791 \centering
{\scalefont{0.77} 748 792 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 749 793 \begin{tabular}{|c|ccccc|c|c|}
\hline 750 794 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 751 795 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 752 796 \hline
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 753 797 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ 754 798 2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\
3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ 755 799 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 \\ 756 800 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 \\ 757 801 5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\
\hline 758 802 \hline
\end{tabular} 759 803 \end{tabular}
} 760 804 }
\end{table} 761 805 \end{table}
762 806
\begin{table}[h!tb] 763 807 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 764 808 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\label{tbl:gurobi_min_80} 765 809 \label{tbl:gurobi_min_80}
\centering 766 810 \centering
{\scalefont{0.77} 767 811 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 768 812 \begin{tabular}{|c|ccccc|c|c|}
\hline 769 813 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 770 814 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 771 815 \hline
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 772 816 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ 773 817 2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\
3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ 774 818 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 \\ 775 819 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 \\ 776 820 5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\
\hline 777 821 \hline
\end{tabular} 778 822 \end{tabular}
} 779 823 }
\end{table} 780 824 \end{table}
781 825
\begin{table}[h!tb] 782 826 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 783 827 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\label{tbl:gurobi_min_100} 784 828 \label{tbl:gurobi_min_100}
\centering 785 829 \centering
{\scalefont{0.77} 786 830 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 787 831 \begin{tabular}{|c|ccccc|c|c|}
\hline 788 832 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 789 833 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 790 834 \hline
1 & - & - & - & - & - & - & - \\ 791 835 1 & - & - & - & - & - & - & - \\
2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\ 792 836 2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\
3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\ 793 837 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 \\ 794 838 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 \\ 795 839 5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\
\hline 796 840 \hline
\end{tabular} 797 841 \end{tabular}
} 798 842 }
\end{table} 799 843 \end{table}
\renewcommand{\arraystretch}{1} 800 844 \renewcommand{\arraystretch}{1}
801 845
From these tables, we can first state that almost all configurations reach the targeted rejection 802 846 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 803 847 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 804 848 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. 805 849 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 806 850 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 807 851 (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. 808 852 respectively). More generally, the more filters are cascaded, the lower the occupied area.
809 853
Like in previous section, the solver chooses always a little filter as first 810 854 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 811 855 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 812 856 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 813 857 signal and in the second filter selecting a better filter to improve rejection without
having too many bits in the output data. 814 858 having too many bits in the output data.
815 859
For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 816 860 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 817 861 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 $n = 4$. 818 862 solution is equivalent to the result for $n = 4$.
819 863
The following graphs present the rejection for real data on the FPGA. In all the following 820 864 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 821 865 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 822 866 data on the FPGA as measured experimentally and the dashed line is the noise level
given by the quadratic solver. 823 867 given by the quadratic solver.
824 868
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 825 869 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. 826 870 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. 827 871 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. 828 872 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
829 873
\begin{figure} 830 874 % \begin{figure}
\centering 831 875 % \centering
\includegraphics[width=\linewidth]{images/min_40} 832 876 % \includegraphics[width=\linewidth]{images/min_40}
\caption{Signal spectrum for MIN/40} 833 877 % \caption{Signal spectrum for MIN/40}
\label{fig:min_40} 834 878 % \label{fig:min_40}
\end{figure} 835 879 % \end{figure}
880 %
881 % \begin{figure}
882 % \centering
883 % \includegraphics[width=\linewidth]{images/min_60}
884 % \caption{Signal spectrum for MIN/60}
885 % \label{fig:min_60}
886 % \end{figure}
887 %
888 % \begin{figure}
889 % \centering
890 % \includegraphics[width=\linewidth]{images/min_80}
891 % \caption{Signal spectrum for MIN/80}
892 % \label{fig:min_80}
893 % \end{figure}
894 %
895 % \begin{figure}
896 % \centering
897 % \includegraphics[width=\linewidth]{images/min_100}
898 % \caption{Signal spectrum for MIN/100}
899 % \label{fig:min_100}
900 % \end{figure}
836 901
902 % r2.14 et r2.15 et r2.16
\begin{figure} 837 903 \begin{figure}
\centering 838 904 \centering
\includegraphics[width=\linewidth]{images/min_60} 839 905 \begin{subfigure}{\linewidth}
\caption{Signal spectrum for MIN/60} 840 906 \includegraphics[width=\linewidth]{images/min_40}
\label{fig:min_60} 841 907 \caption{Signal spectrum for MIN/40}
\end{figure} 842 908 \label{fig:min_40}
909 \end{subfigure}
843 910
\begin{figure} 844 911 \begin{subfigure}{\linewidth}
\centering 845 912 \includegraphics[width=\linewidth]{images/min_60}
\includegraphics[width=\linewidth]{images/min_80} 846 913 \caption{Signal spectrum for MIN/60}
\caption{Signal spectrum for MIN/80} 847 914 \label{fig:min_60}
\label{fig:min_80} 848 915 \end{subfigure}
\end{figure} 849
850 916
\begin{figure} 851 917 \begin{subfigure}{\linewidth}
\centering 852 918 \includegraphics[width=\linewidth]{images/min_80}
\includegraphics[width=\linewidth]{images/min_100} 853 919 \caption{Signal spectrum for MIN/80}
\caption{Signal spectrum for MIN/100} 854 920 \label{fig:min_80}
\label{fig:min_100} 855 921 \end{subfigure}
922
923 \begin{subfigure}{\linewidth}
924 \includegraphics[width=\linewidth]{images/min_100}
925 \caption{Signal spectrum for MIN/100}
926 \label{fig:min_100}
927 \end{subfigure}
928 \caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100}
\end{figure} 856 929 \end{figure}
857 930
We observe that all rejections given by the quadratic solver are close to the experimentally 858 931 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 859 932 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. 860 933 respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.
