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% fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee 1 1 % fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee
% demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de 2 2 % demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de
% rejection par bit et perte si moins de bits que rejection/6 3 3 % rejection par bit et perte si moins de bits que rejection/6
% developper programme lineaire en incluant le decalage de bits 4 4 % developper programme lineaire en incluant le decalage de bits
% insister que avant on etait synthetisable mais pas implementable, alors que maintenant on 5 5 % insister que avant on etait synthetisable mais pas implementable, alors que maintenant on
% implemente et on demontre que ca tourne 6 6 % implemente et on demontre que ca tourne
% gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ? 7 7 % gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ?
% Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer 8 8 % Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer
% (zedboard ou redpit) 9 9 % (zedboard ou redpit)
10 10
% label schema : verifier que "argumenter de la cascade de FIR" est fait 11 11 % label schema : verifier que "argumenter de la cascade de FIR" est fait
12 12
\documentclass[a4paper,journal]{IEEEtran/IEEEtran} 13 13 \documentclass[a4paper,journal]{IEEEtran/IEEEtran}
\usepackage{graphicx,color,hyperref} 14 14 \usepackage{graphicx,color,hyperref}
\usepackage{amsfonts} 15 15 \usepackage{amsfonts}
\usepackage{amsthm} 16 16 \usepackage{amsthm}
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\usepackage{amsmath} 18 18 \usepackage{amsmath}
\usepackage{algorithm2e} 19 19 \usepackage{algorithm2e}
\usepackage{url,balance} 20 20 \usepackage{url,balance}
\usepackage[normalem]{ulem} 21 21 \usepackage[normalem]{ulem}
\usepackage{tikz} 22 22 \usepackage{tikz}
\usetikzlibrary{positioning,fit} 23 23 \usetikzlibrary{positioning,fit}
\usepackage{multirow} 24 24 \usepackage{multirow}
\usepackage{scalefnt} 25 25 \usepackage{scalefnt}
\usepackage{caption} 26 26 \usepackage{caption}
\usepackage{subcaption} 27 27 \usepackage{subcaption}
28 28
% correct bad hyphenation here 29 29 % correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor} 30 30 \hyphenation{op-tical net-works semi-conduc-tor}
\textheight=26cm 31 31 \textheight=26cm
\setlength{\footskip}{30pt} 32 32 \setlength{\footskip}{30pt}
\pagenumbering{gobble} 33 33 \pagenumbering{gobble}
\begin{document} 34 34 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 35 35 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 36 36 to oscillator metrology}
37 37
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 38 38 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 39 39 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\ 40 40 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\ 41 41 \IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\
\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\ 42 42 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 43 43 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 44 44 }
\maketitle 45 45 \maketitle
\thispagestyle{plain} 46 46 \thispagestyle{plain}
\pagestyle{plain} 47 47 \pagestyle{plain}
\newtheorem{definition}{Definition} 48 48 \newtheorem{definition}{Definition}
49 49
\begin{abstract} 50 50 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 51 51 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 52 52 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 53 53 of ultrastable clocks, stringent filtering requirements are defined by spurious signal or
noise rejection needs. Since real time radiofrequency processing must be performed in a 54 54 noise rejection needs. Since real time radiofrequency processing must be performed in a
Field Programmable Array to meet timing constraints, we investigate optimization strategies 55 55 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 56 56 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. The 57 57 required and keeping timing constraints within the targeted measurement bandwidths. The
presented technique is applicable to scheduling any sequence of processing blocks characterized 58 58 presented technique is applicable to scheduling any sequence of processing blocks characterized
by a throughput, resource occupation and performance tabulated as a function of configuration 59 59 by a throughput, resource occupation and performance tabulated as a function of configuration
characateristics, as is the case for filters with their coefficients and resolution yielding 60 60 characateristics, as is the case for filters with their coefficients and resolution yielding
rejection and number of multipliers. 61 61 rejection and number of multipliers.
\end{abstract} 62 62 \end{abstract}
63 63
\begin{IEEEkeywords} 64 64 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 65 65 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 66 66 \end{IEEEkeywords}
67 67
\section{Digital signal processing of ultrastable clock signals} 68 68 \section{Digital signal processing of ultrastable clock signals}
69 69
Analog oscillator phase noise characteristics are classically performed by downconverting 70 70 Analog oscillator phase noise characteristics are classically performed by downconverting
the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband, 71 71 the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,
followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In 72 72 followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In
a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by 73 73 a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by
multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}. 74 74 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
75 75
\begin{figure}[h!tb] 76 76 \begin{figure}[h!tb]
\begin{center} 77 77 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 78 78 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 79 79 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 80 80 \caption{Fully digital oscillator phase noise characterization: the Device Under Test
(DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and 81 81 (DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and
downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals 82 82 downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals
and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite 83 83 and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite
Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays 84 84 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 85 85 the spectral characteristics of the phase fluctuations.}
\label{schema} 86 86 \label{schema}
\end{figure} 87 87 \end{figure}
88 88
As with the analog mixer, 89 89 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 90 90 the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as
well as the generation of the frequency sum signal in addition to the frequency difference. 91 91 well as the generation of the frequency sum signal in addition to the frequency difference.
These unwanted spectral characteristics must be rejected before decimating the data stream 92 92 These unwanted spectral characteristics must be rejected before decimating the data stream
for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the 93 93 for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
downconverter 94 94 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 95 95 and the decimation processing blocks are core characteristics of an oscillator characterization
system, and must reject out-of-band signals below the targeted phase noise -- typically in the 96 96 system, and must reject out-of-band signals below the targeted phase noise -- typically in the
sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will 97 97 sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will
use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency 98 98 use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency
datastream: optimizing the performance of the filter while reducing the needed resources is 99 99 datastream: optimizing the performance of the filter while reducing the needed resources is
hence tackled in a systematic approach using optimization techniques. Most significantly, we 100 100 hence tackled in a systematic approach using optimization techniques. Most significantly, we
tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with 101 101 tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with
tunable number of coefficients and tunable number of bits representing the coefficients and the 102 102 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 103 103 data being processed.
104 104
\section{Finite impulse response filter} 105 105 \section{Finite impulse response filter}
106 106
We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined 107 107 We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined
by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the 108 108 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 109 109 outputs $y_k$
\begin{align} 110 110 \begin{align}
y_n=\sum_{k=0}^N b_k x_{n-k} 111 111 y_n=\sum_{k=0}^N b_k x_{n-k}
\label{eq:fir_equation} 112 112 \label{eq:fir_equation}
\end{align} 113 113 \end{align}
114 114
As opposed to an implementation on a general purpose processor in which word size is defined by the 115 115 As opposed to an implementation on a general purpose processor in which word size is defined by the
processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since 116 116 processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since
not only the coefficient values and number of taps must be defined, but also the number of bits 117 117 not only the coefficient values and number of taps must be defined, but also the number of bits
defining the coefficients and the sample size. For this reason, and because we consider pipeline 118 118 defining the coefficients and the sample size. For this reason, and because we consider pipeline
processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency 119 119 processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency
signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but 120 120 signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but
the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language 121 121 the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language
(VHDL) level. 122 122 (VHDL) level.
Since latency is not an issue in a openloop phase noise characterization instrument, 123 123 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. 126 126 is not considered as an issue as would be in a closed loop system.
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, making the large number of coefficients irrelevant: processing 147 147 taps become null, making the large number of coefficients irrelevant: processing
resources 148 148 resources
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 performance (i.e. rejection or ripples in the bandpass for filters) and 166 166 of DSP blocks such as performance (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 performance. In our approach, the solution of the 170 170 minimizing resource occupation for a given performance. 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 performance and resource occupation can be determined. 187 187 of skeleton blocks as long as performance and resource occupation can be determined.
