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% fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee 1 1 % fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee
% demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de 2 2 % demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de
% rejection par bit et perte si moins de bits que rejection/6 3 3 % rejection par bit et perte si moins de bits que rejection/6
% developper programme lineaire en incluant le decalage de bits 4 4 % developper programme lineaire en incluant le decalage de bits
% insister que avant on etait synthetisable mais pas implementable, alors que maintenant on 5 5 % insister que avant on etait synthetisable mais pas implementable, alors que maintenant on
% implemente et on demontre que ca tourne 6 6 % implemente et on demontre que ca tourne
% gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ? 7 7 % gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ?
% Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer 8 8 % Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer
% (zedboard ou redpit) 9 9 % (zedboard ou redpit)
10 10
% label schema : verifier que "argumenter de la cascade de FIR" est fait 11 11 % label schema : verifier que "argumenter de la cascade de FIR" est fait
12 12
\documentclass[a4paper,journal]{IEEEtran/IEEEtran} 13 13 \documentclass[a4paper,journal]{IEEEtran/IEEEtran}
\usepackage{graphicx,color,hyperref} 14 14 \usepackage{graphicx,color,hyperref}
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\usepackage[normalem]{ulem} 21 21 \usepackage[normalem]{ulem}
\usepackage{tikz} 22 22 \usepackage{tikz}
\usetikzlibrary{positioning,fit} 23 23 \usetikzlibrary{positioning,fit}
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\usepackage{caption} 26 26 \usepackage{caption}
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% correct bad hyphenation here 29 29 % correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor} 30 30 \hyphenation{op-tical net-works semi-conduc-tor}
\textheight=26cm 31 31 \textheight=26cm
\setlength{\footskip}{30pt} 32 32 \setlength{\footskip}{30pt}
\pagenumbering{gobble} 33 33 \pagenumbering{gobble}
\begin{document} 34 34 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 35 35 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 36 36 to oscillator metrology}
37 37
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 38 38 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 39 39 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\ 40 40 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\ 41 41 \IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\
\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\ 42 42 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 43 43 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 44 44 }
\maketitle 45 45 \maketitle
\thispagestyle{plain} 46 46 \thispagestyle{plain}
\pagestyle{plain} 47 47 \pagestyle{plain}
\newtheorem{definition}{Definition} 48 48 \newtheorem{definition}{Definition}
49 49
\begin{abstract} 50 50 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 51 51 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 52 52 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 53 53 of ultrastable clocks, stringent filtering requirements are defined by spurious signal or
noise rejection needs. Since real time radiofrequency processing must be performed in a 54 54 noise rejection needs. Since real time radiofrequency processing must be performed in a
Field Programmable Array to meet timing constraints, we investigate optimization strategies 55 55 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 56 56 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. The 57 57 required and keeping timing constraints within the targeted measurement bandwidths. The
presented technique is applicable to scheduling any sequence of processing blocks characterized 58 58 presented technique is applicable to scheduling any sequence of processing blocks characterized
by a throughput, resource occupation and performance tabulated as a function of configuration 59 59 by a throughput, resource occupation and performance tabulated as a function of configuration
characateristics, as is the case for filters with their coefficients and resolution yielding 60 60 characateristics, as is the case for filters with their coefficients and resolution yielding
rejection and number of multipliers. 61 61 rejection and number of multipliers.
\end{abstract} 62 62 \end{abstract}
63 63
\begin{IEEEkeywords} 64 64 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 65 65 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 66 66 \end{IEEEkeywords}
67 67
\section{Digital signal processing of ultrastable clock signals} 68 68 \section{Digital signal processing of ultrastable clock signals}
69 69
Analog oscillator phase noise characteristics are classically performed by downconverting 70 70 Analog oscillator phase noise characteristics are classically performed by downconverting
the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband, 71 71 the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,
followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In 72 72 followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In
a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by 73 73 a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by
multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}. 74 74 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
75 75
\begin{figure}[h!tb] 76 76 \begin{figure}[h!tb]
\begin{center} 77 77 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 78 78 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 79 79 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 80 80 \caption{Fully digital oscillator phase noise characterization: the Device Under Test
(DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and 81 81 (DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and
downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals 82 82 downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals
and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite 83 83 and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite
Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays 84 84 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 85 85 the spectral characteristics of the phase fluctuations.}
\label{schema} 86 86 \label{schema}
\end{figure} 87 87 \end{figure}
88 88
As with the analog mixer, 89 89 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 90 90 the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as
well as the generation of the frequency sum signal in addition to the frequency difference. 91 91 well as the generation of the frequency sum signal in addition to the frequency difference.
These unwanted spectral characteristics must be rejected before decimating the data stream 92 92 These unwanted spectral characteristics must be rejected before decimating the data stream
for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the 93 93 for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
downconverter 94 94 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 95 95 and the decimation processing blocks are core characteristics of an oscillator characterization
system, and must reject out-of-band signals below the targeted phase noise -- typically in the 96 96 system, and must reject out-of-band signals below the targeted phase noise -- typically in the
sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will 97 97 sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will
use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency 98 98 use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency
datastream: optimizing the performance of the filter while reducing the needed resources is 99 99 datastream: optimizing the performance of the filter while reducing the needed resources is
hence tackled in a systematic approach using optimization techniques. Most significantly, we 100 100 hence tackled in a systematic approach using optimization techniques. Most significantly, we
tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with 101 101 tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with
tunable number of coefficients and tunable number of bits representing the coefficients and the 102 102 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 103 103 data being processed.
104 104
\section{Finite impulse response filter} 105 105 \section{Finite impulse response filter}
106 106
We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined 107 107 We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined
by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the 108 108 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 109 109 outputs $y_k$
\begin{align} 110 110 \begin{align}
y_n=\sum_{k=0}^N b_k x_{n-k} 111 111 y_n=\sum_{k=0}^N b_k x_{n-k}
\label{eq:fir_equation} 112 112 \label{eq:fir_equation}
\end{align} 113 113 \end{align}
114 114
As opposed to an implementation on a general purpose processor in which word size is defined by the 115 115 As opposed to an implementation on a general purpose processor in which word size is defined by the
processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since 116 116 processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since
not only the coefficient values and number of taps must be defined, but also the number of bits 117 117 not only the coefficient values and number of taps must be defined, but also the number of bits
defining the coefficients and the sample size. For this reason, and because we consider pipeline 118 118 defining the coefficients and the sample size. For this reason, and because we consider pipeline
processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency 119 119 processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency
signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but 120 120 signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but
the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language 121 121 the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language
(VHDL) level. 122 122 (VHDL) level.
{\color{red}Since latency is not an issue in a openloop phase noise characterization instrument, 123 123 {\color{red}Since latency is not an issue in a openloop phase noise characterization instrument,
the large 124 124 the large
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 125 125 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
is not considered as an issue as would be in a closed loop system.} % r2.4 126 126 is not considered as an issue as would be in a closed loop system.} % r2.4
127 127
The coefficients are classically expressed as floating point values. However, this binary 128 128 The coefficients are classically expressed as floating point values. However, this binary
number representation is not efficient for fast arithmetic computation by an FPGA. Instead, 129 129 number representation is not efficient for fast arithmetic computation by an FPGA. Instead,
we select to quantify these floating point values into integer values. This quantization 130 130 we select to quantify these floating point values into integer values. This quantization
will result in some precision loss. 131 131 will result in some precision loss.
132 132
\begin{figure}[h!tb] 133 133 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/zero_values} 134 134 \includegraphics[width=\linewidth]{images/zero_values}
\caption{Impact of the quantization resolution of the coefficients: the quantization is 135 135 \caption{Impact of the quantization resolution of the coefficients: the quantization is
set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting 136 136 set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting
the 30~first and 30~last coefficients out of the initial 128~band-pass 137 137 the 30~first and 30~last coefficients out of the initial 128~band-pass
filter coefficients to 0 (red dots).} 138 138 filter coefficients to 0 (red dots).}
\label{float_vs_int} 139 139 \label{float_vs_int}
\end{figure} 140 140 \end{figure}
141 141
The tradeoff between quantization resolution and number of coefficients when considering 142 142 The tradeoff between quantization resolution and number of coefficients when considering
integer operations is not trivial. As an illustration of the issue related to the 143 143 integer operations is not trivial. As an illustration of the issue related to the
relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits 144 144 relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits
a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon 145 145 a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon
quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the 146 146 quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the
taps become null, {\color{red}making the large number of coefficients irrelevant: processing 147 147 taps become null, {\color{red}making the large number of coefficients irrelevant: processing
resources % r1.1 148 148 resources % r1.1
are hence saved by shrinking the filter length.} This tradeoff aimed at minimizing resources 149 149 are hence saved by shrinking the filter length.} This tradeoff aimed at minimizing resources
to reach a given rejection level, or maximizing out of band rejection for a given computational 150 150 to reach a given rejection level, or maximizing out of band rejection for a given computational
resource, will drive the investigation on cascading filters designed with varying tap resolution 151 151 resource, will drive the investigation on cascading filters designed with varying tap resolution
and tap length, as will be shown in the next section. Indeed, our development strategy closely 152 152 and tap length, as will be shown in the next section. Indeed, our development strategy closely
follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards} 153 153 follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
in which basic blocks are defined and characterized before being assembled \cite{hide} 154 154 in which basic blocks are defined and characterized before being assembled \cite{hide}
in a complete processing chain. In our case, assembling the filter blocks is a simpler block 155 155 in a complete processing chain. In our case, assembling the filter blocks is a simpler block
combination process since we assume a single value to be processed and a single value to be 156 156 combination process since we assume a single value to be processed and a single value to be
generated at each clock cycle. The FIR filters will not be considered to decimate in the 157 157 generated at each clock cycle. The FIR filters will not be considered to decimate in the
current implementation: the decimation is assumed to be located after the FIR cascade at the 158 158 current implementation: the decimation is assumed to be located after the FIR cascade at the
moment. 159 159 moment.
160 160
\section{Methodology description} 161 161 \section{Methodology description}
162 162
Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP) 163 163 Our objective is to develop a new methodology applicable to any Digital Signal Processing (DSP)
chain obtained by assembling basic processing blocks, with hardware and manufacturer independence. 164 164 chain obtained by assembling basic processing blocks, with hardware and manufacturer independence.
Achieving such a target requires defining an abstract model to represent some basic properties 165 165 Achieving such a target requires defining an abstract model to represent some basic properties
of DSP blocks such as perfomance (i.e. rejection or ripples in the bandpass for filters) and 166 166 of DSP blocks such as perfomance (i.e. rejection or ripples in the bandpass for filters) and
resource occupation. These abstract properties, not necessarily related to the detailed hardware 167 167 resource occupation. These abstract properties, not necessarily related to the detailed hardware
implementation of a given platform, will feed a scheduler solver aimed at assembling the optimum 168 168 implementation of a given platform, will feed a scheduler solver aimed at assembling the optimum
target, whether in terms of maximizing performance for a given arbitrary resource occupation, or 169 169 target, whether in terms of maximizing performance for a given arbitrary resource occupation, or
minimizing resource occupation for a given perfomance. In our approach, the solution of the 170 170 minimizing resource occupation for a given perfomance. In our approach, the solution of the
solver is then synthesized using the dedicated tool provided by each platform manufacturer 171 171 solver is then synthesized using the dedicated tool provided by each platform manufacturer
to assess the validity of our abstract resource occupation indicator, and the result of running 172 172 to assess the validity of our abstract resource occupation indicator, and the result of running
the DSP chain on the FPGA allows for assessing the performance of the scheduler. We emphasize 173 173 the DSP chain on the FPGA allows for assessing the performance of the scheduler. We emphasize
that all solutions found by the solver are synthesized and executed on hardware at the end 174 174 that all solutions found by the solver are synthesized and executed on hardware at the end
of the analysis. 175 175 of the analysis.
176 176
In this demonstration , we focus on only two operations: filtering and shifting the number of 177 177 In this demonstration , we focus on only two operations: filtering and shifting the number of
bits needed to represent the data along the processing chain. 178 178 bits needed to represent the data along the processing chain.
