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% fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee 1 1 % fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee
% demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de 2 2 % demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de
% rejection par bit et perte si moins de bits que rejection/6 3 3 % rejection par bit et perte si moins de bits que rejection/6
% developper programme lineaire en incluant le decalage de bits 4 4 % developper programme lineaire en incluant le decalage de bits
% insister que avant on etait synthetisable mais pas implementable, alors que maintenant on 5 5 % insister que avant on etait synthetisable mais pas implementable, alors que maintenant on
% implemente et on demontre que ca tourne 6 6 % implemente et on demontre que ca tourne
% gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ? 7 7 % gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ?
% Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer 8 8 % Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer
% (zedboard ou redpit) 9 9 % (zedboard ou redpit)
10 10
% label schema : verifier que "argumenter de la cascade de FIR" est fait 11 11 % label schema : verifier que "argumenter de la cascade de FIR" est fait
12 12
\documentclass[a4paper,journal]{IEEEtran/IEEEtran} 13 13 \documentclass[a4paper,journal]{IEEEtran/IEEEtran}
\usepackage{graphicx,color,hyperref} 14 14 \usepackage{graphicx,color,hyperref}
\usepackage{amsfonts} 15 15 \usepackage{amsfonts}
\usepackage{amsthm} 16 16 \usepackage{amsthm}
\usepackage{amssymb} 17 17 \usepackage{amssymb}
\usepackage{amsmath} 18 18 \usepackage{amsmath}
\usepackage{algorithm2e} 19 19 \usepackage{algorithm2e}
\usepackage{url,balance} 20 20 \usepackage{url,balance}
\usepackage[normalem]{ulem} 21 21 \usepackage[normalem]{ulem}
\usepackage{tikz} 22 22 \usepackage{tikz}
\usetikzlibrary{positioning,fit} 23 23 \usetikzlibrary{positioning,fit}
\usepackage{multirow} 24 24 \usepackage{multirow}
\usepackage{scalefnt} 25 25 \usepackage{scalefnt}
\usepackage{caption} 26 26 \usepackage{caption}
\usepackage{subcaption} 27 27 \usepackage{subcaption}
28 28
% correct bad hyphenation here 29 29 % correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor} 30 30 \hyphenation{op-tical net-works semi-conduc-tor}
\textheight=26cm 31 31 \textheight=26cm
\setlength{\footskip}{30pt} 32 32 \setlength{\footskip}{30pt}
\pagenumbering{gobble} 33 33 \pagenumbering{gobble}
\begin{document} 34 34 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 35 35 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 36 36 to oscillator metrology}
37 37
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 38 38 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 39 39 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\ 40 40 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\ 41 41 \IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\
\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\ 42 42 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 43 43 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 44 44 }
\maketitle 45 45 \maketitle
\thispagestyle{plain} 46 46 \thispagestyle{plain}
\pagestyle{plain} 47 47 \pagestyle{plain}
\newtheorem{definition}{Definition} 48 48 \newtheorem{definition}{Definition}
49 49
\begin{abstract} 50 50 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 51 51 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 52 52 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 53 53 of ultrastable clocks, stringent filtering requirements are defined by spurious signal or
noise rejection needs. Since real time radiofrequency processing must be performed in a 54 54 noise rejection needs. Since real time radiofrequency processing must be performed in a
Field Programmable Array to meet timing constraints, we investigate optimization strategies 55 55 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 56 56 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. The 57 57 required and keeping timing constraints within the targeted measurement bandwidths. The
presented technique is applicable to scheduling any sequence of processing blocks characterized 58 58 presented technique is applicable to scheduling any sequence of processing blocks characterized
by a throughput, resource occupation and performance tabulated as a function of configuration 59 59 by a throughput, resource occupation and performance tabulated as a function of configuration
characateristics, as is the case for filters with their coefficients and resolution yielding 60 60 characateristics, as is the case for filters with their coefficients and resolution yielding
rejection and number of multipliers. 61 61 rejection and number of multipliers.
\end{abstract} 62 62 \end{abstract}
63 63
\begin{IEEEkeywords} 64 64 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 65 65 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 66 66 \end{IEEEkeywords}
67 67
\section{Digital signal processing of ultrastable clock signals} 68 68 \section{Digital signal processing of ultrastable clock signals}
69 69
Analog oscillator phase noise characteristics are classically performed by downconverting 70 70 Analog oscillator phase noise characteristics are classically performed by downconverting
the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband, 71 71 the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,
followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In 72 72 followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In
a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by 73 73 a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by
multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}. 74 74 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
75 75
\begin{figure}[h!tb] 76 76 \begin{figure}[h!tb]
\begin{center} 77 77 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 78 78 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 79 79 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 80 80 \caption{Fully digital oscillator phase noise characterization: the Device Under Test
(DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and 81 81 (DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and
downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals 82 82 downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals
and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite 83 83 and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite
Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays 84 84 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 85 85 the spectral characteristics of the phase fluctuations.}
\label{schema} 86 86 \label{schema}
\end{figure} 87 87 \end{figure}
88 88
As with the analog mixer, 89 89 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 90 90 the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as
well as the generation of the frequency sum signal in addition to the frequency difference. 91 91 well as the generation of the frequency sum signal in addition to the frequency difference.
These unwanted spectral characteristics must be rejected before decimating the data stream 92 92 These unwanted spectral characteristics must be rejected before decimating the data stream
for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the 93 93 for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
downconverter 94 94 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 95 95 and the decimation processing blocks are core characteristics of an oscillator characterization
system, and must reject out-of-band signals below the targeted phase noise -- typically in the 96 96 system, and must reject out-of-band signals below the targeted phase noise -- typically in the
sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will 97 97 sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will
use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency 98 98 use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency
datastream: optimizing the performance of the filter while reducing the needed resources is 99 99 datastream: optimizing the performance of the filter while reducing the needed resources is
hence tackled in a systematic approach using optimization techniques. Most significantly, we 100 100 hence tackled in a systematic approach using optimization techniques. Most significantly, we
tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with 101 101 tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with
tunable number of coefficients and tunable number of bits representing the coefficients and the 102 102 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 103 103 data being processed.
104 104
\section{Finite impulse response filter} 105 105 \section{Finite impulse response filter}
106 106
We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined 107 107 We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined
by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the 108 108 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 109 109 outputs $y_k$
\begin{align} 110 110 \begin{align}
y_n=\sum_{k=0}^N b_k x_{n-k} 111 111 y_n=\sum_{k=0}^N b_k x_{n-k}
\label{eq:fir_equation} 112 112 \label{eq:fir_equation}
\end{align} 113 113 \end{align}
114 114
As opposed to an implementation on a general purpose processor in which word size is defined by the 115 115 As opposed to an implementation on a general purpose processor in which word size is defined by the
processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since 116 116 processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since
not only the coefficient values and number of taps must be defined, but also the number of bits 117 117 not only the coefficient values and number of taps must be defined, but also the number of bits
defining the coefficients and the sample size. For this reason, and because we consider pipeline 118 118 defining the coefficients and the sample size. For this reason, and because we consider pipeline
processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency 119 119 processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency
signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but 120 120 signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but
the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language 121 121 the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language
(VHDL) level. 122 122 (VHDL) level.