Hence, 188 188 Hence,
in this paper we will apply our methodology on simple DSP chains: a white noise input signal 189 189 in this paper we will apply our methodology on simple DSP chains: a white noise input signal
is generated using a Pseudo-Random Number (PRN) generator or 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
The first step of our approach is to model the DSP chain. Since we aim at only optimizing 198 198 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 199 199 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. 200 200 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 201 201 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 202 202 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. 203 203 cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter.
204 204
After each filter we leave the possibility of shifting the filtered data to consume 205 205 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 206 206 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). 207 207 and a shifter (the shift could be omitted if we do not need to divide the filtered data).
208 208
\subsection{Model of a FIR filter} 209 209 \subsection{Model of a FIR filter}
210 210
A cascade of filters is composed of $n$ FIR stages. In stage $i$ ($1 \leq i \leq n$) 211 211 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 $\pi^C_i$ 212 212 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 213 213 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} 214 214 the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage}
shows a filtering stage. 215 215 shows a filtering stage.
216 216
\begin{figure} 217 217 \begin{figure}
\centering 218 218 \centering
\begin{tikzpicture}[node distance=2cm] 219 219 \begin{tikzpicture}[node distance=2cm]
\node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ; 220 220 \node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;
\node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ; 221 221 \node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;
\node (Start) [left of=FIR] { } ; 222 222 \node (Start) [left of=FIR] { } ;
\node (End) [right of=Shift] { } ; 223 223 \node (End) [right of=Shift] { } ;
224 224
\node[draw,fit=(FIR) (Shift)] (Filter) { } ; 225 225 \node[draw,fit=(FIR) (Shift)] (Filter) { } ;
226 226
\draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ; 227 227 \draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;
\draw[->] (FIR) -- (Shift) ; 228 228 \draw[->] (FIR) -- (Shift) ;
\draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ; 229 229 \draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;
\end{tikzpicture} 230 230 \end{tikzpicture}
\caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)} 231 231 \caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}
\label{fig:fir_stage} 232 232 \label{fig:fir_stage}
\end{figure} 233 233 \end{figure}
234 234
FIR $i$ has been characterized through numerical simulation as able to reject $F(C_i, \pi_i^C)$ dB. 235 235 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 236 236 This rejection has been computed using GNU Octave software FIR coefficient design functions
(\texttt{firls} and \texttt{fir1}). 237 237 (\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. 238 238 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, 239 239 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. 240 240 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. 241 241 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
With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter 243 243 With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter
transfer function. 244 244 transfer function.
Comparing the performance between FIRs requires however defining a unique criterion. As shown in figure~\ref{fig:fir_mag}, 245 245 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 246 246 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}. Throughout this demonstration, 247 247 bandpass ripples as emphasized in \cite{lim_1988,lim_1996}. Throughout this demonstration,
we arbitrarily set a bandpass of 40\% of the Nyquist frequency and a bandstop from 60\% 248 248 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 249 249 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 250 250 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 modeling steps: Fig. \ref{fig:rejection_pyramid} 251 251 shape as long as it is defined from the initial modeling steps: Fig. \ref{fig:rejection_pyramid}
as described below is indeed unique for each filter shape. 252 252 as described below is indeed unique for each filter shape.
253 253
\begin{figure} 254 254 \begin{figure}
\begin{center} 255 255 \begin{center}
\scalebox{0.8}{ 256 256 \scalebox{0.8}{
\centering 257 257 \centering
\begin{tikzpicture}[scale=0.3] 258 258 \begin{tikzpicture}[scale=0.3]
\draw[<->] (0,15) -- (0,0) -- (21,0) ; 259 259 \draw[<->] (0,15) -- (0,0) -- (21,0) ;
\draw[thick] (0,12) -- (8,12) -- (20,0) ; 260 260 \draw[thick] (0,12) -- (8,12) -- (20,0) ;
261 261
\draw (0,14) node [left] { $P$ } ; 262 262 \draw (0,14) node [left] { $P$ } ;
\draw (20,0) node [below] { $f$ } ; 263 263 \draw (20,0) node [below] { $f$ } ;
264 264
\draw[>=latex,<->] (0,14) -- (8,14) ; 265 265 \draw[>=latex,<->] (0,14) -- (8,14) ;
\draw (4,14) node [above] { passband } node [below] { $40\%$ } ; 266 266 \draw (4,14) node [above] { passband } node [below] { $40\%$ } ;
267 267
\draw[>=latex,<->] (8,14) -- (12,14) ; 268 268 \draw[>=latex,<->] (8,14) -- (12,14) ;
\draw (10,14) node [above] { transition } node [below] { $20\%$ } ; 269 269 \draw (10,14) node [above] { transition } node [below] { $20\%$ } ;
270 270
\draw[>=latex,<->] (12,14) -- (20,14) ; 271 271 \draw[>=latex,<->] (12,14) -- (20,14) ;
\draw (16,14) node [above] { stopband } node [below] { $40\%$ } ; 272 272 \draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;
273 273
\draw[>=latex,<->] (16,12) -- (16,8) ; 274 274 \draw[>=latex,<->] (16,12) -- (16,8) ;
\draw (16,10) node [right] { rejection } ; 275 275 \draw (16,10) node [right] { rejection } ;
276 276
\draw[dashed] (8,-1) -- (8,14) ; 277 277 \draw[dashed] (8,-1) -- (8,14) ;
\draw[dashed] (12,-1) -- (12,14) ; 278 278 \draw[dashed] (12,-1) -- (12,14) ;
279 279
\draw[dashed] (8,12) -- (16,12) ; 280 280 \draw[dashed] (8,12) -- (16,12) ;
\draw[dashed] (12,8) -- (16,8) ; 281 281 \draw[dashed] (12,8) -- (16,8) ;
282 282
\end{tikzpicture} 283 283 \end{tikzpicture}
} 284 284 }
\end{center} 285 285 \end{center}
\caption{Shape of the filter transmitted power $P$ as a function of frequency $f$: 286 286 \caption{Shape of the filter transmitted power $P$ as a function of frequency $f$:
the passband is considered to occupy the initial 40\% of the Nyquist frequency range, 287 287 the passband is considered to occupy the initial 40\% of the Nyquist frequency range,
the stopband the last 40\%, allowing 20\% transition width.} 288 288 the stopband the last 40\%, allowing 20\% transition width.}
\label{fig:fir_mag} 289 289 \label{fig:fir_mag}
\end{figure} 290 290 \end{figure}
291 291
In the transition band, the behavior of the filter is left free, we only define the passband and the stopband characteristics. 292 292 In the transition band, the behavior of the filter is left free, we only define the passband and the stopband characteristics.
% r2.7 293 293 % r2.7
Initial considered criteria include the mean value of the stopband rejection which yields unacceptable results since notches 294 294 Initial considered criteria include the mean value of the stopband rejection which yields unacceptable results since notches
overestimate the rejection capability of the filter. 295 295 overestimate the rejection capability of the filter.
% Furthermore, the losses within 296 296 % 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.
{\color{red} In intermediate criterion considered the minimal rejection within the stopband, to which the sum of the absolute values 298 298 {\color{red} An intermediate criterion considered the maximal rejection within the stopband, to which the sum of the absolute values
299 % JMF : je fais le choix de remplacer minimal par maximal rejection pour etre coherent avec caption de Fig custom_criterion mais surtout parceque
300 % rejection me semble plus convaincant si on la maximise (il me semble que -120 dB de S21 signifie 120 dB de rejection donc on veut maximiser)
within the passband is subtracted to avoid filters with excessive ripples, normalized to the 299 301 within the passband is subtracted to avoid filters with excessive ripples, normalized to the
bin width to remain consistent with the passband criterion (dBc/Hz units in all cases). 300 302 bin width to remain consistent with the passband criterion (dBc/Hz units in all cases).