We have chosen these basic operations because shifting and the filtering have already been studied 179 179 We have chosen these basic operations because shifting and the filtering have already been studied
in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for 180 180 in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for
assessing our results. Furthermore, filtering is a core step in any radiofrequency frontend 181 181 assessing our results. Furthermore, filtering is a core step in any radiofrequency frontend
requiring pipelined processing at full bandwidth for the earliest steps, including for 182 182 requiring pipelined processing at full bandwidth for the earliest steps, including for
time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}. 183 183 time and frequency transfer or characterization \cite{carolina1,carolina2,rsi}.
184 184
Addressing only two operations allows for demonstrating the methodology but should not be 185 185 Addressing only two operations allows for demonstrating the methodology but should not be
considered as a limitation of the framework which can be extended to assembling any number 186 186 considered as a limitation of the framework which can be extended to assembling any number
of skeleton blocks as long as perfomance and resource occupation can be determined. {\color{red} 187 187 of skeleton blocks as long as perfomance and resource occupation can be determined. {\color{red}
Hence, 188 188 Hence,
in this paper we will apply our methodology on simple DSP chains: a white noise input signal % r1.2 189 189 in this paper we will apply our methodology on simple DSP chains: a white noise input signal % r1.2
is generated using a Pseudo-Random Number (PRN) generator or by sampling a wideband (125~MS/s) 190 190 is generated using a Pseudo-Random Number (PRN) generator or by sampling a wideband (125~MS/s)
14-bit Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.} Once samples have been 191 191 14-bit Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.} Once samples have been
digitized at a rate of 125~MS/s, filtering is applied to qualify the processing block performance -- 192 192 digitized at a rate of 125~MS/s, filtering is applied to qualify the processing block performance --
practically meeting the radiofrequency frontend requirement of noise and bandwidth reduction 193 193 practically meeting the radiofrequency frontend requirement of noise and bandwidth reduction
by filtering and decimating. Finally, bursts of filtered samples are stored for post-processing, 194 194 by filtering and decimating. Finally, bursts of filtered samples are stored for post-processing,
allowing to assess either filter rejection for a given resource usage, or validating the rejection 195 195 allowing to assess either filter rejection for a given resource usage, or validating the rejection
when implementing a solution minimizing resource occupation. 196 196 when implementing a solution minimizing resource occupation.
197 197
{\color{red} 198 198 {\color{red}
The first step of our approach is to model the DSP chain. Since we aim at only optimizing % r1.3 199 199 The first step of our approach is to model the DSP chain. Since we aim at only optimizing % r1.3
the filtering part of the signal processing chain, we have not included the PRN generator or the 200 200 the filtering part of the signal processing chain, we have not included the PRN generator or the
ADC in the model: the input data size and rate are considered fixed and defined by the hardware. 201 201 ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
The filtering can be done in two ways, either by considering a single monolithic FIR filter 202 202 The filtering can be done in two ways, either by considering a single monolithic FIR filter
requiring many coefficients to reach the targeted noise rejection ratio, or by 203 203 requiring many coefficients to reach the targeted noise rejection ratio, or by
cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.} 204 204 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.}
205 205
After each filter we leave the possibility of shifting the filtered data to consume 206 206 After each filter we leave the possibility of shifting the filtered data to consume
less resources. Hence in the case of cascaded filter, we define a stage as a filter 207 207 less resources. Hence in the case of cascaded filter, we define a stage as a filter
and a shifter (the shift could be omitted if we do not need to divide the filtered data). 208 208 and a shifter (the shift could be omitted if we do not need to divide the filtered data).
209 209
\subsection{Model of a FIR filter} 210 210 \subsection{Model of a FIR filter}
211 211
A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$) 212 212 A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$)
the FIR has $C_i$ coefficients and each coefficient is an integer value with $\pi^C_i$ 213 213 the FIR has $C_i$ coefficients and each coefficient is an integer value with $\pi^C_i$
bits while the filtered data are shifted by $\pi^S_i$ bits. We define also $\pi^-_i$ as 214 214 bits while the filtered data are shifted by $\pi^S_i$ bits. We define also $\pi^-_i$ as
the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage} 215 215 the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage}
shows a filtering stage. 216 216 shows a filtering stage.
217 217
\begin{figure} 218 218 \begin{figure}
\centering 219 219 \centering
\begin{tikzpicture}[node distance=2cm] 220 220 \begin{tikzpicture}[node distance=2cm]
\node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ; 221 221 \node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;
\node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ; 222 222 \node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;
\node (Start) [left of=FIR] { } ; 223 223 \node (Start) [left of=FIR] { } ;
\node (End) [right of=Shift] { } ; 224 224 \node (End) [right of=Shift] { } ;
225 225
\node[draw,fit=(FIR) (Shift)] (Filter) { } ; 226 226 \node[draw,fit=(FIR) (Shift)] (Filter) { } ;
227 227
\draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ; 228 228 \draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;
\draw[->] (FIR) -- (Shift) ; 229 229 \draw[->] (FIR) -- (Shift) ;
\draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ; 230 230 \draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;
\end{tikzpicture} 231 231 \end{tikzpicture}
\caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)} 232 232 \caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}
\label{fig:fir_stage} 233 233 \label{fig:fir_stage}
\end{figure} 234 234 \end{figure}
235 235
FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB. 236 236 FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB.
This rejection has been computed using GNU Octave software FIR coefficient design functions 237 237 This rejection has been computed using GNU Octave software FIR coefficient design functions
(\texttt{firls} and \texttt{fir1}). 238 238 (\texttt{firls} and \texttt{fir1}).
For each configuration $(C_i, \pi_i^C)$, we first create a FIR with floating point coefficients and a given $C_i$ number of coefficients. 239 239 For each configuration $(C_i, \pi_i^C)$, we first create a FIR with floating point coefficients and a given $C_i$ number of coefficients.
Then, the floating point coefficients are discretized into integers. In order to ensure that the coefficients are coded on $\pi_i^C$~bits effectively, 240 240 Then, the floating point coefficients are discretized into integers. In order to ensure that the coefficients are coded on $\pi_i^C$~bits effectively,
the coefficients are normalized by their absolute maximum before being scaled to integer coefficients. 241 241 the coefficients are normalized by their absolute maximum before being scaled to integer coefficients.
At least one coefficient is coded on $\pi_i^C$~bits, and in practice only $b_{C_i/2}$ is coded on $\pi_i^C$~bits while the others are coded on much fewer bits. 242 242 At least one coefficient is coded on $\pi_i^C$~bits, and in practice only $b_{C_i/2}$ is coded on $\pi_i^C$~bits while the others are coded on much fewer bits.
243 243
With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter 244 244 With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter
transfer function. 245 245 transfer function.
Comparing the performance between FIRs requires however defining a unique criterion. As shown in figure~\ref{fig:fir_mag}, 246 246 Comparing the performance between FIRs requires however defining a unique criterion. As shown in figure~\ref{fig:fir_mag},
the FIR magnitude exhibits two parts: we focus here on the transitions width and the rejection rather than on the 247 247 the FIR magnitude exhibits two parts: we focus here on the transitions width and the rejection rather than on the
bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. {\color{red}Throughout this demonstration, 248 248 bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. {\color{red}Throughout this demonstration,
we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\% 249 249 we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%
of the Nyquist frequency to the end of the band, as would be typically selected to prevent 250 250 of the Nyquist frequency to the end of the band, as would be typically selected to prevent
aliasing before decimating the dataflow by 2. The method is however generalized to any filter 251 251 aliasing before decimating the dataflow by 2. The method is however generalized to any filter
shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid} 252 252 shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid}
as described below is indeed unique for each filter shape.} 253 253 as described below is indeed unique for each filter shape.}
254 254
\begin{figure} 255 255 \begin{figure}
\begin{center} 256 256 \begin{center}
\scalebox{0.8}{ 257 257 \scalebox{0.8}{
\centering 258 258 \centering
\begin{tikzpicture}[scale=0.3] 259 259 \begin{tikzpicture}[scale=0.3]
\draw[<->] (0,15) -- (0,0) -- (21,0) ; 260 260 \draw[<->] (0,15) -- (0,0) -- (21,0) ;
\draw[thick] (0,12) -- (8,12) -- (20,0) ; 261 261 \draw[thick] (0,12) -- (8,12) -- (20,0) ;
262 262
\draw (0,14) node [left] { $P$ } ; 263 263 \draw (0,14) node [left] { $P$ } ;
\draw (20,0) node [below] { $f$ } ; 264 264 \draw (20,0) node [below] { $f$ } ;
265 265
\draw[>=latex,<->] (0,14) -- (8,14) ; 266 266 \draw[>=latex,<->] (0,14) -- (8,14) ;
\draw (4,14) node [above] { passband } node [below] { $40\%$ } ; 267 267 \draw (4,14) node [above] { passband } node [below] { $40\%$ } ;
268 268
\draw[>=latex,<->] (8,14) -- (12,14) ; 269 269 \draw[>=latex,<->] (8,14) -- (12,14) ;
\draw (10,14) node [above] { transition } node [below] { $20\%$ } ; 270 270 \draw (10,14) node [above] { transition } node [below] { $20\%$ } ;
271 271
\draw[>=latex,<->] (12,14) -- (20,14) ; 272 272 \draw[>=latex,<->] (12,14) -- (20,14) ;
\draw (16,14) node [above] { stopband } node [below] { $40\%$ } ; 273 273 \draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;
274 274
\draw[>=latex,<->] (16,12) -- (16,8) ; 275 275 \draw[>=latex,<->] (16,12) -- (16,8) ;
\draw (16,10) node [right] { rejection } ; 276 276 \draw (16,10) node [right] { rejection } ;
277 277
\draw[dashed] (8,-1) -- (8,14) ; 278 278 \draw[dashed] (8,-1) -- (8,14) ;
\draw[dashed] (12,-1) -- (12,14) ; 279 279 \draw[dashed] (12,-1) -- (12,14) ;
280 280
\draw[dashed] (8,12) -- (16,12) ; 281 281 \draw[dashed] (8,12) -- (16,12) ;
\draw[dashed] (12,8) -- (16,8) ; 282 282 \draw[dashed] (12,8) -- (16,8) ;
283 283
\end{tikzpicture} 284 284 \end{tikzpicture}
} 285 285 }
\end{center} 286 286 \end{center}
\caption{Shape of the filter transmitted power $P$ as a function of frequency $f$: 287 287 \caption{Shape of the filter transmitted power $P$ as a function of frequency $f$:
the passband is considered to occupy the initial 40\% of the Nyquist frequency range, 288 288 the passband is considered to occupy the initial 40\% of the Nyquist frequency range,
the stopband the last 40\%, allowing 20\% transition width.} 289 289 the stopband the last 40\%, allowing 20\% transition width.}
\label{fig:fir_mag} 290 290 \label{fig:fir_mag}
\end{figure} 291 291 \end{figure}
292 292
In the transition band, the behavior of the filter is left free, we only {\color{red}define} the passband and the stopband characteristics. 293 293 In the transition band, the behavior of the filter is left free, we only {\color{red}define} the passband and the stopband characteristics.
% r2.7 294 294 % r2.7
% Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion 295 295 % Our initial criterion considered the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion
% yields unacceptable results since notches overestimate the rejection capability of the filter. Furthermore, the losses within 296 296 % yields unacceptable results since notches overestimate the rejection capability of the filter. Furthermore, the losses within
% the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients. 297 297 % the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients.
