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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}
\usepackage{amsfonts} 15 15 \usepackage{amsfonts}
\usepackage{amsthm} 16 16 \usepackage{amsthm}
\usepackage{amssymb} 17 17 \usepackage{amssymb}
\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}
\usepackage{caption} 26 26 \usepackage{caption}
\usepackage{subcaption} 27 27 \usepackage{subcaption}
28 28
% correct bad hyphenation here 29 29 % correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor} 30 30 \hyphenation{op-tical net-works semi-conduc-tor}
\textheight=26cm 31 31 \textheight=26cm
\setlength{\footskip}{30pt} 32 32 \setlength{\footskip}{30pt}
\pagenumbering{gobble} 33 33 \pagenumbering{gobble}
\begin{document} 34 34 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 35 35 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 36 36 to oscillator metrology}
37 37
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 38 38 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 39 39 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\ 40 40 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\ 41 41 \IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\
\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\ 42 42 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 43 43 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 44 44 }
\maketitle 45 45 \maketitle
\thispagestyle{plain} 46 46 \thispagestyle{plain}
\pagestyle{plain} 47 47 \pagestyle{plain}
\newtheorem{definition}{Definition} 48 48 \newtheorem{definition}{Definition}
49 49
\begin{abstract} 50 50 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 51 51 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 52 52 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 53 53 of ultrastable clocks, stringent filtering requirements are defined by spurious signal or
noise rejection needs. Since real time radiofrequency processing must be performed in a 54 54 noise rejection needs. Since real time radiofrequency processing must be performed in a
Field Programmable Array to meet timing constraints, we investigate optimization strategies 55 55 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 56 56 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. The 57 57 required and keeping timing constraints within the targeted measurement bandwidths. The
presented technique is applicable to scheduling any sequence of processing blocks characterized 58 58 presented technique is applicable to scheduling any sequence of processing blocks characterized
by a throughput, resource occupation and performance tabulated as a function of configuration 59 59 by a throughput, resource occupation and performance tabulated as a function of configuration
characateristics, as is the case for filters with their coefficients and resolution yielding 60 60 characateristics, as is the case for filters with their coefficients and resolution yielding
rejection and number of multipliers. 61 61 rejection and number of multipliers.
\end{abstract} 62 62 \end{abstract}
63 63
\begin{IEEEkeywords} 64 64 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 65 65 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 66 66 \end{IEEEkeywords}
67 67
\section{Digital signal processing of ultrastable clock signals} 68 68 \section{Digital signal processing of ultrastable clock signals}
69 69
Analog oscillator phase noise characteristics are classically performed by downconverting 70 70 Analog oscillator phase noise characteristics are classically performed by downconverting
the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband, 71 71 the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,
followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In 72 72 followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In
a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by 73 73 a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by
multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}. 74 74 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
75 75
\begin{figure}[h!tb] 76 76 \begin{figure}[h!tb]
\begin{center} 77 77 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 78 78 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 79 79 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 80 80 \caption{Fully digital oscillator phase noise characterization: the Device Under Test
(DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and 81 81 (DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and
downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals 82 82 downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals
and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite 83 83 and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite
Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays 84 84 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 85 85 the spectral characteristics of the phase fluctuations.}
\label{schema} 86 86 \label{schema}
\end{figure} 87 87 \end{figure}
88 88
As with the analog mixer, 89 89 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 90 90 the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as
well as the generation of the frequency sum signal in addition to the frequency difference. 91 91 well as the generation of the frequency sum signal in addition to the frequency difference.
These unwanted spectral characteristics must be rejected before decimating the data stream 92 92 These unwanted spectral characteristics must be rejected before decimating the data stream
for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the 93 93 for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
downconverter 94 94 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 95 95 and the decimation processing blocks are core characteristics of an oscillator characterization
system, and must reject out-of-band signals below the targeted phase noise -- typically in the 96 96 system, and must reject out-of-band signals below the targeted phase noise -- typically in the
sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will 97 97 sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will
use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency 98 98 use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency
datastream: optimizing the performance of the filter while reducing the needed resources is 99 99 datastream: optimizing the performance of the filter while reducing the needed resources is
hence tackled in a systematic approach using optimization techniques. Most significantly, we 100 100 hence tackled in a systematic approach using optimization techniques. Most significantly, we
tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with 101 101 tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with
tunable number of coefficients and tunable number of bits representing the coefficients and the 102 102 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 103 103 data being processed.
104 104
\section{Finite impulse response filter} 105 105 \section{Finite impulse response filter}
106 106
We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined 107 107 We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined
by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the 108 108 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 109 109 outputs $y_k$
\begin{align} 110 110 \begin{align}
y_n=\sum_{k=0}^N b_k x_{n-k} 111 111 y_n=\sum_{k=0}^N b_k x_{n-k}
\label{eq:fir_equation} 112 112 \label{eq:fir_equation}
\end{align} 113 113 \end{align}
114 114
As opposed to an implementation on a general purpose processor in which word size is defined by the 115 115 As opposed to an implementation on a general purpose processor in which word size is defined by the
processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since 116 116 processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since
not only the coefficient values and number of taps must be defined, but also the number of bits 117 117 not only the coefficient values and number of taps must be defined, but also the number of bits
defining the coefficients and the sample size. For this reason, and because we consider pipeline 118 118 defining the coefficients and the sample size. For this reason, and because we consider pipeline
processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency 119 119 processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency
signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but 120 120 signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but
the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language 121 121 the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language
(VHDL) level. 122 122 (VHDL) level.
{\color{red}Since latency is not an issue in a openloop phase noise characterization instrument, 123 123 {\color{red}Since latency is not an issue in a openloop phase noise characterization instrument,
the large 124 124 the large
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 125 125 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
is not considered as an issue as would be in a closed loop system.} % r2.4 126 126 is not considered as an issue as would be in a closed loop system.} % r2.4
127 127
The coefficients are classically expressed as floating point values. However, this binary 128 128 The coefficients are classically expressed as floating point values. However, this binary
number representation is not efficient for fast arithmetic computation by an FPGA. Instead, 129 129 number representation is not efficient for fast arithmetic computation by an FPGA. Instead,
we select to quantify these floating point values into integer values. This quantization 130 130 we select to quantify these floating point values into integer values. This quantization
will result in some precision loss. 131 131 will result in some precision loss.
132 132
\begin{figure}[h!tb] 133 133 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/zero_values} 134 134 \includegraphics[width=\linewidth]{images/zero_values}
\caption{Impact of the quantization resolution of the coefficients: the quantization is 135 135 \caption{Impact of the quantization resolution of the coefficients: the quantization is
set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting 136 136 set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting
the 30~first and 30~last coefficients out of the initial 128~band-pass 137 137 the 30~first and 30~last coefficients out of the initial 128~band-pass
filter coefficients to 0 (red dots).} 138 138 filter coefficients to 0 (red dots).}
\label{float_vs_int} 139 139 \label{float_vs_int}
\end{figure} 140 140 \end{figure}
141 141
The tradeoff between quantization resolution and number of coefficients when considering 142 142 The tradeoff between quantization resolution and number of coefficients when considering
integer operations is not trivial. As an illustration of the issue related to the 143 143 integer operations is not trivial. As an illustration of the issue related to the
relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits 144 144 relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits
a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon 145 145 a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon
quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the 146 146 quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the
taps become null, {\color{red}making the large number of coefficients irrelevant: processing 147 147 taps become null, {\color{red}making the large number of coefficients irrelevant: processing
resources % r1.1 148 148 resources % r1.1
are hence saved by shrinking the filter length.} This tradeoff aimed at minimizing resources 149 149 are hence saved by shrinking the filter length.} This tradeoff aimed at minimizing resources
to reach a given rejection level, or maximizing out of band rejection for a given computational 150 150 to reach a given rejection level, or maximizing out of band rejection for a given computational
resource, will drive the investigation on cascading filters designed with varying tap resolution 151 151 resource, will drive the investigation on cascading filters designed with varying tap resolution
and tap length, as will be shown in the next section. Indeed, our development strategy closely 152 152 and tap length, as will be shown in the next section. Indeed, our development strategy closely
follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards} 153 153 follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
in which basic blocks are defined and characterized before being assembled \cite{hide} 154 154 in which basic blocks are defined and characterized before being assembled \cite{hide}
in a complete processing chain. In our case, assembling the filter blocks is a simpler block 155 155 in a complete processing chain. In our case, assembling the filter blocks is a simpler block
combination process since we assume a single value to be processed and a single value to be 156 156 combination process since we assume a single value to be processed and a single value to be
generated at each clock cycle. The FIR filters will not be considered to decimate in the 157 157 generated at each clock cycle. The FIR filters will not be considered to decimate in the
current implementation: the decimation is assumed to be located after the FIR cascade at the 158 158 current implementation: the decimation is assumed to be located after the FIR cascade at the
moment. 159 159 moment.
160 160
\section{Methodology description} 161 161 \section{Methodology description}
162 162
Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP) 163 163 Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP)
chain obtained by assembling basic processing blocks, with hardware and manufacturer independence. 164 164 chain obtained by assembling basic processing blocks, with hardware and manufacturer independence.
Achieving such a target requires defining an abstract model to represent some basic properties 165 165 Achieving such a target requires defining an abstract model to represent some basic properties
of DSP blocks such as perfomance (i.e. rejection or ripples in the bandpass for filters) and 166 166 of DSP blocks such as perfomance (i.e. rejection or ripples in the bandpass for filters) and
resource occupation. These abstract properties, not necessarily related to the detailed hardware 167 167 resource occupation. These abstract properties, not necessarily related to the detailed hardware
implementation of a given platform, will feed a scheduler solver aimed at assembling the optimum 168 168 implementation of a given platform, will feed a scheduler solver aimed at assembling the optimum
target, whether in terms of maximizing performance for a given arbitrary resource occupation, or 169 169 target, whether in terms of maximizing performance for a given arbitrary resource occupation, or
minimizing resource occupation for a given perfomance. In our approach, the solution of the 170 170 minimizing resource occupation for a given perfomance. In our approach, the solution of the
solver is then synthesized using the dedicated tool provided by each platform manufacturer 171 171 solver is then synthesized using the dedicated tool provided by each platform manufacturer
to assess the validity of our abstract resource occupation indicator, and the result of running 172 172 to assess the validity of our abstract resource occupation indicator, and the result of running
the DSP chain on the FPGA allows for assessing the performance of the scheduler. We emphasize 173 173 the DSP chain on the FPGA allows for assessing the performance of the scheduler. We emphasize
that all solutions found by the solver are synthesized and executed on hardware at the end 174 174 that all solutions found by the solver are synthesized and executed on hardware at the end
of the analysis. 175 175 of the analysis.
176 176
In this demonstration , we focus on only two operations: filtering and shifting the number of 177 177 In this demonstration, we focus on only two operations: filtering and shifting the number of
bits needed to represent the data along the processing chain. 178 178 bits needed to represent the data along the processing chain.
We have chosen these basic operations because shifting and the filtering have already been studied 179 179 We have chosen these basic operations because shifting and the filtering have already been studied
in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for 180 180 in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for
assessing our results. Furthermore, filtering is a core step in any radiofrequency frontend 181 181 assessing our results. Furthermore, filtering is a core step in any radiofrequency frontend
requiring pipelined processing at full bandwidth for the earliest steps, including for 182 182 requiring pipelined processing at full bandwidth for the earliest steps, including for
time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}. 183 183 time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}.
184 184
Addressing only two operations allows for demonstrating the methodology but should not be 185 185 Addressing only two operations allows for demonstrating the methodology but should not be
considered as a limitation of the framework which can be extended to assembling any number 186 186 considered as a limitation of the framework which can be extended to assembling any number
of skeleton blocks as long as perfomance and resource occupation can be determined. {\color{red} 187 187 of skeleton blocks as long as perfomance and resource occupation can be determined. {\color{red}
Hence, 188 188 Hence,
in this paper we will apply our methodology on simple DSP chains: a white noise input signal % r1.2 189 189 in this paper we will apply our methodology on simple DSP chains: a white noise input signal % r1.2
is generated using a Pseudo-Random Number (PRN) generator or by sampling a wideband (125~MS/s) 190 190 is generated using a Pseudo-Random Number (PRN) generator or by sampling a wideband (125~MS/s)
14-bit Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.} Once samples have been 191 191 14-bit Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.} Once samples have been
digitized at a rate of 125~MS/s, filtering is applied to qualify the processing block performance -- 192 192 digitized at a rate of 125~MS/s, filtering is applied to qualify the processing block performance --
practically meeting the radiofrequency frontend requirement of noise and bandwidth reduction 193 193 practically meeting the radiofrequency frontend requirement of noise and bandwidth reduction
by filtering and decimating. Finally, bursts of filtered samples are stored for post-processing, 194 194 by filtering and decimating. Finally, bursts of filtered samples are stored for post-processing,
allowing to assess either filter rejection for a given resource usage, or validating the rejection 195 195 allowing to assess either filter rejection for a given resource usage, or validating the rejection
when implementing a solution minimizing resource occupation. 196 196 when implementing a solution minimizing resource occupation.