Since latency is not an issue in a openloop phase noise characterization instrument, the large 123 123 {\color{red}Since latency is not an issue in a openloop phase noise characterization instrument,
124 the large
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 124 125 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. 125 126 is not considered as an issue as would be in a closed loop system.} % r2.4
126 127
The coefficients are classically expressed as floating point values. However, this binary 127 128 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, 128 129 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 129 130 we select to quantify these floating point values into integer values. This quantization
will result in some precision loss. 130 131 will result in some precision loss.
131 132
\begin{figure}[h!tb] 132 133 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/zero_values} 133 134 \includegraphics[width=\linewidth]{images/zero_values}
\caption{Impact of the quantization resolution of the coefficients: the quantization is 134 135 \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 135 136 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 136 137 the 30~first and 30~last coefficients out of the initial 128~band-pass
filter coefficients to 0 (red dots).} 137 138 filter coefficients to 0 (red dots).}
\label{float_vs_int} 138 139 \label{float_vs_int}
\end{figure} 139 140 \end{figure}
140 141
The tradeoff between quantization resolution and number of coefficients when considering 141 142 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 142 143 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 143 144 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 144 145 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 145 146 quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the
taps become null, making the large number of coefficients irrelevant and allowing to save 146 147 taps become null, {\color{red}making the large number of coefficients irrelevant: processing
processing resource by shrinking the filter length. This tradeoff aimed at minimizing resources 147 148 resources % r1.1
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 148 150 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 149 151 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 150 152 and tap length, as will be shown in the next section. Indeed, our development strategy closely
follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards} 151 153 follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
in which basic blocks are defined and characterized before being assembled \cite{hide} 152 154 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 153 155 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 154 156 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 155 157 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 156 158 current implementation: the decimation is assumed to be located after the FIR cascade at the
moment. 157 159 moment.
158 160
\section{Methodology description} 159 161 \section{Methodology description}
160 162
Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP) 161 163 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. 162 164 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 163 165 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 164 166 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 165 167 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 166 168 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 167 169 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 168 170 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 169 171 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 170 172 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 171 173 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 172 174 that all solutions found by the solver are synthesized and executed on hardware at the end
of the analysis. 173 175 of the analysis.
174 176
In this demonstration , we focus on only two operations: filtering and shifting the number of 175 177 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. 176 178 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 177 179 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 178 180 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 179 181 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 180 182 requiring pipelined processing at full bandwidth for the earliest steps, including for
time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}. 181 183 time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}.
182 184
Addressing only two operations allows for demonstrating the methodology but should not be 183 185 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 184 186 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. Hence, 185 187 of skeleton blocks as long as perfomance and resource occupation can be determined. {\color{red}
in this paper we will apply our methodology on simple DSP chains: a white noise input signal 186 188 Hence,
is generated using a Pseudo-Random Number (PRN) generator or thanks at a radiofrequency-grade 187 189 in this paper we will apply our methodology on simple DSP chains: a white noise input signal % r1.2
Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor. Once samples have been 188 190 is generated using a Pseudo-Random Number (PRN) generator or by sampling a wideband (125~MS/s)
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 -- 189 192 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 190 193 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, 191 194 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 192 195 allowing to assess either filter rejection for a given resource usage, or validating the rejection
when implementing a solution minimizing resource occupation. 193 196 when implementing a solution minimizing resource occupation.
194 197
The first step of our approach is to model the DSP chain and since we just optimize 195 198 {\color{red}
the filtering, we have not modeling the PRN generator or the ADC. The filtering can be 196 199 The first step of our approach is to model the DSP chain. Since we aim at only optimizing % r1.3
done by two ways. The first one we use only one FIR filter with lot of coefficients 197 200 the filtering part of the signal processing chain, we have not included the PRN generator or the
to rejection the noise, we called this approach a monolithic approach. And the second one 198 201 ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
we select different FIR filters with less coefficients the monolithic filter and we cascaded 199 202 The filtering can be done in two ways, either by considering a single monolithic FIR filter
it to filtering the signal. 200 203 requiring many coefficients to reach the targeted noise rejection ratio, or by
204 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.}
201 205
After each filter we leave the possibility of shifting the filtered data to consume 202 206 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 203 207 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). 204 208 and a shifter (the shift could be omitted if we do not need to divide the filtered data).
205 209
\subsection{Model of a FIR filter} 206 210 \subsection{Model of a FIR filter}
207 211
A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$) 208 212 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 $\pi^C_i$ 209 213 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 210 214 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} 211 215 the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage}
shows a filtering stage. 212 216 shows a filtering stage.
213 217
\begin{figure} 214 218 \begin{figure}
\centering 215 219 \centering
\begin{tikzpicture}[node distance=2cm] 216 220 \begin{tikzpicture}[node distance=2cm]
\node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ; 217 221 \node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;
\node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ; 218 222 \node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;
\node (Start) [left of=FIR] { } ; 219 223 \node (Start) [left of=FIR] { } ;
\node (End) [right of=Shift] { } ; 220 224 \node (End) [right of=Shift] { } ;
221 225
\node[draw,fit=(FIR) (Shift)] (Filter) { } ; 222 226 \node[draw,fit=(FIR) (Shift)] (Filter) { } ;
223 227
\draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ; 224 228 \draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;
\draw[->] (FIR) -- (Shift) ; 225 229 \draw[->] (FIR) -- (Shift) ;
\draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ; 226 230 \draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;
\end{tikzpicture} 227 231 \end{tikzpicture}
\caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)} 228 232 \caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}
\label{fig:fir_stage} 229 233 \label{fig:fir_stage}
\end{figure} 230 234 \end{figure}
231 235
FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB. 232 236 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 233 237 This rejection has been computed using GNU Octave software FIR coefficient design functions
(\texttt{firls} and \texttt{fir1}). 234 238 (\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. 235 239 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, 236 240 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. 237 241 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. 238 242 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.
239 243
With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter 240 244 With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter
transfer function. 241 245 transfer function.