861 934
Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60; 862 935 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 863 936 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 864 937 have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and PL to 865 938 the FIR filters and remove additional processing blocks including FIFO and PL to
PS communication. 866 939 PS communication.
867 940
\renewcommand{\arraystretch}{1.2} 868 941 \renewcommand{\arraystretch}{1.2}
\begin{table} 869 942 \begin{table}
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 870 943 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage_comp} 871 944 \label{tbl:resources_usage_comp}
\centering 872 945 \centering
{\scalefont{0.90} 873 946 {\scalefont{0.90}
\begin{tabular}{|c|c|cccc|c|} 874 947 \begin{tabular}{|c|c|cccc|c|}
\hline 875 948 \hline
$n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline 876 949 $n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 343 & 334 & 772 & - & \emph{17600} \\ 877 950 & LUT & 343 & 334 & 772 & - & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\ 878 951 1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\
& DSP & 27 & 39 & 55 & - & \emph{80} \\ \hline 879 952 & DSP & 27 & 39 & 55 & - & \emph{80} \\ \hline
& LUT & 1252 & 2862 & 5099 & 640 & \emph{17600} \\ 880 953 & LUT & 1252 & 2862 & 5099 & 640 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & 2 & \emph{120} \\ 881 954 2 & BRAM & 2 & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 0 & 66 & \emph{80} \\ \hline 882 955 & DSP & 0 & 0 & 0 & 66 & \emph{80} \\ \hline
& LUT & 891 & 2148 & 2023 & 2448 & \emph{17600} \\ 883 956 & LUT & 891 & 2148 & 2023 & 2448 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & 3 & \emph{120} \\ 884 957 3 & BRAM & 3 & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 0 & 19 & 27 & \emph{80} \\ \hline 885 958 & DSP & 0 & 0 & 19 & 27 & \emph{80} \\ \hline
& LUT & 662 & 1729 & 2451 & 2893 & \emph{17600} \\ 886 959 & LUT & 662 & 1729 & 2451 & 2893 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & 4 & \emph{120} \\ 887 960 4 & BRAM & 4 & 4 & 4 & 4 & \emph{120} \\
& DPS & 0 & 0 & 7 & 19 & \emph{80} \\ \hline 888 961 & DPS & 0 & 0 & 7 & 19 & \emph{80} \\ \hline
& LUT & - & 1259 & 2602 & 2505 & \emph{17600} \\ 889 962 & LUT & - & 1259 & 2602 & 2505 & \emph{17600} \\
5 & BRAM & - & 5 & 5 & 5 & \emph{120} \\ 890 963 5 & BRAM & - & 5 & 5 & 5 & \emph{120} \\
& DPS & - & 0 & 0 & 19 & \emph{80} \\ \hline 891 964 & DPS & - & 0 & 0 & 19 & \emph{80} \\ \hline
\end{tabular} 892 965 \end{tabular}
} 893 966 }
\end{table} 894 967 \end{table}
\renewcommand{\arraystretch}{1} 895 968 \renewcommand{\arraystretch}{1}
896 969
If we keep the previous estimation of cost of one DSP in terms of LUT (1 DSP $\approx$ 100 LUT) 897 970 If we keep the previous estimation of cost of one DSP in terms of LUT (1 DSP $\approx$ 100 LUT)
the real resource consumption decreases as a function of the number of stages in the cascaded 898 971 the real resource consumption decreases as a function of the number of stages in the cascaded
ifcs2018_journal_reponse.tex
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%Minor Revision - TUFFC-09469-2019 1 1 %Minor Revision - TUFFC-09469-2019
%Transactions on Ultrasonics, Ferroelectrics, and Frequency 2 2 %Transactions on Ultrasonics, Ferroelectrics, and Frequency
%Control (July 23, 2019 9:29 PM) 3 3 %Control (July 23, 2019 9:29 PM)
%To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr, 4 4 %To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr,
%gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr, 5 5 %gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr,
%jmfriedt@femto-st.fr 6 6 %jmfriedt@femto-st.fr
%CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu 7 7 %CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu
% 8 8 %
%Dear Mr. Arthur HUGEAT 9 9 %Dear Mr. Arthur HUGEAT
% 10 10 %
%Congratulations! Your manuscript 11 11 %Congratulations! Your manuscript
% 12 12 %
%MANUSCRIPT NO. TUFFC-09469-2019 13 13 %MANUSCRIPT NO. TUFFC-09469-2019
%MANUSCRIPT TYPE: Papers 14 14 %MANUSCRIPT TYPE: Papers
%TITLE: Filter optimization for real time digital processing of radiofrequency 15 15 %TITLE: Filter optimization for real time digital processing of radiofrequency