Our criterion to compute the filter rejection considers 298 298 Our criterion to compute the filter rejection considers
% r2.8 et r2.2 r2.3 299 299 % r2.8 et r2.2 r2.3
the maximum magnitude within the stopband, to which the {\color{red}sum of the absolute values 300 300 the {\color{red}minimal} rejection within the stopband, to which the {\color{red}sum of the absolute values
within the passband is subtracted to avoid filters with excessive ripples}. With this 301 301 within the passband is subtracted to avoid filters with excessive ripples}. With this
criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}. 302 302 criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
303 303
% \begin{figure} 304 304 % \begin{figure}
% \centering 305 305 % \centering
% \includegraphics[width=\linewidth]{images/colored_mean_criterion} 306 306 % \includegraphics[width=\linewidth]{images/colored_mean_criterion}
% \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters} 307 307 % \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}
% \label{fig:mean_criterion} 308 308 % \label{fig:mean_criterion}
% \end{figure} 309 309 % \end{figure}
310 310
\begin{figure} 311 311 \begin{figure}
\centering 312 312 \centering
\includegraphics[width=\linewidth]{images/colored_custom_criterion} 313 313 \includegraphics[width=\linewidth]{images/colored_custom_criterion}
\caption{Custom criterion (maximum rejection in the stopband minus the mean of the absolute value of the passband rejection) 314 314 \caption{Custom criterion (maximum rejection in the stopband minus the mean of the absolute value of the passband rejection)
comparison between monolithic filter and cascaded filters} 315 315 comparison between monolithic filter and cascaded filters}
\label{fig:custom_criterion} 316 316 \label{fig:custom_criterion}
\end{figure} 317 317 \end{figure}
318 318
Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps 319 319 Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps
and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the 320 320 and estimate their rejection. Figure~\ref{fig:rejection_pyramid} exhibits the
rejection as a function of the number of coefficients and the number of bits representing these coefficients. 321 321 rejection as a function of the number of coefficients and the number of bits representing these coefficients.
The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet. 322 322 The curve shaped as a pyramid exhibits optimum configurations sets at the vertex where both edges meet.
Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection. 323 323 Indeed for a given number of coefficients, increasing the number of bits over the edge will not improve the rejection.
Conversely when setting the a given number of bits, increasing the number of coefficients will not improve 324 324 Conversely when setting the a given number of bits, increasing the number of coefficients will not improve
the rejection. Hence the best coefficient set are on the vertex of the pyramid. 325 325 the rejection. Hence the best coefficient set are on the vertex of the pyramid.
326 326
\begin{figure} 327 327 \begin{figure}
\centering 328 328 \centering
\includegraphics[width=\linewidth]{images/rejection_pyramid} 329 329 \includegraphics[width=\linewidth]{images/rejection_pyramid}
\caption{Rejection as a function of number of coefficients and number of bits} 330 330 \caption{Rejection as a function of number of coefficients and number of bits}
\label{fig:rejection_pyramid} 331 331 \label{fig:rejection_pyramid}
\end{figure} 332 332 \end{figure}
333 333
Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps), 334 334 Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps),
we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria. 335 335 we have a problem when we cascade filters and estimate the criterion as a sum two or more individual criteria.
If the FIR filter coefficients are the same between the stages, we have: 336 336 If the FIR filter coefficients are the same between the stages, we have:
$$F_{total} = F_1 + F_2$$ 337 337 $$F_{total} = F_1 + F_2$$
But selecting two different sets of coefficient will yield a more complex situation in which 338 338 But selecting two different sets of coefficient will yield a more complex situation in which
the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves 339 339 the previous relation is no longer valid as illustrated on figure~\ref{fig:sum_rejection}. The red and blue curves
are two different filters with maximums and notches not located at the same frequency offsets. 340 340 are two different filters with maximums and notches not located at the same frequency offsets.
Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved 341 341 Hence when summing the transfer functions, the resulting rejection shown as the dashed yellow line is improved
with respect to a basic sum of the rejection criteria shown as a the dotted yellow line. 342 342 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
% r2.9 343 343 % r2.9
Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection 344 344 Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection
criteria of each filter. However since the this sum underestimates the rejection capability of the cascade, 345 345 criteria of each filter. However since the this sum underestimates the rejection capability of the cascade,
% r2.10 346 346 % r2.10
this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability 347 347 this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
of the filter cascade to meet design criteria. 348 348 of the filter cascade to meet design criteria.
349 349
\begin{figure} 350 350 \begin{figure}
\centering 351 351 \centering
\includegraphics[width=\linewidth]{images/cascaded_criterion} 352 352 \includegraphics[width=\linewidth]{images/cascaded_criterion}
\caption{Rejection of two cascaded filters} 353 353 \caption{Rejection of two cascaded filters}
\label{fig:sum_rejection} 354 354 \label{fig:sum_rejection}
\end{figure} 355 355 \end{figure}
356 356
% r2.6 357 357 % r2.6
Finally in our case, we consider that the input signal are fully known. So the 358 358 Finally in our case, we consider that the input signal are fully known. So the
resolution of the data stream are fixed and still the same for all experiments 359 359 resolution of the data stream are fixed and still the same for all experiments
in this paper. 360 360 in this paper.
361 361
Based on this analysis, we address the estimate of resource consumption (called 362 362 Based on this analysis, we address the estimate of resource consumption (called
% r2.11 363 363 % r2.11
silicon area -- in the case of FPGAs this means processing cells) as a function of 364 364 silicon area -- in the case of FPGAs this means processing cells) as a function of
filter characteristics. As a reminder, we do not aim at matching actual hardware 365 365 filter characteristics. As a reminder, we do not aim at matching actual hardware
configuration but consider an arbitrary silicon area occupied by each processing function, 366 366 configuration but consider an arbitrary silicon area occupied by each processing function,
and will assess after synthesis the adequation of this arbitrary unit with actual 367 367 and will assess after synthesis the adequation of this arbitrary unit with actual
hardware resources provided by FPGA manufacturers. The sum of individual processing 368 368 hardware resources provided by FPGA manufacturers. The sum of individual processing
unit areas is constrained by a total silicon area representative of FPGA global resources. 369 369 unit areas is constrained by a total silicon area representative of FPGA global resources.
Formally, variable $a_i$ is the area taken by filter~$i$ 370 370 Formally, variable $a_i$ is the area taken by filter~$i$
(in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB). 371 371 (in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB).
Constant $\mathcal{A}$ is the total available area. We model our problem as follows: 372 372 Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
373 373
\begin{align} 374 374 \begin{align}
\text{Maximize } & \sum_{i=1}^n r_i \notag \\ 375 375 \text{Maximize } & \sum_{i=1}^n r_i \notag \\
\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\ 376 376 \sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\
a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\ 377 377 a_i & = C_i \times (\pi_i^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef} \\
r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\ 378 378 r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\
\pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\ 379 379 \pi_i^+ & = \pi_i^- + \pi_i^C - \pi_i^S, & \forall i \in [1, n] \label{eq:bits} \\
\pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\ 380 380 \pi_{i - 1}^+ & = \pi_i^-, & \forall i \in [2, n] \label{eq:inout} \\
\pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\ 381 381 \pi_i^+ & \geq 1 + \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right), & \forall i \in [1, n] \label{eq:maxshift} \\
\pi_1^- &= \Pi^I \label{eq:init} 382 382 \pi_1^- &= \Pi^I \label{eq:init}
\end{align} 383 383 \end{align}
384 384
Equation~\ref{eq:area} states that the total area taken by the filters must be 385 385 Equation~\ref{eq:area} states that the total area taken by the filters must be
less than the available area. Equation~\ref{eq:areadef} gives the definition of 386 386 less than the available area. Equation~\ref{eq:areadef} gives the definition of
the area used by a filter, considered as the area of the FIR since the Shifter is 387 387 the area used by a filter, considered as the area of the FIR since the Shifter is
assumed not to require significant resources. We consider that the FIR needs $C_i$ registers of size 388 388 assumed not to require significant resources. We consider that the FIR needs $C_i$ registers of size
$\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the 389 389 $\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the
input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the 390 390 input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the
definition of the rejection of the filter thanks to the tabulated function~$F$ that we defined 391 391 definition of the rejection of the filter thanks to the tabulated function~$F$ that we defined
previously. The Shifter does not introduce negative rejection as we will explain later, 392 392 previously. The Shifter does not introduce negative rejection as we will explain later,
so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the 393 393 so the rejection only comes from the FIR. Equation~\ref{eq:bits} states the
relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add 394 394 relation between $\pi_i^+$ and $\pi_i^-$. The multiplications in the FIR add
$\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes 395 395 $\pi_i^C$ bits as most coefficients are close to zero, and the Shifter removes
$\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of 396 396 $\pi_i^S$ bits. Equation~\ref{eq:inout} states that the output number of bits of
a filter is the same as the input number of bits of the next filter. 397 397 a filter is the same as the input number of bits of the next filter.
Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative 398 398 Equation~\ref{eq:maxshift} ensures that the Shifter does not introduce negative
rejection. Indeed, the results of the FIR can be right shifted without compromising 399 399 rejection. Indeed, the results of the FIR can be right shifted without compromising
the quality of the rejection until a threshold. Each bit of the output data 400 400 the quality of the rejection until a threshold. Each bit of the output data
increases the maximum rejection level by 6~dB. We add one to take the sign bit 401 401 increases the maximum rejection level by 6~dB. We add one to take the sign bit
into account. If equation~\ref{eq:maxshift} was not present, the Shifter could 402 402 into account. If equation~\ref{eq:maxshift} was not present, the Shifter could
shift too much and introduce some noise in the output data. Each supplementary 403 403 shift too much and introduce some noise in the output data. Each supplementary
shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is: 404 404 shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is:
$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. 405 405 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
Finally, equation~\ref{eq:init} gives the number of bits of the global input. 406 406 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
407 407
{\color{red} 408 408 {\color{red}
This model is non-linear since we multiply some variable with another variable 409 409 This model is non-linear since we multiply some variable with another variable
and it is even non-quadratic, as $F$ does not have a known 410 410 and it is even non-quadratic, as $F$ does not have a known
linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations. 411 411 linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
This variable must be defined by the user, it represent the number of different 412 412 This variable must be defined by the user, it represent the number of different
set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1} 413 413 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave). So $C_{ij}$ and $\pi_{ij}^C$ become constant and 414 414 functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid}
we defined $1 \leq j \leq p$ and the function $F$ can be estimate for each configurations 415 415 to restrict the number of configurations. Indeed, it is useless to have too many coefficients or
416 too many bits, hence we take the configurations close to edge of pyramid. Thank to theses
417 configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant
418 and the function $F$ can be estimate for each configurations
thanks our rejection criterion. We also defined binary 416 419 thanks our rejection criterion. We also defined binary
variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 417 420 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
and 0 otherwise. The new equations are as follows: 418 421 and 0 otherwise. The new equations are as follows:
} 419 422 }
420 423
\begin{align} 421 424 \begin{align}
a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\ 422 425 a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\
r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\ 423 426 r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
\pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\ 424 427 \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\
\sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} 425 428 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
\end{align} 426 429 \end{align}
427 430
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 428 431 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 429 432 respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most. 430 433 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
431 434
{\color{red} 432 435 {\color{red}
However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply 433 436 However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply
$\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can 434 437 $\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can
linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size 435 438 linearize this multiplication. The following formula shows how to linearize
we define $0 < \pi_i^- \leq 128$ which is the maximal data size that we can process. 436 439 this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):
440 \begin{equation*}
441 m = x \times y \implies
442 \left \{
443 \begin{split}
444 m & \geq 0 \\
445 m & \leq y \times X^{max} \\
446 m & \leq x \\
447 m & \geq x - (1 - y) \times X^{max} \\
448 \end{split}
449 \right .
450 \end{equation*}
451
452 So if we bound up $\pi_i^-$ by 128~bits to represent the maximum data size tolerated,
453 the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize
454 for us the quadratic problem so the model is left as is.
} 437 455 }
Moreover the Gurobi 438 456 This model has $O(np)$ variables and $O(n)$ constraints.
(\url{www.gurobi.com}) optimization software is used to solve this quadratic 439
model, and since Gurobi is able to linearize, the model is left as is. This model 440
has $O(np)$ variables and $O(n)$ constraints. 441
442 457
% This model is non-linear and even non-quadratic, as $F$ does not have a known 443 458 % This model is non-linear and even non-quadratic, as $F$ does not have a known
% linear or quadratic expression. We introduce $p$ FIR configurations 444 459 % linear or quadratic expression. We introduce $p$ FIR configurations
% $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 445 460 % $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
% % r2.12 446 461 % % r2.12
% This variable must be defined by the user, it represent the number of different 447 462 % This variable must be defined by the user, it represent the number of different
% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1} 448 463 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
% functions from GNU Octave). 449 464 % functions from GNU Octave).