Comparing the performance between FIRs requires however defining a unique criterion. As shown in figure~\ref{fig:fir_mag}, 242 246 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 243 247 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}. 244 248 bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. {\color{red}Throughout this demonstration,
249 we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%
250 of the Nyquist frequency to the end of the band, as would be typically selected to prevent
251 aliasing before decimating the dataflow by 2. The method is however generalized to any filter
252 shape as long as it is defined from the initial modelling steps.}
245 253
\begin{figure} 246 254 \begin{figure}
\begin{center} 247 255 \begin{center}
\scalebox{0.8}{ 248 256 \scalebox{0.8}{
\centering 249 257 \centering
\begin{tikzpicture}[scale=0.3] 250 258 \begin{tikzpicture}[scale=0.3]
\draw[<->] (0,15) -- (0,0) -- (21,0) ; 251 259 \draw[<->] (0,15) -- (0,0) -- (21,0) ;
\draw[thick] (0,12) -- (8,12) -- (20,0) ; 252 260 \draw[thick] (0,12) -- (8,12) -- (20,0) ;
253 261
\draw (0,14) node [left] { $P$ } ; 254 262 \draw (0,14) node [left] { $P$ } ;
\draw (20,0) node [below] { $f$ } ; 255 263 \draw (20,0) node [below] { $f$ } ;
256 264
\draw[>=latex,<->] (0,14) -- (8,14) ; 257 265 \draw[>=latex,<->] (0,14) -- (8,14) ;
\draw (4,14) node [above] { passband } node [below] { $40\%$ } ; 258 266 \draw (4,14) node [above] { passband } node [below] { $40\%$ } ;
259 267
\draw[>=latex,<->] (8,14) -- (12,14) ; 260 268 \draw[>=latex,<->] (8,14) -- (12,14) ;
\draw (10,14) node [above] { transition } node [below] { $20\%$ } ; 261 269 \draw (10,14) node [above] { transition } node [below] { $20\%$ } ;
262 270
\draw[>=latex,<->] (12,14) -- (20,14) ; 263 271 \draw[>=latex,<->] (12,14) -- (20,14) ;
\draw (16,14) node [above] { stopband } node [below] { $40\%$ } ; 264 272 \draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;
265 273
\draw[>=latex,<->] (16,12) -- (16,8) ; 266 274 \draw[>=latex,<->] (16,12) -- (16,8) ;
\draw (16,10) node [right] { rejection } ; 267 275 \draw (16,10) node [right] { rejection } ;
268 276
\draw[dashed] (8,-1) -- (8,14) ; 269 277 \draw[dashed] (8,-1) -- (8,14) ;
\draw[dashed] (12,-1) -- (12,14) ; 270 278 \draw[dashed] (12,-1) -- (12,14) ;
271 279
\draw[dashed] (8,12) -- (16,12) ; 272 280 \draw[dashed] (8,12) -- (16,12) ;
\draw[dashed] (12,8) -- (16,8) ; 273 281 \draw[dashed] (12,8) -- (16,8) ;
274 282
\end{tikzpicture} 275 283 \end{tikzpicture}
} 276 284 }
\end{center} 277 285 \end{center}
\caption{Shape of the filter transmitted power $P$ as a function of frequency $f$: 278 286 \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, 279 287 the passband is considered to occupy the initial 40\% of the Nyquist frequency range,
the stopband the last 40\%, allowing 20\% transition width.} 280 288 the stopband the last 40\%, allowing 20\% transition width.}
\label{fig:fir_mag} 281 289 \label{fig:fir_mag}
\end{figure} 282 290 \end{figure}
283 291
In the transition band, the behavior of the filter is left free, we only care about the passband and the stopband characteristics. 284 292 In the transition band, the behavior of the filter is left free, we only care about the passband and the stopband characteristics.
% r2.7 285 293 % r2.7
% Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion 286 294 % 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 287 295 % 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. 288 296 % 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 289 297 Our criterion to compute the filter rejection takes
% r2.8 et r2.2 r2.3 290 298 % 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}. 291 299 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}.
292 300
% \begin{figure} 293 301 % \begin{figure}
% \centering 294 302 % \centering
% \includegraphics[width=\linewidth]{images/colored_mean_criterion} 295 303 % \includegraphics[width=\linewidth]{images/colored_mean_criterion}
% \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters} 296 304 % \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}
% \label{fig:mean_criterion} 297 305 % \label{fig:mean_criterion}
% \end{figure} 298 306 % \end{figure}
299 307
\begin{figure} 300 308 \begin{figure}
\centering 301 309 \centering
\includegraphics[width=\linewidth]{images/colored_custom_criterion} 302 310 \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) 303 311 \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} 304 312 comparison between monolithic filter and cascaded filters}
\label{fig:custom_criterion} 305 313 \label{fig:custom_criterion}
\end{figure} 306 314 \end{figure}
307 315
Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps 308 316 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 309 317 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. 310 318 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. 311 319 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. 312 320 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 313 321 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. 314 322 the rejection. Hence the best coefficient set are on the vertex of the pyramid.
315 323
\begin{figure} 316 324 \begin{figure}
\centering 317 325 \centering
\includegraphics[width=\linewidth]{images/rejection_pyramid} 318 326 \includegraphics[width=\linewidth]{images/rejection_pyramid}
\caption{Rejection as a function of number of coefficients and number of bits} 319 327 \caption{Rejection as a function of number of coefficients and number of bits}
\label{fig:rejection_pyramid} 320 328 \label{fig:rejection_pyramid}
\end{figure} 321 329 \end{figure}
322 330
Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps), 323 331 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. 324 332 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: 325 333 If the FIR filter coefficients are the same between the stages, we have:
$$F_{total} = F_1 + F_2$$ 326 334 $$F_{total} = F_1 + F_2$$
But selecting two different sets of coefficient will yield a more complex situation in which 327 335 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 328 336 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. 329 337 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 330 338 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. 331 339 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
% r2.9 332 340 % r2.9
Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection 333 341 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, 334 342 criteria of each filter. However since the this sum underestimates the rejection capability of the cascade,
% r2.10 335 343 % r2.10
this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability 336 344 this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
of the filter cascade to meet design criteria. 337 345 of the filter cascade to meet design criteria.
338 346
\begin{figure} 339 347 \begin{figure}
\centering 340 348 \centering
\includegraphics[width=\linewidth]{images/cascaded_criterion} 341 349 \includegraphics[width=\linewidth]{images/cascaded_criterion}
\caption{Rejection of two cascaded filters} 342 350 \caption{Rejection of two cascaded filters}
\label{fig:sum_rejection} 343 351 \label{fig:sum_rejection}
\end{figure} 344 352 \end{figure}
345 353
% r2.6 346 354 % r2.6
Finally in our case, we consider that the input signal are fully known. So the 347 355 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 348 356 resolution of the data stream are fixed and still the same for all experiments
in this paper. 349 357 in this paper.
350 358
Based on this analysis, we address the estimate of resource consumption (called 351 359 Based on this analysis, we address the estimate of resource consumption (called
% r2.11 352 360 % r2.11
silicon area -- in the case of FPGAs this means processing cells) as a function of 353 361 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 354 362 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, 355 363 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 356 364 and will assess after synthesis the adequation of this arbitrary unit with actual
hardware resources provided by FPGA manufacturers. The sum of individual processing 357 365 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. 358 366 unit areas is constrained by a total silicon area representative of FPGA global resources.
Formally, variable $a_i$ is the area taken by filter~$i$ 359 367 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). 360 368 (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: 361 369 Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
362 370
\begin{align} 363 371 \begin{align}
\text{Maximize } & \sum_{i=1}^n r_i \notag \\ 364 372 \text{Maximize } & \sum_{i=1}^n r_i \notag \\
\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\ 365 373 \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} \\ 366 374 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} \\ 367 375 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} \\ 368 376 \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} \\ 369 377 \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} \\ 370 378 \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} 371 379 \pi_1^- &= \Pi^I \label{eq:init}
\end{align} 372 380 \end{align}
373 381
Equation~\ref{eq:area} states that the total area taken by the filters must be 374 382 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 375 383 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 376 384 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 377 385 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 378 386 $\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the
input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the 379 387 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 380 388 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, 381 389 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 382 390 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 383 391 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 384 392 $\pi_i^C$ 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 385 393 $\pi_i^S$ 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. 386 394 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 387 395 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 388 396 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 389 397 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 390 398 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 391 399 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 392 400 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: 393 401 shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is:
$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. 394 402 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
Finally, equation~\ref{eq:init} gives the number of bits of the global input. 395 403 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
396 404
This model is non-linear and even non-quadratic, as $F$ does not have a known 397 405 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 398 406 linear or quadratic expression. We introduce $p$ FIR configurations
$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 399 407 $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
% r2.12 400 408 % r2.12
This variable must be defined by the user, it represent the number of different 401 409 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} 402 410 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave). 403 411 functions from GNU Octave).
