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re-relecture JMF

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% fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee 1 1 % fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee
% demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de 2 2 % demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de
% rejection par bit et perte si moins de bits que rejection/6 3 3 % rejection par bit et perte si moins de bits que rejection/6
% developper programme lineaire en incluant le decalage de bits 4 4 % developper programme lineaire en incluant le decalage de bits
% insister que avant on etait synthetisable mais pas implementable, alors que maintenant on 5 5 % insister que avant on etait synthetisable mais pas implementable, alors que maintenant on
% implemente et on demontre que ca tourne 6 6 % implemente et on demontre que ca tourne
% gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ? 7 7 % gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ?
% Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer 8 8 % Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer
% (zedboard ou redpit) 9 9 % (zedboard ou redpit)
10 10
% label schema : verifier que "argumenter de la cascade de FIR" est fait 11 11 % label schema : verifier que "argumenter de la cascade de FIR" est fait
12 12
\documentclass[a4paper,journal]{IEEEtran/IEEEtran} 13 13 \documentclass[a4paper,journal]{IEEEtran/IEEEtran}
\usepackage{graphicx,color,hyperref} 14 14 \usepackage{graphicx,color,hyperref}
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\usepackage[normalem]{ulem} 21 21 \usepackage[normalem]{ulem}
\usepackage{tikz} 22 22 \usepackage{tikz}
\usetikzlibrary{positioning,fit} 23 23 \usetikzlibrary{positioning,fit}
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\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.} 252 252 shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid}
253 as described below is indeed unique for each filter shape.}
253 254
\begin{figure} 254 255 \begin{figure}
\begin{center} 255 256 \begin{center}
\scalebox{0.8}{ 256 257 \scalebox{0.8}{
\centering 257 258 \centering
\begin{tikzpicture}[scale=0.3] 258 259 \begin{tikzpicture}[scale=0.3]
\draw[<->] (0,15) -- (0,0) -- (21,0) ; 259 260 \draw[<->] (0,15) -- (0,0) -- (21,0) ;
\draw[thick] (0,12) -- (8,12) -- (20,0) ; 260 261 \draw[thick] (0,12) -- (8,12) -- (20,0) ;
261 262
\draw (0,14) node [left] { $P$ } ; 262 263 \draw (0,14) node [left] { $P$ } ;
\draw (20,0) node [below] { $f$ } ; 263 264 \draw (20,0) node [below] { $f$ } ;
264 265
\draw[>=latex,<->] (0,14) -- (8,14) ; 265 266 \draw[>=latex,<->] (0,14) -- (8,14) ;
\draw (4,14) node [above] { passband } node [below] { $40\%$ } ; 266 267 \draw (4,14) node [above] { passband } node [below] { $40\%$ } ;
267 268
\draw[>=latex,<->] (8,14) -- (12,14) ; 268 269 \draw[>=latex,<->] (8,14) -- (12,14) ;
\draw (10,14) node [above] { transition } node [below] { $20\%$ } ; 269 270 \draw (10,14) node [above] { transition } node [below] { $20\%$ } ;
270 271
\draw[>=latex,<->] (12,14) -- (20,14) ; 271 272 \draw[>=latex,<->] (12,14) -- (20,14) ;
\draw (16,14) node [above] { stopband } node [below] { $40\%$ } ; 272 273 \draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;
273 274
\draw[>=latex,<->] (16,12) -- (16,8) ; 274 275 \draw[>=latex,<->] (16,12) -- (16,8) ;
\draw (16,10) node [right] { rejection } ; 275 276 \draw (16,10) node [right] { rejection } ;
276 277
\draw[dashed] (8,-1) -- (8,14) ; 277 278 \draw[dashed] (8,-1) -- (8,14) ;
\draw[dashed] (12,-1) -- (12,14) ; 278 279 \draw[dashed] (12,-1) -- (12,14) ;
279 280
\draw[dashed] (8,12) -- (16,12) ; 280 281 \draw[dashed] (8,12) -- (16,12) ;
\draw[dashed] (12,8) -- (16,8) ; 281 282 \draw[dashed] (12,8) -- (16,8) ;
282 283
\end{tikzpicture} 283 284 \end{tikzpicture}
} 284 285 }
\end{center} 285 286 \end{center}
\caption{Shape of the filter transmitted power $P$ as a function of frequency $f$: 286 287 \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, 287 288 the passband is considered to occupy the initial 40\% of the Nyquist frequency range,
the stopband the last 40\%, allowing 20\% transition width.} 288 289 the stopband the last 40\%, allowing 20\% transition width.}
\label{fig:fir_mag} 289 290 \label{fig:fir_mag}
\end{figure} 290 291 \end{figure}
291 292
In the transition band, the behavior of the filter is left free, we only care about the passband and the stopband characteristics. 292 293 In the transition band, the behavior of the filter is left free, we only {\color{red}define} the passband and the stopband characteristics.
% r2.7 293 294 % r2.7
% Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion 294 295 % 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 295 296 % 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. 296 297 % 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 takes 297 298 Our criterion to compute the filter rejection considers
% r2.8 et r2.2 r2.3 298 299 % r2.8 et r2.2 r2.3
the maximum magnitude within the stopband minus the sum of the absolute value of passband rejection. With this criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}. 299 300 the maximum magnitude within the stopband, to which the {\color{red}sum of the absolute values
301 within the passband rejection is subtracted to avoid filters with excessive ripples}. With this
302 criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
300 303
% \begin{figure} 301 304 % \begin{figure}
% \centering 302 305 % \centering
% \includegraphics[width=\linewidth]{images/colored_mean_criterion} 303 306 % \includegraphics[width=\linewidth]{images/colored_mean_criterion}
% \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters} 304 307 % \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}
% \label{fig:mean_criterion} 305 308 % \label{fig:mean_criterion}
% \end{figure} 306 309 % \end{figure}
307 310
\begin{figure} 308 311 \begin{figure}
\centering 309 312 \centering
\includegraphics[width=\linewidth]{images/colored_custom_criterion} 310 313 \includegraphics[width=\linewidth]{images/colored_custom_criterion}
\caption{Custom criterion (maximum rejection in the stopband minus the mean of the absolute value of the passband rejection) 311 314 \caption{Custom criterion (maximum rejection in the stopband minus the mean of the absolute value of the passband rejection)
comparison between monolithic filter and cascaded filters} 312 315 comparison between monolithic filter and cascaded filters}
\label{fig:custom_criterion} 313 316 \label{fig:custom_criterion}
\end{figure} 314 317 \end{figure}
315 318
Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps 316 319 Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps
and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the 317 320 and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the
rejection as a function of the number of coefficients and the number of bits representing these coefficients. 318 321 rejection as a function of the number of coefficients and the number of bits representing these coefficients.
The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet. 319 322 The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet.
Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection. 320 323 Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection.
Conversely when setting the a given number of bits, increasing the number of coefficients will not improve 321 324 Conversely when setting the a given number of bits, increasing the number of coefficients will not improve
the rejection. Hence the best coefficient set are on the vertex of the pyramid. 322 325 the rejection. Hence the best coefficient set are on the vertex of the pyramid.
323 326
\begin{figure} 324 327 \begin{figure}
\centering 325 328 \centering
\includegraphics[width=\linewidth]{images/rejection_pyramid} 326 329 \includegraphics[width=\linewidth]{images/rejection_pyramid}
\caption{Rejection as a function of number of coefficients and number of bits} 327 330 \caption{Rejection as a function of number of coefficients and number of bits}
\label{fig:rejection_pyramid} 328 331 \label{fig:rejection_pyramid}
\end{figure} 329 332 \end{figure}
330 333
Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps), 331 334 Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps),
we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria. 332 335 we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria.