%signals: application to oscillator metrology 16 16 %signals: application to oscillator metrology
%AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, 17 17 %AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois,
%Pierre-Yves; Friedt, Jean-Michel 18 18 %Pierre-Yves; Friedt, Jean-Michel
% 19 19 %
%has been reviewed and it has been suggested that it be accepted for publication 20 20 %has been reviewed and it has been suggested that it be accepted for publication
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% 23 23 %
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% 68 68 %
%Giorgio Santarelli 69 69 %Giorgio Santarelli
%Associate Editor in Chief 70 70 %Associate Editor in Chief
%Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 71 71 %Transactions on Ultrasonics, Ferroelectrics, and Frequency Control
% 72 72 %
%**************************************************** 73 73 %****************************************************
%REVIEWERS' COMMENTS: 74 74 %REVIEWERS' COMMENTS:
75 75
\documentclass[a4paper]{article} 76 76 \documentclass[a4paper]{article}
\usepackage{fullpage,graphicx} 77 77 \usepackage{fullpage,graphicx}
\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. 85 85 On page 2, "...allowing to save processing resource..." could be improved. % r1.1
86 86
On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what 87 87 On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2
the author meant. 88 88 the author meant.
89 89
One page 2, the whole paragraph "The first step of our approach is to model..." 90 90 One page 2, the whole paragraph "The first step of our approach is to model..." % r1.3
could be improved. 91 91 could be improved.
} 92 92 }
93 93
{\bf 94 94 {\bf
I appreciate that the authors attempted and document two optimizations: that 95 95 I appreciate that the authors attempted and document two optimizations: that % r1.4 - en attente des résultats
of maximum rejection ratio at fixed silicon area, as well as minimum silicon 96 96 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 97 97 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 98 98 useful to compare the results of both optimization paths to the performance and
resource-utilization of generic low-pass filter gateware offered by device 99 99 resource-utilization of generic low-pass filter gateware offered by device
manufacturers. I appreciate also that the authors have presented source code 100 100 manufacturers. I appreciate also that the authors have presented source code
for examination online. 101 101 for examination online.
} 102 102 }
103 103
TODO : FIR Compiler et regarder les ressources pour un FIR comparable a ceux monolithiques 104 104 TODO : FIR Compiler et regarder les ressources pour un FIR comparable a ceux monolithiques
fournis dans l'article (memes coefs et meme nombre de coefs) 105 105 fournis dans l'article (memes coefs et meme nombre de coefs)
106 106
{\bf 107 107 {\bf
Reviewer: 2 108 108 Reviewer: 2
} 109 109 }
110 110
%Comments to the Author 111 111 %Comments to the Author
%In the Manuscript, the Authors describe an optimization methodology for filter 112 112 %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 113 113 %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 114 114 %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, 115 115 %focus on filtering and shifting while the other aspects, in particular decimation,
%will be considered in a future work. The optimization problem is modelled 116 116 %will be considered in a future work. The optimization problem is modelled
%theoretically and then solved by means of a commercial software. The solutions 117 117 %theoretically and then solved by means of a commercial software. The solutions
%are tested experimentally on the Redpitaya platform with synthetic and real 118 118 %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: 119 119 %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 120 120 %maximum rejection given a fixed amount of resources and minimum resource
%utilization given a fixed amount of rejection. 121 121 %utilization given a fixed amount of rejection.