We define binary 404 412 We define binary
variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 405 413 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
and 0 otherwise. The new equations are as follows: 406 414 and 0 otherwise. The new equations are as follows:
407 415
\begin{align} 408 416 \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} \\ 409 417 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} \\ 410 418 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} \\ 411 419 \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} 412 420 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
\end{align} 413 421 \end{align}
414 422
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 415 423 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 416 424 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. 417 425 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
418 426
% r2.13 419 427 % r2.13
This modified model is quadratic since we multiply two variables in the 420 428 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. 421 429 equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
The Gurobi 422 430 The Gurobi
(\url{www.gurobi.com}) optimization software is used to solve this quadratic 423 431 (\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 424 432 model, and since Gurobi is able to linearize, the model is left as is. This model
has $O(np)$ variables and $O(n)$ constraints. 425 433 has $O(np)$ variables and $O(n)$ constraints.
426 434
Two problems will be addressed using the workflow described in the next section: on the one 427 435 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 428 436 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 429 437 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 430 438 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
objective function is replaced with: 431 439 objective function is replaced with:
\begin{align} 432 440 \begin{align}
\text{Minimize } & \sum_{i=1}^n a_i \notag 433 441 \text{Minimize } & \sum_{i=1}^n a_i \notag
\end{align} 434 442 \end{align}
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 435 443 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 436 444 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
rejection required. 437 445 rejection required.
438 446
\begin{align} 439 447 \begin{align}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 440 448 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
\end{align} 441 449 \end{align}
442 450
\section{Design workflow} 443 451 \section{Design workflow}
\label{sec:workflow} 444 452 \label{sec:workflow}
445 453
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 446 454 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 447 455 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
in the computation of the results. 448 456 in the computation of the results.
449 457
\begin{figure} 450 458 \begin{figure}
\centering 451 459 \centering
\begin{tikzpicture}[node distance=0.75cm and 2cm] 452 460 \begin{tikzpicture}[node distance=0.75cm and 2cm]
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 453 461 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\node (Start) [left= 3cm of Solver] { } ; 454 462 \node (Start) [left= 3cm of Solver] { } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 455 463 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\node (Input) [above= of TCL] { } ; 456 464 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 457 465 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 458 466 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 459 467 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 460 468 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node (Results) [left= of Postproc] { } ; 461 469 \node (Results) [left= of Postproc] { } ;
462 470
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 463 471 \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) ; 464 472 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 465 473 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 466 474 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 467 475 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 468 476 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 469 477 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 470 478 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 471 479 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->] (Postproc) -- (Results) ; 472 480 \draw[->] (Postproc) -- (Results) ;
\end{tikzpicture} 473 481 \end{tikzpicture}
\caption{Design workflow from the input parameters to the results} 474 482 \caption{Design workflow from the input parameters to the results}
\label{fig:workflow} 475 483 \label{fig:workflow}
\end{figure} 476 484 \end{figure}
477 485
The filter solver is a C++ program that takes as input the maximum area 478 486 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 $\Pi^I$, 479 487 $\mathcal{A}$, 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 480 488 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. 481 489 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}) 482 490 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
and a deploy script ((1b) on figure~\ref{fig:workflow}). 483 491 and a deploy script ((1b) on figure~\ref{fig:workflow}).
484 492
The TCL script describes the whole digital processing chain from the beginning 485 493 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 486 494 (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 487 495 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) 488 496 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. 489 497 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 490 498 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 491 499 comes from a skeleton library. The generic FIR is highly configurable
with the number of coefficients and the size of the coefficients. The coefficients 492 500 with the number of coefficients and the size of the coefficients. The coefficients
themselves are not stored in the script. 493 501 themselves are not stored in the script.
As the signal is processed in real-time, the output signal is stored as 494 502 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 495 503 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 496 504 implemented FIR cascade transfer function with the design criteria and the expected
transfer function. 497 505 transfer function.
498 506
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 499 507 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 500 508 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 501 509 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 502 510 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
provide a broadband noise source. 503 511 provide a broadband noise source.
The board runs the Linux kernel and surrounding environment produced from the 504 512 The board runs the Linux kernel and surrounding environment produced from the
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 505 513 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 506 514 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
fetching the results is automated. 507 515 fetching the results is automated.
508 516
The deploy script uploads the bitstream to the board ((3) on 509 517 The deploy script uploads the bitstream to the board ((3) on
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 510 518 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
configures the coefficients of the FIR filters. It then waits for the results 511 519 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}). 512 520 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
513 521
Finally, an Octave post-processing script computes the final results thanks to 514 522 Finally, an Octave post-processing script computes the final results thanks to
the output data ((5) on figure~\ref{fig:workflow}). 515 523 the output data ((5) on figure~\ref{fig:workflow}).
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 516 524 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
and the different configurations can be compared. 517 525 and the different configurations can be compared.
518 526
\section{Maximizing the rejection at fixed silicon area} 519 527 \section{Maximizing the rejection at fixed silicon area}
\label{sec:fixed_area} 520 528 \label{sec:fixed_area}
This section presents the output of the filter solver {\em i.e.} the computed 521 529 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. 522 530 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. 523 531 Such results allow for understanding the choices made by the solver to compute its solutions.
524 532
The experimental setup is composed of three cases. The raw input is generated 525 533 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 $\Pi^I$. 526 534 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 527 535 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. 528 536 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 $\pi^C$ 529 537 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 530 538 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. 531 539 result up to five stages ($n = 5$) in the cascaded filter.
532 540
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 533 541 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. 534 542 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. 535 543 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
536 544
\renewcommand{\arraystretch}{1.4} 537 545 \renewcommand{\arraystretch}{1.4}
538 546
\begin{table} 539 547 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 540 548 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
\label{tbl:gurobi_max_500} 541 549 \label{tbl:gurobi_max_500}
\centering 542 550 \centering
{\scalefont{0.77} 543 551 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 544 552 \begin{tabular}{|c|ccccc|c|c|}
\hline 545 553 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 546 554 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 547 555 \hline
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 548 556 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ 549 557 2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\
3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ 550 558 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 \\ 551 559 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 \\ 552 560 5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\
\hline 553 561 \hline
\end{tabular} 554 562 \end{tabular}
} 555 563 }
\end{table} 556 564 \end{table}
557 565
\begin{table} 558 566 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 559 567 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
\label{tbl:gurobi_max_1000} 560 568 \label{tbl:gurobi_max_1000}
\centering 561 569 \centering
{\scalefont{0.77} 562 570 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 563 571 \begin{tabular}{|c|ccccc|c|c|}
\hline 564 572 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 565 573 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 566 574 \hline
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 567 575 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ 568 576 2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\
3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ 569 577 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 \\ 570 578 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 \\ 571 579 5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\
\hline 572 580 \hline
\end{tabular} 573 581 \end{tabular}
} 574 582 }
\end{table} 575 583 \end{table}
576 584
\begin{table} 577 585 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 578 586 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
\label{tbl:gurobi_max_1500} 579 587 \label{tbl:gurobi_max_1500}
\centering 580 588 \centering
{\scalefont{0.77} 581 589 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 582 590 \begin{tabular}{|c|ccccc|c|c|}
\hline 583 591 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 584 592 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 585 593 \hline
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 586 594 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ 587 595 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\
3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ 588 596 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 \\ 589 597 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 \\ 590 598 5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\
\hline 591 599 \hline
\end{tabular} 592 600 \end{tabular}
} 593 601 }
\end{table} 594 602 \end{table}
595 603
\renewcommand{\arraystretch}{1} 596 604 \renewcommand{\arraystretch}{1}
597 605
From these tables, we can first state that the more stages are used to define 598 606 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 599 607 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 600 608 been previously observed that many small filters are better than
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 601 609 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 602 610 being hardly used in practice due to the lack of tools for identifying individual filter
coefficients in the cascaded approach. 603 611 coefficients in the cascaded approach.