If the FIR filter coefficients are the same between the stages, we have: 333 336 If the FIR filter coefficients are the same between the stages, we have:
$$F_{total} = F_1 + F_2$$ 334 337 $$F_{total} = F_1 + F_2$$
But selecting two different sets of coefficient will yield a more complex situation in which 335 338 But selecting two different sets of coefficient will yield a more complex situation in which
the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves 336 339 the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves
are two different filters with maximums and notches not located at the same frequency offsets. 337 340 are two different filters with maximums and notches not located at the same frequency offsets.
Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved 338 341 Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved
with respect to a basic sum of the rejection criteria shown as a the dotted yellow line. 339 342 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
% r2.9 340 343 % r2.9
Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection 341 344 Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection
criteria of each filter. However since the this sum underestimates the rejection capability of the cascade, 342 345 criteria of each filter. However since the this sum underestimates the rejection capability of the cascade,
% r2.10 343 346 % r2.10
this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability 344 347 this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
of the filter cascade to meet design criteria. 345 348 of the filter cascade to meet design criteria.
346 349
\begin{figure} 347 350 \begin{figure}
\centering 348 351 \centering
\includegraphics[width=\linewidth]{images/cascaded_criterion} 349 352 \includegraphics[width=\linewidth]{images/cascaded_criterion}
\caption{Rejection of two cascaded filters} 350 353 \caption{Rejection of two cascaded filters}
\label{fig:sum_rejection} 351 354 \label{fig:sum_rejection}
\end{figure} 352 355 \end{figure}
353 356
% r2.6 354 357 % r2.6
Finally in our case, we consider that the input signal are fully known. So the 355 358 Finally in our case, we consider that the input signal are fully known. So the
resolution of the data stream are fixed and still the same for all experiments 356 359 resolution of the data stream are fixed and still the same for all experiments
in this paper. 357 360 in this paper.
358 361
Based on this analysis, we address the estimate of resource consumption (called 359 362 Based on this analysis, we address the estimate of resource consumption (called
% r2.11 360 363 % r2.11
silicon area -- in the case of FPGAs this means processing cells) as a function of 361 364 silicon area -- in the case of FPGAs this means processing cells) as a function of
filter characteristics. As a reminder, we do not aim at matching actual hardware 362 365 filter characteristics. As a reminder, we do not aim at matching actual hardware
configuration but consider an arbitrary silicon area occupied by each processing function, 363 366 configuration but consider an arbitrary silicon area occupied by each processing function,
and will assess after synthesis the adequation of this arbitrary unit with actual 364 367 and will assess after synthesis the adequation of this arbitrary unit with actual
hardware resources provided by FPGA manufacturers. The sum of individual processing 365 368 hardware resources provided by FPGA manufacturers. The sum of individual processing
unit areas is constrained by a total silicon area representative of FPGA global resources. 366 369 unit areas is constrained by a total silicon area representative of FPGA global resources.
Formally, variable $a_i$ is the area taken by filter~$i$ 367 370 Formally, variable $a_i$ is the area taken by filter~$i$
(in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB). 368 371 (in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB).
Constant $\mathcal{A}$ is the total available area. We model our problem as follows: 369 372 Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
370 373
\begin{align} 371 374 \begin{align}
\text{Maximize } & \sum_{i=1}^n r_i \notag \\ 372 375 \text{Maximize } & \sum_{i=1}^n r_i \notag \\
\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\ 373 376 \sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\
a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\ 374 377 a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\
r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\ 375 378 r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\
\pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\ 376 379 \pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\
\pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\ 377 380 \pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\
\pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\ 378 381 \pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\
\pi_1^- &= \Pi^I \label{eq:init} 379 382 \pi_1^- &= \Pi^I \label{eq:init}
\end{align} 380 383 \end{align}
381 384
Equation~\ref{eq:area} states that the total area taken by the filters must be 382 385 Equation~\ref{eq:area} states that the total area taken by the filters must be
less than the available area. Equation~\ref{eq:areadef} gives the definition of 383 386 less than the available area. Equation~\ref{eq:areadef} gives the definition of
the area used by a filter, considered as the area of the FIR since the Shifter is 384 387 the area used by a filter, considered as the area of the FIR since the Shifter is
assumed not to require significant resources. We consider that the FIR needs $C_i$ registers of size 385 388 assumed not to require significant resources. We consider that the FIR needs $C_i$ registers of size
$\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the 386 389 $\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the
input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the 387 390 input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the
definition of the rejection of the filter thanks to the tabulated function~$F$ that we defined 388 391 definition of the rejection of the filter thanks to the tabulated function~$F$ that we defined
previously. The Shifter does not introduce negative rejection as we will explain later, 389 392 previously. The Shifter does not introduce negative rejection as we will explain later,
so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the 390 393 so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the
relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add 391 394 relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add
$\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes 392 395 $\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes
$\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of 393 396 $\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of
a filter is the same as the input number of bits of the next filter. 394 397 a filter is the same as the input number of bits of the next filter.
Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative 395 398 Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative
rejection. Indeed, the results of the FIR can be right shifted without compromising 396 399 rejection. Indeed, the results of the FIR can be right shifted without compromising
the quality of the rejection until a threshold. Each bit of the output data 397 400 the quality of the rejection until a threshold. Each bit of the output data
increases the maximum rejection level by 6~dB. We add one to take the sign bit 398 401 increases the maximum rejection level by 6~dB. We add one to take the sign bit
into account. If equation~\ref{eq:maxshift} was not present, the Shifter could 399 402 into account. If equation~\ref{eq:maxshift} was not present, the Shifter could
shift too much and introduce some noise in the output data. Each supplementary 400 403 shift too much and introduce some noise in the output data. Each supplementary
shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is: 401 404 shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is:
$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. 402 405 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
Finally, equation~\ref{eq:init} gives the number of bits of the global input. 403 406 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
404 407
This model is non-linear and even non-quadratic, as $F$ does not have a known 405 408 This model is non-linear and even non-quadratic, as $F$ does not have a known
linear or quadratic expression. We introduce $p$ FIR configurations 406 409 linear or quadratic expression. We introduce $p$ FIR configurations
$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 407 410 $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
% r2.12 408 411 % r2.12
This variable must be defined by the user, it represent the number of different 409 412 This variable must be defined by the user, it represent the number of different
set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1} 410 413 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave). 411 414 functions from GNU Octave).
We define binary 412 415 We define binary
variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 413 416 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
and 0 otherwise. The new equations are as follows: 414 417 and 0 otherwise. The new equations are as follows:
415 418
\begin{align} 416 419 \begin{align}
a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\ 417 420 a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\
r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\ 418 421 r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
\pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\ 419 422 \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\
\sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} 420 423 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
\end{align} 421 424 \end{align}
422 425
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 423 426 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 424 427 respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most. 425 428 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
426 429
% r2.13 427 430 % r2.13
This modified model is quadratic since we multiply two variables in the 428 431 This modified model is quadratic since we multiply two variables in the
equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary. 429 432 equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
The Gurobi 430 433 The Gurobi
(\url{www.gurobi.com}) optimization software is used to solve this quadratic 431 434 (\url{www.gurobi.com}) optimization software is used to solve this quadratic
model, and since Gurobi is able to linearize, the model is left as is. This model 432 435 model, and since Gurobi is able to linearize, the model is left as is. This model
has $O(np)$ variables and $O(n)$ constraints. 433 436 has $O(np)$ variables and $O(n)$ constraints.
434 437
Two problems will be addressed using the workflow described in the next section: on the one 435 438 Two problems will be addressed using the workflow described in the next section: on the one
hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary 436 439 hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary
silcon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area 437 440 silcon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area
for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the 438 441 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
objective function is replaced with: 439 442 objective function is replaced with:
\begin{align} 440 443 \begin{align}
\text{Minimize } & \sum_{i=1}^n a_i \notag 441 444 \text{Minimize } & \sum_{i=1}^n a_i \notag
\end{align} 442 445 \end{align}
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 443 446 We adapt our constraints of quadratic program to replace equation \ref{eq:area}
with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal 444 447 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
rejection required. 445 448 rejection required.