In this case, when we cascaded too filters with a excessive deviation in passband ($>$ 1~dB), 301 303 In this case, cascading too many filters with individual excessive ($>$ 1~dB) passband ripples
the final deviation in passband may be too considerable ($>$ 10~dB). Hence our final 302 304 led to unacceptable ($>$ 10~dB) final ripple levels, especially close to the transition band.
criterion always take the minimal rejection in stopband but we substract the maximal 303 305 Hence, the final criterion considers the minimal rejection in the stopband to which the
amplitude in passband (maximum value minus the minimum value). If this amplitude is 304 306 the maximal amplitude in the passband (maximum value minus the minimum value) is substracted, with
greater than 1~dB, we discard the filter.} 305 307 a 1~dB threshold on the latter quantity over which the filter is discarded.}
% Our final criterion to compute the filter rejection considers 306 308 % Our final criterion to compute the filter rejection considers
% % r2.8 et r2.2 r2.3 307 309 % % r2.8 et r2.2 r2.3
% the minimal rejection within the stopband, to which the sum of the absolute values 308 310 % the minimal rejection within the stopband, to which the sum of the absolute values
% within the passband is subtracted to avoid filters with excessive ripples, normalized to the 309 311 % within the passband is subtracted to avoid filters with excessive ripples, normalized to the
% bin width to remain consistent with the passband criterion (dBc/Hz units in all cases). 310 312 % bin width to remain consistent with the passband criterion (dBc/Hz units in all cases).
With this 311 313 With this
criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}. 312 314 criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
{\color{red} The best filter has a correct rejection estimation and the worst filter 313 315 {\color{red} The best filter has a correct rejection estimation and the worst filter
is discarded.} % AH 20191609: Utile ? 314 316 is discarded based on the excessive passband ripple criterion.}
315 317
% \begin{figure} 316 318 % \begin{figure}
% \centering 317 319 % \centering
% \includegraphics[width=\linewidth]{images/colored_mean_criterion} 318 320 % \includegraphics[width=\linewidth]{images/colored_mean_criterion}
% \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters} 319 321 % \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}
% \label{fig:mean_criterion} 320 322 % \label{fig:mean_criterion}
% \end{figure} 321 323 % \end{figure}
322 324
\begin{figure} 323 325 \begin{figure}
\centering 324 326 \centering
\includegraphics[width=\linewidth]{images/custom_criterion} 325 327 \includegraphics[width=\linewidth]{images/custom_criterion}
\caption{\color{red}Custom criterion (maximum rejection in the stopband minus the maximal 326 328 \caption{\color{red}Selected filter qualification criterion computed as the maximum rejection in the stopband
amplitude in passband (if $>$ 1~dB the filter is discarded) rejection normalized to the bandwidth) 327 329 minus the maximal ripple amplitude in the passband with a $>$ 1~dB threshold above which the filter is discarded:
comparison between monolithic filter and cascaded filters} 328 330 comparison between monolithic filter (blue, rejected in this case) and cascaded filters (red).}
\label{fig:custom_criterion} 329 331 \label{fig:custom_criterion}
\end{figure} 330 332 \end{figure}
331 333
Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps 332 334 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 333 335 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. 334 336 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. 335 337 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. 336 338 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 337 339 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. 338 340 the rejection. Hence the best coefficient set are on the vertex of the pyramid. {\color{red} Notice that the word length
341 and number of coefficients do not start at 1: filters with too few coefficients or too little tap word size are rejected
342 by the excessive ripple constraint of the criterion. Hence, the size of the pyramid is significantly reduced by discarding
343 these filters and so is the solution search space.} % ajout JMF
339 344
\begin{figure} 340 345 \begin{figure}
\centering 341 346 \centering
\includegraphics[width=\linewidth]{images/rejection_pyramid} 342 347 \includegraphics[width=\linewidth]{images/rejection_pyramid}
\caption{\color{red}Filter rejection as a function of number of coefficients and number of bits 343 348 \caption{\color{red}Filter rejection as a function of number of coefficients and number of bits
: this lookup table will be used to identify which filter parameters -- number of bits 344 349 : this lookup table will be used to identify which filter parameters -- number of bits
representing coefficients and number of coefficients -- best match the targeted transfer function.} 345 350 representing coefficients and number of coefficients -- best match the targeted transfer function. {\color{red}Filters
351 with fewer than 10~taps or with coefficients coded on fewer than 5~bits are discarded due to excessive
352 ripples in the passband.}} % ajout JMF
\label{fig:rejection_pyramid} 346 353 \label{fig:rejection_pyramid}
\end{figure} 347 354 \end{figure}
348 355
Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps), 349 356 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. 350 357 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: 351 358 If the FIR filter coefficients are the same between the stages, we have:
$$F_{total} = F_1 + F_2$$ 352 359 $$F_{total} = F_1 + F_2$$
But selecting two different sets of coefficient will yield a more complex situation in which 353 360 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 354 361 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. 355 362 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 356 363 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. 357 364 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
% r2.9 358 365 % r2.9
Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection 359 366 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 individual filter rejection sum underestimates the rejection capability of the cascade, 360 367 criteria of each filter. However since the individual filter rejection sum underestimates the rejection capability of the cascade,
% r2.10 361 368 % r2.10
this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability 362 369 this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
of the filter cascade to meet design criteria. 363 370 of the filter cascade to meet design criteria.
364 371
\begin{figure} 365 372 \begin{figure}
\centering 366 373 \centering
\includegraphics[width=\linewidth]{images/cascaded_criterion} 367 374 \includegraphics[width=\linewidth]{images/cascaded_criterion}
\caption{Transfer function of individual filters and after cascading the two filters, 368 375 \caption{Transfer function of individual filters and after cascading the two filters,
demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal 369 376 demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal
lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop 370 377 lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop
maximum of each individual filter. 371 378 maximum of each individual filter.
} 372 379 }
\label{fig:sum_rejection} 373 380 \label{fig:sum_rejection}
\end{figure} 374 381 \end{figure}
375 382
Finally in our case, we consider that the input signal are fully known. The 376 383 Finally in our case, we consider that the input signal are fully known. The
resolution of the input data stream are fixed and still the same for all experiments 377 384 resolution of the input data stream are fixed and still the same for all experiments
in this paper. 378 385 in this paper.
379 386
Based on this analysis, we address the estimate of resource consumption (called 380 387 Based on this analysis, we address the estimate of resource consumption (called
% r2.11 381 388 % r2.11
silicon area -- in the case of FPGAs this means processing cells) as a function of 382 389 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 383 390 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, 384 391 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 385 392 and will assess after synthesis the adequation of this arbitrary unit with actual
hardware resources provided by FPGA manufacturers. The sum of individual processing 386 393 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. 387 394 unit areas is constrained by a total silicon area representative of FPGA global resources.