604 612
Second, the larger the silicon area, the better the rejection. This was also an 605 613 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 606 614 expected result as more area means a filter of better quality with more coefficients
or more bits per coefficient. 607 615 or more bits per coefficient.
608 616
Then, we also observe that the first stage can have a larger shift than the other 609 617 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 610 618 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 611 619 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} 612 620 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
gives the relation between both values. 613 621 gives the relation between both values.
614 622
Finally, we note that the solver consumes all the given silicon area. 615 623 Finally, we note that the solver consumes all the given silicon area.
616 624
The following graphs present the rejection for real data on the FPGA. In all the following 617 625 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 618 626 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 619 627 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. 620 628 given by the quadratic solver. The configurations are those computed in the previous section.
621 629
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 622 630 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. 623 631 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. 624 632 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
625 633
% \begin{figure} 626 634 % \begin{figure}
% \centering 627 635 % \centering
% \includegraphics[width=\linewidth]{images/max_500} 628 636 % \includegraphics[width=\linewidth]{images/max_500}
% \caption{Signal spectrum for MAX/500} 629 637 % \caption{Signal spectrum for MAX/500}
% \label{fig:max_500_result} 630 638 % \label{fig:max_500_result}
% \end{figure} 631 639 % \end{figure}
% 632 640 %
% \begin{figure} 633 641 % \begin{figure}
% \centering 634 642 % \centering
% \includegraphics[width=\linewidth]{images/max_1000} 635 643 % \includegraphics[width=\linewidth]{images/max_1000}
% \caption{Signal spectrum for MAX/1000} 636 644 % \caption{Signal spectrum for MAX/1000}
% \label{fig:max_1000_result} 637 645 % \label{fig:max_1000_result}
% \end{figure} 638 646 % \end{figure}
% 639 647 %
% \begin{figure} 640 648 % \begin{figure}
% \centering 641 649 % \centering
% \includegraphics[width=\linewidth]{images/max_1500} 642 650 % \includegraphics[width=\linewidth]{images/max_1500}
% \caption{Signal spectrum for MAX/1500} 643 651 % \caption{Signal spectrum for MAX/1500}
% \label{fig:max_1500_result} 644 652 % \label{fig:max_1500_result}
% \end{figure} 645 653 % \end{figure}
646 654
% r2.14 et r2.15 et r2.16 647 655 % r2.14 et r2.15 et r2.16
\begin{figure} 648 656 \begin{figure}
\centering 649 657 \centering
\begin{subfigure}{\linewidth} 650 658 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_500} 651 659 \includegraphics[width=\linewidth]{images/max_500}
\caption{Signal spectrum for MAX/500} 652 660 \caption{Signal spectrum for MAX/500}
\label{fig:max_500_result} 653 661 \label{fig:max_500_result}
\end{subfigure} 654 662 \end{subfigure}
655 663
\begin{subfigure}{\linewidth} 656 664 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1000} 657 665 \includegraphics[width=\linewidth]{images/max_1000}
\caption{Signal spectrum for MAX/1000} 658 666 \caption{Signal spectrum for MAX/1000}
\label{fig:max_1000_result} 659 667 \label{fig:max_1000_result}
\end{subfigure} 660 668 \end{subfigure}
661 669
\begin{subfigure}{\linewidth} 662 670 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1500} 663 671 \includegraphics[width=\linewidth]{images/max_1500}
\caption{Signal spectrum for MAX/1500} 664 672 \caption{Signal spectrum for MAX/1500}
\label{fig:max_1500_result} 665 673 \label{fig:max_1500_result}
\end{subfigure} 666 674 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500} 667 675 \caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500}
\end{figure} 668 676 \end{figure}
669 677
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 670 678 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
671 679
We compare the actual silicon resources given by Vivado to the 672 680 We compare the actual silicon resources given by Vivado to the
resources in arbitrary units. 673 681 resources in arbitrary units.
The goal is to check that our arbitrary units of silicon area models well enough 674 682 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 675 683 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 676 684 number of arbitrary units, the actual silicon resources do not depend on the
number of stages $n$. Most significantly, our approach aims 677 685 number of stages $n$. Most significantly, our approach aims
at remaining far enough from the practical logic gate implementation used by 678 686 at remaining far enough from the practical logic gate implementation used by
various vendors to remain platform independent and be portable from one 679 687 various vendors to remain platform independent and be portable from one
architecture to another. 680 688 architecture to another.
681 689
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 682 690 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 683 691 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 684 692 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 685 693 the FIR filters and remove additional processing blocks including FIFO and Programmable
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 686 694 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
687 695
\begin{table}[h!tb] 688 696 \begin{table}[h!tb]
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 689 697 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage} 690 698 \label{tbl:resources_usage}
\centering 691 699 \centering
\begin{tabular}{|c|c|ccc|c|} 692 700 \begin{tabular}{|c|c|ccc|c|}
\hline 693 701 \hline
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 694 702 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 249 & 453 & 627 & \emph{17600} \\ 695 703 & LUT & 249 & 453 & 627 & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 696 704 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 697 705 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
& LUT & 2374 & 5494 & 691 & \emph{17600} \\ 698 706 & LUT & 2374 & 5494 & 691 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 699 707 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 70 & \emph{80} \\ \hline 700 708 & DSP & 0 & 0 & 70 & \emph{80} \\ \hline
& LUT & 2443 & 3304 & 3521 & \emph{17600} \\ 701 709 & LUT & 2443 & 3304 & 3521 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 702 710 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 19 & 35 & \emph{80} \\ \hline 703 711 & DSP & 0 & 19 & 35 & \emph{80} \\ \hline
& LUT & 2634 & 3753 & 2557 & \emph{17600} \\ 704 712 & LUT & 2634 & 3753 & 2557 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & \emph{120} \\ 705 713 4 & BRAM & 4 & 4 & 4 & \emph{120} \\
& DPS & 0 & 19 & 46 & \emph{80} \\ \hline 706 714 & DPS & 0 & 19 & 46 & \emph{80} \\ \hline
& LUT & 2423 & 3047 & 2847 & \emph{17600} \\ 707 715 & LUT & 2423 & 3047 & 2847 & \emph{17600} \\
5 & BRAM & 5 & 5 & 5 & \emph{120} \\ 708 716 5 & BRAM & 5 & 5 & 5 & \emph{120} \\
& DPS & 0 & 22 & 46 & \emph{80} \\ \hline 709 717 & DPS & 0 & 22 & 46 & \emph{80} \\ \hline
\end{tabular} 710 718 \end{tabular}
\end{table} 711 719 \end{table}
712 720
In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 713 721 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 714 722 when the filter coefficients are small enough, or when the input size is small
enough, Vivado optimizes resource consumption by selecting multiplexers to 715 723 enough, Vivado optimizes resource consumption by selecting multiplexers to
implement the multiplications instead of a DSP. In this case, it is quite difficult 716 724 implement the multiplications instead of a DSP. In this case, it is quite difficult
to compare the whole silicon budget. 717 725 to compare the whole silicon budget.