446 449
\begin{align} 447 450 \begin{align}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 448 451 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
\end{align} 449 452 \end{align}
450 453
\section{Design workflow} 451 454 \section{Design workflow}
\label{sec:workflow} 452 455 \label{sec:workflow}
453 456
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 454 457 In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area}
and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved 455 458 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
in the computation of the results. 456 459 in the computation of the results.
457 460
\begin{figure} 458 461 \begin{figure}
\centering 459 462 \centering
\begin{tikzpicture}[node distance=0.75cm and 2cm] 460 463 \begin{tikzpicture}[node distance=0.75cm and 2cm]
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 461 464 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\node (Start) [left= 3cm of Solver] { } ; 462 465 \node (Start) [left= 3cm of Solver] { } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 463 466 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\node (Input) [above= of TCL] { } ; 464 467 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 465 468 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 466 469 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 467 470 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 468 471 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node (Results) [left= of Postproc] { } ; 469 472 \node (Results) [left= of Postproc] { } ;
470 473
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 471 474 \draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ;
\draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ; 472 475 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 473 476 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 474 477 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 475 478 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 476 479 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 477 480 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 478 481 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 479 482 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->] (Postproc) -- (Results) ; 480 483 \draw[->] (Postproc) -- (Results) ;
\end{tikzpicture} 481 484 \end{tikzpicture}
\caption{Design workflow from the input parameters to the results} 482 485 \caption{Design workflow from the input parameters to the results}
\label{fig:workflow} 483 486 \label{fig:workflow}
\end{figure} 484 487 \end{figure}
485 488
The filter solver is a C++ program that takes as input the maximum area 486 489 The filter solver is a C++ program that takes as input the maximum area
$\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$, 487 490 $\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$,
the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates 488 491 the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates
the quadratic programs and uses the Gurobi solver to estimate the optimal results. 489 492 the quadratic programs and uses the Gurobi solver to estimate the optimal results.
Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow}) 490 493 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
and a deploy script ((1b) on figure~\ref{fig:workflow}). 491 494 and a deploy script ((1b) on figure~\ref{fig:workflow}).
492 495
The TCL script describes the whole digital processing chain from the beginning 493 496 The TCL script describes the whole digital processing chain from the beginning
(the raw signal data) to the end (the filtered data) in a language compatible 494 497 (the raw signal data) to the end (the filtered data) in a language compatible
with proprietary synthesis software, namely Vivado for Xilinx and Quartus for 495 498 with proprietary synthesis software, namely Vivado for Xilinx and Quartus for
Intel/Altera. The raw input data generated from a 20-bit Pseudo Random Number (PRN) 496 499 Intel/Altera. The raw input data generated from a 20-bit Pseudo Random Number (PRN)
generator inside the FPGA and $\Pi^I$ is fixed at 16~bits. 497 500 generator inside the FPGA and $\Pi^I$ is fixed at 16~bits.
Then the script builds each stage of the chain with a generic FIR task that 498 501 Then the script builds each stage of the chain with a generic FIR task that
comes from a skeleton library. The generic FIR is highly configurable 499 502 comes from a skeleton library. The generic FIR is highly configurable
with the number of coefficients and the size of the coefficients. The coefficients 500 503 with the number of coefficients and the size of the coefficients. The coefficients
themselves are not stored in the script. 501 504 themselves are not stored in the script.
As the signal is processed in real-time, the output signal is stored as 502 505 As the signal is processed in real-time, the output signal is stored as
consecutive bursts of data for post-processing, mainly assessing the consistency of the 503 506 consecutive bursts of data for post-processing, mainly assessing the consistency of the
implemented FIR cascade transfer function with the design criteria and the expected 504 507 implemented FIR cascade transfer function with the design criteria and the expected
transfer function. 505 508 transfer function.
506 509
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 507 510 The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}).
We use the 2018.2 version of Xilinx Vivado and we execute the synthesized 508 511 We use the 2018.2 version of Xilinx Vivado and we execute the synthesized
bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series 509 512 bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series
FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to 510 513 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
provide a broadband noise source. 511 514 provide a broadband noise source.
The board runs the Linux kernel and surrounding environment produced from the 512 515 The board runs the Linux kernel and surrounding environment produced from the
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 513 516 Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring
the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and 514 517 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
fetching the results is automated. 515 518 fetching the results is automated.
516 519
The deploy script uploads the bitstream to the board ((3) on 517 520 The deploy script uploads the bitstream to the board ((3) on
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 518 521 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
configures the coefficients of the FIR filters. It then waits for the results 519 522 configures the coefficients of the FIR filters. It then waits for the results
and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}). 520 523 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
521 524
Finally, an Octave post-processing script computes the final results thanks to 522 525 Finally, an Octave post-processing script computes the final results thanks to
the output data ((5) on figure~\ref{fig:workflow}). 523 526 the output data ((5) on figure~\ref{fig:workflow}).
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 524 527 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
and the different configurations can be compared. 525 528 and the different configurations can be compared.
526 529
\section{Maximizing the rejection at fixed silicon area} 527 530 \section{Maximizing the rejection at fixed silicon area}
\label{sec:fixed_area} 528 531 \label{sec:fixed_area}
This section presents the output of the filter solver {\em i.e.} the computed 529 532 This section presents the output of the filter solver {\em i.e.} the computed
configurations for each stage, the computed rejection and the computed silicon area. 530 533 configurations for each stage, the computed rejection and the computed silicon area.
Such results allow for understanding the choices made by the solver to compute its solutions. 531 534 Such results allow for understanding the choices made by the solver to compute its solutions.
532 535
The experimental setup is composed of three cases. The raw input is generated 533 536 The experimental setup is composed of three cases. The raw input is generated
by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$. 534 537 by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$.
Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500 535 538 Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500
arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500. 536 539 arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500.
The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$ 537 540 The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$
ranging from 2 to 22. In each case, the quadratic program has been able to give a 538 541 ranging from 2 to 22. In each case, the quadratic program has been able to give a
result up to five stages ($n = 5$) in the cascaded filter. 539 542 result up to five stages ($n = 5$) in the cascaded filter.
540 543
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 541 544 Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500.
Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000. 542 545 Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000.
Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500. 543 546 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
544 547
\renewcommand{\arraystretch}{1.4} 545 548 \renewcommand{\arraystretch}{1.4}
546 549
\begin{table} 547 550 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 548 551 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
\label{tbl:gurobi_max_500} 549 552 \label{tbl:gurobi_max_500}
\centering 550 553 \centering
{\scalefont{0.77} 551 554 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 552 555 \begin{tabular}{|c|ccccc|c|c|}
\hline 553 556 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 554 557 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 555 558 \hline
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 556 559 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ 557 560 2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\
3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ 558 561 3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\
4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\ 559 562 4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\
5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\ 560 563 5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\
\hline 561 564 \hline
\end{tabular} 562 565 \end{tabular}
} 563 566 }
\end{table} 564 567 \end{table}
565 568
\begin{table} 566 569 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 567 570 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
\label{tbl:gurobi_max_1000} 568 571 \label{tbl:gurobi_max_1000}
\centering 569 572 \centering
{\scalefont{0.77} 570 573 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 571 574 \begin{tabular}{|c|ccccc|c|c|}
\hline 572 575 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 573 576 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 574 577 \hline
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 575 578 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ 576 579 2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\
3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ 577 580 3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\
4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\ 578 581 4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\
5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\ 579 582 5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\
\hline 580 583 \hline
\end{tabular} 581 584 \end{tabular}
} 582 585 }
\end{table} 583 586 \end{table}
584 587
\begin{table} 585 588 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 586 589 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
\label{tbl:gurobi_max_1500} 587 590 \label{tbl:gurobi_max_1500}
\centering 588 591 \centering
{\scalefont{0.77} 589 592 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 590 593 \begin{tabular}{|c|ccccc|c|c|}
\hline 591 594 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 592 595 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 593 596 \hline
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 594 597 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ 595 598 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\
3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ 596 599 3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\
4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\ 597 600 4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\
5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\ 598 601 5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\
\hline 599 602 \hline
\end{tabular} 600 603 \end{tabular}
} 601 604 }
\end{table} 602 605 \end{table}
603 606
\renewcommand{\arraystretch}{1} 604 607 \renewcommand{\arraystretch}{1}
605 608
From these tables, we can first state that the more stages are used to define 606 609 From these tables, we can first state that the more stages are used to define
the cascaded FIR filters, the better the rejection. It was an expected result as it has 607 610 the cascaded FIR filters, the better the rejection. It was an expected result as it has
been previously observed that many small filters are better than 608 611 been previously observed that many small filters are better than
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 609 612 a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions
being hardly used in practice due to the lack of tools for identifying individual filter 610 613 being hardly used in practice due to the lack of tools for identifying individual filter
coefficients in the cascaded approach. 611 614 coefficients in the cascaded approach.