197 197
{\color{red} 198 198 {\color{red}
The first step of our approach is to model the DSP chain. Since we aim at only optimizing % r1.3 199 199 The first step of our approach is to model the DSP chain. Since we aim at only optimizing % r1.3
the filtering part of the signal processing chain, we have not included the PRN generator or the 200 200 the filtering part of the signal processing chain, we have not included the PRN generator or the
ADC in the model: the input data size and rate are considered fixed and defined by the hardware. 201 201 ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
The filtering can be done in two ways, either by considering a single monolithic FIR filter 202 202 The filtering can be done in two ways, either by considering a single monolithic FIR filter
requiring many coefficients to reach the targeted noise rejection ratio, or by 203 203 requiring many coefficients to reach the targeted noise rejection ratio, or by
cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.} 204 204 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.}
205 205
After each filter we leave the possibility of shifting the filtered data to consume 206 206 After each filter we leave the possibility of shifting the filtered data to consume
less resources. Hence in the case of cascaded filter, we define a stage as a filter 207 207 less resources. Hence in the case of cascaded filter, we define a stage as a filter
and a shifter (the shift could be omitted if we do not need to divide the filtered data). 208 208 and a shifter (the shift could be omitted if we do not need to divide the filtered data).
209 209
\subsection{Model of a FIR filter} 210 210 \subsection{Model of a FIR filter}
211 211
A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$) 212 212 A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$)
the FIR has $C_i$ coefficients and each coefficient is an integer value with $\pi^C_i$ 213 213 the FIR has $C_i$ coefficients and each coefficient is an integer value with $\pi^C_i$
bits while the filtered data are shifted by $\pi^S_i$ bits. We define also $\pi^-_i$ as 214 214 bits while the filtered data are shifted by $\pi^S_i$ bits. We define also $\pi^-_i$ as
the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage} 215 215 the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage}
shows a filtering stage. 216 216 shows a filtering stage.
217 217
\begin{figure} 218 218 \begin{figure}
\centering 219 219 \centering
\begin{tikzpicture}[node distance=2cm] 220 220 \begin{tikzpicture}[node distance=2cm]
\node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ; 221 221 \node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;
\node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ; 222 222 \node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;
\node (Start) [left of=FIR] { } ; 223 223 \node (Start) [left of=FIR] { } ;
\node (End) [right of=Shift] { } ; 224 224 \node (End) [right of=Shift] { } ;
225 225
\node[draw,fit=(FIR) (Shift)] (Filter) { } ; 226 226 \node[draw,fit=(FIR) (Shift)] (Filter) { } ;
227 227
\draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ; 228 228 \draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;
\draw[->] (FIR) -- (Shift) ; 229 229 \draw[->] (FIR) -- (Shift) ;
\draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ; 230 230 \draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;
\end{tikzpicture} 231 231 \end{tikzpicture}
\caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)} 232 232 \caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}
\label{fig:fir_stage} 233 233 \label{fig:fir_stage}
\end{figure} 234 234 \end{figure}
235 235
FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB. 236 236 FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB.
This rejection has been computed using GNU Octave software FIR coefficient design functions 237 237 This rejection has been computed using GNU Octave software FIR coefficient design functions
(\texttt{firls} and \texttt{fir1}). 238 238 (\texttt{firls} and \texttt{fir1}).
For each configuration $(C_i, \pi_i^C)$, we first create a FIR with floating point coefficients and a given $C_i$ number of coefficients. 239 239 For each configuration $(C_i, \pi_i^C)$, we first create a FIR with floating point coefficients and a given $C_i$ number of coefficients.
Then, the floating point coefficients are discretized into integers. In order to ensure that the coefficients are coded on $\pi_i^C$~bits effectively, 240 240 Then, the floating point coefficients are discretized into integers. In order to ensure that the coefficients are coded on $\pi_i^C$~bits effectively,
the coefficients are normalized by their absolute maximum before being scaled to integer coefficients. 241 241 the coefficients are normalized by their absolute maximum before being scaled to integer coefficients.
At least one coefficient is coded on $\pi_i^C$~bits, and in practice only $b_{C_i/2}$ is coded on $\pi_i^C$~bits while the others are coded on much fewer bits. 242 242 At least one coefficient is coded on $\pi_i^C$~bits, and in practice only $b_{C_i/2}$ is coded on $\pi_i^C$~bits while the others are coded on much fewer bits.
243 243
With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter 244 244 With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter
transfer function. 245 245 transfer function.
Comparing the performance between FIRs requires however defining a unique criterion. As shown in figure~\ref{fig:fir_mag}, 246 246 Comparing the performance between FIRs requires however defining a unique criterion. As shown in figure~\ref{fig:fir_mag},
the FIR magnitude exhibits two parts: we focus here on the transitions width and the rejection rather than on the 247 247 the FIR magnitude exhibits two parts: we focus here on the transitions width and the rejection rather than on the
bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. {\color{red}Throughout this demonstration, 248 248 bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. {\color{red}Throughout this demonstration,
we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\% 249 249 we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%
of the Nyquist frequency to the end of the band, as would be typically selected to prevent 250 250 of the Nyquist frequency to the end of the band, as would be typically selected to prevent
aliasing before decimating the dataflow by 2. The method is however generalized to any filter 251 251 aliasing before decimating the dataflow by 2. The method is however generalized to any filter
shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid} 252 252 shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid}
as described below is indeed unique for each filter shape.} 253 253 as described below is indeed unique for each filter shape.}
254 254
\begin{figure} 255 255 \begin{figure}
\begin{center} 256 256 \begin{center}
\scalebox{0.8}{ 257 257 \scalebox{0.8}{
\centering 258 258 \centering
\begin{tikzpicture}[scale=0.3] 259 259 \begin{tikzpicture}[scale=0.3]
\draw[<->] (0,15) -- (0,0) -- (21,0) ; 260 260 \draw[<->] (0,15) -- (0,0) -- (21,0) ;
\draw[thick] (0,12) -- (8,12) -- (20,0) ; 261 261 \draw[thick] (0,12) -- (8,12) -- (20,0) ;
262 262
\draw (0,14) node [left] { $P$ } ; 263 263 \draw (0,14) node [left] { $P$ } ;
\draw (20,0) node [below] { $f$ } ; 264 264 \draw (20,0) node [below] { $f$ } ;
265 265
\draw[>=latex,<->] (0,14) -- (8,14) ; 266 266 \draw[>=latex,<->] (0,14) -- (8,14) ;
\draw (4,14) node [above] { passband } node [below] { $40\%$ } ; 267 267 \draw (4,14) node [above] { passband } node [below] { $40\%$ } ;
268 268
\draw[>=latex,<->] (8,14) -- (12,14) ; 269 269 \draw[>=latex,<->] (8,14) -- (12,14) ;
\draw (10,14) node [above] { transition } node [below] { $20\%$ } ; 270 270 \draw (10,14) node [above] { transition } node [below] { $20\%$ } ;
271 271
\draw[>=latex,<->] (12,14) -- (20,14) ; 272 272 \draw[>=latex,<->] (12,14) -- (20,14) ;
\draw (16,14) node [above] { stopband } node [below] { $40\%$ } ; 273 273 \draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;
274 274
\draw[>=latex,<->] (16,12) -- (16,8) ; 275 275 \draw[>=latex,<->] (16,12) -- (16,8) ;
\draw (16,10) node [right] { rejection } ; 276 276 \draw (16,10) node [right] { rejection } ;
277 277
\draw[dashed] (8,-1) -- (8,14) ; 278 278 \draw[dashed] (8,-1) -- (8,14) ;
\draw[dashed] (12,-1) -- (12,14) ; 279 279 \draw[dashed] (12,-1) -- (12,14) ;
280 280
\draw[dashed] (8,12) -- (16,12) ; 281 281 \draw[dashed] (8,12) -- (16,12) ;
\draw[dashed] (12,8) -- (16,8) ; 282 282 \draw[dashed] (12,8) -- (16,8) ;
283 283
\end{tikzpicture} 284 284 \end{tikzpicture}
} 285 285 }
\end{center} 286 286 \end{center}
\caption{Shape of the filter transmitted power $P$ as a function of frequency $f$: 287 287 \caption{Shape of the filter transmitted power $P$ as a function of frequency $f$:
the passband is considered to occupy the initial 40\% of the Nyquist frequency range, 288 288 the passband is considered to occupy the initial 40\% of the Nyquist frequency range,
the stopband the last 40\%, allowing 20\% transition width.} 289 289 the stopband the last 40\%, allowing 20\% transition width.}
\label{fig:fir_mag} 290 290 \label{fig:fir_mag}
\end{figure} 291 291 \end{figure}
292 292
In the transition band, the behavior of the filter is left free, we only {\color{red}define} the passband and the stopband characteristics. 293 293 In the transition band, the behavior of the filter is left free, we only {\color{red}define} the passband and the stopband characteristics.
% r2.7 294 294 % r2.7
% Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion 295 295 % Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion
% yields unacceptable results since notches overestimate the rejection capability of the filter. Furthermore, the losses within 296 296 % yields unacceptable results since notches overestimate the rejection capability of the filter. Furthermore, the losses within
% the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients. 297 297 % the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients.
Our criterion to compute the filter rejection considers 298 298 Our criterion to compute the filter rejection considers
% r2.8 et r2.2 r2.3 299 299 % r2.8 et r2.2 r2.3
the {\color{red}minimal} rejection within the stopband, to which the {\color{red}sum of the absolute values 300 300 the {\color{red}minimal} rejection within the stopband, to which the {\color{red}sum of the absolute values
within the passband is subtracted to avoid filters with excessive ripples}. With this 301 301 within the passband is subtracted to avoid filters with excessive ripples, normalized to the
criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}. 302 302 bin width to remain consistent with the passband criterion (dBc/Hz units in all cases)}. With this
303 303 criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
% \begin{figure} 304 304
% \centering 305 305 % \begin{figure}
% \includegraphics[width=\linewidth]{images/colored_mean_criterion} 306 306 % \centering
% \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters} 307 307 % \includegraphics[width=\linewidth]{images/colored_mean_criterion}
% \label{fig:mean_criterion} 308 308 % \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}
% \end{figure} 309 309 % \label{fig:mean_criterion}
310 310 % \end{figure}
\begin{figure} 311 311
\centering 312 312 \begin{figure}
\includegraphics[width=\linewidth]{images/colored_custom_criterion} 313 313 \centering
\caption{Custom criterion (maximum rejection in the stopband minus the mean of the absolute value of the passband rejection) 314 314 \includegraphics[width=\linewidth]{images/colored_custom_criterion}
comparison between monolithic filter and cascaded filters} 315 315 \caption{Custom criterion (maximum rejection in the stopband minus the {\color{red} sum of the
\label{fig:custom_criterion} 316 316 absolute values of the passband rejection normalized to the bandwidth})
\end{figure} 317 317 comparison between monolithic filter and cascaded filters}
318 318 \label{fig:custom_criterion}
Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps 319 319 \end{figure}
and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the 320 320
rejection as a function of the number of coefficients and the number of bits representing these coefficients. 321 321 Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps
The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet. 322 322 and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the
Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection. 323 323 rejection as a function of the number of coefficients and the number of bits representing these coefficients.
Conversely when setting the a given number of bits, increasing the number of coefficients will not improve 324 324 The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet.
the rejection. Hence the best coefficient set are on the vertex of the pyramid. 325 325 Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection.
326 326 Conversely when setting the a given number of bits, increasing the number of coefficients will not improve
\begin{figure} 327 327 the rejection. Hence the best coefficient set are on the vertex of the pyramid.