%The Authors find that filtering improves significantly when the number of 122 122 %The Authors find that filtering improves significantly when the number of
%filters increases. 123 123 %filters increases.
%A lot of work has been done in generalizing and automating the procedure so 124 124 %A lot of work has been done in generalizing and automating the procedure so
%that different approaches can be investigated quickly and efficiently. The 125 125 %that different approaches can be investigated quickly and efficiently. The
%results presented in the Manuscript seem to be just a case study based on 126 126 %results presented in the Manuscript seem to be just a case study based on
%the particular criterion chosen by the Authors. Different criteria, in 127 127 %the particular criterion chosen by the Authors. Different criteria, in
%general, could lead to different results and it is important to consider 128 128 %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 129 129 %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 130 130 %is adequate to compare the performance of filters and if multi-stage
%filters are really superior than monolithic filters. 131 131 %filters are really superior than monolithic filters.
132 132
{\bf 133 133 {\bf
By observing the results presented in fig. 10-16, it is clear that the 134 134 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 135 135 performances of multi-stage filters are obtained at the expense of their
selectivity and, in this sense, the filters presented in these figures 136 136 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, 137 137 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 138 138 the attenuation is almost 15 dB for n = 5, while it is not noticeable for
n = 1. 139 139 n = 1.
} 140 140 }
141 141
TODO : ajouter les gabarits 142 142 TODO : ajouter les gabarits
143 143
Peut etre refaire une serie de simulation dans lesquelles on impose une coupure 144 144 Peut etre refaire une serie de simulation dans lesquelles on impose une coupure
non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte 145 145 non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte
au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque 146 146 au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque
a la cascade de filtres. 147 147 a la cascade de filtres.
148 AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de
149 relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai
148 150
{\bf 149 151 {\bf
The reason is in the criterion that considers the average attenuation in 150 152 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 151 153 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 152 154 in this region, which is a very important parameter for specifying a filter
and for evaluating its performance. For example, with this criterion, a 153 155 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 154 156 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 155 157 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. 156 158 and in the results that are obtained and has to be reconsidered.
} 157 159 }
158 160
Je ne pense pas que ca soit le cas : la somme des valeurs absolues des pertes 159 161 Je ne pense pas que ca soit le cas : la somme des valeurs absolues des pertes
dans la bande va defavoriser un filtre avec 10 dB de ripples. Il n'a pas compris que 160 162 dans la bande va defavoriser un filtre avec 10 dB de ripples. Il n'a pas compris que
la bandpass s'arrete a 40\% de la bande, donc mettre le gabarit clarifierait ce point je 161 163 la bandpass s'arrete a 40\% de la bande, donc mettre le gabarit clarifierait ce point je
pense 162 164 pense
165 AH: Il y avait une faute, j'avais mis "mean of absolute value" au lieu de "sum of absolute value". Je pense que je n'ai pas besoin de mettre plus de détail ?
163 166
{\bf 164 167 {\bf
I strongly suggest to re-run the analysis with a criterion that takes also 165 168 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 166 169 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 167 170 fixing its value to a typical one, as it has been done for the transition
bandwidth. 168 171 bandwidth.
} 169 172 }
173 AH: Il y avait une faute, j'avais mis "mean of absolute value" au lieu de "sum of absolute value". Je pense que je n'ai pas besoin de mettre plus de détail ?
170 174
{\bf 171 175 {\bf
In addition, I suggest to address the following points: 172 176 In addition, I suggest to address the following points: % r2.4
- Page 1, line 50: the Authors state that IIR have shorter impulse response 173 177 - 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. 174 178 than FIR. This is not true in general. The sentence should be reconsidered.
} 175 179 }
176 180
J'aurais du dire ``lag'' au lieu de ``impulse response'' je pense 177 181 J'aurais du dire ``lag'' au lieu de ``impulse response'' je pense
182 AH: Je ne comprends pas trop ce qui ne va pas ici
178 183
{\bf 179 184 {\bf
- Fig. 4: the Author should motivate in the text why it has been chosen 180 185 - 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 181 186 this transition bandwidth and if it is a typical requirement for phase-noise
metrology. 182 187 metrology.