Formally, variable $a_i$ is the area taken by filter~$i$ 388 395 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). 389 396 (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: 390 397 Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
391 398
\begin{align} 392 399 \begin{align}
\text{Maximize } & \sum_{i=1}^n r_i \notag \\ 393 400 \text{Maximize } & \sum_{i=1}^n r_i \notag \\
\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\ 394 401 \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} \\ 395 402 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} \\ 396 403 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} \\ 397 404 \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} \\ 398 405 \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} \\ 399 406 \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} 400 407 \pi_1^- &= \Pi^I \label{eq:init}
\end{align} 401 408 \end{align}
402 409
Equation~\ref{eq:area} states that the total area taken by the filters must be 403 410 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 404 411 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 405 412 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 406 413 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 407 414 $\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the
input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the 408 415 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 409 416 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, 410 417 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 411 418 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 412 419 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 413 420 $\pi_i^C$ 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 414 421 $\pi_i^S$ 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. 415 422 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 416 423 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 417 424 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 418 425 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 419 426 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 420 427 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 421 428 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: 422 429 shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is:
$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. 423 430 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
Finally, equation~\ref{eq:init} gives the number of bits of the global input. 424 431 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
425 432
This model is non-linear since we multiply some variable with another variable 426 433 This model is non-linear since we multiply some variable with another variable
and it is even non-quadratic, as the cost function $F$ does not have a known 427 434 and it is even non-quadratic, as the cost function $F$ does not have a known
linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations. 428 435 linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
% AH: conflit merge 429 436 % AH: conflit merge
% This variable must be defined by the user, it represent the number of different 430 437 % 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} 431 438 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
% functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid} 432 439 % functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid}
% to restrict the number of configurations. Indeed, it is useless to have too many coefficients or 433 440 % to restrict the number of configurations. Indeed, it is useless to have too many coefficients or
% too many bits, hence we take the configurations close to edge of pyramid. Thank to theses 434 441 % too many bits, hence we take the configurations close to edge of pyramid. Thank to theses
% configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant 435 442 % configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant
% and the function $F$ can be estimate for each configurations 436 443 % and the function $F$ can be estimate for each configurations
% thanks our rejection criterion. We also defined binary 437 444 % thanks our rejection criterion. We also defined binary
This variable $p$ is defined by the user, and represents the number of different 438 445 This variable $p$ is defined by the user, and represents the number of different
set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1} 439 446 set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave) based on the targeted filter characteristics and implementation 440 447 functions from GNU Octave) based on the targeted filter characteristics and implementation
assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and 441 448 assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and
$\pi_{ij}^C$ become constants and 442 449 $\pi_{ij}^C$ become constants and
we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table) 443 450 we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table)
for each configurations thanks to the rejection criterion. We also define the binary 444 451 for each configurations thanks to the rejection criterion. We also define the binary
variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 445 452 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
and 0 otherwise. The new equations are as follows: 446 453 and 0 otherwise. The new equations are as follows:
447 454
\begin{align} 448 455 \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} \\ 449 456 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} \\ 450 457 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} \\ 451 458 \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} 452 459 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
\end{align} 453 460 \end{align}
454 461
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 455 462 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 456 463 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. 457 464 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
458 465
% JM: conflict merge 459 466 % JM: conflict merge
% However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2} 460 467 % However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
% we multiply 461 468 % we multiply
% $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can 462 469 % $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
% linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size, 463 470 % linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size,
% we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is 464 471 % we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is
% assumed on hardware characteristics. 465 472 % assumed on hardware characteristics.
% The Gurobi (\url{www.gurobi.com}) optimization software used to solve this quadratic 466 473 % The Gurobi (\url{www.gurobi.com}) optimization software used to solve this quadratic
% model is able to linearize the model provided as is. This model 467 474 % model is able to linearize the model provided as is. This model
% has $O(np)$ variables and $O(n)$ constraints.} 468 475 % has $O(np)$ variables and $O(n)$ constraints.}
The problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2} 469 476 The problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
we multiply 470 477 we multiply
$\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can 471 478 $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
linearize this multiplication. The following formula shows how to linearize 472 479 linearize this multiplication. The following formula shows how to linearize
this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$): 473 480 this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):
\begin{equation*} 474 481 \begin{equation*}
m = x \times y \implies 475 482 m = x \times y \implies
\left \{ 476 483 \left \{
\begin{split} 477 484 \begin{split}
m & \geq 0 \\ 478 485 m & \geq 0 \\
m & \leq y \times X^{max} \\ 479 486 m & \leq y \times X^{max} \\
m & \leq x \\ 480 487 m & \leq x \\
m & \geq x - (1 - y) \times X^{max} \\ 481 488 m & \geq x - (1 - y) \times X^{max} \\
\end{split} 482 489 \end{split}
\right . 483 490 \right .
\end{equation*} 484 491 \end{equation*}
So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is 485 492 So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is
assumed on hardware characteristics, 486 493 assumed on hardware characteristics,
the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize 487 494 the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize
for us the quadratic problem so the model is left as is. This model 488 495 for us the quadratic problem so the model is left as is. This model
has $O(np)$ variables and $O(n)$ constraints. 489 496 has $O(np)$ variables and $O(n)$ constraints.
490 497
% This model is non-linear and even non-quadratic, as $F$ does not have a known 491 498 % 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 492 499 % linear or quadratic expression. We introduce $p$ FIR configurations
% $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 493 500 % $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
% % r2.12 494 501 % % r2.12
% This variable must be defined by the user, it represent the number of different 495 502 % 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} 496 503 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
% functions from GNU Octave). 497 504 % functions from GNU Octave).
% We define binary 498 505 % We define binary
% variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 499 506 % variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
% and 0 otherwise. The new equations are as follows: 500 507 % and 0 otherwise. The new equations are as follows:
% 501 508 %
% \begin{align} 502 509 % \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} \\ 503 510 % 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} \\ 504 511 % 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} \\ 505 512 % \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} 506 513 % \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
% \end{align} 507 514 % \end{align}
% 508 515 %
% Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 509 516 % Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
% respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 510 517 % 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. 511 518 % Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
% 512 519 %
% % r2.13 513 520 % % r2.13
% This modified model is quadratic since we multiply two variables in the 514 521 % 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. 515 522 % equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
% The Gurobi 516 523 % The Gurobi
% (\url{www.gurobi.com}) optimization software is used to solve this quadratic 517 524 % (\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 518 525 % model, and since Gurobi is able to linearize, the model is left as is. This model
% has $O(np)$ variables and $O(n)$ constraints. 519 526 % has $O(np)$ variables and $O(n)$ constraints.
520 527
Two problems will be addressed using the workflow described in the next section: on the one 521 528 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 522 529 hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary
silicon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area 523 530 silicon 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 524 531 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
objective function is replaced with: 525 532 objective function is replaced with:
\begin{align} 526 533 \begin{align}
\text{Minimize } & \sum_{i=1}^n a_i \notag 527 534 \text{Minimize } & \sum_{i=1}^n a_i \notag
\end{align} 528 535 \end{align}
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 529 536 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 530 537 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
rejection required. 531 538 rejection required.
532 539
\begin{align} 533 540 \begin{align}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 534 541 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
\end{align} 535 542 \end{align}
536 543
\section{Design workflow} 537 544 \section{Design workflow}
\label{sec:workflow} 538 545 \label{sec:workflow}
539 546
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 540 547 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 541 548 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
in the computation of the results. 542 549 in the computation of the results.
543 550
\begin{figure} 544 551 \begin{figure}
\centering 545 552 \centering
\begin{tikzpicture}[node distance=0.75cm and 2cm] 546 553 \begin{tikzpicture}[node distance=0.75cm and 2cm]
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 547 554 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\node (Start) [left= 3cm of Solver] { } ; 548 555 \node (Start) [left= 3cm of Solver] { } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 549 556 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\node (Input) [above= of TCL] { } ; 550 557 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 551 558 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 552 559 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 553 560 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 554 561 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node (Results) [left= of Postproc] { } ; 555 562 \node (Results) [left= of Postproc] { } ;
556 563
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 557 564 \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) ; 558 565 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 559 566 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 560 567 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 561 568 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 562 569 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 563 570 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 564 571 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 565 572 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->] (Postproc) -- (Results) ; 566 573 \draw[->] (Postproc) -- (Results) ;
\end{tikzpicture} 567 574 \end{tikzpicture}
\caption{Design workflow from the input parameters to the results allowing for 568 575 \caption{Design workflow from the input parameters to the results allowing for
a fully automated optimal solution search.} 569 576 a fully automated optimal solution search.}
\label{fig:workflow} 570 577 \label{fig:workflow}
\end{figure} 571 578 \end{figure}
572 579
The filter solver is a C++ program that takes as input the maximum area 573 580 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 $\Pi^I$, 574 581 $\mathcal{A}$, 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 575 582 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. 576 583 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}) 577 584 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
and a deploy script ((1b) on figure~\ref{fig:workflow}). 578 585 and a deploy script ((1b) on figure~\ref{fig:workflow}).