718 726
However, a rough estimation can be made with a simple equivalence: looking at 719 727 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 $n \geq 2$, 720 728 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 721 729 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, 722 730 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 723 731 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 724 732 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 725 733 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. 726 734 by the optimizations done by Vivado based on the detailed map of available processing resources.
727 735
We now present the computation time needed to solve the quadratic problem. 728 736 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 729 737 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 730 738 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 731 739 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. 732 740 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
733 741
\begin{table}[h!tb] 734 742 \begin{table}[h!tb]
\caption{Time needed to solve the quadratic program with Gurobi} 735 743 \caption{Time needed to solve the quadratic program with Gurobi}
\label{tbl:area_time} 736 744 \label{tbl:area_time}
\centering 737 745 \centering
\begin{tabular}{|c|c|c|c|}\hline 738 746 \begin{tabular}{|c|c|c|c|}\hline
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 739 747 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
1 & 0.1~s & 0.1~s & 0.3~s \\ 740 748 1 & 0.1~s & 0.1~s & 0.3~s \\
2 & 1.1~s & 2.2~s & 12~s \\ 741 749 2 & 1.1~s & 2.2~s & 12~s \\
3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ 742 750 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) \\ 743 751 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 744 752 5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline
\end{tabular} 745 753 \end{tabular}
\end{table} 746 754 \end{table}
747 755
As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? 748 756 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 749 757 When the area is limited, the design exploration space is more limited and the solver is able to
find an optimal solution faster. 750 758 find an optimal solution faster.
751 759
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 752 760 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
753 761
This section presents the results of the complementary quadratic program aimed at 754 762 This section presents the results of the complementary quadratic program aimed at
minimizing the area occupation for a targeted rejection level. 755 763 minimizing the area occupation for a targeted rejection level.
756 764
The experimental setup is composed of four cases. The raw input is the same 757 765 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 $\Pi^I$. 758 766 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. 759 767 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. 760 768 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. 761 769 The number of configurations $p$ is the same as previous section.
762 770
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 763 771 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. 764 772 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. 765 773 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. 766 774 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
767 775
\renewcommand{\arraystretch}{1.4} 768 776 \renewcommand{\arraystretch}{1.4}
769 777
\begin{table}[h!tb] 770 778 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 771 779 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\label{tbl:gurobi_min_40} 772 780 \label{tbl:gurobi_min_40}
\centering 773 781 \centering
{\scalefont{0.77} 774 782 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 775 783 \begin{tabular}{|c|ccccc|c|c|}
\hline 776 784 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 777 785 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 778 786 \hline
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 779 787 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ 780 788 2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\
3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ 781 789 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 \\ 782 790 4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\
\hline 783 791 \hline
\end{tabular} 784 792 \end{tabular}
} 785 793 }
\end{table} 786 794 \end{table}
787 795
\begin{table}[h!tb] 788 796 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 789 797 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\label{tbl:gurobi_min_60} 790 798 \label{tbl:gurobi_min_60}
\centering 791 799 \centering
{\scalefont{0.77} 792 800 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 793 801 \begin{tabular}{|c|ccccc|c|c|}
\hline 794 802 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 795 803 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 796 804 \hline
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 797 805 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ 798 806 2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\
3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ 799 807 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 \\ 800 808 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 \\ 801 809 5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\
\hline 802 810 \hline
\end{tabular} 803 811 \end{tabular}
} 804 812 }
\end{table} 805 813 \end{table}
806 814
\begin{table}[h!tb] 807 815 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 808 816 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\label{tbl:gurobi_min_80} 809 817 \label{tbl:gurobi_min_80}
\centering 810 818 \centering
{\scalefont{0.77} 811 819 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 812 820 \begin{tabular}{|c|ccccc|c|c|}
\hline 813 821 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 814 822 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 815 823 \hline
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 816 824 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ 817 825 2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\
3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ 818 826 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 \\ 819 827 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 \\ 820 828 5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\
\hline 821 829 \hline
\end{tabular} 822 830 \end{tabular}
} 823 831 }
\end{table} 824 832 \end{table}
825 833
\begin{table}[h!tb] 826 834 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 827 835 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\label{tbl:gurobi_min_100} 828 836 \label{tbl:gurobi_min_100}
\centering 829 837 \centering
{\scalefont{0.77} 830 838 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 831 839 \begin{tabular}{|c|ccccc|c|c|}
\hline 832 840 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 833 841 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 834 842 \hline
1 & - & - & - & - & - & - & - \\ 835 843 1 & - & - & - & - & - & - & - \\
2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\ 836 844 2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\
3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\ 837 845 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 \\ 838 846 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 \\ 839 847 5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\
\hline 840 848 \hline
\end{tabular} 841 849 \end{tabular}
} 842 850 }
\end{table} 843 851 \end{table}
\renewcommand{\arraystretch}{1} 844 852 \renewcommand{\arraystretch}{1}
845 853
From these tables, we can first state that almost all configurations reach the targeted rejection 846 854 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 847 855 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 848 856 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. 849 857 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 850 858 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 851 859 (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. 852 860 respectively). More generally, the more filters are cascaded, the lower the occupied area.
853 861
Like in previous section, the solver chooses always a little filter as first 854 862 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 855 863 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 856 864 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 857 865 signal and in the second filter selecting a better filter to improve rejection without
having too many bits in the output data. 858 866 having too many bits in the output data.
859 867
For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 860 868 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 861 869 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 $n = 4$. 862 870 solution is equivalent to the result for $n = 4$.
863 871
The following graphs present the rejection for real data on the FPGA. In all the following 864 872 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 865 873 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 866 874 data on the FPGA as measured experimentally and the dashed line is the noise level
given by the quadratic solver. 867 875 given by the quadratic solver.