\centering 328 328
\includegraphics[width=\linewidth]{images/rejection_pyramid} 329 329 \begin{figure}
\caption{Rejection as a function of number of coefficients and number of bits} 330 330 \centering
\label{fig:rejection_pyramid} 331 331 \includegraphics[width=\linewidth]{images/rejection_pyramid}
\end{figure} 332 332 \caption{{\color{red}{Filter}} rejection as a function of number of coefficients and number of bits
333 333 {\color{red}: this lookup table will be used to identify which filter parameters -- number of bits
Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps), 334 334 representing coefficients and number of coefficients -- best match the targeted transfer function.}}
we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria. 335 335 \label{fig:rejection_pyramid}
If the FIR filter coefficients are the same between the stages, we have: 336 336 \end{figure}
$$F_{total} = F_1 + F_2$$ 337 337
But selecting two different sets of coefficient will yield a more complex situation in which 338 338 Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps),
the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves 339 339 we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria.
are two different filters with maximums and notches not located at the same frequency offsets. 340 340 If the FIR filter coefficients are the same between the stages, we have:
Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved 341 341 $$F_{total} = F_1 + F_2$$
with respect to a basic sum of the rejection criteria shown as a the dotted yellow line. 342 342 But selecting two different sets of coefficient will yield a more complex situation in which
% r2.9 343 343 the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves
Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection 344 344 are two different filters with maximums and notches not located at the same frequency offsets.
criteria of each filter. However since the this sum underestimates the rejection capability of the cascade, 345 345 Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved
% r2.10 346 346 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability 347 347 % r2.9
of the filter cascade to meet design criteria. 348 348 Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection
349 349 criteria of each filter. However since the {\color{red}individual filter rejection} sum underestimates the rejection capability of the cascade,
\begin{figure} 350 350 % r2.10
\centering 351 351 this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
\includegraphics[width=\linewidth]{images/cascaded_criterion} 352 352 of the filter cascade to meet design criteria.
\caption{Rejection of two cascaded filters} 353 353
\label{fig:sum_rejection} 354 354 \begin{figure}
\end{figure} 355 355 \centering
356 356 \includegraphics[width=\linewidth]{images/cascaded_criterion}
% r2.6 357 357 \caption{{\color{red}Transfer function of individual filters and after cascading} the two filters,
Finally in our case, we consider that the input signal are fully known. So the 358 358 {\color{red}demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal
resolution of the data stream are fixed and still the same for all experiments 359 359 lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop
in this paper. 360 360 maximum of each individual filter.}
361 361 }
Based on this analysis, we address the estimate of resource consumption (called 362 362 \label{fig:sum_rejection}
% r2.11 363 363 \end{figure}
silicon area -- in the case of FPGAs this means processing cells) as a function of 364 364
filter characteristics. As a reminder, we do not aim at matching actual hardware 365 365 % r2.6
configuration but consider an arbitrary silicon area occupied by each processing function, 366 366 {\color{red}
and will assess after synthesis the adequation of this arbitrary unit with actual 367 367 Finally in our case, we consider that the input signal are fully known. The
hardware resources provided by FPGA manufacturers. The sum of individual processing 368 368 resolution of the input data stream are fixed and still the same for all experiments
unit areas is constrained by a total silicon area representative of FPGA global resources. 369 369 in this paper.}
Formally, variable $a_i$ is the area taken by filter~$i$ 370 370
(in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB). 371 371 Based on this analysis, we address the estimate of resource consumption (called
Constant $\mathcal{A}$ is the total available area. We model our problem as follows: 372 372 % r2.11
373 373 silicon area -- in the case of FPGAs this means processing cells) as a function of
\begin{align} 374 374 filter characteristics. As a reminder, we do not aim at matching actual hardware
\text{Maximize } & \sum_{i=1}^n r_i \notag \\ 375 375 configuration but consider an arbitrary silicon area occupied by each processing function,
\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\ 376 376 and will assess after synthesis the adequation of this arbitrary unit with actual
a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\ 377 377 hardware resources provided by FPGA manufacturers. The sum of individual processing
r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\ 378 378 unit areas is constrained by a total silicon area representative of FPGA global resources.
\pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\ 379 379 Formally, variable $a_i$ is the area taken by filter~$i$
\pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\ 380 380 (in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB).
\pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\ 381 381 Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
\pi_1^- &= \Pi^I \label{eq:init} 382 382
\end{align} 383 383 \begin{align}
384 384 \text{Maximize } & \sum_{i=1}^n r_i \notag \\
Equation~\ref{eq:area} states that the total area taken by the filters must be 385 385 \sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\
less than the available area. Equation~\ref{eq:areadef} gives the definition of 386 386 a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\
the area used by a filter, considered as the area of the FIR since the Shifter is 387 387 r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\
assumed not to require significant resources. We consider that the FIR needs $C_i$ registers of size 388 388 \pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\
$\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the 389 389 \pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\
input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the 390 390 \pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\
definition of the rejection of the filter thanks to the tabulated function~$F$ that we defined 391 391 \pi_1^- &= \Pi^I \label{eq:init}
previously. The Shifter does not introduce negative rejection as we will explain later, 392 392 \end{align}
so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the 393 393
relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add 394 394 Equation~\ref{eq:area} states that the total area taken by the filters must be
$\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes 395 395 less than the available area. Equation~\ref{eq:areadef} gives the definition of
$\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of 396 396 the area used by a filter, considered as the area of the FIR since the Shifter is
a filter is the same as the input number of bits of the next filter. 397 397 assumed not to require significant resources. We consider that the FIR needs $C_i$ registers of size
Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative 398 398 $\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the
rejection. Indeed, the results of the FIR can be right shifted without compromising 399 399 input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the
the quality of the rejection until a threshold. Each bit of the output data 400 400 definition of the rejection of the filter thanks to the tabulated function~$F$ that we defined
increases the maximum rejection level by 6~dB. We add one to take the sign bit 401 401 previously. The Shifter does not introduce negative rejection as we will explain later,
into account. If equation~\ref{eq:maxshift} was not present, the Shifter could 402 402 so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the
shift too much and introduce some noise in the output data. Each supplementary 403 403 relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add
shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is: 404 404 $\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes
$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. 405 405 $\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of
Finally, equation~\ref{eq:init} gives the number of bits of the global input. 406 406 a filter is the same as the input number of bits of the next filter.
407 407 Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative
{\color{red} 408 408 rejection. Indeed, the results of the FIR can be right shifted without compromising
This model is non-linear since we multiply some variable with another variable 409 409 the quality of the rejection until a threshold. Each bit of the output data
and it is even non-quadratic, as $F$ does not have a known 410 410 increases the maximum rejection level by 6~dB. We add one to take the sign bit
linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations. 411 411 into account. If equation~\ref{eq:maxshift} was not present, the Shifter could
This variable must be defined by the user, it represent the number of different 412 412 shift too much and introduce some noise in the output data. Each supplementary
set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1} 413 413 shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is:
functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid} 414 414 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
to restrict the number of configurations. Indeed, it is useless to have too many coefficients or 415 415 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
too many bits, hence we take the configurations close to edge of pyramid. Thank to theses 416 416
configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant 417 417 {\color{red}
and the function $F$ can be estimate for each configurations 418 418 This model is non-linear since we multiply some variable with another variable
thanks our rejection criterion. We also defined binary 419 419 and it is even non-quadratic, as the cost function $F$ does not have a known
variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 420 420 linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
421 % AH: conflit merge
422 % This variable must be defined by the user, it represent the number of different
423 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
424 % functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid}
425 % to restrict the number of configurations. Indeed, it is useless to have too many coefficients or
426 % too many bits, hence we take the configurations close to edge of pyramid. Thank to theses
427 % configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant
428 % and the function $F$ can be estimate for each configurations
429 % thanks our rejection criterion. We also defined binary
and 0 otherwise. The new equations are as follows: 421 430 This variable $p$ is defined by the user, and represents the number of different
} 422 431 set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}
423 432 functions from GNU Octave) based on the targeted filter characteristics and implementation
\begin{align} 424 433 assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and
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} \\ 425 434 $\pi_{ij}^C$ become constants and
r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\ 426 435 we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table)
\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} \\ 427 436 for each configurations thanks to the rejection criterion. We also define the binary
\sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} 428 437 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
\end{align} 429 438 and 0 otherwise. The new equations are as follows:
430 439 }
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 431 440
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 432 441 \begin{align}
Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most. 433 442 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} \\
434 443 r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
{\color{red} 435 444 \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} \\
However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply 436 445 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
$\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can 437 446 \end{align}
linearize this multiplication. The following formula shows how to linearize 438 447
this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$): 439 448 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
\begin{equation*} 440 449 respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
m = x \times y \implies 441 450 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
\left \{ 442 451
\begin{split} 443 452 {\color{red}
m & \geq 0 \\ 444 453 % JM: conflict merge
454 % However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
455 % we multiply
456 % $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
457 % linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size,
458 % we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is
459 % assumed on hardware characteristics.
460 % The Gurobi (\url{www.gurobi.com}) optimization software used to solve this quadratic
461 % model is able to linearize the model provided as is. This model
462 % has $O(np)$ variables and $O(n)$ constraints.}
463 However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
m & \leq y \times X^{max} \\ 445 464 we multiply
m & \leq x \\ 446 465 $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
m & \geq x - (1 - y) \times X^{max} \\ 447 466 linearise linearize this multiplication. The following formula shows how to linearize
\end{split} 448 467 this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):
\right . 449 468 \begin{equation*}
\end{equation*} 450 469 m = x \times y \implies
451 470 \left \{
471 \begin{split}
472 m & \geq 0 \\
473 m & \leq y \times X^{max} \\
474 m & \leq x \\
475 m & \geq x - (1 - y) \times X^{max} \\
476 \end{split}
477 \right .
478 \end{equation*}
479 So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is
480 assumed on hardware characteristics,
481 the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize
482 for us the quadratic problem so the model is left as is. This model
So if we bound up $\pi_i^-$ by 128~bits to represent the maximum data size tolerated, 452 483 has $O(np)$ variables and $O(n)$ constraints.}
the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize 453 484
for us the quadratic problem so the model is left as is. 454 485 % This model is non-linear and even non-quadratic, as $F$ does not have a known
} 455 486 % linear or quadratic expression. We introduce $p$ FIR configurations
This model has $O(np)$ variables and $O(n)$ constraints. 456 487 % $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
457 488 % % r2.12
% This model is non-linear and even non-quadratic, as $F$ does not have a known 458 489 % This variable must be defined by the user, it represent the number of different
% linear or quadratic expression. We introduce $p$ FIR configurations 459 490 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
% $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 460 491 % functions from GNU Octave).
% % r2.12 461 492 % We define binary
% This variable must be defined by the user, it represent the number of different 462 493 % variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1} 463 494 % and 0 otherwise. The new equations are as follows:
% functions from GNU Octave). 464 495 %
% We define binary 465 496 % \begin{align}
% variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 466 497 % 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} \\
% and 0 otherwise. The new equations are as follows: 467 498 % r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
% 468 499 % \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} \\
% \begin{align} 469 500 % \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
% 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} \\ 470 501 % \end{align}
% r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\ 471 502 %
% \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} \\ 472 503 % Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
% \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} 473 504 % respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
% \end{align} 474 505 % Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
% 475 506 %
% Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 476 507 % % r2.13
% respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 477 508 % This modified model is quadratic since we multiply two variables in the
% Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most. 478 509 % equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
% 479 510 % The Gurobi
% % r2.13 480 511 % (\url{www.gurobi.com}) optimization software is used to solve this quadratic
% This modified model is quadratic since we multiply two variables in the 481 512 % model, and since Gurobi is able to linearize, the model is left as is. This model
% equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary. 482 513 % has $O(np)$ variables and $O(n)$ constraints.
% The Gurobi 483 514
% (\url{www.gurobi.com}) optimization software is used to solve this quadratic 484 515 Two problems will be addressed using the workflow described in the next section: on the one
% model, and since Gurobi is able to linearize, the model is left as is. This model 485 516 hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary
% has $O(np)$ variables and $O(n)$ constraints. 486 517 silcon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area
487 518 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
Two problems will be addressed using the workflow described in the next section: on the one 488 519 objective function is replaced with:
hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary 489 520 \begin{align}
silcon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area 490 521 \text{Minimize } & \sum_{i=1}^n a_i \notag
for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the 491 522 \end{align}
objective function is replaced with: 492 523 We adapt our constraints of quadratic program to replace equation \ref{eq:area}
\begin{align} 493 524 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
\text{Minimize } & \sum_{i=1}^n a_i \notag 494 525 rejection required.
\end{align} 495 526
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 496 527 \begin{align}
with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal 497 528 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
rejection required. 498 529 \end{align}
499 530
\begin{align} 500 531 \section{Design workflow}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 501 532 \label{sec:workflow}
\end{align} 502 533
503 534 In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area}
\section{Design workflow} 504 535 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
\label{sec:workflow} 505 536 in the computation of the results.