% We define binary 450 465 % We define binary
% variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 451 466 % variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
% and 0 otherwise. The new equations are as follows: 452 467 % and 0 otherwise. The new equations are as follows:
% 453 468 %
% \begin{align} 454 469 % \begin{align}
% a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\ 455 470 % a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\
% r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\ 456 471 % r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\
% \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\ 457 472 % \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\
% \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} 458 473 % \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
% \end{align} 459 474 % \end{align}
% 460 475 %
% Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 461 476 % Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
% respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 462 477 % respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
% Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most. 463 478 % Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
% 464 479 %
% % r2.13 465 480 % % r2.13
% This modified model is quadratic since we multiply two variables in the 466 481 % This modified model is quadratic since we multiply two variables in the
% equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary. 467 482 % equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
% The Gurobi 468 483 % The Gurobi
% (\url{www.gurobi.com}) optimization software is used to solve this quadratic 469 484 % (\url{www.gurobi.com}) optimization software is used to solve this quadratic
% model, and since Gurobi is able to linearize, the model is left as is. This model 470 485 % model, and since Gurobi is able to linearize, the model is left as is. This model
% has $O(np)$ variables and $O(n)$ constraints. 471 486 % has $O(np)$ variables and $O(n)$ constraints.
472 487
Two problems will be addressed using the workflow described in the next section: on the one 473 488 Two problems will be addressed using the workflow described in the next section: on the one
hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary 474 489 hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary
silcon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area 475 490 silcon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area
for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the 476 491 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
objective function is replaced with: 477 492 objective function is replaced with:
\begin{align} 478 493 \begin{align}
\text{Minimize } & \sum_{i=1}^n a_i \notag 479 494 \text{Minimize } & \sum_{i=1}^n a_i \notag
\end{align} 480 495 \end{align}
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 481 496 We adapt our constraints of quadratic program to replace equation \ref{eq:area}
with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal 482 497 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
rejection required. 483 498 rejection required.
484 499
\begin{align} 485 500 \begin{align}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 486 501 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
\end{align} 487 502 \end{align}
488 503
\section{Design workflow} 489 504 \section{Design workflow}
\label{sec:workflow} 490 505 \label{sec:workflow}
491 506
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 492 507 In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area}
and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved 493 508 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
in the computation of the results. 494 509 in the computation of the results.
495 510
\begin{figure} 496 511 \begin{figure}
\centering 497 512 \centering
\begin{tikzpicture}[node distance=0.75cm and 2cm] 498 513 \begin{tikzpicture}[node distance=0.75cm and 2cm]
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 499 514 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\node (Start) [left= 3cm of Solver] { } ; 500 515 \node (Start) [left= 3cm of Solver] { } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 501 516 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\node (Input) [above= of TCL] { } ; 502 517 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 503 518 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 504 519 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 505 520 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 506 521 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node (Results) [left= of Postproc] { } ; 507 522 \node (Results) [left= of Postproc] { } ;
508 523
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 509 524 \draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ;
\draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ; 510 525 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 511 526 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 512 527 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 513 528 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 514 529 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 515 530 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 516 531 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 517 532 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->] (Postproc) -- (Results) ; 518 533 \draw[->] (Postproc) -- (Results) ;
\end{tikzpicture} 519 534 \end{tikzpicture}
\caption{Design workflow from the input parameters to the results} 520 535 \caption{Design workflow from the input parameters to the results}
\label{fig:workflow} 521 536 \label{fig:workflow}
\end{figure} 522 537 \end{figure}
523 538
The filter solver is a C++ program that takes as input the maximum area 524 539 The filter solver is a C++ program that takes as input the maximum area
$\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$, 525 540 $\mathcal{A}$, the number of stages $n$, the size of the input signal $\Pi^I$,
the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates 526 541 the FIR configurations $(C_{ij}, \pi_{ij}^C)$ and the function $F$. It creates
the quadratic programs and uses the Gurobi solver to estimate the optimal results. 527 542 the quadratic programs and uses the Gurobi solver to estimate the optimal results.
Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow}) 528 543 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
and a deploy script ((1b) on figure~\ref{fig:workflow}). 529 544 and a deploy script ((1b) on figure~\ref{fig:workflow}).
530 545
The TCL script describes the whole digital processing chain from the beginning 531 546 The TCL script describes the whole digital processing chain from the beginning
(the raw signal data) to the end (the filtered data) in a language compatible 532 547 (the raw signal data) to the end (the filtered data) in a language compatible
with proprietary synthesis software, namely Vivado for Xilinx and Quartus for 533 548 with proprietary synthesis software, namely Vivado for Xilinx and Quartus for
Intel/Altera. The raw input data generated from a 20-bit Pseudo Random Number (PRN) 534 549 Intel/Altera. The raw input data generated from a 20-bit Pseudo Random Number (PRN)
generator inside the FPGA and $\Pi^I$ is fixed at 16~bits. 535 550 generator inside the FPGA and $\Pi^I$ is fixed at 16~bits.
Then the script builds each stage of the chain with a generic FIR task that 536 551 Then the script builds each stage of the chain with a generic FIR task that
comes from a skeleton library. The generic FIR is highly configurable 537 552 comes from a skeleton library. The generic FIR is highly configurable
with the number of coefficients and the size of the coefficients. The coefficients 538 553 with the number of coefficients and the size of the coefficients. The coefficients
themselves are not stored in the script. 539 554 themselves are not stored in the script.
As the signal is processed in real-time, the output signal is stored as 540 555 As the signal is processed in real-time, the output signal is stored as
consecutive bursts of data for post-processing, mainly assessing the consistency of the 541 556 consecutive bursts of data for post-processing, mainly assessing the consistency of the
implemented FIR cascade transfer function with the design criteria and the expected 542 557 implemented FIR cascade transfer function with the design criteria and the expected
transfer function. 543 558 transfer function.
544 559
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 545 560 The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}).
We use the 2018.2 version of Xilinx Vivado and we execute the synthesized 546 561 We use the 2018.2 version of Xilinx Vivado and we execute the synthesized
bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series 547 562 bitstream on a Redpitaya board fitted with a Xilinx Zynq-7010 series
FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to 548 563 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
provide a broadband noise source. 549 564 provide a broadband noise source.
The board runs the Linux kernel and surrounding environment produced from the 550 565 The board runs the Linux kernel and surrounding environment produced from the
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 551 566 Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring
the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and 552 567 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
fetching the results is automated. 553 568 fetching the results is automated.
554 569
The deploy script uploads the bitstream to the board ((3) on 555 570 The deploy script uploads the bitstream to the board ((3) on
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 556 571 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
configures the coefficients of the FIR filters. It then waits for the results 557 572 configures the coefficients of the FIR filters. It then waits for the results
and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}). 558 573 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
559 574
Finally, an Octave post-processing script computes the final results thanks to 560 575 Finally, an Octave post-processing script computes the final results thanks to
the output data ((5) on figure~\ref{fig:workflow}). 561 576 the output data ((5) on figure~\ref{fig:workflow}).
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 562 577 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
and the different configurations can be compared. 563 578 and the different configurations can be compared.
564 579
\section{Maximizing the rejection at fixed silicon area} 565 580 \section{Maximizing the rejection at fixed silicon area}
\label{sec:fixed_area} 566 581 \label{sec:fixed_area}
This section presents the output of the filter solver {\em i.e.} the computed 567 582 This section presents the output of the filter solver {\em i.e.} the computed
configurations for each stage, the computed rejection and the computed silicon area. 568 583 configurations for each stage, the computed rejection and the computed silicon area.
Such results allow for understanding the choices made by the solver to compute its solutions. 569 584 Such results allow for understanding the choices made by the solver to compute its solutions.
570 585
The experimental setup is composed of three cases. The raw input is generated 571 586 The experimental setup is composed of three cases. The raw input is generated
by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$. 572 587 by a Pseudo Random Number (PRN) generator, which fixes the input data size $\Pi^I$.
Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500 573 588 Then the total silicon area $\mathcal{A}$ has been fixed to either 500, 1000 or 1500
arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500. 574 589 arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500.
The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$ 575 590 The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$
ranging from 2 to 22. In each case, the quadratic program has been able to give a 576 591 ranging from 2 to 22. In each case, the quadratic program has been able to give a
result up to five stages ($n = 5$) in the cascaded filter. 577 592 result up to five stages ($n = 5$) in the cascaded filter.
578 593
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 579 594 Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500.
Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000. 580 595 Table~\ref{tbl:gurobi_max_1000} shows the results obtained by the filter solver for MAX/1000.
Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500. 581 596 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
582 597
\renewcommand{\arraystretch}{1.4} 583 598 \renewcommand{\arraystretch}{1.4}
584 599
\begin{table} 585 600 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 586 601 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
\label{tbl:gurobi_max_500} 587 602 \label{tbl:gurobi_max_500}
\centering 588 603 \centering
{\scalefont{0.77} 589 604 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 590 605 \begin{tabular}{|c|ccccc|c|c|}
\hline 591 606 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 592 607 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 593 608 \hline
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 594 609 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ 595 610 2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\
3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ 596 611 3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\
4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\ 597 612 4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\
5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\ 598 613 5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\
\hline 599 614 \hline
\end{tabular} 600 615 \end{tabular}
} 601 616 }
\end{table} 602 617 \end{table}
603 618
\begin{table} 604 619 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 605 620 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
\label{tbl:gurobi_max_1000} 606 621 \label{tbl:gurobi_max_1000}
\centering 607 622 \centering
{\scalefont{0.77} 608 623 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 609 624 \begin{tabular}{|c|ccccc|c|c|}
\hline 610 625 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 611 626 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 612 627 \hline
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 613 628 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ 614 629 2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\
3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ 615 630 3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\
4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\ 616 631 4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\
5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\ 617 632 5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\
\hline 618 633 \hline
\end{tabular} 619 634 \end{tabular}
} 620 635 }
\end{table} 621 636 \end{table}
622 637
\begin{table} 623 638 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 624 639 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
\label{tbl:gurobi_max_1500} 625 640 \label{tbl:gurobi_max_1500}
\centering 626 641 \centering
{\scalefont{0.77} 627 642 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 628 643 \begin{tabular}{|c|ccccc|c|c|}
\hline 629 644 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 630 645 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 631 646 \hline
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 632 647 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ 633 648 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\
3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ 634 649 3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\
4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\ 635 650 4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\
5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\ 636 651 5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\
\hline 637 652 \hline
\end{tabular} 638 653 \end{tabular}
} 639 654 }
\end{table} 640 655 \end{table}
641 656
\renewcommand{\arraystretch}{1} 642 657 \renewcommand{\arraystretch}{1}
643 658
From these tables, we can first state that the more stages are used to define 644 659 From these tables, we can first state that the more stages are used to define
the cascaded FIR filters, the better the rejection. It was an expected result as it has 645 660 the cascaded FIR filters, the better the rejection. It was an expected result as it has
been previously observed that many small filters are better than 646 661 been previously observed that many small filters are better than
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 647 662 a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions
being hardly used in practice due to the lack of tools for identifying individual filter 648 663 being hardly used in practice due to the lack of tools for identifying individual filter
coefficients in the cascaded approach. 649 664 coefficients in the cascaded approach.
650 665
Second, the larger the silicon area, the better the rejection. This was also an 651 666 Second, the larger the silicon area, the better the rejection. This was also an
expected result as more area means a filter of better quality with more coefficients 652 667 expected result as more area means a filter of better quality with more coefficients
or more bits per coefficient. 653 668 or more bits per coefficient.
654 669
Then, we also observe that the first stage can have a larger shift than the other 655 670 Then, we also observe that the first stage can have a larger shift than the other
stages. This is explained by the fact that the solver tries to use just enough 656 671 stages. This is explained by the fact that the solver tries to use just enough
bits for the computed rejection after each stage. In the first stage, a 657 672 bits for the computed rejection after each stage. In the first stage, a
balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift} 658 673 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
gives the relation between both values. 659 674 gives the relation between both values.