868 876
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 869 877 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. 870 878 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. 871 879 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. 872 880 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
873 881
% \begin{figure} 874 882 % \begin{figure}
% \centering 875 883 % \centering
% \includegraphics[width=\linewidth]{images/min_40} 876 884 % \includegraphics[width=\linewidth]{images/min_40}
% \caption{Signal spectrum for MIN/40} 877 885 % \caption{Signal spectrum for MIN/40}
% \label{fig:min_40} 878 886 % \label{fig:min_40}
% \end{figure} 879 887 % \end{figure}
% 880 888 %
% \begin{figure} 881 889 % \begin{figure}
% \centering 882 890 % \centering
% \includegraphics[width=\linewidth]{images/min_60} 883 891 % \includegraphics[width=\linewidth]{images/min_60}
% \caption{Signal spectrum for MIN/60} 884 892 % \caption{Signal spectrum for MIN/60}
% \label{fig:min_60} 885 893 % \label{fig:min_60}
% \end{figure} 886 894 % \end{figure}
% 887 895 %
% \begin{figure} 888 896 % \begin{figure}
% \centering 889 897 % \centering
% \includegraphics[width=\linewidth]{images/min_80} 890 898 % \includegraphics[width=\linewidth]{images/min_80}
% \caption{Signal spectrum for MIN/80} 891 899 % \caption{Signal spectrum for MIN/80}
% \label{fig:min_80} 892 900 % \label{fig:min_80}
% \end{figure} 893 901 % \end{figure}
% 894 902 %
% \begin{figure} 895 903 % \begin{figure}
% \centering 896 904 % \centering
% \includegraphics[width=\linewidth]{images/min_100} 897 905 % \includegraphics[width=\linewidth]{images/min_100}
% \caption{Signal spectrum for MIN/100} 898 906 % \caption{Signal spectrum for MIN/100}
% \label{fig:min_100} 899 907 % \label{fig:min_100}
% \end{figure} 900 908 % \end{figure}
901 909
% r2.14 et r2.15 et r2.16 902 910 % r2.14 et r2.15 et r2.16
\begin{figure} 903 911 \begin{figure}
\centering 904 912 \centering
\begin{subfigure}{\linewidth} 905 913 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_40} 906 914 \includegraphics[width=\linewidth]{images/min_40}
\caption{Signal spectrum for MIN/40} 907 915 \caption{Signal spectrum for MIN/40}
\label{fig:min_40} 908 916 \label{fig:min_40}
\end{subfigure} 909 917 \end{subfigure}
910 918
\begin{subfigure}{\linewidth} 911 919 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_60} 912 920 \includegraphics[width=\linewidth]{images/min_60}
\caption{Signal spectrum for MIN/60} 913 921 \caption{Signal spectrum for MIN/60}
\label{fig:min_60} 914 922 \label{fig:min_60}
\end{subfigure} 915 923 \end{subfigure}
916 924
\begin{subfigure}{\linewidth} 917 925 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_80} 918 926 \includegraphics[width=\linewidth]{images/min_80}
\caption{Signal spectrum for MIN/80} 919 927 \caption{Signal spectrum for MIN/80}
\label{fig:min_80} 920 928 \label{fig:min_80}
\end{subfigure} 921 929 \end{subfigure}
922 930
\begin{subfigure}{\linewidth} 923 931 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_100} 924 932 \includegraphics[width=\linewidth]{images/min_100}
\caption{Signal spectrum for MIN/100} 925 933 \caption{Signal spectrum for MIN/100}
\label{fig:min_100} 926 934 \label{fig:min_100}
\end{subfigure} 927 935 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100} 928 936 \caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100}
\end{figure} 929 937 \end{figure}
ifcs2018_journal_reponse.tex
Diff comments   View file @ 90c5584
%Minor Revision - TUFFC-09469-2019 1 1 %Minor Revision - TUFFC-09469-2019
%Transactions on Ultrasonics, Ferroelectrics, and Frequency 2 2 %Transactions on Ultrasonics, Ferroelectrics, and Frequency
%Control (July 23, 2019 9:29 PM) 3 3 %Control (July 23, 2019 9:29 PM)
%To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr, 4 4 %To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr,
%gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr, 5 5 %gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr,
%jmfriedt@femto-st.fr 6 6 %jmfriedt@femto-st.fr
%CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu 7 7 %CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu
% 8 8 %
%Dear Mr. Arthur HUGEAT 9 9 %Dear Mr. Arthur HUGEAT
% 10 10 %
%Congratulations! Your manuscript 11 11 %Congratulations! Your manuscript
% 12 12 %
%MANUSCRIPT NO. TUFFC-09469-2019 13 13 %MANUSCRIPT NO. TUFFC-09469-2019
%MANUSCRIPT TYPE: Papers 14 14 %MANUSCRIPT TYPE: Papers
%TITLE: Filter optimization for real time digital processing of radiofrequency 15 15 %TITLE: Filter optimization for real time digital processing of radiofrequency
%signals: application to oscillator metrology 16 16 %signals: application to oscillator metrology
%AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, 17 17 %AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois,
%Pierre-Yves; Friedt, Jean-Michel 18 18 %Pierre-Yves; Friedt, Jean-Michel
% 19 19 %
%has been reviewed and it has been suggested that it be accepted for publication 20 20 %has been reviewed and it has been suggested that it be accepted for publication
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%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
88 The sentence was split and now reads ``number of coefficients irrelevant: processing
89 resources are hence saved by shrinking the filter length.''
90
91 {\bf
On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 87 92 On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2
the author meant. 88 93 the author meant.}
89 94
One page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 90 95 Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s)
96 Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
97
98 {\bf
99 On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3
could be improved. 91 100 could be improved.
} 92 101 }
93 102
103 Indeed this paragraph has be written again and now reads as\\
104 ``The first step of our approach is to model the DSP chain. Since we aim at only optimizing
105 the filtering part of the signal processing chain, we have not included the PRN generator or the
106 ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
107 The filtering can be done in two ways, either by considering a single monolithic FIR filter
108 requiring many coefficients to reach the targeted noise rejection ratio, or by
109 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.
110 ''
111
{\bf 94 112 {\bf
I appreciate that the authors attempted and document two optimizations: that % r1.4 - en attente des résultats 95 113 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 96 114 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 97 115 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 98 116 useful to compare the results of both optimization paths to the performance and
resource-utilization of generic low-pass filter gateware offered by device 99 117 resource-utilization of generic low-pass filter gateware offered by device
manufacturers. I appreciate also that the authors have presented source code 100 118 manufacturers. I appreciate also that the authors have presented source code
for examination online. 101 119 for examination online.
} 102 120 }
103 121
TODO : FIR Compiler et regarder les ressources pour un FIR comparable a ceux monolithiques 104 122 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) 105 123 fournis dans l'article (memes coefs et meme nombre de coefs)
106 124
{\bf 107 125 {\bf
Reviewer: 2 108 126 Reviewer: 2
} 109 127 }
110 128
%Comments to the Author 111 129 %Comments to the Author
%In the Manuscript, the Authors describe an optimization methodology for filter 112 130 %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 113 131 %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 114 132 %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, 115 133 %focus on filtering and shifting while the other aspects, in particular decimation,
%will be considered in a future work. The optimization problem is modelled 116 134 %will be considered in a future work. The optimization problem is modelled
%theoretically and then solved by means of a commercial software. The solutions 117 135 %theoretically and then solved by means of a commercial software. The solutions
%are tested experimentally on the Redpitaya platform with synthetic and real 118 136 %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: 119 137 %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 120 138 %maximum rejection given a fixed amount of resources and minimum resource
%utilization given a fixed amount of rejection. 121 139 %utilization given a fixed amount of rejection.
%The Authors find that filtering improves significantly when the number of 122 140 %The Authors find that filtering improves significantly when the number of
%filters increases. 123 141 %filters increases.