579 586
The TCL script describes the whole digital processing chain from the beginning 580 587 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 581 588 (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 582 589 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) 583 590 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. 584 591 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 585 592 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 586 593 comes from a skeleton library. The generic FIR is highly configurable
with the number of coefficients and the size of the coefficients. The coefficients 587 594 with the number of coefficients and the size of the coefficients. The coefficients
themselves are not stored in the script. 588 595 themselves are not stored in the script.
As the signal is processed in real-time, the output signal is stored as 589 596 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 590 597 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 591 598 implemented FIR cascade transfer function with the design criteria and the expected
transfer function. 592 599 transfer function.
593 600
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 594 601 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 595 602 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 596 603 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 597 604 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
provide a broadband noise source. 598 605 provide a broadband noise source.
The board runs the Linux kernel and surrounding environment produced from the 599 606 The board runs the Linux kernel and surrounding environment produced from the
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 600 607 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 601 608 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
fetching the results is automated. 602 609 fetching the results is automated.
603 610
The deploy script uploads the bitstream to the board ((3) on 604 611 The deploy script uploads the bitstream to the board ((3) on
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 605 612 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
configures the coefficients of the FIR filters. It then waits for the results 606 613 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}). 607 614 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
608 615
Finally, an Octave post-processing script computes the final results thanks to 609 616 Finally, an Octave post-processing script computes the final results thanks to
the output data ((5) on figure~\ref{fig:workflow}). 610 617 the output data ((5) on figure~\ref{fig:workflow}).
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 611 618 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
and the different configurations can be compared. 612 619 and the different configurations can be compared.
613 620
\section{Maximizing the rejection at fixed silicon area} 614 621 \section{Maximizing the rejection at fixed silicon area}
\label{sec:fixed_area} 615 622 \label{sec:fixed_area}
This section presents the output of the filter solver {\em i.e.} the computed 616 623 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. 617 624 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. 618 625 Such results allow for understanding the choices made by the solver to compute its solutions.
619 626
The experimental setup is composed of three cases. The raw input is generated 620 627 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 $\Pi^I$. 621 628 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 622 629 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. 623 630 arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500.
The number of configurations $p$ is \color{1133}, with $C_i$ ranging from 3 to 60 and $\pi^C$ 624 631 The number of configurations $p$ is {\color{red}1133}, 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 625 632 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. 626 633 result up to five stages ($n = 5$) in the cascaded filter.
627 634
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 628 635 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. 629 636 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. 630 637 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
631 638
\renewcommand{\arraystretch}{1.4} 632 639 \renewcommand{\arraystretch}{1.4}
633 640
\begin{table} 634 641 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 635 642 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
\label{tbl:gurobi_max_500} 636 643 \label{tbl:gurobi_max_500}
\centering 637 644 \centering
{\color{red} 638 645 {\color{red}
\scalefont{0.77} 639 646 \scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 640 647 \begin{tabular}{|c|ccccc|c|c|}
\hline 641 648 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 642 649 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 643 650 \hline
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 644 651 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
2 & (3, 5, 18) & (33, 10, 0) & - & - & - & 48~dB & 492 \\ 645 652 2 & (3, 5, 18) & (33, 10, 0) & - & - & - & 48~dB & 492 \\
3 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\ 646 653 3 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\
4 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\ 647 654 4 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\
5 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\ 648 655 5 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\
\hline 649 656 \hline
\end{tabular} 650 657 \end{tabular}
} 651 658 }
\end{table} 652 659 \end{table}
653 660
\begin{table} 654 661 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 655 662 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
\label{tbl:gurobi_max_1000} 656 663 \label{tbl:gurobi_max_1000}
\centering 657 664 \centering
{\color{red}\scalefont{0.77} 658 665 {\color{red}\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 659 666 \begin{tabular}{|c|ccccc|c|c|}
\hline 660 667 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 661 668 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 662 669 \hline
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 663 670 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\ 664 671 2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\
3 & (3, 13, 26) & (31, 9, 1) & (27, 9, 0) & - & - & 92~dB & 999 \\ 665 672 3 & (3, 13, 26) & (31, 9, 1) & (27, 9, 0) & - & - & 92~dB & 999 \\
4 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\ 666 673 4 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\
5 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\ 667 674 5 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\
\hline 668 675 \hline
\end{tabular} 669 676 \end{tabular}
} 670 677 }
\end{table} 671 678 \end{table}
672 679
\begin{table} 673 680 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 674 681 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
\label{tbl:gurobi_max_1500} 675 682 \label{tbl:gurobi_max_1500}
\centering 676 683 \centering
{\color{red}\scalefont{0.77} 677 684 {\color{red}\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 678 685 \begin{tabular}{|c|ccccc|c|c|}
\hline 679 686 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 680 687 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 681 688 \hline
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 682 689 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 102~dB & 1489 \\ 683 690 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 102~dB & 1489 \\
3 & (15, 9, 18) & (31, 8, 0) & (27, 9, 0) & - & - & 116~dB & 1488 \\ 684 691 3 & (15, 9, 18) & (31, 8, 0) & (27, 9, 0) & - & - & 116~dB & 1488 \\
4 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\ 685 692 4 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\
5 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\ 686 693 5 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\
\hline 687 694 \hline
\end{tabular} 688 695 \end{tabular}
} 689 696 }
\end{table} 690 697 \end{table}
691 698
\renewcommand{\arraystretch}{1} 692 699 \renewcommand{\arraystretch}{1}
693 700
% From these tables, we can first state that the more stages are used to define 694 701 % From these tables, we can first state that the more stages are used to define
% the cascaded FIR filters, the better the rejection. 695 702 % the cascaded FIR filters, the better the rejection.
{\color{red} From these tables, we can first state that we reach an optimal solution 696 703 {\color{red} By analyzing these tables, we can first state that we reach an optimal solution
for each case : $n = 3$ for MAX/500 and $n = 4$ for MAX/1000 and MAX/1500. Moreover 697 704 for each case : $n = 3$ for MAX/500, and $n = 4$ for MAX/1000 and MAX/1500. Moreover
the cascade filters always are better than monolithic solution.} 698 705 the cascaded filters always exhibit better performance than the monolithic solution.}
It was an expected result as it has 699 706 It was an expected result as it has
been previously observed that many small filters are better than 700 707 been previously observed that many small filters are better than
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 701 708 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 702 709 being hardly used in practice due to the lack of tools for identifying individual filter
coefficients in the cascaded approach. 703 710 coefficients in the cascaded approach.
704 711
Second, the larger the silicon area, the better the rejection. This was also an 705 712 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 706 713 expected result as more area means a filter of better quality with more coefficients
or more bits per coefficient. 707 714 or more bits per coefficient.
708 715
Then, we also observe that the first stage can have a larger shift than the other 709 716 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 710 717 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 711 718 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} 712 719 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
gives the relation between both values. 713 720 gives the relation between both values.
714 721
Finally, we note that the solver consumes all the given silicon area. 715 722 Finally, we note that the solver consumes all the given silicon area.
716 723
The following graphs present the rejection for real data on the FPGA. In all the following 717 724 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 718 725 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 719 726 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. 720 727 given by the quadratic solver. The configurations are those computed in the previous section.