660 675
Finally, we note that the solver consumes all the given silicon area. 661 676 Finally, we note that the solver consumes all the given silicon area.
662 677
The following graphs present the rejection for real data on the FPGA. In all the following 663 678 The following graphs present the rejection for real data on the FPGA. In all the following
figures, the solid line represents the actual rejection of the filtered 664 679 figures, the solid line represents the actual rejection of the filtered
data on the FPGA as measured experimentally and the dashed line are the noise levels 665 680 data on the FPGA as measured experimentally and the dashed line are the noise levels
given by the quadratic solver. The configurations are those computed in the previous section. 666 681 given by the quadratic solver. The configurations are those computed in the previous section.
667 682
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 668 683 Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500.
Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000. 669 684 Figure~\ref{fig:max_1000_result} shows the rejection of the different configurations in the case of MAX/1000.
Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500. 670 685 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
671 686
% \begin{figure} 672 687 % \begin{figure}
% \centering 673 688 % \centering
% \includegraphics[width=\linewidth]{images/max_500} 674 689 % \includegraphics[width=\linewidth]{images/max_500}
% \caption{Signal spectrum for MAX/500} 675 690 % \caption{Signal spectrum for MAX/500}
% \label{fig:max_500_result} 676 691 % \label{fig:max_500_result}
% \end{figure} 677 692 % \end{figure}
% 678 693 %
% \begin{figure} 679 694 % \begin{figure}
% \centering 680 695 % \centering
% \includegraphics[width=\linewidth]{images/max_1000} 681 696 % \includegraphics[width=\linewidth]{images/max_1000}
% \caption{Signal spectrum for MAX/1000} 682 697 % \caption{Signal spectrum for MAX/1000}
% \label{fig:max_1000_result} 683 698 % \label{fig:max_1000_result}
% \end{figure} 684 699 % \end{figure}
% 685 700 %
% \begin{figure} 686 701 % \begin{figure}
% \centering 687 702 % \centering
% \includegraphics[width=\linewidth]{images/max_1500} 688 703 % \includegraphics[width=\linewidth]{images/max_1500}
% \caption{Signal spectrum for MAX/1500} 689 704 % \caption{Signal spectrum for MAX/1500}
% \label{fig:max_1500_result} 690 705 % \label{fig:max_1500_result}
% \end{figure} 691 706 % \end{figure}
692 707
% r2.14 et r2.15 et r2.16 693 708 % r2.14 et r2.15 et r2.16
\begin{figure} 694 709 \begin{figure}
\centering 695 710 \centering
\begin{subfigure}{\linewidth} 696 711 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_500} 697 712 \includegraphics[width=\linewidth]{images/max_500}
\caption{Signal spectrum for MAX/500} 698 713 \caption{Signal spectrum for MAX/500}
\label{fig:max_500_result} 699 714 \label{fig:max_500_result}
\end{subfigure} 700 715 \end{subfigure}
701 716
\begin{subfigure}{\linewidth} 702 717 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1000} 703 718 \includegraphics[width=\linewidth]{images/max_1000}
\caption{Signal spectrum for MAX/1000} 704 719 \caption{Signal spectrum for MAX/1000}
\label{fig:max_1000_result} 705 720 \label{fig:max_1000_result}
\end{subfigure} 706 721 \end{subfigure}
707 722
\begin{subfigure}{\linewidth} 708 723 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1500} 709 724 \includegraphics[width=\linewidth]{images/max_1500}
\caption{Signal spectrum for MAX/1500} 710 725 \caption{Signal spectrum for MAX/1500}
\label{fig:max_1500_result} 711 726 \label{fig:max_1500_result}
\end{subfigure} 712 727 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500} 713 728 \caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500}
\end{figure} 714 729 \end{figure}
715 730
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 716 731 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
717 732
We compare the actual silicon resources given by Vivado to the 718 733 We compare the actual silicon resources given by Vivado to the
resources in arbitrary units. 719 734 resources in arbitrary units.
The goal is to check that our arbitrary units of silicon area models well enough 720 735 The goal is to check that our arbitrary units of silicon area models well enough
the real resources on the FPGA. Especially we want to verify that, for a given 721 736 the real resources on the FPGA. Especially we want to verify that, for a given
number of arbitrary units, the actual silicon resources do not depend on the 722 737 number of arbitrary units, the actual silicon resources do not depend on the
number of stages $n$. Most significantly, our approach aims 723 738 number of stages $n$. Most significantly, our approach aims
at remaining far enough from the practical logic gate implementation used by 724 739 at remaining far enough from the practical logic gate implementation used by
various vendors to remain platform independent and be portable from one 725 740 various vendors to remain platform independent and be portable from one
architecture to another. 726 741 architecture to another.
727 742
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 728 743 Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and
MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000 729 744 MAX/1500 \emph{i.e.} when the maximum allowed silicon area is fixed to 500, 1000
and 1500 arbitrary units. We have taken care to extract solely the resources used by 730 745 and 1500 arbitrary units. We have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and Programmable 731 746 the FIR filters and remove additional processing blocks including FIFO and Programmable
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 732 747 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
733 748
\begin{table}[h!tb] 734 749 \begin{table}[h!tb]
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 735 750 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage} 736 751 \label{tbl:resources_usage}
\centering 737 752 \centering
\begin{tabular}{|c|c|ccc|c|} 738 753 \begin{tabular}{|c|c|ccc|c|}
\hline 739 754 \hline
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 740 755 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 249 & 453 & 627 & \emph{17600} \\ 741 756 & LUT & 249 & 453 & 627 & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 742 757 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 743 758 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
& LUT & 2374 & 5494 & 691 & \emph{17600} \\ 744 759 & LUT & 2374 & 5494 & 691 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 745 760 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 70 & \emph{80} \\ \hline 746 761 & DSP & 0 & 0 & 70 & \emph{80} \\ \hline
& LUT & 2443 & 3304 & 3521 & \emph{17600} \\ 747 762 & LUT & 2443 & 3304 & 3521 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 748 763 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 19 & 35 & \emph{80} \\ \hline 749 764 & DSP & 0 & 19 & 35 & \emph{80} \\ \hline
& LUT & 2634 & 3753 & 2557 & \emph{17600} \\ 750 765 & LUT & 2634 & 3753 & 2557 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & \emph{120} \\ 751 766 4 & BRAM & 4 & 4 & 4 & \emph{120} \\
& DPS & 0 & 19 & 46 & \emph{80} \\ \hline 752 767 & DPS & 0 & 19 & 46 & \emph{80} \\ \hline
& LUT & 2423 & 3047 & 2847 & \emph{17600} \\ 753 768 & LUT & 2423 & 3047 & 2847 & \emph{17600} \\
5 & BRAM & 5 & 5 & 5 & \emph{120} \\ 754 769 5 & BRAM & 5 & 5 & 5 & \emph{120} \\
& DPS & 0 & 22 & 46 & \emph{80} \\ \hline 755 770 & DPS & 0 & 22 & 46 & \emph{80} \\ \hline
\end{tabular} 756 771 \end{tabular}
\end{table} 757 772 \end{table}
758 773
In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 759 774 In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that,
when the filter coefficients are small enough, or when the input size is small 760 775 when the filter coefficients are small enough, or when the input size is small
enough, Vivado optimizes resource consumption by selecting multiplexers to 761 776 enough, Vivado optimizes resource consumption by selecting multiplexers to
implement the multiplications instead of a DSP. In this case, it is quite difficult 762 777 implement the multiplications instead of a DSP. In this case, it is quite difficult
to compare the whole silicon budget. 763 778 to compare the whole silicon budget.
764 779
However, a rough estimation can be made with a simple equivalence: looking at 765 780 However, a rough estimation can be made with a simple equivalence: looking at
the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$, 766 781 the first column (MAX/500), where the number of LUTs is quite stable for $n \geq 2$,
we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon 767 782 we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon
area use. With this equivalence, our 500 arbitraty units correspond to 2500 LUTs, 768 783 area use. With this equivalence, our 500 arbitraty units correspond to 2500 LUTs,
1000 arbitrary units correspond to 5000 LUTs and 1500 arbitrary units correspond 769 784 1000 arbitrary units correspond to 5000 LUTs and 1500 arbitrary units correspond
to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary 770 785 to 7300 LUTs. The conclusion is that the orders of magnitude of our arbitrary
unit map well to actual hardware resources. The relatively small differences can probably be explained 771 786 unit map well to actual hardware resources. The relatively small differences can probably be explained
by the optimizations done by Vivado based on the detailed map of available processing resources. 772 787 by the optimizations done by Vivado based on the detailed map of available processing resources.
773 788
We now present the computation time needed to solve the quadratic problem. 774 789 We now present the computation time needed to solve the quadratic problem.
For each case, the filter solver software is executed on a Intel(R) Xeon(R) CPU E5606 775 790 For each case, the filter solver software is executed on a Intel(R) Xeon(R) CPU E5606
clocked at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve 776 791 clocked at 2.13~GHz. The CPU has 8 cores that are used by Gurobi to solve
the quadratic problem. Table~\ref{tbl:area_time} shows the time needed to solve the quadratic 777 792 the quadratic problem. Table~\ref{tbl:area_time} shows the time needed to solve the quadratic
problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units. 778 793 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
779 794
\begin{table}[h!tb] 780 795 \begin{table}[h!tb]
\caption{Time needed to solve the quadratic program with Gurobi} 781 796 \caption{Time needed to solve the quadratic program with Gurobi}
\label{tbl:area_time} 782 797 \label{tbl:area_time}
\centering 783 798 \centering
\begin{tabular}{|c|c|c|c|}\hline 784 799 \begin{tabular}{|c|c|c|c|}\hline
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 785 800 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
1 & 0.1~s & 0.1~s & 0.3~s \\ 786 801 1 & 0.1~s & 0.1~s & 0.3~s \\
2 & 1.1~s & 2.2~s & 12~s \\ 787 802 2 & 1.1~s & 2.2~s & 12~s \\
3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ 788 803 3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\
4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\ 789 804 4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\
5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline 790 805 5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline
\end{tabular} 791 806 \end{tabular}
\end{table} 792 807 \end{table}
793 808
As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? 794 809 As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ?
When the area is limited, the design exploration space is more limited and the solver is able to 795 810 When the area is limited, the design exploration space is more limited and the solver is able to
find an optimal solution faster. 796 811 find an optimal solution faster.
797 812
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 798 813 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
799 814
This section presents the results of the complementary quadratic program aimed at 800 815 This section presents the results of the complementary quadratic program aimed at
minimizing the area occupation for a targeted rejection level. 801 816 minimizing the area occupation for a targeted rejection level.
802 817
The experimental setup is composed of four cases. The raw input is the same 803 818 The experimental setup is composed of four cases. The raw input is the same
as in the previous section, from a PRN generator, which fixes the input data size $\Pi^I$. 804 819 as in the previous section, from a PRN generator, which fixes the input data size $\Pi^I$.
Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB. 805 820 Then the targeted rejection $\mathcal{R}$ has been fixed to either 40, 60, 80 or 100~dB.
Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100. 806 821 Hence, the three cases have been named: MIN/40, MIN/60, MIN/80 and MIN/100.
The number of configurations $p$ is the same as previous section. 807 822 The number of configurations $p$ is the same as previous section.
808 823
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 809 824 Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40.
Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60. 810 825 Table~\ref{tbl:gurobi_min_60} shows the results obtained by the filter solver for MIN/60.
Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80. 811 826 Table~\ref{tbl:gurobi_min_80} shows the results obtained by the filter solver for MIN/80.
Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100. 812 827 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
813 828
\renewcommand{\arraystretch}{1.4} 814 829 \renewcommand{\arraystretch}{1.4}
815 830
\begin{table}[h!tb] 816 831 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 817 832 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\label{tbl:gurobi_min_40} 818 833 \label{tbl:gurobi_min_40}
\centering 819 834 \centering
{\scalefont{0.77} 820 835 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 821 836 \begin{tabular}{|c|ccccc|c|c|}
\hline 822 837 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 823 838 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 824 839 \hline
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 825 840 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ 826 841 2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\
3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ 827 842 3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\
4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\ 828 843 4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\
\hline 829 844 \hline
\end{tabular} 830 845 \end{tabular}
} 831 846 }
\end{table} 832 847 \end{table}
833 848
\begin{table}[h!tb] 834 849 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 835 850 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\label{tbl:gurobi_min_60} 836 851 \label{tbl:gurobi_min_60}
\centering 837 852 \centering
{\scalefont{0.77} 838 853 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 839 854 \begin{tabular}{|c|ccccc|c|c|}
\hline 840 855 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 841 856 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 842 857 \hline
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 843 858 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ 844 859 2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\
3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ 845 860 3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\
4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\ 846 861 4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\
5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\ 847 862 5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\
\hline 848 863 \hline
\end{tabular} 849 864 \end{tabular}
} 850 865 }
\end{table} 851 866 \end{table}
852 867
\begin{table}[h!tb] 853 868 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 854 869 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\label{tbl:gurobi_min_80} 855 870 \label{tbl:gurobi_min_80}
\centering 856 871 \centering
{\scalefont{0.77} 857 872 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 858 873 \begin{tabular}{|c|ccccc|c|c|}
\hline 859 874 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 860 875 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 861 876 \hline
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 862 877 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ 863 878 2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\
3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ 864 879 3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\
4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\ 865 880 4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\
5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\ 866 881 5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\
\hline 867 882 \hline
\end{tabular} 868 883 \end{tabular}
} 869 884 }
\end{table} 870 885 \end{table}
871 886
\begin{table}[h!tb] 872 887 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 873 888 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\label{tbl:gurobi_min_100} 874 889 \label{tbl:gurobi_min_100}
\centering 875 890 \centering
{\scalefont{0.77} 876 891 {\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 877 892 \begin{tabular}{|c|ccccc|c|c|}
\hline 878 893 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 879 894 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 880 895 \hline
1 & - & - & - & - & - & - & - \\ 881 896 1 & - & - & - & - & - & - & - \\
2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\ 882 897 2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\
3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\ 883 898 3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\
4 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 0) & - & 101~dB & 909 \\ 884 899 4 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 0) & - & 101~dB & 909 \\
5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\ 885 900 5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\
\hline 886 901 \hline
\end{tabular} 887 902 \end{tabular}
} 888 903 }
\end{table} 889 904 \end{table}
\renewcommand{\arraystretch}{1} 890 905 \renewcommand{\arraystretch}{1}
891 906
From these tables, we can first state that almost all configurations reach the targeted rejection 892 907 From these tables, we can first state that almost all configurations reach the targeted rejection
level or even better thanks to our underestimate of the cascade rejection as the sum of the 893 908 level or even better thanks to our underestimate of the cascade rejection as the sum of the
individual filter rejection. The only exception is for the monolithic case ($n = 1$) in 894 909 individual filter rejection. The only exception is for the monolithic case ($n = 1$) in
MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection. 895 910 MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection.
Futhermore, the area of the monolithic filter is twice as big as the two cascaded filters 896 911 Futhermore, the area of the monolithic filter is twice as big as the two cascaded filters
(1131 and 1760 arbitrary units v.s 547 and 903 arbitrary units for 60 and 80~dB rejection 897 912 (1131 and 1760 arbitrary units v.s 547 and 903 arbitrary units for 60 and 80~dB rejection
respectively). More generally, the more filters are cascaded, the lower the occupied area. 898 913 respectively). More generally, the more filters are cascaded, the lower the occupied area.
899 914
Like in previous section, the solver chooses always a little filter as first 900 915 Like in previous section, the solver chooses always a little filter as first
filter stage and the second one is often the biggest filter. This choice can be explained 901 916 filter stage and the second one is often the biggest filter. This choice can be explained
as in the previous section, with the solver using just enough bits not to degrade the input 902 917 as in the previous section, with the solver using just enough bits not to degrade the input
signal and in the second filter selecting a better filter to improve rejection without 903 918 signal and in the second filter selecting a better filter to improve rejection without
having too many bits in the output data. 904 919 having too many bits in the output data.
905 920
For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 906 921 For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal
number of filters is 4 so it did not chose any configuration for the last filter. Hence this 907 922 number of filters is 4 so it did not chose any configuration for the last filter. Hence this
solution is equivalent to the result for $n = 4$. 908 923 solution is equivalent to the result for $n = 4$.
909 924
The following graphs present the rejection for real data on the FPGA. In all the following 910 925 The following graphs present the rejection for real data on the FPGA. In all the following
figures, the solid line represents the actual rejection of the filtered 911 926 figures, the solid line represents the actual rejection of the filtered
data on the FPGA as measured experimentally and the dashed line is the noise level 912 927 data on the FPGA as measured experimentally and the dashed line is the noise level
given by the quadratic solver. 913 928 given by the quadratic solver.
914 929
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 915 930 Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40.
Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60. 916 931 Figure~\ref{fig:min_60} shows the rejection of the different configurations in the case of MIN/60.
Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80. 917 932 Figure~\ref{fig:min_80} shows the rejection of the different configurations in the case of MIN/80.
Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100. 918 933 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
919 934
% \begin{figure} 920 935 % \begin{figure}
% \centering 921 936 % \centering
% \includegraphics[width=\linewidth]{images/min_40} 922 937 % \includegraphics[width=\linewidth]{images/min_40}
% \caption{Signal spectrum for MIN/40} 923 938 % \caption{Signal spectrum for MIN/40}
% \label{fig:min_40} 924 939 % \label{fig:min_40}
% \end{figure} 925 940 % \end{figure}
% 926 941 %
% \begin{figure} 927 942 % \begin{figure}
% \centering 928 943 % \centering
% \includegraphics[width=\linewidth]{images/min_60} 929 944 % \includegraphics[width=\linewidth]{images/min_60}
% \caption{Signal spectrum for MIN/60} 930 945 % \caption{Signal spectrum for MIN/60}
% \label{fig:min_60} 931 946 % \label{fig:min_60}
% \end{figure} 932 947 % \end{figure}
% 933 948 %
% \begin{figure} 934 949 % \begin{figure}
% \centering 935 950 % \centering
% \includegraphics[width=\linewidth]{images/min_80} 936 951 % \includegraphics[width=\linewidth]{images/min_80}
% \caption{Signal spectrum for MIN/80} 937 952 % \caption{Signal spectrum for MIN/80}
% \label{fig:min_80} 938 953 % \label{fig:min_80}
% \end{figure} 939 954 % \end{figure}
% 940 955 %
% \begin{figure} 941 956 % \begin{figure}
% \centering 942 957 % \centering
% \includegraphics[width=\linewidth]{images/min_100} 943 958 % \includegraphics[width=\linewidth]{images/min_100}
% \caption{Signal spectrum for MIN/100} 944 959 % \caption{Signal spectrum for MIN/100}
% \label{fig:min_100} 945 960 % \label{fig:min_100}
% \end{figure} 946 961 % \end{figure}
947 962
% r2.14 et r2.15 et r2.16 948 963 % r2.14 et r2.15 et r2.16
\begin{figure} 949 964 \begin{figure}
\centering 950 965 \centering
\begin{subfigure}{\linewidth} 951 966 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_40} 952 967 \includegraphics[width=\linewidth]{images/min_40}
\caption{Signal spectrum for MIN/40} 953 968 \caption{Signal spectrum for MIN/40}
\label{fig:min_40} 954 969 \label{fig:min_40}
\end{subfigure} 955 970 \end{subfigure}
956 971
\begin{subfigure}{\linewidth} 957 972 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_60} 958 973 \includegraphics[width=\linewidth]{images/min_60}
\caption{Signal spectrum for MIN/60} 959 974 \caption{Signal spectrum for MIN/60}
\label{fig:min_60} 960 975 \label{fig:min_60}
\end{subfigure} 961 976 \end{subfigure}
962 977
\begin{subfigure}{\linewidth} 963 978 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_80} 964 979 \includegraphics[width=\linewidth]{images/min_80}
\caption{Signal spectrum for MIN/80} 965 980 \caption{Signal spectrum for MIN/80}
\label{fig:min_80} 966 981 \label{fig:min_80}
\end{subfigure} 967 982 \end{subfigure}
968 983
\begin{subfigure}{\linewidth} 969 984 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/min_100} 970 985 \includegraphics[width=\linewidth]{images/min_100}
\caption{Signal spectrum for MIN/100} 971 986 \caption{Signal spectrum for MIN/100}
\label{fig:min_100} 972 987 \label{fig:min_100}
\end{subfigure} 973 988 \end{subfigure}
\caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100} 974 989 \caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100}
\end{figure} 975 990 \end{figure}
976 991
We observe that all rejections given by the quadratic solver are close to the experimentally 977 992 We observe that all rejections given by the quadratic solver are close to the experimentally
measured rejection. All curves prove that the constraint to reach the target rejection is 978 993 measured rejection. All curves prove that the constraint to reach the target rejection is
respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters. 979 994 respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.
980 995
Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60; 981 996 Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60;
MIN/80 and MIN/100 \emph{i.e.} when the target rejection is fixed to 40, 60, 80 and 100~dB. We 982 997 MIN/80 and MIN/100 \emph{i.e.} when the target rejection is fixed to 40, 60, 80 and 100~dB. We
have taken care to extract solely the resources used by 983 998 have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and PL to 984 999 the FIR filters and remove additional processing blocks including FIFO and PL to
PS communication. 985 1000 PS communication.
986 1001
\renewcommand{\arraystretch}{1.2} 987 1002 \renewcommand{\arraystretch}{1.2}
\begin{table} 988 1003 \begin{table}
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 989 1004 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage_comp} 990 1005 \label{tbl:resources_usage_comp}
\centering 991 1006 \centering
{\scalefont{0.90} 992 1007 {\scalefont{0.90}
\begin{tabular}{|c|c|cccc|c|} 993 1008 \begin{tabular}{|c|c|cccc|c|}
\hline 994 1009 \hline
$n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline 995 1010 $n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 343 & 334 & 772 & - & \emph{17600} \\ 996 1011 & LUT & 343 & 334 & 772 & - & \emph{17600} \\
ifcs2018_journal_reponse.tex
Diff comments   View file @ 9c253d6
%Minor Revision - TUFFC-09469-2019 1 1 %Minor Revision - TUFFC-09469-2019
%Transactions on Ultrasonics, Ferroelectrics, and Frequency 2 2 %Transactions on Ultrasonics, Ferroelectrics, and Frequency
%Control (July 23, 2019 9:29 PM) 3 3 %Control (July 23, 2019 9:29 PM)
%To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr, 4 4 %To: arthur.hugeat@femto-st.fr, julien.bernard@femto-st.fr,
%gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr, 5 5 %gwenhael.goavec@femto-st.fr, pyb2@femto-st.fr, pierre-yves.bourgeois@femto-st.fr,
%jmfriedt@femto-st.fr 6 6 %jmfriedt@femto-st.fr
%CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu 7 7 %CC: giorgio.santarelli@institutoptique.fr, lewin@ece.drexel.edu
% 8 8 %
%Dear Mr. Arthur HUGEAT 9 9 %Dear Mr. Arthur HUGEAT
% 10 10 %
%Congratulations! Your manuscript 11 11 %Congratulations! Your manuscript
% 12 12 %
%MANUSCRIPT NO. TUFFC-09469-2019 13 13 %MANUSCRIPT NO. TUFFC-09469-2019
%MANUSCRIPT TYPE: Papers 14 14 %MANUSCRIPT TYPE: Papers
%TITLE: Filter optimization for real time digital processing of radiofrequency 15 15 %TITLE: Filter optimization for real time digital processing of radiofrequency
%signals: application to oscillator metrology 16 16 %signals: application to oscillator metrology
%AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, 17 17 %AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois,
%Pierre-Yves; Friedt, Jean-Michel 18 18 %Pierre-Yves; Friedt, Jean-Michel
% 19 19 %
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%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,amsmath} 77 77 \usepackage{fullpage,graphicx,amsmath, subcaption}
\begin{document} 78 78 \begin{document}
{\bf Reviewer: 1} 79 79 {\bf Reviewer: 1}
80 80
%Comments to the Author 81 81 %Comments to the Author
%In general, the language/grammar is adequate. 82 82 %In general, the language/grammar is adequate.