%A lot of work has been done in generalizing and automating the procedure so 124 142 %A lot of work has been done in generalizing and automating the procedure so
%that different approaches can be investigated quickly and efficiently. The 125 143 %that different approaches can be investigated quickly and efficiently. The
%results presented in the Manuscript seem to be just a case study based on 126 144 %results presented in the Manuscript seem to be just a case study based on
%the particular criterion chosen by the Authors. Different criteria, in 127 145 %the particular criterion chosen by the Authors. Different criteria, in
%general, could lead to different results and it is important to consider 128 146 %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 129 147 %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 130 148 %is adequate to compare the performance of filters and if multi-stage
%filters are really superior than monolithic filters. 131 149 %filters are really superior than monolithic filters.
132 150
{\bf 133 151 {\bf
By observing the results presented in fig. 10-16, it is clear that the % r2.1 - fait 134 152 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 135 153 performances of multi-stage filters are obtained at the expense of their
selectivity and, in this sense, the filters presented in these figures 136 154 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, 137 155 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 138 156 the attenuation is almost 15 dB for n = 5, while it is not noticeable for
n = 1. 139 157 n = 1.
} 140 158 }
141 159
TODO : ajouter les gabarits 142 160 TODO : ajouter les gabarits
143 161
Peut etre refaire une serie de simulation dans lesquelles on impose une coupure 144 162 Peut etre refaire une serie de simulation dans lesquelles on impose une coupure
non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte 145 163 non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte
au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque 146 164 au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque
a la cascade de filtres. 147 165 a la cascade de filtres.
AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de 148 166 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 149 167 relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai
150 168
{\bf 151 169 {\bf
The reason is in the criterion that considers the average attenuation in % r2.2 - fait 152 170 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 153 171 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 154 172 in this region, which is a very important parameter for specifying a filter
and for evaluating its performance. For example, with this criterion, a 155 173 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 156 174 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 157 175 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. 158 176 and in the results that are obtained and has to be reconsidered.
} 159 177 }
160 178
Je ne pense pas que ca soit le cas : la somme des valeurs absolues des pertes 161 179 Je ne pense pas que ca soit le cas : la somme des valeurs absolues des pertes
dans la bande va defavoriser un filtre avec 10 dB de ripples. Il n'a pas compris que 162 180 dans la bande va defavoriser un filtre avec 10 dB de ripples. Il n'a pas compris que
la bandpass s'arrete a 40\% de la bande, donc mettre le gabarit clarifierait ce point je 163 181 la bandpass s'arrete a 40\% de la bande, donc mettre le gabarit clarifierait ce point je
pense 164 182 pense
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 ? 165 183 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 ?
166 184
{\bf 167 185 {\bf
I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait 168 186 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 169 187 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 170 188 fixing its value to a typical one, as it has been done for the transition
bandwidth. 171 189 bandwidth.
} 172 190 }
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 ? 173 191 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 ?
174 192
{\bf 175 193 {\bf
In addition, I suggest to address the following points: % r2.4 176 194 In addition, I suggest to address the following points: % r2.4
- Page 1, line 50: the Authors state that IIR have shorter impulse response 177 195 - 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. 178 196 than FIR. This is not true in general. The sentence should be reconsidered.
} 179 197 }
180 198
J'aurais du dire ``lag'' au lieu de ``impulse response'' je pense 181 199 We have not stated that the IIR has a shorter impulse response but a shorter lag.
AH: Je ne comprends pas trop ce qui ne va pas ici 182 200 Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filters
201 have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing
202 the first output, an IIR will start providing outputs only 5 time steps after the initial
203 input starts feeding the IIR. Hence, the issue we address here is lag and not impulse
204 response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue
205 in a openloop phase noise characterization instrument, the large
206 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
207 is not considered as an issue as would be in a closed loop system in which lag aims at being
208 minimized to avoid oscillation conditions.
209 ''
183 210
{\bf 184 211 {\bf
- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 185 212 - 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 186 213 this transition bandwidth and if it is a typical requirement for phase-noise
metrology. 187 214 metrology.
} 188 215 }
AH: Je ne sais pas comment justifier ça. Je dois dire que comme ça on peut éventuellement 189
décimer par deux le flux ? 190
191 216
217 The purpose of the paper is to demonstrate how a given filter shape can be achieved by
218 minimizing varous resource criteria. Indeed the stopband and bandpass boundaries can
219 be questioned: we have selected this filter shape as a typical anti-aliasing filter considering
220 the the dataflow is to be halved. Hence, selecting a cutoff frequency of 40\% the initial
221 Nyquist frequency prevents noise from reaching baseband after decimating the dataflow by a
222 factor of 2. Such ideas are now stated explicitly in the text as ``Throughout this demonstration,
223 we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%
224 of the Nyquist frequency to the end of the band, as would be typically selected to prevent
225 aliasing before decimating the dataflow by 2. The method is however generalized to any filter
226 shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid}
227 as described below is indeed unique for each filter shape.''
228
{\bf 192 229 {\bf
- The impact of the coefficient resolution is discussed. What about the % r2.6 - fait 193 230 - 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 194 231 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 195 232 used in the analysis? If not, how is it changed with respect to the
coefficient resolution? 196 233 coefficient resolution?
} 197 234 }
198 235
Pr\'eciser que le flux de donn\'ees en entr\'ees est de r\'esolution fixe 199 236 We have now stated in the beginning of the document that ``we have not included the PRN generator
237 or the ADC in the model: the input data size and rate are considered fixed and defined by the
238 hardware.'' so indeed the input datastream resolution is considered as a given.
200 239
{\bf 201 240 {\bf
- Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait 202 241 - Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait
Fig. 5 can be removed. 203 242 Fig. 5 can be removed.
- Page 3, line 55: “maximum rejection” is not compatible with fig. 4. % r2.8 - fait 204 243 - Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait
It should be “minimum” 205 244 It should be ``minimum''
} 206 245 }
AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max 207 246 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. 208 247 de la magnitude. J'ai corrigé en ce sens.
209 248
{\bf 210 249 {\bf
- Page e, line 55, second column: “takin” % r2.9 - fait 211 250 - Page e, line 55, second column: “takin” % r2.9 - fait
- Page 3, line 58: “pessimistic” should be replaced with “conservative” % r2.10 - fait 212 251 - Page 3, line 58: “pessimistic” should be replaced with “conservative” % r2.10 - fait
- Page 4, line 17: “meaning” --> “this means” % r2.11 - fait 213 252 - Page 4, line 17: “meaning” --> “this means” % r2.11 - fait
- Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait 214 253 - 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? 215 254 these particular configurations? Are they chosen automatically?
- Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait 216 255 - Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait
and non-quadratic to a quadratic? 217 256 and non-quadratic to a quadratic?}
257
258 JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
259 je ne comprends pas comment ca repond a la question
260
261 {\bf
- Captions of figure and tables are too minimal. % r2.14 218 262 - 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 219 263 - 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. 220 264 of a single figure. Same for fig. 13-16.
} 221 265 }
222 266
{\bf 223 267 {\bf
- Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait 224 268 - 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 225 269 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. 226 270 differences among the curves. I suggest to reduce the noise below 1 dBpk-pk.
} 227 271 }
228 272
Comment as tu fait tes spectres Arthur ? Si tu as fait une FFT sur e.g. 2048 points 229 273 Indeed averaging had been omitted during post-processing and figure generation: we