612 615
Second, the larger the silicon area, the better the rejection. This was also an 613 616 Second, the larger the silicon area, the better the rejection. This was also an
expected result as more area means a filter of better quality with more coefficients 614 617 expected result as more area means a filter of better quality with more coefficients
or more bits per coefficient. 615 618 or more bits per coefficient.
616 619
Then, we also observe that the first stage can have a larger shift than the other 617 620 Then, we also observe that the first stage can have a larger shift than the other
stages. This is explained by the fact that the solver tries to use just enough 618 621 stages. This is explained by the fact that the solver tries to use just enough
bits for the computed rejection after each stage. In the first stage, a 619 622 bits for the computed rejection after each stage. In the first stage, a
balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift} 620 623 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
gives the relation between both values. 621 624 gives the relation between both values.
622 625
Finally, we note that the solver consumes all the given silicon area. 623 626 Finally, we note that the solver consumes all the given silicon area.
624 627
The following graphs present the rejection for real data on the FPGA. In all the following 625 628 The following graphs present the rejection for real data on the FPGA. In all the following
figures, the solid line represents the actual rejection of the filtered 626 629 figures, the solid line represents the actual rejection of the filtered
data on the FPGA as measured experimentally and the dashed line are the noise levels 627 630 data on the FPGA as measured experimentally and the dashed line are the noise levels
given by the quadratic solver. The configurations are those computed in the previous section. 628 631 given by the quadratic solver. The configurations are those computed in the previous section.
629 632
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 630 633 Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500.
Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000. 631 634 Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000.
Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500. 632 635 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
633 636
% \begin{figure} 634 637 % \begin{figure}
% \centering 635 638 % \centering
% \includegraphics[width=\linewidth]{images/max_500} 636 639 % \includegraphics[width=\linewidth]{images/max_500}
% \caption{Signal spectrum for MAX/500} 637 640 % \caption{Signal spectrum for MAX/500}
% \label{fig:max_500_result} 638 641 % \label{fig:max_500_result}
% \end{figure} 639 642 % \end{figure}
% 640 643 %
% \begin{figure} 641 644 % \begin{figure}
% \centering 642 645 % \centering
% \includegraphics[width=\linewidth]{images/max_1000} 643 646 % \includegraphics[width=\linewidth]{images/max_1000}
% \caption{Signal spectrum for MAX/1000} 644 647 % \caption{Signal spectrum for MAX/1000}
% \label{fig:max_1000_result} 645 648 % \label{fig:max_1000_result}
% \end{figure} 646 649 % \end{figure}
% 647 650 %
% \begin{figure} 648 651 % \begin{figure}
% \centering 649 652 % \centering
% \includegraphics[width=\linewidth]{images/max_1500} 650 653 % \includegraphics[width=\linewidth]{images/max_1500}
% \caption{Signal spectrum for MAX/1500} 651 654 % \caption{Signal spectrum for MAX/1500}
% \label{fig:max_1500_result} 652 655 % \label{fig:max_1500_result}
% \end{figure} 653 656 % \end{figure}
654 657
% r2.14 et r2.15 et r2.16 655 658 % r2.14 et r2.15 et r2.16
\begin{figure} 656 659 \begin{figure}
\centering 657 660 \centering
\begin{subfigure}{\linewidth} 658 661 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_500} 659 662 \includegraphics[width=\linewidth]{images/max_500}
\caption{Signal spectrum for MAX/500} 660 663 \caption{Signal spectrum for MAX/500}
\label{fig:max_500_result} 661 664 \label{fig:max_500_result}
\end{subfigure} 662 665 \end{subfigure}
663 666
\begin{subfigure}{\linewidth} 664 667 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1000} 665 668 \includegraphics[width=\linewidth]{images/max_1000}
\caption{Signal spectrum for MAX/1000} 666 669 \caption{Signal spectrum for MAX/1000}
\label{fig:max_1000_result} 667 670 \label{fig:max_1000_result}
\end{subfigure} 668 671 \end{subfigure}
669 672
\begin{subfigure}{\linewidth} 670 673 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1500} 671 674 \includegraphics[width=\linewidth]{images/max_1500}
\caption{Signal spectrum for MAX/1500} 672 675 \caption{Signal spectrum for MAX/1500}
\label{fig:max_1500_result} 673 676 \label{fig:max_1500_result}
\end{subfigure} 674 677 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500} 675 678 \caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500}
\end{figure} 676 679 \end{figure}
677 680
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 678 681 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
679 682
We compare the actual silicon resources given by Vivado to the 680 683 We compare the actual silicon resources given by Vivado to the
resources in arbitrary units. 681 684 resources in arbitrary units.
The goal is to check that our arbitrary units of silicon area models well enough 682 685 The goal is to check that our arbitrary units of silicon area models well enough
the real resources on the FPGA. Especially we want to verify that, for a given 683 686 the real resources on the FPGA. Especially we want to verify that, for a given
number of arbitrary units, the actual silicon resources do not depend on the 684 687 number of arbitrary units, the actual silicon resources do not depend on the
number of stages $n$. Most significantly, our approach aims 685 688 number of stages $n$. Most significantly, our approach aims
at remaining far enough from the practical logic gate implementation used by 686 689 at remaining far enough from the practical logic gate implementation used by
various vendors to remain platform independent and be portable from one 687 690 various vendors to remain platform independent and be portable from one
architecture to another. 688 691 architecture to another.
689 692
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 690 693 Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and
MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000 691 694 MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000
and 1500 arbitrary units. We have taken care to extract solely the resources used by 692 695 and 1500 arbitrary units. We have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and Programmable 693 696 the FIR filters and remove additional processing blocks including FIFO and Programmable
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 694 697 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
695 698
\begin{table}[h!tb] 696 699 \begin{table}[h!tb]
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 697 700 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage} 698 701 \label{tbl:resources_usage}
\centering 699 702 \centering
\begin{tabular}{|c|c|ccc|c|} 700 703 \begin{tabular}{|c|c|ccc|c|}
\hline 701 704 \hline
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 702 705 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 249 & 453 & 627 & \emph{17600} \\ 703 706 & LUT & 249 & 453 & 627 & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 704 707 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 705 708 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
& LUT & 2374 & 5494 & 691 & \emph{17600} \\ 706 709 & LUT & 2374 & 5494 & 691 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 707 710 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 70 & \emph{80} \\ \hline 708 711 & DSP & 0 & 0 & 70 & \emph{80} \\ \hline
& LUT & 2443 & 3304 & 3521 & \emph{17600} \\ 709 712 & LUT & 2443 & 3304 & 3521 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 710 713 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 19 & 35 & \emph{80} \\ \hline 711 714 & DSP & 0 & 19 & 35 & \emph{80} \\ \hline
& LUT & 2634 & 3753 & 2557 & \emph{17600} \\ 712 715 & LUT & 2634 & 3753 & 2557 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & \emph{120} \\ 713 716 4 & BRAM & 4 & 4 & 4 & \emph{120} \\
& DPS & 0 & 19 & 46 & \emph{80} \\ \hline 714 717 & DPS & 0 & 19 & 46 & \emph{80} \\ \hline
& LUT & 2423 & 3047 & 2847 & \emph{17600} \\ 715 718 & LUT & 2423 & 3047 & 2847 & \emph{17600} \\
5 & BRAM & 5 & 5 & 5 & \emph{120} \\ 716 719 5 & BRAM & 5 & 5 & 5 & \emph{120} \\
& DPS & 0 & 22 & 46 & \emph{80} \\ \hline 717 720 & DPS & 0 & 22 & 46 & \emph{80} \\ \hline
\end{tabular} 718 721 \end{tabular}
\end{table} 719 722 \end{table}
720 723
In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 721 724 In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that,
when the filter coefficients are small enough, or when the input size is small 722 725 when the filter coefficients are small enough, or when the input size is small
enough, Vivado optimizes resource consumption by selecting multiplexers to 723 726 enough, Vivado optimizes resource consumption by selecting multiplexers to
implement the multiplications instead of a DSP. In this case, it is quite difficult 724 727 implement the multiplications instead of a DSP. In this case, it is quite difficult
to compare the whole silicon budget. 725 728 to compare the whole silicon budget.