506 537
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 507 538 \begin{figure}
and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved 508 539 \centering
in the computation of the results. 509 540 \begin{tikzpicture}[node distance=0.75cm and 2cm]
510 541 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\begin{figure} 511 542 \node (Start) [left= 3cm of Solver] { } ;
\centering 512 543 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\begin{tikzpicture}[node distance=0.75cm and 2cm] 513 544 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 514 545 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node (Start) [left= 3cm of Solver] { } ; 515 546 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 516 547 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node (Input) [above= of TCL] { } ; 517 548 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 518 549 \node (Results) [left= of Postproc] { } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 519 550
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 520 551 \draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 521 552 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\node (Results) [left= of Postproc] { } ; 522 553 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
523 554 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 524 555 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ; 525 556 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 526 557 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 527 558 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 528 559 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 529 560 \draw[->] (Postproc) -- (Results) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 530 561 \end{tikzpicture}
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 531 562 \caption{Design workflow from the input parameters to the results {\color{red} allowing for
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 532 563 a fully automated optimal solution search.}}
\draw[->] (Postproc) -- (Results) ; 533 564 \label{fig:workflow}
\end{tikzpicture} 534 565 \end{figure}
\caption{Design workflow from the input parameters to the results} 535 566
\label{fig:workflow} 536 567 The filter solver is a C++ program that takes as input the maximum area
\end{figure} 537 568 $\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$,
538 569 the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates
The filter solver is a C++ program that takes as input the maximum area 539 570 the quadratic programs and uses the Gurobi solver to estimate the optimal results.
$\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$, 540 571 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates 541 572 and a deploy script ((1b) on figure~\ref{fig:workflow}).
the quadratic programs and uses the Gurobi solver to estimate the optimal results. 542 573
Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow}) 543 574 The TCL script describes the whole digital processing chain from the beginning
and a deploy script ((1b) on figure~\ref{fig:workflow}). 544 575 (the raw signal data) to the end (the filtered data) in a language compatible
545 576 with proprietary synthesis software, namely Vivado for Xilinx and Quartus for
The TCL script describes the whole digital processing chain from the beginning 546 577 Intel/Altera. The raw input data generated from a 20-bit Pseudo Random Number (PRN)
(the raw signal data) to the end (the filtered data) in a language compatible 547 578 generator inside the FPGA and $\Pi^I$ is fixed at 16~bits.
with proprietary synthesis software, namely Vivado for Xilinx and Quartus for 548 579 Then the script builds each stage of the chain with a generic FIR task that
Intel/Altera. The raw input data generated from a 20-bit Pseudo Random Number (PRN) 549 580 comes from a skeleton library. The generic FIR is highly configurable
generator inside the FPGA and $\Pi^I$ is fixed at 16~bits. 550 581 with the number of coefficients and the size of the coefficients. The coefficients
Then the script builds each stage of the chain with a generic FIR task that 551 582 themselves are not stored in the script.
comes from a skeleton library. The generic FIR is highly configurable 552 583 As the signal is processed in real-time, the output signal is stored as
with the number of coefficients and the size of the coefficients. The coefficients 553 584 consecutive bursts of data for post-processing, mainly assessing the consistency of the
themselves are not stored in the script. 554 585 implemented FIR cascade transfer function with the design criteria and the expected
As the signal is processed in real-time, the output signal is stored as 555 586 transfer function.
consecutive bursts of data for post-processing, mainly assessing the consistency of the 556 587
implemented FIR cascade transfer function with the design criteria and the expected 557 588 The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}).
transfer function. 558 589 We use the 2018.2 version of Xilinx Vivado and we execute the synthesized
559 590 bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 560 591 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
We use the 2018.2 version of Xilinx Vivado and we execute the synthesized 561 592 provide a broadband noise source.
bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series 562 593 The board runs the Linux kernel and surrounding environment produced from the
FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to 563 594 Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring
provide a broadband noise source. 564 595 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
The board runs the Linux kernel and surrounding environment produced from the 565 596 fetching the results is automated.
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 566 597
the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and 567 598 The deploy script uploads the bitstream to the board ((3) on
fetching the results is automated. 568 599 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
569 600 configures the coefficients of the FIR filters. It then waits for the results
The deploy script uploads the bitstream to the board ((3) on 570 601 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 571 602
configures the coefficients of the FIR filters. It then waits for the results 572 603 Finally, an Octave post-processing script computes the final results thanks to
and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}). 573 604 the output data ((5) on figure~\ref{fig:workflow}).
574 605 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
Finally, an Octave post-processing script computes the final results thanks to 575 606 and the different configurations can be compared.
the output data ((5) on figure~\ref{fig:workflow}). 576 607
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 577 608 \section{Maximizing the rejection at fixed silicon area}
and the different configurations can be compared. 578 609 \label{sec:fixed_area}
579 610 This section presents the output of the filter solver {\em i.e.} the computed
\section{Maximizing the rejection at fixed silicon area} 580 611 configurations for each stage, the computed rejection and the computed silicon area.
\label{sec:fixed_area} 581 612 Such results allow for understanding the choices made by the solver to compute its solutions.
This section presents the output of the filter solver {\em i.e.} the computed 582 613
configurations for each stage, the computed rejection and the computed silicon area. 583 614 The experimental setup is composed of three cases. The raw input is generated
Such results allow for understanding the choices made by the solver to compute its solutions. 584 615 by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$.
585 616 Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500
The experimental setup is composed of three cases. The raw input is generated 586 617 arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500.
by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$. 587 618 The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$
Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500 588 619 ranging from 2 to 22. In each case, the quadratic program has been able to give a
arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500. 589 620 result up to five stages ($n = 5$) in the cascaded filter.
The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$ 590 621
ranging from 2 to 22. In each case, the quadratic program has been able to give a 591 622 Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500.
result up to five stages ($n = 5$) in the cascaded filter. 592 623 Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000.
593 624 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 594 625
Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000. 595 626 \renewcommand{\arraystretch}{1.4}
Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500. 596 627
597 628 \begin{table}
\renewcommand{\arraystretch}{1.4} 598 629 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
599 630 \label{tbl:gurobi_max_500}
\begin{table} 600 631 \centering
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 601 632 {\scalefont{0.77}
\label{tbl:gurobi_max_500} 602 633 \begin{tabular}{|c|ccccc|c|c|}
\centering 603 634 \hline
{\scalefont{0.77} 604 635 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\begin{tabular}{|c|ccccc|c|c|} 605 636 \hline
\hline 606 637 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 607 638 2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\
\hline 608 639 3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 609 640 4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\
2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ 610 641 5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\
3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ 611 642 \hline
4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\ 612 643 \end{tabular}
5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\ 613 644 }
\hline 614 645 \end{table}
\end{tabular} 615 646
} 616 647 \begin{table}
\end{table} 617 648 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
618 649 \label{tbl:gurobi_max_1000}
\begin{table} 619 650 \centering
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 620 651 {\scalefont{0.77}
\label{tbl:gurobi_max_1000} 621 652 \begin{tabular}{|c|ccccc|c|c|}
\centering 622 653 \hline
{\scalefont{0.77} 623 654 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\begin{tabular}{|c|ccccc|c|c|} 624 655 \hline
\hline 625 656 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 626 657 2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\
\hline 627 658 3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 628 659 4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\
2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ 629 660 5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\
3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ 630 661 \hline
4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\ 631 662 \end{tabular}
5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\ 632 663 }
\hline 633 664 \end{table}
\end{tabular} 634 665
} 635 666 \begin{table}
\end{table} 636 667 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
637 668 \label{tbl:gurobi_max_1500}
\begin{table} 638 669 \centering
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 639 670 {\scalefont{0.77}
\label{tbl:gurobi_max_1500} 640 671 \begin{tabular}{|c|ccccc|c|c|}
\centering 641 672 \hline
{\scalefont{0.77} 642 673 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\begin{tabular}{|c|ccccc|c|c|} 643 674 \hline
\hline 644 675 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 645 676 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\
\hline 646 677 3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 647 678 4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ 648 679 5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\
3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ 649 680 \hline
4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\ 650 681 \end{tabular}
5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\ 651 682 }
\hline 652 683 \end{table}
\end{tabular} 653 684
} 654 685 \renewcommand{\arraystretch}{1}
\end{table} 655 686
656 687 From these tables, we can first state that the more stages are used to define
\renewcommand{\arraystretch}{1} 657 688 the cascaded FIR filters, the better the rejection. It was an expected result as it has
658 689 been previously observed that many small filters are better than
From these tables, we can first state that the more stages are used to define 659 690 a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions
the cascaded FIR filters, the better the rejection. It was an expected result as it has 660 691 being hardly used in practice due to the lack of tools for identifying individual filter
been previously observed that many small filters are better than 661 692 coefficients in the cascaded approach.
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 662 693
being hardly used in practice due to the lack of tools for identifying individual filter 663 694 Second, the larger the silicon area, the better the rejection. This was also an
coefficients in the cascaded approach. 664 695 expected result as more area means a filter of better quality with more coefficients
665 696 or more bits per coefficient.
Second, the larger the silicon area, the better the rejection. This was also an 666 697
expected result as more area means a filter of better quality with more coefficients 667 698 Then, we also observe that the first stage can have a larger shift than the other
or more bits per coefficient. 668 699 stages. This is explained by the fact that the solver tries to use just enough
669 700 bits for the computed rejection after each stage. In the first stage, a
Then, we also observe that the first stage can have a larger shift than the other 670 701 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
stages. This is explained by the fact that the solver tries to use just enough 671 702 gives the relation between both values.
bits for the computed rejection after each stage. In the first stage, a 672 703
balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift} 673 704 Finally, we note that the solver consumes all the given silicon area.
gives the relation between both values. 674 705
675 706 The following graphs present the rejection for real data on the FPGA. In all the following
Finally, we note that the solver consumes all the given silicon area. 676 707 figures, the solid line represents the actual rejection of the filtered
677 708 data on the FPGA as measured experimentally and the dashed line are the noise levels
The following graphs present the rejection for real data on the FPGA. In all the following 678 709 given by the quadratic solver. The configurations are those computed in the previous section.
figures, the solid line represents the actual rejection of the filtered 679 710
data on the FPGA as measured experimentally and the dashed line are the noise levels 680 711 Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500.
given by the quadratic solver. The configurations are those computed in the previous section. 681 712 Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000.
682 713 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 683 714
Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000. 684 715 % \begin{figure}
Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500. 685 716 % \centering
686 717 % \includegraphics[width=\linewidth]{images/max_500}
% \begin{figure} 687 718 % \caption{Signal spectrum for MAX/500}
% \centering 688 719 % \label{fig:max_500_result}
% \includegraphics[width=\linewidth]{images/max_500} 689 720 % \end{figure}
% \caption{Signal spectrum for MAX/500} 690 721 %
% \label{fig:max_500_result} 691 722 % \begin{figure}
% \end{figure} 692 723 % \centering
% 693 724 % \includegraphics[width=\linewidth]{images/max_1000}
% \begin{figure} 694 725 % \caption{Signal spectrum for MAX/1000}
% \centering 695 726 % \label{fig:max_1000_result}
% \includegraphics[width=\linewidth]{images/max_1000} 696 727 % \end{figure}
% \caption{Signal spectrum for MAX/1000} 697 728 %
% \label{fig:max_1000_result} 698 729 % \begin{figure}
% \end{figure} 699 730 % \centering
% 700 731 % \includegraphics[width=\linewidth]{images/max_1500}
% \begin{figure} 701 732 % \caption{Signal spectrum for MAX/1500}
% \centering 702 733 % \label{fig:max_1500_result}
% \includegraphics[width=\linewidth]{images/max_1500} 703 734 % \end{figure}
% \caption{Signal spectrum for MAX/1500} 704 735
% \label{fig:max_1500_result} 705 736 % r2.14 et r2.15 et r2.16
% \end{figure} 706 737 \begin{figure}
707 738 \centering
% r2.14 et r2.15 et r2.16 708 739 \begin{subfigure}{\linewidth}
\begin{figure} 709 740 \includegraphics[width=\linewidth]{images/max_500}
\centering 710 741 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
\begin{subfigure}{\linewidth} 711 742 the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).}
\includegraphics[width=\linewidth]{images/max_500} 712 743 \label{fig:max_500_result}
\caption{Signal spectrum for MAX/500} 713 744 \end{subfigure}
\label{fig:max_500_result} 714 745
\end{subfigure} 715 746 \begin{subfigure}{\linewidth}
716 747 \includegraphics[width=\linewidth]{images/max_1000}
\begin{subfigure}{\linewidth} 717 748 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
\includegraphics[width=\linewidth]{images/max_1000} 718 749 the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).}
\caption{Signal spectrum for MAX/1000} 719 750 \label{fig:max_1000_result}
\label{fig:max_1000_result} 720 751 \end{subfigure}
\end{subfigure} 721 752
722 753 \begin{subfigure}{\linewidth}
\begin{subfigure}{\linewidth} 723 754 \includegraphics[width=\linewidth]{images/max_1500}
\includegraphics[width=\linewidth]{images/max_1500} 724 755 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
\caption{Signal spectrum for MAX/1500} 725 756 the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).}
\label{fig:max_1500_result} 726 757 \label{fig:max_1500_result}
\end{subfigure} 727 758 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500} 728 759 \caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing
\end{figure} 729 760 rejection for a given resource allocation.