721 728
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 722 729 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. 723 730 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. 724 731 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
725 732
% \begin{figure} 726 733 % \begin{figure}
% \centering 727 734 % \centering
% \includegraphics[width=\linewidth]{images/max_500} 728 735 % \includegraphics[width=\linewidth]{images/max_500}
% \caption{Signal spectrum for MAX/500} 729 736 % \caption{Signal spectrum for MAX/500}
% \label{fig:max_500_result} 730 737 % \label{fig:max_500_result}
% \end{figure} 731 738 % \end{figure}
% 732 739 %
% \begin{figure} 733 740 % \begin{figure}
% \centering 734 741 % \centering
% \includegraphics[width=\linewidth]{images/max_1000} 735 742 % \includegraphics[width=\linewidth]{images/max_1000}
% \caption{Signal spectrum for MAX/1000} 736 743 % \caption{Signal spectrum for MAX/1000}
% \label{fig:max_1000_result} 737 744 % \label{fig:max_1000_result}
% \end{figure} 738 745 % \end{figure}
% 739 746 %
% \begin{figure} 740 747 % \begin{figure}
% \centering 741 748 % \centering
% \includegraphics[width=\linewidth]{images/max_1500} 742 749 % \includegraphics[width=\linewidth]{images/max_1500}
% \caption{Signal spectrum for MAX/1500} 743 750 % \caption{Signal spectrum for MAX/1500}
% \label{fig:max_1500_result} 744 751 % \label{fig:max_1500_result}
% \end{figure} 745 752 % \end{figure}
746 753
% r2.14 et r2.15 et r2.16 747 754 % r2.14 et r2.15 et r2.16
\begin{figure} 748 755 \begin{figure}
\centering 749 756 \centering
\begin{subfigure}{\linewidth} 750 757 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_500} 751 758 \includegraphics[width=\linewidth]{images/max_500}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 752 759 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).} 753 760 the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).}
\label{fig:max_500_result} 754 761 \label{fig:max_500_result}
\end{subfigure} 755 762 \end{subfigure}
756 763
\begin{subfigure}{\linewidth} 757 764 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1000} 758 765 \includegraphics[width=\linewidth]{images/max_1000}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 759 766 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).} 760 767 the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).}
\label{fig:max_1000_result} 761 768 \label{fig:max_1000_result}
\end{subfigure} 762 769 \end{subfigure}
763 770
\begin{subfigure}{\linewidth} 764 771 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1500} 765 772 \includegraphics[width=\linewidth]{images/max_1500}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 766 773 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).} 767 774 the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).}
\label{fig:max_1500_result} 768 775 \label{fig:max_1500_result}
\end{subfigure} 769 776 \end{subfigure}
\caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing 770 777 \caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing
rejection for a given resource allocation. 771 778 rejection for a given resource allocation.
The filter shape constraint (bandpass and bandstop) is shown as thick 772 779 The filter shape constraint (bandpass and bandstop) is shown as thick
horizontal lines on each chart.} 773 780 horizontal lines on each chart.}
\end{figure} 774 781 \end{figure}
775 782
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 776 783 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
777 784
We compare the actual silicon resources given by Vivado to the 778 785 We compare the actual silicon resources given by Vivado to the
resources in arbitrary units. 779 786 resources in arbitrary units.
The goal is to check that our arbitrary units of silicon area models well enough 780 787 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 781 788 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 782 789 number of arbitrary units, the actual silicon resources do not depend on the
number of stages $n$. Most significantly, our approach aims 783 790 number of stages $n$. Most significantly, our approach aims
at remaining far enough from the practical logic gate implementation used by 784 791 at remaining far enough from the practical logic gate implementation used by
various vendors to remain platform independent and be portable from one 785 792 various vendors to remain platform independent and be portable from one
architecture to another. 786 793 architecture to another.
787 794
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 788 795 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 789 796 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 790 797 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 791 798 the FIR filters and remove additional processing blocks including FIFO and Programmable
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 792 799 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
793 800
\begin{table}[h!tb] 794 801 \begin{table}[h!tb]
\caption{Resource occupation following synthesis of the solutions found for 795 802 \caption{Resource occupation following synthesis of the solutions found for
the problem of maximizing rejection for a given resource allocation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 796 803 the problem of maximizing rejection for a given resource allocation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage} 797 804 \label{tbl:resources_usage}
\color{red} 798 805 \color{red}
\centering 799 806 \centering
\begin{tabular}{|c|c|ccc|c|} 800 807 \begin{tabular}{|c|c|ccc|c|}
\hline 801 808 \hline
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 802 809 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 249 & 453 & 627 & \emph{17600} \\ 803 810 & LUT & 249 & 453 & 627 & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 804 811 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 805 812 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
& LUT & 2253 & 474 & 691 & \emph{17600} \\ 806 813 & LUT & 2253 & 474 & 691 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 807 814 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 50 & 70 & \emph{80} \\ \hline 808 815 & DSP & 0 & 50 & 70 & \emph{80} \\ \hline
& LUT & 1329 & 2006 & 3158 & \emph{17600} \\ 809 816 & LUT & 1329 & 2006 & 3158 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 810 817 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& DSP & 15 & 30 & 42 & \emph{80} \\ \hline 811 818 & DSP & 15 & 30 & 42 & \emph{80} \\ \hline
& LUT & 1329 & 1600 & 2260 & \emph{17600} \\ 812 819 & LUT & 1329 & 1600 & 2260 & \emph{17600} \\
4 & BRAM & 3 & 4 & 4 & \emph{120} \\ 813 820 4 & BRAM & 3 & 4 & 4 & \emph{120} \\
& DPS & 15 & 38 & 49 & \emph{80} \\ \hline 814 821 & DPS & 15 & 38 & 49 & \emph{80} \\ \hline
& LUT & 1329 & 1600 & 2260 & \emph{17600} \\ 815 822 & LUT & 1329 & 1600 & 2260 & \emph{17600} \\
5 & BRAM & 3 & 4 & 4 & \emph{120} \\ 816 823 5 & BRAM & 3 & 4 & 4 & \emph{120} \\
& DPS & 15 & 38 & 49 & \emph{80} \\ \hline 817 824 & DPS & 15 & 38 & 49 & \emph{80} \\ \hline
\end{tabular} 818 825 \end{tabular}
\end{table} 819 826 \end{table}
820 827
{\color{red} In case $n = 2$ for MAX/500}, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 821 828 {\color{red} In case $n = 2$ for MAX/500}, 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 822 829 when the filter coefficients are small enough, or when the input size is small
enough, Vivado optimizes resource consumption by selecting multiplexers to 823 830 enough, Vivado optimizes resource consumption by selecting multiplexers to
implement the multiplications instead of a DSP. In this case, it is quite difficult 824 831 implement the multiplications instead of a DSP. In this case, it is quite difficult
to compare the whole silicon budget. 825 832 to compare the whole silicon budget.
826 833
However, a rough estimation can be made with a simple equivalence: looking at 827 834 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 $n \geq 2$, 828 835 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 829 836 we can deduce that a DSP is roughly equivalent to 100~LUTs in terms of silicon
area use. With this equivalence, our 500 arbitrary units correspond to 2500 LUTs, 830 837 area use. With this equivalence, our 500 arbitrary units correspond to 2500 LUTs,
1000 arbitrary units correspond to 5000 LUTs and 1500 arbitrary units correspond 831 838 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 832 839 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 833 840 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. 834 841 by the optimizations done by Vivado based on the detailed map of available processing resources.