726 729
However, a rough estimation can be made with a simple equivalence: looking at 727 730 However, a rough estimation can be made with a simple equivalence: looking at
the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$, 728 731 the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$,
we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon 729 732 we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon
area use. With this equivalence, our 500 arbitraty units correspond to 2500 LUTs, 730 733 area use. With this equivalence, our 500 arbitraty units correspond to 2500 LUTs,
1000 arbitrary units correspond to 5000 LUTs and 1500 arbitrary units correspond 731 734 1000 arbitrary units correspond to 5000 LUTs and 1500 arbitrary units correspond
to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary 732 735 to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary
unit map well to actual hardware resources. The relatively small differences can probably be explained 733 736 unit map well to actual hardware resources. The relatively small differences can probably be explained
by the optimizations done by Vivado based on the detailed map of available processing resources. 734 737 by the optimizations done by Vivado based on the detailed map of available processing resources.
735 738
We now present the computation time needed to solve the quadratic problem. 736 739 We now present the computation time needed to solve the quadratic problem.
For each case, the filter solver software is executed on a Intel(R) Xeon(R) CPU E5606 737 740 For each case, the filter solver software is executed on a Intel(R) Xeon(R) CPU E5606
clocked at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve 738 741 clocked at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve
the quadratic problem. Table~\ref{tbl:area_time} shows the time needed to solve the quadratic 739 742 the quadratic problem. Table~\ref{tbl:area_time} shows the time needed to solve the quadratic
problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units. 740 743 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
741 744
\begin{table}[h!tb] 742 745 \begin{table}[h!tb]
\caption{Time needed to solve the quadratic program with Gurobi} 743 746 \caption{Time needed to solve the quadratic program with Gurobi}
\label{tbl:area_time} 744 747 \label{tbl:area_time}
\centering 745 748 \centering
\begin{tabular}{|c|c|c|c|}\hline 746 749 \begin{tabular}{|c|c|c|c|}\hline
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 747 750 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
1 & 0.1~s & 0.1~s & 0.3~s \\ 748 751 1 & 0.1~s & 0.1~s & 0.3~s \\
2 & 1.1~s & 2.2~s & 12~s \\ 749 752 2 & 1.1~s & 2.2~s & 12~s \\
3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ 750 753 3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\
4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\ 751 754 4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\
5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline 752 755 5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline
\end{tabular} 753 756 \end{tabular}
\end{table} 754 757 \end{table}
755 758
As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? 756 759 As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ?
When the area is limited, the design exploration space is more limited and the solver is able to 757 760 When the area is limited, the design exploration space is more limited and the solver is able to
find an optimal solution faster. 758 761 find an optimal solution faster.
759 762
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 760 763 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
761 764
This section presents the results of the complementary quadratic program aimed at 762 765 This section presents the results of the complementary quadratic program aimed at
minimizing the area occupation for a targeted rejection level. 763 766 minimizing the area occupation for a targeted rejection level.
764 767
The experimental setup is composed of four cases. The raw input is the same 765 768 The experimental setup is composed of four cases. The raw input is the same
as in the previous section, from a PRN generator, which fixes the input data size $\Pi^I$. 766 769 as in the previous section, from a PRN generator, which fixes the input data size $\Pi^I$.
Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB. 767 770 Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB.
Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100. 768 771 Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100.
The number of configurations $p$ is the same as previous section. 769 772 The number of configurations $p$ is the same as previous section.
770 773
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 771 774 Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40.
Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60. 772 775 Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60.
Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80. 773 776 Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80.
Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100. 774 777 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
775 778
\renewcommand{\arraystretch}{1.4} 776 779 \renewcommand{\arraystretch}{1.4}
777 780
\begin{table}[h!tb] 778 781 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 779 782 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\label{tbl:gurobi_min_40} 780 783 \label{tbl:gurobi_min_40}
\centering 781 784 \centering
{\scalefont{0.77} 782 785 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 783 786 \begin{tabular}{|c|ccccc|c|c|}
\hline 784 787 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 785 788 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 786 789 \hline
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 787 790 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ 788 791 2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\
3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ 789 792 3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\
4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\ 790 793 4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\
\hline 791 794 \hline
\end{tabular} 792 795 \end{tabular}
} 793 796 }
\end{table} 794 797 \end{table}
795 798
\begin{table}[h!tb] 796 799 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 797 800 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\label{tbl:gurobi_min_60} 798 801 \label{tbl:gurobi_min_60}
\centering 799 802 \centering
{\scalefont{0.77} 800 803 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 801 804 \begin{tabular}{|c|ccccc|c|c|}
\hline 802 805 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 803 806 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 804 807 \hline
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 805 808 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ 806 809 2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\
3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ 807 810 3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\
4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\ 808 811 4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\
5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\ 809 812 5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\
\hline 810 813 \hline
\end{tabular} 811 814 \end{tabular}
} 812 815 }
\end{table} 813 816 \end{table}
814 817
\begin{table}[h!tb] 815 818 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 816 819 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\label{tbl:gurobi_min_80} 817 820 \label{tbl:gurobi_min_80}
\centering 818 821 \centering
{\scalefont{0.77} 819 822 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 820 823 \begin{tabular}{|c|ccccc|c|c|}
\hline 821 824 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 822 825 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 823 826 \hline
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 824 827 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ 825 828 2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\
3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ 826 829 3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\
4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\ 827 830 4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\
5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\ 828 831 5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\
\hline 829 832 \hline
\end{tabular} 830 833 \end{tabular}
} 831 834 }
\end{table} 832 835 \end{table}
833 836
\begin{table}[h!tb] 834 837 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 835 838 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\label{tbl:gurobi_min_100} 836 839 \label{tbl:gurobi_min_100}
\centering 837 840 \centering
{\scalefont{0.77} 838 841 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 839 842 \begin{tabular}{|c|ccccc|c|c|}
\hline 840 843 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 841 844 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 842 845 \hline
1 & - & - & - & - & - & - & - \\ 843 846 1 & - & - & - & - & - & - & - \\
2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\ 844 847 2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\
3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\ 845 848 3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\
4 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 0) & - & 101~dB & 909 \\ 846 849 4 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 0) & - & 101~dB & 909 \\
5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\ 847 850 5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\
\hline 848 851 \hline
\end{tabular} 849 852 \end{tabular}
} 850 853 }
\end{table} 851 854 \end{table}
\renewcommand{\arraystretch}{1} 852 855 \renewcommand{\arraystretch}{1}
853 856
From these tables, we can first state that almost all configurations reach the targeted rejection 854 857 From these tables, we can first state that almost all configurations reach the targeted rejection
level or even better thanks to our underestimate of the cascade rejection as the sum of the 855 858 level or even better thanks to our underestimate of the cascade rejection as the sum of the
individual filter rejection. The only exception is for the monolithic case ($n = 1$) in 856 859 individual filter rejection. The only exception is for the monolithic case ($n = 1$) in
MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection. 857 860 MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection.