730 761 The filter shape constraint (bandpass and bandstop) is shown as thick
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 731 762 horizontal lines on each chart.}
732 763 \end{figure}
We compare the actual silicon resources given by Vivado to the 733 764
resources in arbitrary units. 734 765 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
The goal is to check that our arbitrary units of silicon area models well enough 735 766
the real resources on the FPGA. Especially we want to verify that, for a given 736 767 We compare the actual silicon resources given by Vivado to the
number of arbitrary units, the actual silicon resources do not depend on the 737 768 resources in arbitrary units.
number of stages $n$. Most significantly, our approach aims 738 769 The goal is to check that our arbitrary units of silicon area models well enough
at remaining far enough from the practical logic gate implementation used by 739 770 the real resources on the FPGA. Especially we want to verify that, for a given
various vendors to remain platform independent and be portable from one 740 771 number of arbitrary units, the actual silicon resources do not depend on the
architecture to another. 741 772 number of stages $n$. Most significantly, our approach aims
742 773 at remaining far enough from the practical logic gate implementation used by
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 743 774 various vendors to remain platform independent and be portable from one
MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000 744 775 architecture to another.
and 1500 arbitrary units. We have taken care to extract solely the resources used by 745 776
the FIR filters and remove additional processing blocks including FIFO and Programmable 746 777 Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 747 778 MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000
748 779 and 1500 arbitrary units. We have taken care to extract solely the resources used by
\begin{table}[h!tb] 749 780 the FIR filters and remove additional processing blocks including FIFO and Programmable
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 750 781 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
\label{tbl:resources_usage} 751 782
\centering 752 783 \begin{table}[h!tb]
\begin{tabular}{|c|c|ccc|c|} 753 784 \caption{Resource occupation {\color{red}following synthesis of the solutions found for
\hline 754 785 the problem of maximizing rejection for a given resource allocation}. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 755 786 \label{tbl:resources_usage}
& LUT & 249 & 453 & 627 & \emph{17600} \\ 756 787 \centering
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 757 788 \begin{tabular}{|c|c|ccc|c|}
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 758 789 \hline
& LUT & 2374 & 5494 & 691 & \emph{17600} \\ 759 790 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 760 791 & LUT & 249 & 453 & 627 & \emph{17600} \\
& DSP & 0 & 0 & 70 & \emph{80} \\ \hline 761 792 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& LUT & 2443 & 3304 & 3521 & \emph{17600} \\ 762 793 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 763 794 & LUT & 2374 & 5494 & 691 & \emph{17600} \\
& DSP & 0 & 19 & 35 & \emph{80} \\ \hline 764 795 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& LUT & 2634 & 3753 & 2557 & \emph{17600} \\ 765 796 & DSP & 0 & 0 & 70 & \emph{80} \\ \hline
4 & BRAM & 4 & 4 & 4 & \emph{120} \\ 766 797 & LUT & 2443 & 3304 & 3521 & \emph{17600} \\
& DPS & 0 & 19 & 46 & \emph{80} \\ \hline 767 798 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& LUT & 2423 & 3047 & 2847 & \emph{17600} \\ 768 799 & DSP & 0 & 19 & 35 & \emph{80} \\ \hline
5 & BRAM & 5 & 5 & 5 & \emph{120} \\ 769 800 & LUT & 2634 & 3753 & 2557 & \emph{17600} \\
& DPS & 0 & 22 & 46 & \emph{80} \\ \hline 770 801 4 & BRAM & 4 & 4 & 4 & \emph{120} \\
\end{tabular} 771 802 & DPS & 0 & 19 & 46 & \emph{80} \\ \hline
\end{table} 772 803 & LUT & 2423 & 3047 & 2847 & \emph{17600} \\
773 804 5 & BRAM & 5 & 5 & 5 & \emph{120} \\
In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 774 805 & DPS & 0 & 22 & 46 & \emph{80} \\ \hline
when the filter coefficients are small enough, or when the input size is small 775 806 \end{tabular}
enough, Vivado optimizes resource consumption by selecting multiplexers to 776 807 \end{table}
implement the multiplications instead of a DSP. In this case, it is quite difficult 777 808
to compare the whole silicon budget. 778 809 In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that,
779 810 when the filter coefficients are small enough, or when the input size is small
However, a rough estimation can be made with a simple equivalence: looking at 780 811 enough, Vivado optimizes resource consumption by selecting multiplexers to
the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$, 781 812 implement the multiplications instead of a DSP. In this case, it is quite difficult
we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon 782 813 to compare the whole silicon budget.
area use. With this equivalence, our 500 arbitraty units correspond to 2500 LUTs, 783 814
1000 arbitrary units correspond to 5000 LUTs and 1500 arbitrary units correspond 784 815 However, a rough estimation can be made with a simple equivalence: looking at
to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary 785 816 the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$,
unit map well to actual hardware resources. The relatively small differences can probably be explained 786 817 we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon
by the optimizations done by Vivado based on the detailed map of available processing resources. 787 818 area use. With this equivalence, our 500 arbitraty units correspond to 2500 LUTs,
788 819 1000 arbitrary units correspond to 5000 LUTs and 1500 arbitrary units correspond
We now present the computation time needed to solve the quadratic problem. 789 820 to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary
For each case, the filter solver software is executed on a Intel(R) Xeon(R) CPU E5606 790 821 unit map well to actual hardware resources. The relatively small differences can probably be explained
clocked at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve 791 822 by the optimizations done by Vivado based on the detailed map of available processing resources.
the quadratic problem. Table~\ref{tbl:area_time} shows the time needed to solve the quadratic 792 823
problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units. 793 824 We now present the computation time needed to solve the quadratic problem.
794 825 For each case, the filter solver software is executed on a Intel(R) Xeon(R) CPU E5606
\begin{table}[h!tb] 795 826 clocked at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve
\caption{Time needed to solve the quadratic program with Gurobi} 796 827 the quadratic problem. Table~\ref{tbl:area_time} shows the time needed to solve the quadratic
\label{tbl:area_time} 797 828 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
\centering 798 829
\begin{tabular}{|c|c|c|c|}\hline 799 830 \begin{table}[h!tb]
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 800 831 \caption{Time needed to solve the quadratic program with Gurobi}
1 & 0.1~s & 0.1~s & 0.3~s \\ 801 832 \label{tbl:area_time}
2 & 1.1~s & 2.2~s & 12~s \\ 802 833 \centering
3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ 803 834 \begin{tabular}{|c|c|c|c|}\hline
4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\ 804 835 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline 805 836 1 & 0.1~s & 0.1~s & 0.3~s \\
\end{tabular} 806 837 2 & 1.1~s & 2.2~s & 12~s \\
\end{table} 807 838 3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\
808 839 4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\
As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? 809 840 5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline
When the area is limited, the design exploration space is more limited and the solver is able to 810 841 \end{tabular}
find an optimal solution faster. 811 842 \end{table}
812 843
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 813 844 As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ?
814 845 When the area is limited, the design exploration space is more limited and the solver is able to
This section presents the results of the complementary quadratic program aimed at 815 846 find an optimal solution faster.
minimizing the area occupation for a targeted rejection level. 816 847
817 848 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
The experimental setup is composed of four cases. The raw input is the same 818 849
as in the previous section, from a PRN generator, which fixes the input data size $\Pi^I$. 819 850 This section presents the results of the complementary quadratic program aimed at
Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB. 820 851 minimizing the area occupation for a targeted rejection level.
Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100. 821 852
The number of configurations $p$ is the same as previous section. 822 853 The experimental setup is composed of four cases. The raw input is the same
823 854 as in the previous section, from a PRN generator, which fixes the input data size $\Pi^I$.
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 824 855 Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB.
Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60. 825 856 Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100.
Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80. 826 857 The number of configurations $p$ is the same as previous section.
Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100. 827 858
828 859 Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40.
\renewcommand{\arraystretch}{1.4} 829 860 Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60.
830 861 Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80.
\begin{table}[h!tb] 831 862 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 832 863
\label{tbl:gurobi_min_40} 833 864 \renewcommand{\arraystretch}{1.4}
\centering 834 865
{\scalefont{0.77} 835 866 \begin{table}[h!tb]
\begin{tabular}{|c|ccccc|c|c|} 836 867 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\hline 837 868 \label{tbl:gurobi_min_40}
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 838 869 \centering
\hline 839 870 {\scalefont{0.77}
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 840 871 \begin{tabular}{|c|ccccc|c|c|}
2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ 841 872 \hline
3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ 842 873 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\ 843 874 \hline
\hline 844 875 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
\end{tabular} 845 876 2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\
} 846 877 3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\
\end{table} 847 878 4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\
848 879 \hline
\begin{table}[h!tb] 849 880 \end{tabular}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 850 881 }
\label{tbl:gurobi_min_60} 851 882 \end{table}
\centering 852 883
{\scalefont{0.77} 853 884 \begin{table}[h!tb]
\begin{tabular}{|c|ccccc|c|c|} 854 885 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\hline 855 886 \label{tbl:gurobi_min_60}
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 856 887 \centering
\hline 857 888 {\scalefont{0.77}
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 858 889 \begin{tabular}{|c|ccccc|c|c|}
2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ 859 890 \hline
3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ 860 891 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\ 861 892 \hline
5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\ 862 893 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
\hline 863 894 2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\
\end{tabular} 864 895 3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\
} 865 896 4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\
\end{table} 866 897 5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\
867 898 \hline
\begin{table}[h!tb] 868 899 \end{tabular}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 869 900 }
\label{tbl:gurobi_min_80} 870 901 \end{table}
\centering 871 902
{\scalefont{0.77} 872 903 \begin{table}[h!tb]
\begin{tabular}{|c|ccccc|c|c|} 873 904 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\hline 874 905 \label{tbl:gurobi_min_80}
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 875 906 \centering
\hline 876 907 {\scalefont{0.77}
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 877 908 \begin{tabular}{|c|ccccc|c|c|}
2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ 878 909 \hline
3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ 879 910 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\ 880 911 \hline
5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\ 881 912 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
\hline 882 913 2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\
\end{tabular} 883 914 3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\
} 884 915 4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\
\end{table} 885 916 5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\
886 917 \hline
\begin{table}[h!tb] 887 918 \end{tabular}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 888 919 }
\label{tbl:gurobi_min_100} 889 920 \end{table}
\centering 890 921
{\scalefont{0.77} 891 922 \begin{table}[h!tb]
\begin{tabular}{|c|ccccc|c|c|} 892 923 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\hline 893 924 \label{tbl:gurobi_min_100}
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 894 925 \centering
\hline 895 926 {\scalefont{0.77}
1 & - & - & - & - & - & - & - \\ 896 927 \begin{tabular}{|c|ccccc|c|c|}
2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\ 897 928 \hline
3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\ 898 929 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
4 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 0) & - & 101~dB & 909 \\ 899 930 \hline
5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\ 900 931 1 & - & - & - & - & - & - & - \\
\hline 901 932 2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\
\end{tabular} 902 933 3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\
} 903 934 4 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 0) & - & 101~dB & 909 \\
\end{table} 904 935 5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\
\renewcommand{\arraystretch}{1} 905 936 \hline
906 937 \end{tabular}
From these tables, we can first state that almost all configurations reach the targeted rejection 907 938 }
level or even better thanks to our underestimate of the cascade rejection as the sum of the 908 939 \end{table}
individual filter rejection. The only exception is for the monolithic case ($n = 1$) in 909 940 \renewcommand{\arraystretch}{1}
MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection. 910 941
Futhermore, the area of the monolithic filter is twice as big as the two cascaded filters 911 942 From these tables, we can first state that almost all configurations reach the targeted rejection
(1131 and 1760 arbitrary units v.s 547 and 903 arbitrary units for 60 and 80~dB rejection 912 943 level or even better thanks to our underestimate of the cascade rejection as the sum of the
respectively). More generally, the more filters are cascaded, the lower the occupied area. 913 944 individual filter rejection. The only exception is for the monolithic case ($n = 1$) in
914 945 MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection.