835 842
We now present the computation time needed to solve the quadratic problem. 836 843 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 837 844 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 838 845 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 839 846 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. 840 847 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
841 848
\begin{table}[h!tb] 842 849 \begin{table}[h!tb]
\caption{Time needed to solve the quadratic program with Gurobi} 843 850 \caption{Time needed to solve the quadratic program with Gurobi}
\label{tbl:area_time} 844 851 \label{tbl:area_time}
\centering 845 852 \centering
\color{red} 846 853 \color{red}
\begin{tabular}{|c|c|c|c|}\hline 847 854 \begin{tabular}{|c|c|c|c|}\hline
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 848 855 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
1 & 0.01~s & 0.02~s & 0.03~s \\ 849 856 1 & 0.01~s & 0.02~s & 0.03~s \\
2 & 0.1~s & 1~s & 2~s \\ 850 857 2 & 0.1~s & 1~s & 2~s \\
3 & 5~s & 27~s & 351~s ($\approx$ 6~min) \\ 851 858 3 & 5~s & 27~s & 351~s ($\approx$ 6~min) \\
4 & 4~s & 141~s ($\approx$ 3~min) & 1134~s ($\approx$ 18~min) \\ 852 859 4 & 4~s & 141~s ($\approx$ 3~min) & 1134~s ($\approx$ 18~min) \\
5 & 6~s & 630~s ($\approx$ 10~min) & 49400~s ($\approx$ 13~h) \\\hline 853 860 5 & 6~s & 630~s ($\approx$ 10~min) & 49400~s ($\approx$ 13~h) \\\hline
\end{tabular} 854 861 \end{tabular}
\end{table} 855 862 \end{table}
856 863
As expected, the computation time seems to rise exponentially with the number of stages. 857 864 As expected, the computation time seems to rise exponentially with the number of stages.
When the area is limited, the design exploration space is more limited and the solver is able to 858 865 When the area is limited, the design exploration space is more limited and the solver is able to
find an optimal solution faster. 859 866 find an optimal solution faster.
{\color{red} We can also notice that the solution with $n$ greater than the optimal $n$ 860 867 {\color{red} We also notice that the solution with $n$ greater than the optimal value
take more time than the optimal one. This can be explain since the search space is 861 868 takes more time to be found than the optimal one. This can be explained since the search space is
more important and we need more time to ensure that the previous solution (from the 862 869 larger and we need more time to ensure that the previous solution (from the
smaller value of $n$) still the optimal solution.} 863 870 smaller value of $n$) still remains the optimal solution.}
864 871
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 865 872 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
866 873
This section presents the results of the complementary quadratic program aimed at 867 874 This section presents the results of the complementary quadratic program aimed at
minimizing the area occupation for a targeted rejection level. 868 875 minimizing the area occupation for a targeted rejection level.
869 876
The experimental setup is composed of four cases. The raw input is the same 870 877 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 $\Pi^I$. 871 878 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. 872 879 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. 873 880 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. 874 881 The number of configurations $p$ is the same as previous section.
875 882
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 876 883 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. 877 884 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. 878 885 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. 879 886 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
880 887
\renewcommand{\arraystretch}{1.4} 881 888 \renewcommand{\arraystretch}{1.4}
882 889
\begin{table}[h!tb] 883 890 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 884 891 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\label{tbl:gurobi_min_40} 885 892 \label{tbl:gurobi_min_40}
\centering 886 893 \centering
{\scalefont{0.77}\color{red} 887 894 {\scalefont{0.77}\color{red}
\begin{tabular}{|c|ccccc|c|c|} 888 895 \begin{tabular}{|c|ccccc|c|c|}
\hline 889 896 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 890 897 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 891 898 \hline
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 892 899 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
2 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\ 893 900 2 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\
3 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\ 894 901 3 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\
4 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\ 895 902 4 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\
5 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\ 896 903 5 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\
\hline 897 904 \hline
\end{tabular} 898 905 \end{tabular}
} 899 906 }
\end{table} 900 907 \end{table}
901 908
\begin{table}[h!tb] 902 909 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 903 910 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\label{tbl:gurobi_min_60} 904 911 \label{tbl:gurobi_min_60}
\centering 905 912 \centering
{\scalefont{0.77}\color{red} 906 913 {\scalefont{0.77}\color{red}
\begin{tabular}{|c|ccccc|c|c|} 907 914 \begin{tabular}{|c|ccccc|c|c|}
\hline 908 915 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 909 916 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 910 917 \hline
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 911 918 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
2 & (15, 6, 16) & (23, 9, 0) & - & - & - & 60~dB & 675 \\ 912 919 2 & (15, 6, 16) & (23, 9, 0) & - & - & - & 60~dB & 675 \\
3 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\ 913 920 3 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\
4 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\ 914 921 4 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\
5 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\ 915 922 5 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\
\hline 916 923 \hline
\end{tabular} 917 924 \end{tabular}
} 918 925 }
\end{table} 919 926 \end{table}
920 927
\begin{table}[h!tb] 921 928 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 922 929 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\label{tbl:gurobi_min_80} 923 930 \label{tbl:gurobi_min_80}
\centering 924 931 \centering
{\scalefont{0.77}\color{red} 925 932 {\scalefont{0.77}\color{red}
\begin{tabular}{|c|ccccc|c|c|} 926 933 \begin{tabular}{|c|ccccc|c|c|}
\hline 927 934 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 928 935 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 929 936 \hline
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 930 937 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\ 931 938 2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\
3 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\ 932 939 3 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\
4 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\ 933 940 4 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\
5 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\ 934 941 5 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\
\hline 935 942 \hline
\end{tabular} 936 943 \end{tabular}
} 937 944 }
\end{table} 938 945 \end{table}
939 946
\begin{table}[h!tb] 940 947 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 941 948 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\label{tbl:gurobi_min_100} 942 949 \label{tbl:gurobi_min_100}
\centering 943 950 \centering
{\scalefont{0.77}\color{red} 944 951 {\scalefont{0.77}\color{red}
\begin{tabular}{|c|ccccc|c|c|} 945 952 \begin{tabular}{|c|ccccc|c|c|}
\hline 946 953 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 947 954 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 948 955 \hline
1 & - & - & - & - & - & - & - \\ 949 956 1 & - & - & - & - & - & - & - \\
2 & (27, 9, 15) & (35, 11, 0) & - & - & - & 100~dB & 1410 \\ 950 957 2 & (27, 9, 15) & (35, 11, 0) & - & - & - & 100~dB & 1410 \\
3 & (3, 5, 18) & (35, 11, 1) & (27, 9, 0) & - & - & 100~dB & 1147 \\ 951 958 3 & (3, 5, 18) & (35, 11, 1) & (27, 9, 0) & - & - & 100~dB & 1147 \\
4 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\ 952 959 4 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\
5 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\ 953 960 5 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\
\hline 954 961 \hline
\end{tabular} 955 962 \end{tabular}
} 956 963 }
\end{table} 957 964 \end{table}
\renewcommand{\arraystretch}{1} 958 965 \renewcommand{\arraystretch}{1}
959 966
From these tables, we can first state that almost all configurations reach the targeted rejection 960 967 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 961 968 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 962 969 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. 963 970 MIN/100: no solution is found for a single monolithic filter reach a 100~dB rejection.
Furthermore, the area of the monolithic filter is twice as big as the two cascaded filters 964 971 Furthermore, the area of the monolithic filter is twice as big as the two cascaded filters
{\color{red}(675 and 1131 arbitrary units v.s 990 and 1760 arbitrary units for 60 and 80~dB rejection} 965 972 {\color{red}(675 and 1131 arbitrary units v.s 990 and 1760 arbitrary units for 60 and 80~dB rejection}
respectively). More generally, the more filters are cascaded, the lower the occupied area. 966 973 respectively). More generally, the more filters are cascaded, the lower the occupied area.
967 974
Like in previous section, the solver chooses always a little filter as first 968 975 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 969 976 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 970 977 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 971 978 signal and in the second filter selecting a better filter to improve rejection without
having too many bits in the output data. 972 979 having too many bits in the output data.
973 980
{\color{red} For each case, we found an optimal solution with $n < 5$: for MIN/40 $n=2$, 974 981 {\color{red} For each case, we found an optimal solution with $n < 5$: for MIN/40 $n=2$,
for MIN/60 and MIN/80 $n = 3$ and for MIN/100 $n = 4$. In all cases, the solutions 975 982 for MIN/60 and MIN/80 $n = 3$ and for MIN/100 $n = 4$. In all cases, the solutions
when $n$ is greater than the optimal $n$ they remain identical to the optimal one.} 976 983 when $n$ is greater than this optimal $n$ remain identical to the optimal one.}
% For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 977 984 % 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 978 985 % 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 $n = 4$. 979 986 % solution is equivalent to the result for $n = 4$.