Futhermore, the area of the monolithic filter is twice as big as the two cascaded filters 858 861 Futhermore, the area of the monolithic filter is twice as big as the two cascaded filters
(1131 and 1760 arbitrary units v.s 547 and 903 arbitrary units for 60 and 80~dB rejection 859 862 (1131 and 1760 arbitrary units v.s 547 and 903 arbitrary units for 60 and 80~dB rejection
respectively). More generally, the more filters are cascaded, the lower the occupied area. 860 863 respectively). More generally, the more filters are cascaded, the lower the occupied area.
861 864
Like in previous section, the solver chooses always a little filter as first 862 865 Like in previous section, the solver chooses always a little filter as first
filter stage and the second one is often the biggest filter. This choice can be explained 863 866 filter stage and the second one is often the biggest filter. This choice can be explained
as in the previous section, with the solver using just enough bits not to degrade the input 864 867 as in the previous section, with the solver using just enough bits not to degrade the input
signal and in the second filter selecting a better filter to improve rejection without 865 868 signal and in the second filter selecting a better filter to improve rejection without
having too many bits in the output data. 866 869 having too many bits in the output data.
867 870
For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 868 871 For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal
number of filters is 4 so it did not chose any configuration for the last filter. Hence this 869 872 number of filters is 4 so it did not chose any configuration for the last filter. Hence this
solution is equivalent to the result for $n = 4$. 870 873 solution is equivalent to the result for $n = 4$.
871 874
The following graphs present the rejection for real data on the FPGA. In all the following 872 875 The following graphs present the rejection for real data on the FPGA. In all the following
figures, the solid line represents the actual rejection of the filtered 873 876 figures, the solid line represents the actual rejection of the filtered
data on the FPGA as measured experimentally and the dashed line is the noise level 874 877 data on the FPGA as measured experimentally and the dashed line is the noise level
given by the quadratic solver. 875 878 given by the quadratic solver.
876 879
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 877 880 Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40.
Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60. 878 881 Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60.
Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80. 879 882 Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80.
Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100. 880 883 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
881 884
% \begin{figure} 882 885 % \begin{figure}
% \centering 883 886 % \centering
% \includegraphics[width=\linewidth]{images/min_40} 884 887 % \includegraphics[width=\linewidth]{images/min_40}
% \caption{Signal spectrum for MIN/40} 885 888 % \caption{Signal spectrum for MIN/40}
% \label{fig:min_40} 886 889 % \label{fig:min_40}
% \end{figure} 887 890 % \end{figure}
% 888 891 %
% \begin{figure} 889 892 % \begin{figure}
% \centering 890 893 % \centering
% \includegraphics[width=\linewidth]{images/min_60} 891 894 % \includegraphics[width=\linewidth]{images/min_60}
% \caption{Signal spectrum for MIN/60} 892 895 % \caption{Signal spectrum for MIN/60}
% \label{fig:min_60} 893 896 % \label{fig:min_60}
% \end{figure} 894 897 % \end{figure}
% 895 898 %
% \begin{figure} 896 899 % \begin{figure}
% \centering 897 900 % \centering
% \includegraphics[width=\linewidth]{images/min_80} 898 901 % \includegraphics[width=\linewidth]{images/min_80}
% \caption{Signal spectrum for MIN/80} 899 902 % \caption{Signal spectrum for MIN/80}
% \label{fig:min_80} 900 903 % \label{fig:min_80}
% \end{figure} 901 904 % \end{figure}
% 902 905 %
% \begin{figure} 903 906 % \begin{figure}
% \centering 904 907 % \centering
% \includegraphics[width=\linewidth]{images/min_100} 905 908 % \includegraphics[width=\linewidth]{images/min_100}
% \caption{Signal spectrum for MIN/100} 906 909 % \caption{Signal spectrum for MIN/100}
% \label{fig:min_100} 907 910 % \label{fig:min_100}
% \end{figure} 908 911 % \end{figure}
909 912
% r2.14 et r2.15 et r2.16 910 913 % r2.14 et r2.15 et r2.16
\begin{figure} 911 914 \begin{figure}
\centering 912 915 \centering
\begin{subfigure}{\linewidth} 913 916 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_40} 914 917 \includegraphics[width=\linewidth]{images/min_40}
\caption{Signal spectrum for MIN/40} 915 918 \caption{Signal spectrum for MIN/40}
\label{fig:min_40} 916 919 \label{fig:min_40}
\end{subfigure} 917 920 \end{subfigure}
918 921
\begin{subfigure}{\linewidth} 919 922 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_60} 920 923 \includegraphics[width=\linewidth]{images/min_60}
\caption{Signal spectrum for MIN/60} 921 924 \caption{Signal spectrum for MIN/60}
\label{fig:min_60} 922 925 \label{fig:min_60}
\end{subfigure} 923 926 \end{subfigure}
924 927
\begin{subfigure}{\linewidth} 925 928 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_80} 926 929 \includegraphics[width=\linewidth]{images/min_80}
ifcs2018_journal_reponse.tex
Diff comments   View file @ 959bbc5
%Minor Revision - TUFFC-09469-2019 1 1 %Minor Revision - TUFFC-09469-2019
%Transactions on Ultrasonics, Ferroelectrics, and Frequency 2 2 %Transactions on Ultrasonics, Ferroelectrics, and Frequency
%Control (July 23, 2019 9:29 PM) 3 3 %Control (July 23, 2019 9:29 PM)
%To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr, 4 4 %To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr,
%gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr, 5 5 %gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr,
%jmfriedt@femto-st.fr 6 6 %jmfriedt@femto-st.fr
%CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu 7 7 %CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu
% 8 8 %
%Dear Mr. Arthur HUGEAT 9 9 %Dear Mr. Arthur HUGEAT
% 10 10 %
%Congratulations! Your manuscript 11 11 %Congratulations! Your manuscript
% 12 12 %
%MANUSCRIPT NO. TUFFC-09469-2019 13 13 %MANUSCRIPT NO. TUFFC-09469-2019
%MANUSCRIPT TYPE: Papers 14 14 %MANUSCRIPT TYPE: Papers
%TITLE: Filter optimization for real time digital processing of radiofrequency 15 15 %TITLE: Filter optimization for real time digital processing of radiofrequency
%signals: application to oscillator metrology 16 16 %signals: application to oscillator metrology
%AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, 17 17 %AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois,
%Pierre-Yves; Friedt, Jean-Michel 18 18 %Pierre-Yves; Friedt, Jean-Michel
% 19 19 %
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%Sincerely, 67 67 %Sincerely,
% 68 68 %
%Giorgio Santarelli 69 69 %Giorgio Santarelli
%Associate Editor in Chief 70 70 %Associate Editor in Chief
%Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 71 71 %Transactions on Ultrasonics, Ferroelectrics, and Frequency Control
% 72 72 %
%**************************************************** 73 73 %****************************************************
%REVIEWERS' COMMENTS: 74 74 %REVIEWERS' COMMENTS:
75 75
\documentclass[a4paper]{article} 76 76 \documentclass[a4paper]{article}
\usepackage{fullpage,graphicx} 77 77 \usepackage{fullpage,graphicx}
\begin{document} 78 78 \begin{document}
{\bf Reviewer: 1} 79 79 {\bf Reviewer: 1}
80 80
%Comments to the Author 81 81 %Comments to the Author
%In general, the language/grammar is adequate. 82 82 %In general, the language/grammar is adequate.