Like in previous section, the solver chooses always a little filter as first 915 946 Futhermore, the area of the monolithic filter is twice as big as the two cascaded filters
filter stage and the second one is often the biggest filter. This choice can be explained 916 947 (1131 and 1760 arbitrary units v.s 547 and 903 arbitrary units for 60 and 80~dB rejection
as in the previous section, with the solver using just enough bits not to degrade the input 917 948 respectively). More generally, the more filters are cascaded, the lower the occupied area.
signal and in the second filter selecting a better filter to improve rejection without 918 949
having too many bits in the output data. 919 950 Like in previous section, the solver chooses always a little filter as first
920 951 filter stage and the second one is often the biggest filter. This choice can be explained
For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 921 952 as in the previous section, with the solver using just enough bits not to degrade the input
number of filters is 4 so it did not chose any configuration for the last filter. Hence this 922 953 signal and in the second filter selecting a better filter to improve rejection without
solution is equivalent to the result for $n = 4$. 923 954 having too many bits in the output data.
924 955
The following graphs present the rejection for real data on the FPGA. In all the following 925 956 For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal
figures, the solid line represents the actual rejection of the filtered 926 957 number of filters is 4 so it did not chose any configuration for the last filter. Hence this
data on the FPGA as measured experimentally and the dashed line is the noise level 927 958 solution is equivalent to the result for $n = 4$.
given by the quadratic solver. 928 959
929 960 The following graphs present the rejection for real data on the FPGA. In all the following
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 930 961 figures, the solid line represents the actual rejection of the filtered
Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60. 931 962 data on the FPGA as measured experimentally and the dashed line is the noise level
Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80. 932 963 given by the quadratic solver.
Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100. 933 964
934 965 Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40.
% \begin{figure} 935 966 Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60.
% \centering 936 967 Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80.
% \includegraphics[width=\linewidth]{images/min_40} 937 968 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
% \caption{Signal spectrum for MIN/40} 938 969
% \label{fig:min_40} 939 970 % \begin{figure}
% \end{figure} 940 971 % \centering
% 941 972 % \includegraphics[width=\linewidth]{images/min_40}
% \begin{figure} 942 973 % \caption{Signal spectrum for MIN/40}
% \centering 943 974 % \label{fig:min_40}
% \includegraphics[width=\linewidth]{images/min_60} 944 975 % \end{figure}
% \caption{Signal spectrum for MIN/60} 945 976 %
% \label{fig:min_60} 946 977 % \begin{figure}
% \end{figure} 947 978 % \centering
% 948 979 % \includegraphics[width=\linewidth]{images/min_60}
% \begin{figure} 949 980 % \caption{Signal spectrum for MIN/60}
% \centering 950 981 % \label{fig:min_60}
% \includegraphics[width=\linewidth]{images/min_80} 951 982 % \end{figure}
% \caption{Signal spectrum for MIN/80} 952 983 %
% \label{fig:min_80} 953 984 % \begin{figure}
% \end{figure} 954 985 % \centering
% 955 986 % \includegraphics[width=\linewidth]{images/min_80}
% \begin{figure} 956 987 % \caption{Signal spectrum for MIN/80}
% \centering 957 988 % \label{fig:min_80}
% \includegraphics[width=\linewidth]{images/min_100} 958 989 % \end{figure}
% \caption{Signal spectrum for MIN/100} 959 990 %
% \label{fig:min_100} 960 991 % \begin{figure}
% \end{figure} 961 992 % \centering
962 993 % \includegraphics[width=\linewidth]{images/min_100}
% r2.14 et r2.15 et r2.16 963 994 % \caption{Signal spectrum for MIN/100}
\begin{figure} 964 995 % \label{fig:min_100}
\centering 965 996 % \end{figure}
\begin{subfigure}{\linewidth} 966 997
\includegraphics[width=\linewidth]{images/min_40} 967 998 % r2.14 et r2.15 et r2.16
\caption{Signal spectrum for MIN/40} 968 999 \begin{figure}
\label{fig:min_40} 969 1000 \centering
\end{subfigure} 970 1001 \begin{subfigure}{\linewidth}
971 1002 \includegraphics[width=.91\linewidth]{images/min_40}
\begin{subfigure}{\linewidth} 972 1003 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
\includegraphics[width=\linewidth]{images/min_60} 973 1004 the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.}
\caption{Signal spectrum for MIN/60} 974 1005 \label{fig:min_40}
\label{fig:min_60} 975 1006 \end{subfigure}
\end{subfigure} 976 1007
977 1008 \begin{subfigure}{\linewidth}
\begin{subfigure}{\linewidth} 978 1009 \includegraphics[width=.91\linewidth]{images/min_60}
\includegraphics[width=\linewidth]{images/min_80} 979 1010 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
\caption{Signal spectrum for MIN/80} 980 1011 the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.}
\label{fig:min_80} 981 1012 \label{fig:min_60}
\end{subfigure} 982 1013 \end{subfigure}
983 1014
\begin{subfigure}{\linewidth} 984 1015 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_100} 985 1016 \includegraphics[width=.91\linewidth]{images/min_80}
\caption{Signal spectrum for MIN/100} 986 1017 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
\label{fig:min_100} 987 1018 the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.}
\end{subfigure} 988 1019 \label{fig:min_80}
\caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100} 989 1020 \end{subfigure}
\end{figure} 990 1021
991 1022 \begin{subfigure}{\linewidth}
We observe that all rejections given by the quadratic solver are close to the experimentally 992 1023 \includegraphics[width=.91\linewidth]{images/min_100}
measured rejection. All curves prove that the constraint to reach the target rejection is 993 1024 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters. 994 1025 the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}
995 1026 \label{fig:min_100}
Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60; 996 1027 \end{subfigure}
MIN/80 and MIN/100 \emph{i.e.} when the target rejection is fixed to 40, 60, 80 and 100~dB. We 997 1028 \caption{\color{red}Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a
have taken care to extract solely the resources used by 998 1029 given rejection while minimizing resource allocation. The filter shape constraint (bandpass and
the FIR filters and remove additional processing blocks including FIFO and PL to 999 1030 bandstop) is shown as thick
PS communication. 1000 1031 horizontal lines on each chart.}
1001 1032 \end{figure}
\renewcommand{\arraystretch}{1.2} 1002 1033
\begin{table} 1003 1034 We observe that all rejections given by the quadratic solver are close to the experimentally
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 1004 1035 measured rejection. All curves prove that the constraint to reach the target rejection is
\label{tbl:resources_usage_comp} 1005 1036 respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.
\centering 1006 1037
{\scalefont{0.90} 1007 1038 Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60;
\begin{tabular}{|c|c|cccc|c|} 1008 1039 MIN/80 and MIN/100 \emph{i.e.} when the target rejection is fixed to 40, 60, 80 and 100~dB. We
\hline 1009 1040 have taken care to extract solely the resources used by
$n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline 1010 1041 the FIR filters and remove additional processing blocks including FIFO and PL to
& LUT & 343 & 334 & 772 & - & \emph{17600} \\ 1011 1042 PS communication.
1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\ 1012 1043
& DSP & 27 & 39 & 55 & - & \emph{80} \\ \hline 1013 1044 \renewcommand{\arraystretch}{1.2}
& LUT & 1252 & 2862 & 5099 & 640 & \emph{17600} \\ 1014 1045 \begin{table}
2 & BRAM & 2 & 2 & 2 & 2 & \emph{120} \\ 1015 1046 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
& DSP & 0 & 0 & 0 & 66 & \emph{80} \\ \hline 1016 1047 \label{tbl:resources_usage_comp}
& LUT & 891 & 2148 & 2023 & 2448 & \emph{17600} \\ 1017 1048 \centering
3 & BRAM & 3 & 3 & 3 & 3 & \emph{120} \\ 1018 1049 {\scalefont{0.90}
& DSP & 0 & 0 & 19 & 27 & \emph{80} \\ \hline 1019 1050 \begin{tabular}{|c|c|cccc|c|}
& LUT & 662 & 1729 & 2451 & 2893 & \emph{17600} \\ 1020 1051 \hline
4 & BRAM & 4 & 4 & 4 & 4 & \emph{120} \\ 1021 1052 $n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline
& DPS & 0 & 0 & 7 & 19 & \emph{80} \\ \hline 1022 1053 & LUT & 343 & 334 & 772 & - & \emph{17600} \\
& LUT & - & 1259 & 2602 & 2505 & \emph{17600} \\ 1023 1054 1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\
ifcs2018_journal_reponse.tex
Diff comments   View file @ efde7e8
%Minor Revision - TUFFC-09469-2019 1 1 %Minor Revision - TUFFC-09469-2019
%Transactions on Ultrasonics, Ferroelectrics, and Frequency 2 2 %Transactions on Ultrasonics, Ferroelectrics, and Frequency
%Control (July 23, 2019 9:29 PM) 3 3 %Control (July 23, 2019 9:29 PM)
%To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr, 4 4 %To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr,
%gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr, 5 5 %gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr,
%jmfriedt@femto-st.fr 6 6 %jmfriedt@femto-st.fr
%CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu 7 7 %CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu
% 8 8 %
%Dear Mr. Arthur HUGEAT 9 9 %Dear Mr. Arthur HUGEAT
% 10 10 %
%Congratulations! Your manuscript 11 11 %Congratulations! Your manuscript
% 12 12 %
%MANUSCRIPT NO. TUFFC-09469-2019 13 13 %MANUSCRIPT NO. TUFFC-09469-2019
%MANUSCRIPT TYPE: Papers 14 14 %MANUSCRIPT TYPE: Papers
%TITLE: Filter optimization for real time digital processing of radiofrequency 15 15 %TITLE: Filter optimization for real time digital processing of radiofrequency
%signals: application to oscillator metrology 16 16 %signals: application to oscillator metrology
%AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, 17 17 %AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois,
%Pierre-Yves; Friedt, Jean-Michel 18 18 %Pierre-Yves; Friedt, Jean-Michel
% 19 19 %
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%**************************************************** 73 73 %****************************************************
%REVIEWERS' COMMENTS: 74 74 %REVIEWERS' COMMENTS:
75 75
\documentclass[a4paper]{article} 76 76 \documentclass[a4paper]{article}
\usepackage{fullpage,graphicx,amsmath, subcaption} 77 77 \usepackage{fullpage,graphicx,amsmath, subcaption}
\begin{document} 78 78 \begin{document}
{\bf Reviewer: 1} 79 79 {\bf Reviewer: 1}
80 80
%Comments to the Author 81 81 %Comments to the Author
%In general, the language/grammar is adequate. 82 82 %In general, the language/grammar is adequate.
83 83
{\bf 84 84 {\bf
On page 2, "...allowing to save processing resource..." could be improved. % r1.1 - fait 85 85 On page 2, "...allowing to save processing resource..." could be improved. % r1.1 - fait
} 86 86 }
87 87
The sentence was split and now reads ``number of coefficients irrelevant: processing 88 88 The sentence was split and now reads ``number of coefficients irrelevant: processing
resources are hence saved by shrinking the filter length.'' 89 89 resources are hence saved by shrinking the filter length.''
90 90
{\bf 91 91 {\bf
On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 - fait 92 92 On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 - fait
the author meant.} 93 93 the author meant.}
94 94
Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s) 95 95 Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s)
Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.'' 96 96 Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
97 97
{\bf 98 98 {\bf
On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 - fait 99 99 On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 - fait
could be improved. 100 100 could be improved.
} 101 101 }
102 102
Indeed this paragraph has be written again and now reads as\\ 103 103 Indeed this paragraph has be written again and now reads as\\
``The first step of our approach is to model the DSP chain. Since we aim at only optimizing 104 104 ``The first step of our approach is to model the DSP chain. Since we aim at only optimizing
the filtering part of the signal processing chain, we have not included the PRN generator or the 105 105 the filtering part of the signal processing chain, we have not included the PRN generator or the
ADC in the model: the input data size and rate are considered fixed and defined by the hardware. 106 106 ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
The filtering can be done in two ways, either by considering a single monolithic FIR filter 107 107 The filtering can be done in two ways, either by considering a single monolithic FIR filter
requiring many coefficients to reach the targeted noise rejection ratio, or by 108 108 requiring many coefficients to reach the targeted noise rejection ratio, or by
cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter. 109 109 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.