- The impact of the coefficient resolution is discussed. What about the 183 188 }
resolution of the data stream? Is it fixed? If so, which value has been 184 189 AH: Je ne sais pas comment justifier ça. Je dois dire que comme ça on peut éventuellement
used in the analysis? If not, how is it changed with respect to the 185 190 décimer par deux le flux ?
191
192 {\bf
193 - The impact of the coefficient resolution is discussed. What about the % r2.6 - fait
194 resolution of the data stream? Is it fixed? If so, which value has been
195 used in the analysis? If not, how is it changed with respect to the
coefficient resolution? 186 196 coefficient resolution?
} 187 197 }
188 198
Pr\'eciser que le flux de donn\'ees en entr\'ees est de r\'esolution fixe 189 199 Pr\'eciser que le flux de donn\'ees en entr\'ees est de r\'esolution fixe
190 200
{\bf 191 201 {\bf
- Page 3, line 47: the initial criterion can be omitted and, consequently, 192 202 - Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait
Fig. 5 can be removed. 193 203 Fig. 5 can be removed.
- Page 3, line 55: “maximum rejection” is not compatible with fig. 4. 194 204 - Page 3, line 55: “maximum rejection” is not compatible with fig. 4. % r2.8 - fait
It should be “minimum” 195 205 It should be “minimum”
- Page e, line 55, second column: “takin” 196 206 }
- Page 3, line 58: “pessimistic” should be replaced with “conservative” 197 207 AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max
- Page 4, line 17: “meaning” --> “this means” 198 208 de la magnitude. J'ai corrigé en ce sens.
- Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose 199 209
210 {\bf
211 - Page e, line 55, second column: “takin” % r2.9 - fait
212 - Page 3, line 58: “pessimistic” should be replaced with “conservative” % r2.10 - fait
213 - Page 4, line 17: “meaning” --> “this means” % r2.11 - fait
214 - 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? 200 215 these particular configurations? Are they chosen automatically?
- Page 4, line 31: how does the delta function transform model from non-linear 201 216 - Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait
and non-quadratic to a quadratic? 202 217 and non-quadratic to a quadratic?
- Captions of figure and tables are too minimal. 203 218 - Captions of figure and tables are too minimal. % r2.14
- Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) 204 219 - Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait
of a single figure. Same for fig. 13-16. 205 220 of a single figure. Same for fig. 13-16.
} 206 221 }
207 222
{\bf 208 223 {\bf
- Please increase the number of averages for the spectrum. Currently the noise 209 224 - Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait
of the curves is about 20 dBpk-pk and it doesn’t allow to appreciate the 210 225 of the curves is about 20 dBpk-pk and it doesn’t allow to appreciate the
differences among the curves. I suggest to reduce the noise below 1 dBpk-pk. 211 226 differences among the curves. I suggest to reduce the noise below 1 dBpk-pk.
} 212 227 }
213 228
Comment as tu fait tes spectres Arthur ? Si tu as fait une FFT sur e.g. 2048 points 214 229 Comment as tu fait tes spectres Arthur ? Si tu as fait une FFT sur e.g. 2048 points
mais que tu as des jeux de donnees de e.g. 10000 points, on peut faire des moyennes 215 230 mais que tu as des jeux de donnees de e.g. 10000 points, on peut faire des moyennes
sur les sequences successives. Au pire si pas possible, une moyenne glissante sur 216 231 sur les sequences successives. Au pire si pas possible, une moyenne glissante sur
chaque spectre pour affiner les traits ? 217 232 chaque spectre pour affiner les traits ?
218 233
%In conclusion, my opinion is that the methodology presented in the Manuscript 219 234 %In conclusion, my opinion is that the methodology presented in the Manuscript
%deserve to be published, provided that the criterion is changed according 220 235 %deserve to be published, provided that the criterion is changed according
%the indications mentioned above. 221 236 %the indications mentioned above.
\end{document} 222 237 \end{document}
%**************************************************** 223 238 %****************************************************
% 224 239 %
%For information about the IEEE Ultrasonics, Ferroelectrics, and Frequency 225 240 %For information about the IEEE Ultrasonics, Ferroelectrics, and Frequency
%Control Society, please visit the website: http://www.ieee-uffc.org. The 226 241 %Control Society, please visit the website: http://www.ieee-uffc.org. The
%website of the Transactions on Ultrasonics, Ferroelectrics, and Frequency 227 242 %website of the Transactions on Ultrasonics, Ferroelectrics, and Frequency
%Control is at: http://ieee-uffc.org/publications/transactions-on-uffc 228 243 %Control is at: http://ieee-uffc.org/publications/transactions-on-uffc
229 244
230
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