83 83
{\bf 84 84 {\bf
On page 2, "...allowing to save processing resource..." could be improved. % r1.1 85 85 On page 2, "...allowing to save processing resource..." could be improved. % r1.1
} 86 86 }
87 87
The sentence was split and now reads ``number of coefficients irrelevant: processing 88 88 The sentence was split and now reads ``number of coefficients irrelevant: processing
resources are hence saved by shrinking the filter length.'' 89 89 resources are hence saved by shrinking the filter length.''
90 90
{\bf 91 91 {\bf
On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 92 92 On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2
the author meant.} 93 93 the author meant.}
94 94
Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s) 95 95 Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s)
Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.'' 96 96 Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
97 97
{\bf 98 98 {\bf
On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 99 99 On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3
could be improved. 100 100 could be improved.
} 101 101 }
102 102
Indeed this paragraph has be written again and now reads as\\ 103 103 Indeed this paragraph has be written again and now reads as\\
``The first step of our approach is to model the DSP chain. Since we aim at only optimizing 104 104 ``The first step of our approach is to model the DSP chain. Since we aim at only optimizing
the filtering part of the signal processing chain, we have not included the PRN generator or the 105 105 the filtering part of the signal processing chain, we have not included the PRN generator or the
ADC in the model: the input data size and rate are considered fixed and defined by the hardware. 106 106 ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
The filtering can be done in two ways, either by considering a single monolithic FIR filter 107 107 The filtering can be done in two ways, either by considering a single monolithic FIR filter
requiring many coefficients to reach the targeted noise rejection ratio, or by 108 108 requiring many coefficients to reach the targeted noise rejection ratio, or by
cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter. 109 109 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.
'' 110 110 ''
111 111
{\bf 112 112 {\bf
I appreciate that the authors attempted and document two optimizations: that % r1.4 - en attente des résultats 113 113 I appreciate that the authors attempted and document two optimizations: that % r1.4 - en attente des résultats
of maximum rejection ratio at fixed silicon area, as well as minimum silicon 114 114 of maximum rejection ratio at fixed silicon area, as well as minimum silicon
area for a fixed minimum rejection ratio. For non-experts, it might be very 115 115 area for a fixed minimum rejection ratio. For non-experts, it might be very
useful to compare the results of both optimization paths to the performance and 116 116 useful to compare the results of both optimization paths to the performance and
resource-utilization of generic low-pass filter gateware offered by device 117 117 resource-utilization of generic low-pass filter gateware offered by device
manufacturers. I appreciate also that the authors have presented source code 118 118 manufacturers. I appreciate also that the authors have presented source code
for examination online. 119 119 for examination online.
} 120 120 }
121 121
TODO : FIR Compiler et regarder les ressources pour un FIR comparable a ceux monolithiques 122 122 TODO : FIR Compiler et regarder les ressources pour un FIR comparable a ceux monolithiques
fournis dans l'article (memes coefs et meme nombre de coefs) 123 123 fournis dans l'article (memes coefs et meme nombre de coefs)
124 124
{\bf 125 125 {\bf
Reviewer: 2 126 126 Reviewer: 2
} 127 127 }
128 128
%Comments to the Author 129 129 %Comments to the Author
%In the Manuscript, the Authors describe an optimization methodology for filter 130 130 %In the Manuscript, the Authors describe an optimization methodology for filter
%design to be used in phase noise metrology. The methodology is general and can 131 131 %design to be used in phase noise metrology. The methodology is general and can
%be used for many aspects of the processing chain. In the Manuscript, the Authors 132 132 %be used for many aspects of the processing chain. In the Manuscript, the Authors
%focus on filtering and shifting while the other aspects, in particular decimation, 133 133 %focus on filtering and shifting while the other aspects, in particular decimation,
%will be considered in a future work. The optimization problem is modelled 134 134 %will be considered in a future work. The optimization problem is modelled
%theoretically and then solved by means of a commercial software. The solutions 135 135 %theoretically and then solved by means of a commercial software. The solutions
%are tested experimentally on the Redpitaya platform with synthetic and real 136 136 %are tested experimentally on the Redpitaya platform with synthetic and real
%white noises. Two cases are considered as a function of the number of filters: 137 137 %white noises. Two cases are considered as a function of the number of filters:
%maximum rejection given a fixed amount of resources and minimum resource 138 138 %maximum rejection given a fixed amount of resources and minimum resource
%utilization given a fixed amount of rejection. 139 139 %utilization given a fixed amount of rejection.
%The Authors find that filtering improves significantly when the number of 140 140 %The Authors find that filtering improves significantly when the number of
%filters increases. 141 141 %filters increases.
%A lot of work has been done in generalizing and automating the procedure so 142 142 %A lot of work has been done in generalizing and automating the procedure so
%that different approaches can be investigated quickly and efficiently. The 143 143 %that different approaches can be investigated quickly and efficiently. The
%results presented in the Manuscript seem to be just a case study based on 144 144 %results presented in the Manuscript seem to be just a case study based on
%the particular criterion chosen by the Authors. Different criteria, in 145 145 %the particular criterion chosen by the Authors. Different criteria, in
%general, could lead to different results and it is important to consider 146 146 %general, could lead to different results and it is important to consider
%carefully the criterion adopted by the Authors, in order to check if it 147 147 %carefully the criterion adopted by the Authors, in order to check if it
%is adequate to compare the performance of filters and if multi-stage 148 148 %is adequate to compare the performance of filters and if multi-stage
%filters are really superior than monolithic filters. 149 149 %filters are really superior than monolithic filters.
150 150
{\bf 151 151 {\bf
By observing the results presented in fig. 10-16, it is clear that the % r2.1 - fait 152 152 By observing the results presented in fig. 10-16, it is clear that the % r2.1 - fait
performances of multi-stage filters are obtained at the expense of their 153 153 performances of multi-stage filters are obtained at the expense of their
selectivity and, in this sense, the filters presented in these figures 154 154 selectivity and, in this sense, the filters presented in these figures
are not equivalent. For example, in Fig. 14, at the limit of the pass band, 155 155 are not equivalent. For example, in Fig. 14, at the limit of the pass band,
the attenuation is almost 15 dB for n = 5, while it is not noticeable for 156 156 the attenuation is almost 15 dB for n = 5, while it is not noticeable for
n = 1. 157 157 n = 1.
} 158 158 }
159 159
TODO : ajouter les gabarits 160 160 We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined
161 the bandpass and the bandstop of the filter.
161 162
Peut etre refaire une serie de simulation dans lesquelles on impose une coupure 162 163 %Peut etre refaire une serie de simulation dans lesquelles on impose une coupure
non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte 163 164 %non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte
au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque 164 165 %au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque
a la cascade de filtres. 165 166 %a la cascade de filtres.
AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de 166 167 %AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de
relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai 167 168 %relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai
168 169
170 JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait
171 moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB
172 quand on cascade 5 filtres !
173
{\bf 169 174 {\bf
The reason is in the criterion that considers the average attenuation in % r2.2 - fait 170 175 The reason is in the criterion that considers the average attenuation in % r2.2 - fait
the pass band. This criterion does not take into account the maximum attenuation 171 176 the pass band. This criterion does not take into account the maximum attenuation
in this region, which is a very important parameter for specifying a filter 172 177 in this region, which is a very important parameter for specifying a filter
and for evaluating its performance. For example, with this criterion, a 173 178 and for evaluating its performance. For example, with this criterion, a
filter with 0.1 dB of ripple is considered equivalent to a filter with 174 179 filter with 0.1 dB of ripple is considered equivalent to a filter with
10 dB of ripple. This point has a strong impact in the optimization process 175 180 10 dB of ripple. This point has a strong impact in the optimization process
and in the results that are obtained and has to be reconsidered. 176 181 and in the results that are obtained and has to be reconsidered.