83 83
{\bf 84 84 {\bf
On page 2, "...allowing to save processing resource..." could be improved. % r1.1 85 85 On page 2, "...allowing to save processing resource..." could be improved. % r1.1 - fait
} 86 86 }
87 87
The sentence was split and now reads ``number of coefficients irrelevant: processing 88 88 The sentence was split and now reads ``number of coefficients irrelevant: processing
resources are hence saved by shrinking the filter length.'' 89 89 resources are hence saved by shrinking the filter length.''
90 90
{\bf 91 91 {\bf
On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 92 92 On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 - fait
the author meant.} 93 93 the author meant.}
94 94
Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s) 95 95 Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s)
Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.'' 96 96 Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
97 97
{\bf 98 98 {\bf
On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 99 99 On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 - fait
could be improved. 100 100 could be improved.
} 101 101 }
102 102
Indeed this paragraph has be written again and now reads as\\ 103 103 Indeed this paragraph has be written again and now reads as\\
``The first step of our approach is to model the DSP chain. Since we aim at only optimizing 104 104 ``The first step of our approach is to model the DSP chain. Since we aim at only optimizing
the filtering part of the signal processing chain, we have not included the PRN generator or the 105 105 the filtering part of the signal processing chain, we have not included the PRN generator or the
ADC in the model: the input data size and rate are considered fixed and defined by the hardware. 106 106 ADC in the model: the input data size and rate are considered fixed and defined by the hardware.
The filtering can be done in two ways, either by considering a single monolithic FIR filter 107 107 The filtering can be done in two ways, either by considering a single monolithic FIR filter
requiring many coefficients to reach the targeted noise rejection ratio, or by 108 108 requiring many coefficients to reach the targeted noise rejection ratio, or by
cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter. 109 109 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.
'' 110 110 ''
111 111
{\bf 112 112 {\bf
I appreciate that the authors attempted and document two optimizations: that % r1.4 - fait 113 113 I appreciate that the authors attempted and document two optimizations: that % r1.4 - fait
of maximum rejection ratio at fixed silicon area, as well as minimum silicon 114 114 of maximum rejection ratio at fixed silicon area, as well as minimum silicon
area for a fixed minimum rejection ratio. For non-experts, it might be very 115 115 area for a fixed minimum rejection ratio. For non-experts, it might be very
useful to compare the results of both optimization paths to the performance and 116 116 useful to compare the results of both optimization paths to the performance and
resource-utilization of generic low-pass filter gateware offered by device 117 117 resource-utilization of generic low-pass filter gateware offered by device
manufacturers. I appreciate also that the authors have presented source code 118 118 manufacturers. I appreciate also that the authors have presented source code
for examination online. 119 119 for examination online.
} 120 120 }
121 121
To compare the performance of our FIR filters and the performance of device 122 122 To compare the performance of our FIR filters and the performance of device
manufacturers generic filter, we have added a paragraph and a table at the 123 123 manufacturers generic filter, we have added a paragraph and a table at the
end of experiments section. We compare the resources consumption with the same 124 124 end of experiments section. We compare the resources consumption with the same
FIR coefficients set. 125 125 FIR coefficients set.
126 126
{\bf 127 127 {\bf
Reviewer: 2 128 128 Reviewer: 2
} 129 129 }
130 130
%Comments to the Author 131 131 %Comments to the Author
%In the Manuscript, the Authors describe an optimization methodology for filter 132 132 %In the Manuscript, the Authors describe an optimization methodology for filter
%design to be used in phase noise metrology. The methodology is general and can 133 133 %design to be used in phase noise metrology. The methodology is general and can
%be used for many aspects of the processing chain. In the Manuscript, the Authors 134 134 %be used for many aspects of the processing chain. In the Manuscript, the Authors
%focus on filtering and shifting while the other aspects, in particular decimation, 135 135 %focus on filtering and shifting while the other aspects, in particular decimation,
%will be considered in a future work. The optimization problem is modelled 136 136 %will be considered in a future work. The optimization problem is modelled
%theoretically and then solved by means of a commercial software. The solutions 137 137 %theoretically and then solved by means of a commercial software. The solutions
%are tested experimentally on the Redpitaya platform with synthetic and real 138 138 %are tested experimentally on the Redpitaya platform with synthetic and real
%white noises. Two cases are considered as a function of the number of filters: 139 139 %white noises. Two cases are considered as a function of the number of filters:
%maximum rejection given a fixed amount of resources and minimum resource 140 140 %maximum rejection given a fixed amount of resources and minimum resource
%utilization given a fixed amount of rejection. 141 141 %utilization given a fixed amount of rejection.
%The Authors find that filtering improves significantly when the number of 142 142 %The Authors find that filtering improves significantly when the number of
%filters increases. 143 143 %filters increases.
%A lot of work has been done in generalizing and automating the procedure so 144 144 %A lot of work has been done in generalizing and automating the procedure so
%that different approaches can be investigated quickly and efficiently. The 145 145 %that different approaches can be investigated quickly and efficiently. The
%results presented in the Manuscript seem to be just a case study based on 146 146 %results presented in the Manuscript seem to be just a case study based on
%the particular criterion chosen by the Authors. Different criteria, in 147 147 %the particular criterion chosen by the Authors. Different criteria, in
%general, could lead to different results and it is important to consider 148 148 %general, could lead to different results and it is important to consider
%carefully the criterion adopted by the Authors, in order to check if it 149 149 %carefully the criterion adopted by the Authors, in order to check if it
%is adequate to compare the performance of filters and if multi-stage 150 150 %is adequate to compare the performance of filters and if multi-stage
%filters are really superior than monolithic filters. 151 151 %filters are really superior than monolithic filters.
152 152
{\bf 153 153 {\bf
By observing the results presented in fig. 10-16, it is clear that the % r2.1 - fait 154 154 By observing the results presented in fig. 10-16, it is clear that the % r2.1
performances of multi-stage filters are obtained at the expense of their 155 155 performances of multi-stage filters are obtained at the expense of their
selectivity and, in this sense, the filters presented in these figures 156 156 selectivity and, in this sense, the filters presented in these figures
are not equivalent. For example, in Fig. 14, at the limit of the pass band, 157 157 are not equivalent. For example, in Fig. 14, at the limit of the pass band,
the attenuation is almost 15 dB for n = 5, while it is not noticeable for 158 158 the attenuation is almost 15 dB for n = 5, while it is not noticeable for
n = 1. 159 159 n = 1.
} 160 160 }
161 161
We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined 162 162 We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined
the bandpass and the bandstop of the filter. 163 163 the bandpass and the bandstop of the filter.
164 164
We are aware of this non equivalence but we think that difference is not due to 165 165 % We are aware of this non equivalence but we think that difference is not due to
the cascaded filters but due to the definition of rejection criterion on the passband. 166 166 % the cascaded filters but due to the definition of rejection criterion on the passband.
Indeed, in this article we have choose to take the summation of absolute values divide 167 167 % Indeed, in this article we have choose to take the summation of absolute values divide
by the bandwidth but this criterion is maybe too permissive and when we cascade 168 168 % by the bandwidth but this criterion is maybe too permissive and when we cascade
some filters this impact is more important. 169 169 % some filters this impact is more important.
170 %
171 % However if we change the passband
172 % criterion by the summation of absolute value in passband, weighting given to the
173 % passband ripples are too strong and the solver are too restricted to provide
174 % any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in
175 % passband, the rejection evaluation are too close form the original criterion and
176 % the result will not be improved.
177 %
178 % In this article, we will highlight the methodology instead of the filter conception.
179 % Even if our rejection criterion is not the best, our methodology was not impacted
180 % by this. So to improve the results, we can choose another criterion to be more
181 % selective in passband but it is not the main objective of our article.
170 182
However if we change the passband 171 183 We are aware of this equivalence but to limit this ripples in passband we need to
criterion by the summation of absolute value in passband, weighting given to the 172 184 enforce the criterion in passband. If we takes a strong constraint like the sum of
passband ripples are too strong and the solver are too restricted to provide 173 185 absolute values in passband. This criterion si too selective because it considers
any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in 174 186 all bin on passband while on stopband we consider only the bin with the minimal
passband, the rejection evaluation are too close form the original criterion and 175 187 rejection. The figure~\ref{fig:letter_sum_criterion} exhibits the results with this
the result will not be improved. 176 188 criterion for the case MAX/1000. With this criterion, the solver find an optimal
189 solution with only two filters in expend of the resource consumption.
177 190
In this article, we will highlight the methodology instead of the filter conception. 178
Even if our rejection criterion is not the best, our methodology was not impacted 179
by this. So to improve the results, we can choose another criterion to be more 180
selective in passband but it is not the main objective of our article. 181
182 191
192
193 If we relax a little the criterion on passband with taking only the maximum absolute
194 value, we will penalize the ripple peak on passband. The figure~\ref{fig:letter_max_criterion}
195 shows the results for the case MAX/1000. There as almost no difference with the
196 article results. Indeed the only little change are on the case $i = 4$ and $i = 5$
197 which they have some minor differences on coefficients choices.
198
199 \begin{figure}[h!tb]
200 \centering
201 \begin{subfigure}{0.48\linewidth}
202 \includegraphics[width=\linewidth]{images/letter_sum_criterion}
203 \caption{Results for the case MAX/1000 with as criterion on passband the sum absolute values}
204 \label{fig:letter_sum_criterion}
205 \end{subfigure}
206 \begin{subfigure}{0.48\linewidth}
207 \includegraphics[width=\linewidth]{images/letter_max_criterion}
208 \caption{Results for the case MAX/1000 with as criterion on passband the maximum absolute value}
209 \label{fig:letter_max_criterion}
210 \end{subfigure}
211 \end{figure}
212
213 Finally, if we ponder the maximum absolute on passband, we should improve the result.
214 We have arbitrary pondered by 5 the maximum. Even with this weighting, the solver
215 choose the same coefficient set.
216
217 To conclude, find a better criterion to avoid the ripples on the passband is difficult.
218 In this article we are focused on the methodology so even if our criterion could
219 be improved, our methodology still the same and it works independently of rejection criterion.
220
% %Peut etre refaire une serie de simulation dans lesquelles on impose une coupure 183 221 % %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 184 222 % %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 185 223 % %au critere qu'on lui impose, et que la coupure moins raide n'est pas intrinseque
% %a la cascade de filtres. 186 224 % %a la cascade de filtres.
% %AH: Je finis les corrections, je poste l'article revu et pendant ce temps j'essaie de 187 225 % %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 188 226 % %relancer des expérimentations. Si j'arrive à les finir à temps, je les intégrerai
% 189 227 %
% densité spectrale de la bande passante 190 228 % densité spectrale de la bande passante
% sum des valeurs absolues / largeur de la bande passante (1/N) vs max dans la bande de coupure 191 229 % sum des valeurs absolues / largeur de la bande passante (1/N) vs max dans la bande de coupure
% 192 230 %
% JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait 193 231 % JMF : il n'a pas tord, la coupure est bcp moins franche a 5 filtres qu'a 1. Ca se voyait
% moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB 194 232 % moins avant de moyenner les fonctions de transfert, mais il y a bien une 15aine de dB
% quand on cascade 5 filtres ! 195 233 % quand on cascade 5 filtres !
% 196 234 %
% Dire que la chute n'est pas du à la casacade mais à notre critère de rejection 197 235 % Dire que la chute n'est pas du à la casacade mais à notre critère de rejection
198 236
{\bf 199 237 {\bf
The reason is in the criterion that considers the average attenuation in % r2.2 - fait 200 238 The reason is in the criterion that considers the average attenuation in % r2.2
the pass band. This criterion does not take into account the maximum attenuation 201 239 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 202 240 in this region, which is a very important parameter for specifying a filter
and for evaluating its performance. For example, with this criterion, a 203 241 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 204 242 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 205 243 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. 206 244 and in the results that are obtained and has to be reconsidered.