'' 110 110 ''
111 111
{\bf 112 112 {\bf
I appreciate that the authors attempted and document two optimizations: that % r1.4 - fait 113 113 I appreciate that the authors attempted and document two optimizations: that % r1.4 - fait
of maximum rejection ratio at fixed silicon area, as well as minimum silicon 114 114 of maximum rejection ratio at fixed silicon area, as well as minimum silicon
area for a fixed minimum rejection ratio. For non-experts, it might be very 115 115 area for a fixed minimum rejection ratio. For non-experts, it might be very
useful to compare the results of both optimization paths to the performance and 116 116 useful to compare the results of both optimization paths to the performance and
resource-utilization of generic low-pass filter gateware offered by device 117 117 resource-utilization of generic low-pass filter gateware offered by device
manufacturers. I appreciate also that the authors have presented source code 118 118 manufacturers. I appreciate also that the authors have presented source code
for examination online. 119 119 for examination online.
} 120 120 }
121 121
To compare the performance of our FIR filters and the performance of device 122 122 To compare the performance of our FIR filters and the performance of device
manufacturers generic filter, we have added a paragraph and a table at the 123 123 manufacturers generic filter, we have added a paragraph and a table at the
end of experiments section. We compare the resources consumption with the same 124 124 end of experiments section. We compare the resources consumption with the same
FIR coefficients set. 125 125 FIR coefficients set.
126 126
{\bf 127 127 {\bf
Reviewer: 2 128 128 Reviewer: 2
} 129 129 }
130 130
%Comments to the Author 131 131 %Comments to the Author
%In the Manuscript, the Authors describe an optimization methodology for filter 132 132 %In the Manuscript, the Authors describe an optimization methodology for filter
%design to be used in phase noise metrology. The methodology is general and can 133 133 %design to be used in phase noise metrology. The methodology is general and can
%be used for many aspects of the processing chain. In the Manuscript, the Authors 134 134 %be used for many aspects of the processing chain. In the Manuscript, the Authors
%focus on filtering and shifting while the other aspects, in particular decimation, 135 135 %focus on filtering and shifting while the other aspects, in particular decimation,
%will be considered in a future work. The optimization problem is modelled 136 136 %will be considered in a future work. The optimization problem is modelled
%theoretically and then solved by means of a commercial software. The solutions 137 137 %theoretically and then solved by means of a commercial software. The solutions
%are tested experimentally on the Redpitaya platform with synthetic and real 138 138 %are tested experimentally on the Redpitaya platform with synthetic and real
%white noises. Two cases are considered as a function of the number of filters: 139 139 %white noises. Two cases are considered as a function of the number of filters:
%maximum rejection given a fixed amount of resources and minimum resource 140 140 %maximum rejection given a fixed amount of resources and minimum resource
%utilization given a fixed amount of rejection. 141 141 %utilization given a fixed amount of rejection.
%The Authors find that filtering improves significantly when the number of 142 142 %The Authors find that filtering improves significantly when the number of
%filters increases. 143 143 %filters increases.
%A lot of work has been done in generalizing and automating the procedure so 144 144 %A lot of work has been done in generalizing and automating the procedure so
%that different approaches can be investigated quickly and efficiently. The 145 145 %that different approaches can be investigated quickly and efficiently. The
%results presented in the Manuscript seem to be just a case study based on 146 146 %results presented in the Manuscript seem to be just a case study based on
%the particular criterion chosen by the Authors. Different criteria, in 147 147 %the particular criterion chosen by the Authors. Different criteria, in
%general, could lead to different results and it is important to consider 148 148 %general, could lead to different results and it is important to consider
%carefully the criterion adopted by the Authors, in order to check if it 149 149 %carefully the criterion adopted by the Authors, in order to check if it
%is adequate to compare the performance of filters and if multi-stage 150 150 %is adequate to compare the performance of filters and if multi-stage
%filters are really superior than monolithic filters. 151 151 %filters are really superior than monolithic filters.
152 152
{\bf 153 153 {\bf
By observing the results presented in fig. 10-16, it is clear that the % r2.1 154 154 By observing the results presented in fig. 10-16, it is clear that the % r2.1
performances of multi-stage filters are obtained at the expense of their 155 155 performances of multi-stage filters are obtained at the expense of their
selectivity and, in this sense, the filters presented in these figures 156 156 selectivity and, in this sense, the filters presented in these figures
are not equivalent. For example, in Fig. 14, at the limit of the pass band, 157 157 are not equivalent. For example, in Fig. 14, at the limit of the pass band,
the attenuation is almost 15 dB for n = 5, while it is not noticeable for 158 158 the attenuation is almost 15 dB for n = 5, while it is not noticeable for
n = 1. 159 159 n = 1.
} 160 160 }
161 161
We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined 162 162 We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined
the bandpass and the bandstop of the filter. 163 163 the bandpass and the bandstop of the filter.
164 164
% We are aware of this non equivalence but we think that difference is not due to 165 165 % We are aware of this non equivalence but we think that difference is not due to
% the cascaded filters but due to the definition of rejection criterion on the passband. 166 166 % the cascaded filters but due to the definition of rejection criterion on the passband.
% Indeed, in this article we have choose to take the summation of absolute values divide 167 167 % Indeed, in this article we have choose to take the summation of absolute values divide
% by the bandwidth but this criterion is maybe too permissive and when we cascade 168 168 % by the bandwidth but this criterion is maybe too permissive and when we cascade
% some filters this impact is more important. 169 169 % some filters this impact is more important.
% 170 170 %
% However if we change the passband 171 171 % However if we change the passband
% criterion by the summation of absolute value in passband, weighting given to the 172 172 % criterion by the summation of absolute value in passband, weighting given to the
% passband ripples are too strong and the solver are too restricted to provide 173 173 % passband ripples are too strong and the solver are too restricted to provide
% any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in 174 174 % any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in
% passband, the rejection evaluation are too close form the original criterion and 175 175 % passband, the rejection evaluation are too close form the original criterion and
% the result will not be improved. 176 176 % the result will not be improved.
% 177 177 %
% In this article, we will highlight the methodology instead of the filter conception. 178 178 % In this article, we will highlight the methodology instead of the filter conception.
% Even if our rejection criterion is not the best, our methodology was not impacted 179 179 % Even if our rejection criterion is not the best, our methodology was not impacted
% by this. So to improve the results, we can choose another criterion to be more 180 180 % by this. So to improve the results, we can choose another criterion to be more
% selective in passband but it is not the main objective of our article. 181 181 % selective in passband but it is not the main objective of our article.
182
183 We are aware of this equivalence but to limit this ripples in passband we need to
184 enforce the criterion in passband. If we takes a strong constraint like the sum of
185 absolute values in passband. This criterion si too selective because it considers
186 all bin on passband while on stopband we consider only the bin with the minimal
187 rejection. The figure~\ref{fig:letter_sum_criterion} exhibits the results with this
188 criterion for the case MAX/1000. With this criterion, the solver find an optimal
189 solution with only two filters in expend of the resource consumption.
190
191
192
193 If we relax a little the criterion on passband with taking only the maximum absolute
194 value, we will penalize the ripple peak on passband. The figure~\ref{fig:letter_max_criterion}
195 shows the results for the case MAX/1000. There as almost no difference with the
196 article results. Indeed the only little change are on the case $i = 4$ and $i = 5$
197 which they have some minor differences on coefficients choices.
198
199 \begin{figure}[h!tb]
200 \centering
201 \begin{subfigure}{0.48\linewidth}
202 \includegraphics[width=\linewidth]{images/letter_sum_criterion}
203 \caption{Results for the case MAX/1000 with as criterion on passband the sum absolute values}
204 \label{fig:letter_sum_criterion}
205 \end{subfigure}
206 \begin{subfigure}{0.48\linewidth}
207 \includegraphics[width=\linewidth]{images/letter_max_criterion}
208 \caption{Results for the case MAX/1000 with as criterion on passband the maximum absolute value}
209 \label{fig:letter_max_criterion}
210 \end{subfigure}
211 \end{figure}
212
213 Finally, if we ponder the maximum absolute on passband, we should improve the result.
214 We have arbitrary pondered by 5 the maximum. Even with this weighting, the solver
215 choose the same coefficient set.
216
217 To conclude, find a better criterion to avoid the ripples on the passband is difficult.
218 In this article we are focused on the methodology so even if our criterion could
219 be improved, our methodology still the same and it works independently of rejection criterion.
182 220
We are aware of this equivalence but to limit this ripples in passband we need to 183 221 % %Peut etre refaire une serie de simulation dans lesquelles on impose une coupure
enforce the criterion in passband. If we takes a strong constraint like the sum of 184 222 % %non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte
absolute values in passband. This criterion si too selective because it considers 185 223 % %au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque
all bin on passband while on stopband we consider only the bin with the minimal 186 224 % %a la cascade de filtres.
rejection. The figure~\ref{fig:letter_sum_criterion} exhibits the results with this 187 225 % %AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de
criterion for the case MAX/1000. With this criterion, the solver find an optimal 188 226 % %relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai
solution with only two filters in expend of the resource consumption. 189 227 %
190 228 % densité spectrale de la bande passante
191 229 % sum des valeurs absolues / largeur de la bande passante (1/N) vs max dans la bande de coupure
192 230 %
If we relax a little the criterion on passband with taking only the maximum absolute 193 231 % JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait
value, we will penalize the ripple peak on passband. The figure~\ref{fig:letter_max_criterion} 194 232 % moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB
shows the results for the case MAX/1000. There as almost no difference with the 195 233 % quand on cascade 5 filtres !
article results. Indeed the only little change are on the case $i = 4$ and $i = 5$ 196 234 %
which they have some minor differences on coefficients choices. 197 235 % Dire que la chute n'est pas du à la casacade mais à notre critère de rejection
198 236
\begin{figure}[h!tb] 199 237 {\bf
\centering 200 238 The reason is in the criterion that considers the average attenuation in % r2.2
\begin{subfigure}{0.48\linewidth} 201 239 the pass band. This criterion does not take into account the maximum attenuation
\includegraphics[width=\linewidth]{images/letter_sum_criterion} 202 240 in this region, which is a very important parameter for specifying a filter
\caption{Results for the case MAX/1000 with as criterion on passband the sum absolute values} 203 241 and for evaluating its performance. For example, with this criterion, a
\label{fig:letter_sum_criterion} 204 242 filter with 0.1 dB of ripple is considered equivalent to a filter with
\end{subfigure} 205 243 10 dB of ripple. This point has a strong impact in the optimization process
\begin{subfigure}{0.48\linewidth} 206 244 and in the results that are obtained and has to be reconsidered.
\includegraphics[width=\linewidth]{images/letter_max_criterion} 207 245 }
\caption{Results for the case MAX/1000 with as criterion on passband the maximum absolute value} 208 246
\label{fig:letter_max_criterion} 209 247 See above: Choose a criterion is difficult and depending on the context. The main
\end{subfigure} 210 248 contribution on this paper is the methodology not the criterion to quantify the
249 rejection.
\end{figure} 211 250
212 251 % The manuscript erroneously stated that we considered the mean of the absolute
Finally, if we ponder the maximum absolute on passband, we should improve the result. 213 252 % value within the bandpass: the manuscript has now been corrected to properly state
We have arbitrary pondered by 5 the maximum. Even with this weighting, the solver 214 253 % the selected criterion, namely the {\em sum} of the absolute value, so that any
choose the same coefficient set. 215 254 % ripple in the bandpass will reduce the chances of a given filter set from being
216 255 % selected. The manuscript now states ``Our criterion to compute the filter rejection considers
To conclude, find a better criterion to avoid the ripples on the passband is difficult. 217 256 % % r2.8 et r2.2 r2.3
In this article we are focused on the methodology so even if our criterion could 218 257 % the maximum magnitude within the stopband, to which the {sum of the absolute values
be improved, our methodology still the same and it works independently of rejection criterion. 219 258 % within the passband is subtracted to avoid filters with excessive ripples}.''
220 259
% %Peut etre refaire une serie de simulation dans lesquelles on impose une coupure 221 260 {\bf
% %non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte 222 261 I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait
% %au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque 223 262 into account the maximum allowed attenuation in pass band, for example by
% %a la cascade de filtres. 224 263 fixing its value to a typical one, as it has been done for the transition
% %AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de 225 264 bandwidth.
% %relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai 226 265 }
% 227 266
% densité spectrale de la bande passante 228 267 See above: the absolute value within the passband will reject filters with
% sum des valeurs absolues / largeur de la bande passante (1/N) vs max dans la bande de coupure 229 268 excessive ripples, including excessive attenuation, within the passband.
% 230 269
% JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait 231 270 % TODO: test max(stopband) - max(abs(passband))
% moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB 232 271
% quand on cascade 5 filtres ! 233 272 {\bf
% 234 273 In addition, I suggest to address the following points: % r2.4 - fait
% Dire que la chute n'est pas du à la casacade mais à notre critère de rejection 235 274 - Page 1, line 50: the Authors state that IIR have shorter impulse response
236 275 than FIR. This is not true in general. The sentence should be reconsidered.