} 177 182 }
178 183
Je ne pense pas que ca soit le cas : la somme des valeurs absolues des pertes 179 184 The manuscript erroneously stated that we considered the mean of the absolute
dans la bande va defavoriser un filtre avec 10 dB de ripples. Il n'a pas compris que 180 185 value within the bandpass: the manuscript has now been corrected to properly state
la bandpass s'arrete a 40\% de la bande, donc mettre le gabarit clarifierait ce point je 181 186 the selected criterion, namely the {\em sum} of the absolute value, so that any
pense 182 187 ripple in the bandpass will reduce the chances of a given filter set from being
AH: Il y avait une faute, j'avais mis "mean of absolute value" au lieu de "sum of absolute value". Je pense que je n'ai pas besoin de mettre plus de détail ? 183 188 selected. The manuscript now states ``Our criterion to compute the filter rejection considers
189 % r2.8 et r2.2 r2.3
190 the maximum magnitude within the stopband, to which the {sum of the absolute values
191 within the passband is subtracted to avoid filters with excessive ripples}.''
184 192
{\bf 185 193 {\bf
I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait 186 194 I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait
into account the maximum allowed attenuation in pass band, for example by 187 195 into account the maximum allowed attenuation in pass band, for example by
fixing its value to a typical one, as it has been done for the transition 188 196 fixing its value to a typical one, as it has been done for the transition
bandwidth. 189 197 bandwidth.
} 190 198 }
AH: Il y avait une faute, j'avais mis "mean of absolute value" au lieu de "sum of absolute value". Je pense que je n'ai pas besoin de mettre plus de détail ? 191
192 199
200 See above: the absolute value within the passband will reject filters with
201 excessive ripples, including excessive attenuation, within the passband.
202
{\bf 193 203 {\bf
In addition, I suggest to address the following points: % r2.4 194 204 In addition, I suggest to address the following points: % r2.4
- Page 1, line 50: the Authors state that IIR have shorter impulse response 195 205 - Page 1, line 50: the Authors state that IIR have shorter impulse response
than FIR. This is not true in general. The sentence should be reconsidered. 196 206 than FIR. This is not true in general. The sentence should be reconsidered.
} 197 207 }
198 208
We have not stated that the IIR has a shorter impulse response but a shorter lag. 199 209 We have not stated that the IIR has a shorter impulse response but a shorter lag.
Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filters 200 210 Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filters
have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing 201 211 have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing
the first output, an IIR will start providing outputs only 5 time steps after the initial 202 212 the first output, an IIR will start providing outputs only 5 time steps after the initial
input starts feeding the IIR. Hence, the issue we address here is lag and not impulse 203 213 input starts feeding the IIR. Hence, the issue we address here is lag and not impulse
response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue 204 214 response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue
in a openloop phase noise characterization instrument, the large 205 215 in a openloop phase noise characterization instrument, the large
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 206 216 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
is not considered as an issue as would be in a closed loop system in which lag aims at being 207 217 is not considered as an issue as would be in a closed loop system in which lag aims at being
minimized to avoid oscillation conditions. 208 218 minimized to avoid oscillation conditions.
'' 209 219 ''
210 220
{\bf 211 221 {\bf
- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 212 222 - Fig. 4: the Author should motivate in the text why it has been chosen % r2.5
this transition bandwidth and if it is a typical requirement for phase-noise 213 223 this transition bandwidth and if it is a typical requirement for phase-noise
metrology. 214 224 metrology.
} 215 225 }
216 226
The purpose of the paper is to demonstrate how a given filter shape can be achieved by 217 227 The purpose of the paper is to demonstrate how a given filter shape can be achieved by
minimizing varous resource criteria. Indeed the stopband and bandpass boundaries can 218 228 minimizing varous resource criteria. Indeed the stopband and bandpass boundaries can
be questioned: we have selected this filter shape as a typical anti-aliasing filter considering 219 229 be questioned: we have selected this filter shape as a typical anti-aliasing filter considering
the the dataflow is to be halved. Hence, selecting a cutoff frequency of 40\% the initial 220 230 the the dataflow is to be halved. Hence, selecting a cutoff frequency of 40\% the initial
Nyquist frequency prevents noise from reaching baseband after decimating the dataflow by a 221 231 Nyquist frequency prevents noise from reaching baseband after decimating the dataflow by a
factor of 2. Such ideas are now stated explicitly in the text as ``Throughout this demonstration, 222 232 factor of 2. Such ideas are now stated explicitly in the text as ``Throughout this demonstration,
we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\% 223 233 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 224 234 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 225 235 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} 226 236 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.'' 227 237 as described below is indeed unique for each filter shape.''
228 238
{\bf 229 239 {\bf
- The impact of the coefficient resolution is discussed. What about the % r2.6 - fait 230 240 - The impact of the coefficient resolution is discussed. What about the % r2.6 - fait
resolution of the data stream? Is it fixed? If so, which value has been 231 241 resolution of the data stream? Is it fixed? If so, which value has been
used in the analysis? If not, how is it changed with respect to the 232 242 used in the analysis? If not, how is it changed with respect to the
coefficient resolution? 233 243 coefficient resolution?
} 234 244 }
235 245
We have now stated in the beginning of the document that ``we have not included the PRN generator 236 246 We have now stated in the beginning of the document that ``we have not included the PRN generator
or the ADC in the model: the input data size and rate are considered fixed and defined by the 237 247 or the ADC in the model: the input data size and rate are considered fixed and defined by the
hardware.'' so indeed the input datastream resolution is considered as a given. 238 248 hardware.'' so indeed the input datastream resolution is considered as a given.
239 249
{\bf 240 250 {\bf
- Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait 241 251 - Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait
Fig. 5 can be removed. 242 252 Fig. 5 can be removed.
- Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait 243 253 - Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait
It should be ``minimum'' 244 254 It should be ``minimum''
} 245 255 }
AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max 246 256 AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max
de la magnitude. J'ai corrigé en ce sens. 247 257 de la magnitude. J'ai corrigé en ce sens.
248 258
{\bf 249 259 {\bf
- Page e, line 55, second column: “takin” % r2.9 - fait 250 260 - Page e, line 55, second column: ``takin'' % r2.9 - fait
- Page 3, line 58: “pessimistic” should be replaced with “conservative” % r2.10 - fait 251 261 - Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait
- Page 4, line 17: “meaning” --> “this means” % r2.11 - fait 252 262 - Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait
263 }
264
265 All typos and grammatical errors have been corrected.
266
267 {\bf
- Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait 253 268 - Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait
these particular configurations? Are they chosen automatically? 254 269 these particular configurations? Are they chosen automatically?
270 }
271
272 JMF : repondre
273
274 {\bf
- Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait 255 275 - Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait
and non-quadratic to a quadratic?} 256 276 and non-quadratic to a quadratic?}
257 277
JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article 258 278 JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
je ne comprends pas comment ca repond a la question 259 279 je ne comprends pas comment ca repond a la question
260 280
{\bf 261 281 {\bf
- Captions of figure and tables are too minimal. % r2.14 262 282 - Captions of figure and tables are too minimal. % r2.14
- Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait 263 283 - Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait
of a single figure. Same for fig. 13-16. 264 284 of a single figure. Same for fig. 13-16.
} 265 285 }
266 286
{\bf 267 287 {\bf
- Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait 268 288 - Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait
of the curves is about 20 dBpk-pk and it doesn’t allow to appreciate the 269 289 of the curves is about 20 dBpk-pk and it doesn’t allow to appreciate the
differences among the curves. I suggest to reduce the noise below 1 dBpk-pk. 270 290 differences among the curves. I suggest to reduce the noise below 1 dBpk-pk.
} 271 291 }
272 292
Indeed averaging had been omitted during post-processing and figure generation: we 273 293 Indeed averaging had been omitted during post-processing and figure generation: we
are grateful to the reviewer for emphasizing this point which has now been corrected. All spectra 274 294 are grateful to the reviewer for emphasizing this point which has now been corrected. All spectra
now exhibit sub-dBpk-pl line thickness. 275 295 now exhibit sub-dBpk-pl line thickness.
296