} 207 245 }
208 246
See above: If we choose the maximum absolute value in passband, we penalize the 209 247 See above: Choose a criterion is difficult and depending on the context. The main
case with 10 dB of ripple. 210 248 contribution on this paper is the methodology not the criterion to quantify the
249 rejection.
211 250
% The manuscript erroneously stated that we considered the mean of the absolute 212 251 % The manuscript erroneously stated that we considered the mean of the absolute
% value within the bandpass: the manuscript has now been corrected to properly state 213 252 % value within the bandpass: the manuscript has now been corrected to properly state
% the selected criterion, namely the {\em sum} of the absolute value, so that any 214 253 % the selected criterion, namely the {\em sum} of the absolute value, so that any
% ripple in the bandpass will reduce the chances of a given filter set from being 215 254 % ripple in the bandpass will reduce the chances of a given filter set from being
% selected. The manuscript now states ``Our criterion to compute the filter rejection considers 216 255 % selected. The manuscript now states ``Our criterion to compute the filter rejection considers
% % r2.8 et r2.2 r2.3 217 256 % % r2.8 et r2.2 r2.3
% the maximum magnitude within the stopband, to which the {sum of the absolute values 218 257 % the maximum magnitude within the stopband, to which the {sum of the absolute values
% within the passband is subtracted to avoid filters with excessive ripples}.'' 219 258 % within the passband is subtracted to avoid filters with excessive ripples}.''
220 259
{\bf 221 260 {\bf
I strongly suggest to re-run the analysis with a criterion that takes also % r2.3 -fait 222 261 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 223 262 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 224 263 fixing its value to a typical one, as it has been done for the transition
bandwidth. 225 264 bandwidth.
} 226 265 }
227 266
See above: the absolute value within the passband will reject filters with 228 267 See above: the absolute value within the passband will reject filters with
excessive ripples, including excessive attenuation, within the passband. 229 268 excessive ripples, including excessive attenuation, within the passband.
230 269
% TODO: test max(stopband) - max(abs(passband)) 231 270 % TODO: test max(stopband) - max(abs(passband))
232 271
{\bf 233 272 {\bf
In addition, I suggest to address the following points: % r2.4 234 273 In addition, I suggest to address the following points: % r2.4 - fait
- Page 1, line 50: the Authors state that IIR have shorter impulse response 235 274 - 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. 236 275 than FIR. This is not true in general. The sentence should be reconsidered.
} 237 276 }
238 277
We have not stated that the IIR has a shorter impulse response but a shorter lag. 239 278 We have not stated that the IIR has a shorter impulse response but a shorter lag.
Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filters 240 279 Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filters
have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing 241 280 have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing
the first output, an IIR will start providing outputs only 5 time steps after the initial 242 281 the first output, an IIR will start providing outputs only 5 time steps after the initial
input starts feeding the IIR. Hence, the issue we address here is lag and not impulse 243 282 input starts feeding the IIR. Hence, the issue we address here is lag and not impulse
response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue 244 283 response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue
in a openloop phase noise characterization instrument, the large 245 284 in a openloop phase noise characterization instrument, the large
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 246 285 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
is not considered as an issue as would be in a closed loop system in which lag aims at being 247 286 is not considered as an issue as would be in a closed loop system in which lag aims at being
minimized to avoid oscillation conditions.'' 248 287 minimized to avoid oscillation conditions.''
249 288
{\bf 250 289 {\bf
- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 251 290 - Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 - fait
this transition bandwidth and if it is a typical requirement for phase-noise 252 291 this transition bandwidth and if it is a typical requirement for phase-noise
metrology. 253 292 metrology.
} 254 293 }
255 294
The purpose of the paper is to demonstrate how a given filter shape can be achieved by 256 295 The purpose of the paper is to demonstrate how a given filter shape can be achieved by
minimizing varous resource criteria. Indeed the stopband and bandpass boundaries can 257 296 minimizing varous resource criteria. Indeed the stopband and bandpass boundaries can
be questioned: we have selected this filter shape as a typical anti-aliasing filter considering 258 297 be questioned: we have selected this filter shape as a typical anti-aliasing filter considering
the the dataflow is to be halved. Hence, selecting a cutoff frequency of 40\% the initial 259 298 the the dataflow is to be halved. Hence, selecting a cutoff frequency of 40\% the initial
Nyquist frequency prevents noise from reaching baseband after decimating the dataflow by a 260 299 Nyquist frequency prevents noise from reaching baseband after decimating the dataflow by a
factor of 2. Such ideas are now stated explicitly in the text as ``Throughout this demonstration, 261 300 factor of 2. Such ideas are now stated explicitly in the text as ``Throughout this demonstration,
we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\% 262 301 we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\%
of the Nyquist frequency to the end of the band, as would be typically selected to prevent 263 302 of the Nyquist frequency to the end of the band, as would be typically selected to prevent
aliasing before decimating the dataflow by 2. The method is however generalized to any filter 264 303 aliasing before decimating the dataflow by 2. The method is however generalized to any filter
shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid} 265 304 shape as long as it is defined from the initial modelling steps: Fig. \ref{fig:rejection_pyramid}
as described below is indeed unique for each filter shape.'' 266 305 as described below is indeed unique for each filter shape.''
267 306
{\bf 268 307 {\bf
- The impact of the coefficient resolution is discussed. What about the % r2.6 - fait 269 308 - 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 270 309 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 271 310 used in the analysis? If not, how is it changed with respect to the
coefficient resolution? 272 311 coefficient resolution?
} 273 312 }
274 313
We have now stated in the beginning of the document that ``we have not included the PRN generator 275 314 We have now stated in the beginning of the document that ``we have not included the PRN generator
or the ADC in the model: the input data size and rate are considered fixed and defined by the 276 315 or the ADC in the model: the input data size and rate are considered fixed and defined by the
hardware.'' so indeed the input datastream resolution is considered as a given. 277 316 hardware.'' so indeed the input datastream resolution is considered as a given.
278 317
{\bf 279 318 {\bf
- Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait 280 319 - Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait
Fig. 5 can be removed. 281 320 Fig. 5 can be removed.
321 }
322
323 Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats
324
325 {\bf
- Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait 282 326 - Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait
It should be ``minimum'' 283 327 It should be ``minimum''
} 284 328 }
AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max 285
de la magnitude. J'ai corrigé en ce sens. 286
287 329
Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats 288 330 This typo has been corrected.
289 331
{\bf 290 332 {\bf
- Page e, line 55, second column: ``takin'' % r2.9 - fait 291 333 - Page e, line 55, second column: ``takin'' % r2.9 - fait
- Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait 292 334 - Page 3, line 58: ``pessimistic'' should be replaced with ``conservative'' % r2.10 - fait
- Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait 293 335 - Page 4, line 17: ``meaning'' $\rightarrow$ ``this means'' % r2.11 - fait
} 294 336 }
295 337
All typos and grammatical errors have been corrected. 296 338 All typos and grammatical errors have been corrected.
297 339
{\bf 298 340 {\bf
- Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait 299 341 - 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? 300 342 these particular configurations? Are they chosen automatically?
} 301 343 }
C'est le nombre de coefficients et un taille raisonnable 302 344 C'est le nombre de coefficients et un taille raisonnable
Troncature de la pyramide 303 345 Troncature de la pyramide
304 346
See below: we have added a better description of $p$ during the transformation explanation. 305 347 See below: we have added a better description of $p$ during the transformation explanation.
``we introduce $p$ FIR configurations. 306 348 ``we introduce $p$ FIR configurations.
This variable must be defined by the user, it represent the number of different 307 349 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} 308 350 set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave)'' 309 351 functions from GNU Octave)''
310 352
{\bf 311 353 {\bf
- Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait 312 354 - Page 4, line 31: how does the delta function transform model from non-linear % r2.13
and non-quadratic to a quadratic?} 313 355 and non-quadratic to a quadratic?}
314 356
The first model is non-quadratic but when we introduce the $p$ configurations, 315 357 The first model is non-quadratic but when we introduce the $p$ configurations,
we can estimate the function $F$ by computing 316 358 we can estimate the function $F$ by computing
the rejection for each configuration, so the model become quadratic because we have 317 359 the rejection for each configuration, so the model become quadratic because we have
some multiplication between variables. With the definition of $\delta_{ij}$ we can 318 360 some multiplication between variables. With the definition of $\delta_{ij}$ we can
replace the multiplication between variables by multiplication with binary variable and 319 361 replace the multiplication between variables by multiplication with binary variable and
this one can be linearise as follow:\\ 320 362 this one can be linearise as follow:\\
$y$ is a binary variable \\ 321 363 $y$ is a binary variable \\
$x$ is a real variable bounded by $X^{max}$ \\ 322 364 $x$ is a real variable bounded by $X^{max}$ \\
\begin{equation*} 323 365 \begin{equation*}
m = x \times y \implies 324 366 m = x \times y \implies
\left \{ 325 367 \left \{
\begin{split} 326 368 \begin{split}
m & \geq 0 \\ 327 369 m & \geq 0 \\
m & \leq y \times X^{max} \\ 328 370 m & \leq y \times X^{max} \\
m & \leq x \\ 329 371 m & \leq x \\
m & \geq x - (1 - y) \times X^{max} \\ 330 372 m & \geq x - (1 - y) \times X^{max} \\
\end{split} 331 373 \end{split}
\right . 332 374 \right .
\end{equation*} 333 375 \end{equation*}
Gurobi does the linearization so we don't explain this step to keep the model more 334 376 Gurobi does the linearization so we don't explain this step to keep the model more
simple. However, to improve the transformation explanation we have rewrote the 335 377 simple. However, to improve the transformation explanation we have rewrote the
paragraph ``This model is non-linear and even non-quadratic...''. 336 378 paragraph ``This model is non-linear and even non-quadratic...''.
337 379
JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article 338 380 % JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
je ne comprends pas comment ca repond a la question 339 381 % je ne comprends pas comment ca repond a la question
340 382 %
AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux. 341 383 % AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
342 384 %
Le problème n'est pas linéaire car nous multiplions des variables 343 385 % Le problème n'est pas linéaire car nous multiplions des variables
entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent 344 386 % entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique 345 387 % des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore 346 388 % quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre 347 389 % quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
$\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible 348 390 % $\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre 349 391 % de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée, 350 392 % cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur 351 393 % nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la 352 394 % qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
linéarisation pour nous. 353 395 % linéarisation pour nous.
354 396
355 397
{\bf 356 398 {\bf
- Captions of figure and tables are too minimal. % r2.14 357 399 - Captions of figure and tables are too minimal. % r2.14
} 358 400 }
We have change the captions of fig 10-16. 359 401 We have change the captions of fig 10-16.
360 402
{\bf 361 403 {\bf
- Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait 362 404 - 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. 363 405 of a single figure. Same for fig. 13-16.
} 364 406 }
We add two sub figure to group the fig.10-12 and fig. 13-16 365 407 We add two sub figure to group the fig.10-12 and fig. 13-16
366 408
{\bf 367 409 {\bf
- Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait 368 410 - 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 369 411 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. 370 412 differences among the curves. I suggest to reduce the noise below 1 dBpk-pk.
} 371 413 }
372 414
Indeed averaging had been omitted during post-processing and figure generation: we 373 415 Indeed averaging had been omitted during post-processing and figure generation: we
are grateful to the reviewer for emphasizing this point which has now been corrected. All spectra 374 416 are grateful to the reviewer for emphasizing this point which has now been corrected. All spectra
now exhibit sub-dBpk-pl line thickness. 375 417 now exhibit sub-dBpk-pl line thickness.
376 418
We believe these updates to the manuscript have improved the presentation and made clearer 377 419 We believe these updates to the manuscript have improved the presentation and made clearer
some of the shortcomings of the initial draft: we are greatful to the reviewers for pointing 378 420 some of the shortcomings of the initial draft: we are greatful to the reviewers for pointing
out these issues. 379 421 out these issues.
380 422
Best wishes, A. Hugeat 381 423 Best wishes, A. Hugeat
382 424
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