{\bf 237 276 }
The reason is in the criterion that considers the average attenuation in % r2.2 238 277
the pass band. This criterion does not take into account the maximum attenuation 239 278 We have not stated that the IIR has a shorter impulse response but a shorter lag.
in this region, which is a very important parameter for specifying a filter 240 279 Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filters
and for evaluating its performance. For example, with this criterion, a 241 280 have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing
filter with 0.1 dB of ripple is considered equivalent to a filter with 242 281 the first output, an IIR will start providing outputs only 5 time steps after the initial
10 dB of ripple. This point has a strong impact in the optimization process 243 282 input starts feeding the IIR. Hence, the issue we address here is lag and not impulse
and in the results that are obtained and has to be reconsidered. 244 283 response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue
} 245 284 in a openloop phase noise characterization instrument, the large
246 285 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
See above: Choose a criterion is difficult and depending on the context. The main 247 286 is not considered as an issue as would be in a closed loop system in which lag aims at being
contribution on this paper is the methodology not the criterion to quantify the 248 287 minimized to avoid oscillation conditions.''
rejection. 249 288
250 289 {\bf
% The manuscript erroneously stated that we considered the mean of the absolute 251 290 - Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 - fait
% value within the bandpass: the manuscript has now been corrected to properly state 252 291 this transition bandwidth and if it is a typical requirement for phase-noise
% the selected criterion, namely the {\em sum} of the absolute value, so that any 253 292 metrology.
% ripple in the bandpass will reduce the chances of a given filter set from being 254 293 }
% selected. The manuscript now states ``Our criterion to compute the filter rejection considers 255 294
% % r2.8 et r2.2 r2.3 256 295 The purpose of the paper is to demonstrate how a given filter shape can be achieved by
% the maximum magnitude within the stopband, to which the {sum of the absolute values 257 296 minimizing varous resource criteria. Indeed the stopband and bandpass boundaries can
% within the passband is subtracted to avoid filters with excessive ripples}.'' 258 297 be questioned: we have selected this filter shape as a typical anti-aliasing filter considering
259 298 the the dataflow is to be halved. Hence, selecting a cutoff frequency of 40\% the initial
{\bf 260 299 Nyquist frequency prevents noise from reaching baseband after decimating the dataflow by a
I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait 261 300 factor of 2. Such ideas are now stated explicitly in the text as ``Throughout this demonstration,
into account the maximum allowed attenuation in pass band, for example by 262 301 we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%
fixing its value to a typical one, as it has been done for the transition 263 302 of the Nyquist frequency to the end of the band, as would be typically selected to prevent
bandwidth. 264 303 aliasing before decimating the dataflow by 2. The method is however generalized to any filter
} 265 304 shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid}
266 305 as described below is indeed unique for each filter shape.''
See above: the absolute value within the passband will reject filters with 267 306
excessive ripples, including excessive attenuation, within the passband. 268 307 {\bf
269 308 - The impact of the coefficient resolution is discussed. What about the % r2.6 - fait
% TODO: test max(stopband) - max(abs(passband)) 270 309 resolution of the data stream? Is it fixed? If so, which value has been
271 310 used in the analysis? If not, how is it changed with respect to the
{\bf 272 311 coefficient resolution?
In addition, I suggest to address the following points: % r2.4 - fait 273 312 }
- Page 1, line 50: the Authors state that IIR have shorter impulse response 274 313
than FIR. This is not true in general. The sentence should be reconsidered. 275 314 We have now stated in the beginning of the document that ``we have not included the PRN generator
} 276 315 or the ADC in the model: the input data size and rate are considered fixed and defined by the
277 316 hardware.'' so indeed the input datastream resolution is considered as a given.
We have not stated that the IIR has a shorter impulse response but a shorter lag. 278 317
Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filters 279 318 {\bf
have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing 280 319 - Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait
the first output, an IIR will start providing outputs only 5 time steps after the initial 281 320 Fig. 5 can be removed.
321 }
322
323 Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats
324
325 {\bf
input starts feeding the IIR. Hence, the issue we address here is lag and not impulse 282 326 - Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait
response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue 283 327 It should be ``minimum''
in a openloop phase noise characterization instrument, the large 284 328 }
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 285
is not considered as an issue as would be in a closed loop system in which lag aims at being 286
minimized to avoid oscillation conditions.'' 287 329
288 330 This typo has been corrected.
{\bf 289 331
- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 - fait 290 332 {\bf
this transition bandwidth and if it is a typical requirement for phase-noise 291 333 - Page e, line 55, second column: ``takin'' % r2.9 - fait
metrology. 292 334 - Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait
} 293 335 - Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait
294 336 }
The purpose of the paper is to demonstrate how a given filter shape can be achieved by 295 337
minimizing varous resource criteria. Indeed the stopband and bandpass boundaries can 296 338 All typos and grammatical errors have been corrected.
be questioned: we have selected this filter shape as a typical anti-aliasing filter considering 297 339
the the dataflow is to be halved. Hence, selecting a cutoff frequency of 40\% the initial 298 340 {\bf
Nyquist frequency prevents noise from reaching baseband after decimating the dataflow by a 299 341 - Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait
factor of 2. Such ideas are now stated explicitly in the text as ``Throughout this demonstration, 300 342 these particular configurations? Are they chosen automatically?
we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\% 301 343 }
of the Nyquist frequency to the end of the band, as would be typically selected to prevent 302 344 C'est le nombre de coefficients et un taille raisonnable
aliasing before decimating the dataflow by 2. The method is however generalized to any filter 303 345 Troncature de la pyramide
shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid} 304 346
as described below is indeed unique for each filter shape.'' 305 347 See below: we have added a better description of $p$ during the transformation explanation.
306 348 ``we introduce $p$ FIR configurations.
{\bf 307 349 This variable must be defined by the user, it represent the number of different
- The impact of the coefficient resolution is discussed. What about the % r2.6 - fait 308 350 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
resolution of the data stream? Is it fixed? If so, which value has been 309 351 functions from GNU Octave)''
used in the analysis? If not, how is it changed with respect to the 310 352
coefficient resolution? 311 353 {\bf
} 312 354 - Page 4, line 31: how does the delta function transform model from non-linear % r2.13
313 355 and non-quadratic to a quadratic?}
We have now stated in the beginning of the document that ``we have not included the PRN generator 314 356
or the ADC in the model: the input data size and rate are considered fixed and defined by the 315 357 The first model is non-quadratic but when we introduce the $p$ configurations,
hardware.'' so indeed the input datastream resolution is considered as a given. 316 358 we can estimate the function $F$ by computing
317 359 the rejection for each configuration, so the model become quadratic because we have
{\bf 318 360 some multiplication between variables. With the definition of $\delta_{ij}$ we can
- Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait 319 361 replace the multiplication between variables by multiplication with binary variable and
Fig. 5 can be removed. 320 362 this one can be linearise as follow:\\
} 321 363 $y$ is a binary variable \\
322 364 $x$ is a real variable bounded by $X^{max}$ \\
Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats 323 365 \begin{equation*}
324 366 m = x \times y \implies
{\bf 325 367 \left \{
- Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait 326 368 \begin{split}
It should be ``minimum'' 327 369 m & \geq 0 \\
} 328 370 m & \leq y \times X^{max} \\
329 371 m & \leq x \\
This typo has been corrected. 330 372 m & \geq x - (1 - y) \times X^{max} \\
331 373 \end{split}
{\bf 332 374 \right .
- Page e, line 55, second column: ``takin'' % r2.9 - fait 333 375 \end{equation*}
- Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait 334 376 Gurobi does the linearization so we don't explain this step to keep the model more
- Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait 335 377 simple. However, to improve the transformation explanation we have rewrote the
} 336 378 paragraph ``This model is non-linear and even non-quadratic...''.
337 379
All typos and grammatical errors have been corrected. 338 380 % JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
339 381 % je ne comprends pas comment ca repond a la question
{\bf 340 382 %
- Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait 341 383 % AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
these particular configurations? Are they chosen automatically? 342 384 %
} 343 385 % Le problème n'est pas linéaire car nous multiplions des variables
C'est le nombre de coefficients et un taille raisonnable 344 386 % entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
Troncature de la pyramide 345 387 % des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
346 388 % quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
See below: we have added a better description of $p$ during the transformation explanation. 347 389 % quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
``we introduce $p$ FIR configurations. 348 390 % $\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
This variable must be defined by the user, it represent the number of different 349 391 % de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1} 350 392 % cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
functions from GNU Octave)'' 351 393 % nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
352 394 % qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
{\bf 353 395 % linéarisation pour nous.
- Page 4, line 31: how does the delta function transform model from non-linear % r2.13 354 396
and non-quadratic to a quadratic?} 355 397
356 398 {\bf
The first model is non-quadratic but when we introduce the $p$ configurations, 357 399 - Captions of figure and tables are too minimal. % r2.14
we can estimate the function $F$ by computing 358 400 }
the rejection for each configuration, so the model become quadratic because we have 359 401 We have change the captions of fig 10-16.
some multiplication between variables. With the definition of $\delta_{ij}$ we can 360 402
replace the multiplication between variables by multiplication with binary variable and 361 403 {\bf
this one can be linearise as follow:\\ 362 404 - Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait
$y$ is a binary variable \\ 363 405 of a single figure. Same for fig. 13-16.
$x$ is a real variable bounded by $X^{max}$ \\ 364 406 }
\begin{equation*} 365 407 We add two sub figure to group the fig.10-12 and fig. 13-16
m = x \times y \implies 366 408
\left \{ 367 409 {\bf
\begin{split} 368 410 - Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait
m & \geq 0 \\ 369 411 of the curves is about 20 dBpk-pk and it doesn’t allow to appreciate the
m & \leq y \times X^{max} \\ 370 412 differences among the curves. I suggest to reduce the noise below 1 dBpk-pk.
m & \leq x \\ 371 413 }
m & \geq x - (1 - y) \times X^{max} \\ 372 414
\end{split} 373 415 Indeed averaging had been omitted during post-processing and figure generation: we
\right . 374 416 are grateful to the reviewer for emphasizing this point which has now been corrected. All spectra
\end{equation*} 375 417 now exhibit sub-dBpk-pl line thickness.
Gurobi does the linearization so we don't explain this step to keep the model more 376 418
simple. However, to improve the transformation explanation we have rewrote the 377 419 We believe these updates to the manuscript have improved the presentation and made clearer
paragraph ``This model is non-linear and even non-quadratic...''. 378 420 some of the shortcomings of the initial draft: we are greatful to the reviewers for pointing
379 421 out these issues.
% JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article 380 422
% je ne comprends pas comment ca repond a la question 381 423 Best wishes, A. Hugeat
% 382 424
% AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux. 383 425 %In conclusion, my opinion is that the methodology presented in the Manuscript
% 384 426 %deserve to be published, provided that the criterion is changed according
% Le problème n'est pas linéaire car nous multiplions des variables 385 427 %the indications mentioned above.
% entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent 386 428 \end{document}
% des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique 387 429 %****************************************************
% quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore 388 430 %
% quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre 389 431 %For information about the IEEE Ultrasonics, Ferroelectrics, and Frequency
% $\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible 390 432 %Control Society, please visit the website: http://www.ieee-uffc.org. The
% de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre 391 433 %website of the Transactions on Ultrasonics, Ferroelectrics, and Frequency
% cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée, 392 434 %Control is at: http://ieee-uffc.org/publications/transactions-on-uffc
% nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur 393 435
% qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la 394
% linéarisation pour nous. 395
396
397
{\bf 398
- Captions of figure and tables are too minimal. % r2.14 399
} 400
We have change the captions of fig 10-16. 401
402
{\bf 403
- Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait 404
of a single figure. Same for fig. 13-16. 405
} 406
We add two sub figure to group the fig.10-12 and fig. 13-16 407
408
{\bf 409
- Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait 410
of the curves is about 20 dBpk-pk and it doesn’t allow to appreciate the 411
differences among the curves. I suggest to reduce the noise below 1 dBpk-pk. 412
} 413
414
Indeed averaging had been omitted during post-processing and figure generation: we 415
are grateful to the reviewer for emphasizing this point which has now been corrected. All spectra 416
now exhibit sub-dBpk-pl line thickness. 417
418
We believe these updates to the manuscript have improved the presentation and made clearer 419
some of the shortcomings of the initial draft: we are greatful to the reviewers for pointing 420
out these issues. 421
422
Best wishes, A. Hugeat 423
424
%In conclusion, my opinion is that the methodology presented in the Manuscript 425
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