980 987
The following graphs present the rejection for real data on the FPGA. In all the following 981 988 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 982 989 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 983 990 data on the FPGA as measured experimentally and the dashed line is the noise level
given by the quadratic solver. 984 991 given by the quadratic solver.
985 992
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 986 993 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. 987 994 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. 988 995 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. 989 996 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
990 997
% \begin{figure} 991 998 % \begin{figure}
% \centering 992 999 % \centering
% \includegraphics[width=\linewidth]{images/min_40} 993 1000 % \includegraphics[width=\linewidth]{images/min_40}
% \caption{Signal spectrum for MIN/40} 994 1001 % \caption{Signal spectrum for MIN/40}
% \label{fig:min_40} 995 1002 % \label{fig:min_40}
% \end{figure} 996 1003 % \end{figure}
% 997 1004 %
% \begin{figure} 998 1005 % \begin{figure}
% \centering 999 1006 % \centering
% \includegraphics[width=\linewidth]{images/min_60} 1000 1007 % \includegraphics[width=\linewidth]{images/min_60}
% \caption{Signal spectrum for MIN/60} 1001 1008 % \caption{Signal spectrum for MIN/60}
% \label{fig:min_60} 1002 1009 % \label{fig:min_60}
% \end{figure} 1003 1010 % \end{figure}
% 1004 1011 %
% \begin{figure} 1005 1012 % \begin{figure}
% \centering 1006 1013 % \centering
% \includegraphics[width=\linewidth]{images/min_80} 1007 1014 % \includegraphics[width=\linewidth]{images/min_80}
% \caption{Signal spectrum for MIN/80} 1008 1015 % \caption{Signal spectrum for MIN/80}
% \label{fig:min_80} 1009 1016 % \label{fig:min_80}
% \end{figure} 1010 1017 % \end{figure}
% 1011 1018 %
% \begin{figure} 1012 1019 % \begin{figure}
% \centering 1013 1020 % \centering
% \includegraphics[width=\linewidth]{images/min_100} 1014 1021 % \includegraphics[width=\linewidth]{images/min_100}
% \caption{Signal spectrum for MIN/100} 1015 1022 % \caption{Signal spectrum for MIN/100}
% \label{fig:min_100} 1016 1023 % \label{fig:min_100}
% \end{figure} 1017 1024 % \end{figure}
1018 1025
% r2.14 et r2.15 et r2.16 1019 1026 % r2.14 et r2.15 et r2.16
\begin{figure} 1020 1027 \begin{figure}
\centering 1021 1028 \centering
\begin{subfigure}{\linewidth} 1022 1029 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_40} 1023 1030 \includegraphics[width=.91\linewidth]{images/min_40}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 1024 1031 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.} 1025 1032 the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.}
\label{fig:min_40} 1026 1033 \label{fig:min_40}
\end{subfigure} 1027 1034 \end{subfigure}
1028 1035
\begin{subfigure}{\linewidth} 1029 1036 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_60} 1030 1037 \includegraphics[width=.91\linewidth]{images/min_60}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 1031 1038 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.} 1032 1039 the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.}
\label{fig:min_60} 1033 1040 \label{fig:min_60}
\end{subfigure} 1034 1041 \end{subfigure}
1035 1042
\begin{subfigure}{\linewidth} 1036 1043 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_80} 1037 1044 \includegraphics[width=.91\linewidth]{images/min_80}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 1038 1045 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.} 1039 1046 the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.}
\label{fig:min_80} 1040 1047 \label{fig:min_80}
\end{subfigure} 1041 1048 \end{subfigure}
1042 1049
\begin{subfigure}{\linewidth} 1043 1050 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_100} 1044 1051 \includegraphics[width=.91\linewidth]{images/min_100}
\caption{\color{red}Filter transfer functions for varying number of cascaded filters solving 1045 1052 \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.} 1046 1053 the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}
\label{fig:min_100} 1047 1054 \label{fig:min_100}
\end{subfigure} 1048 1055 \end{subfigure}
\caption{\color{red}Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a 1049 1056 \caption{\color{red}Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a
given rejection while minimizing resource allocation. The filter shape constraint (bandpass and 1050 1057 given rejection while minimizing resource allocation. The filter shape constraint (bandpass and
bandstop) is shown as thick 1051 1058 bandstop) is shown as thick
horizontal lines on each chart.} 1052 1059 horizontal lines on each chart.}
\end{figure} 1053 1060 \end{figure}
1054 1061
We observe that all rejections given by the quadratic solver are close to the experimentally 1055 1062 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 1056 1063 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. 1057 1064 respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.
1058 1065
Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60; 1059 1066 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 1060 1067 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 1061 1068 have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and PL to 1062 1069 the FIR filters and remove additional processing blocks including FIFO and PL to
PS communication. 1063 1070 PS communication.
1064 1071
\renewcommand{\arraystretch}{1.2} 1065 1072 \renewcommand{\arraystretch}{1.2}
\begin{table} 1066 1073 \begin{table}
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 1067 1074 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage_comp} 1068 1075 \label{tbl:resources_usage_comp}
\centering 1069 1076 \centering
{\scalefont{0.90}\color{red} 1070 1077 {\scalefont{0.90}\color{red}
\begin{tabular}{|c|c|cccc|c|} 1071 1078 \begin{tabular}{|c|c|cccc|c|}
\hline 1072 1079 \hline
$n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline 1073 1080 $n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 343 & 334 & 772 & - & \emph{17600} \\ 1074 1081 & LUT & 343 & 334 & 772 & - & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\ 1075 1082 1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\
& DSP & 27 & 39 & 55 & - & \emph{80} \\ \hline 1076 1083 & DSP & 27 & 39 & 55 & - & \emph{80} \\ \hline
& LUT & 1664 & 2329 & 474 & 620 & \emph{17600} \\ 1077 1084 & LUT & 1664 & 2329 & 474 & 620 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & 2 & \emph{120} \\ 1078 1085 2 & BRAM & 2 & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 15 & 50 & 62 & \emph{80} \\ \hline 1079 1086 & DSP & 0 & 15 & 50 & 62 & \emph{80} \\ \hline
& LUT & 1664 & 3114 & 1884 & 2873 & \emph{17600} \\ 1080 1087 & LUT & 1664 & 3114 & 1884 & 2873 & \emph{17600} \\
3 & BRAM & 2 & 3 & 3 & 3 & \emph{120} \\ 1081 1088 3 & BRAM & 2 & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 0 & 22 & 27 & \emph{80} \\ \hline 1082 1089 & DSP & 0 & 0 & 22 & 27 & \emph{80} \\ \hline
& LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\ 1083 1090 & LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\
4 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\ 1084 1091 4 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\
& DPS & 0 & 15 & 19 & 19 & \emph{80} \\ \hline 1085 1092 & DPS & 0 & 15 & 19 & 19 & \emph{80} \\ \hline
& LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\ 1086 1093 & LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\
5 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\ 1087 1094 5 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\
& DPS & 0 & 0 & 19 & 19 & \emph{80} \\ \hline 1088 1095 & DPS & 0 & 0 & 19 & 19 & \emph{80} \\ \hline
\end{tabular} 1089 1096 \end{tabular}
} 1090 1097 }
\end{table} 1091 1098 \end{table}
\renewcommand{\arraystretch}{1} 1092 1099 \renewcommand{\arraystretch}{1}
1093 1100
If we keep the previous estimation of cost of one DSP in terms of LUT (1 DSP $\approx$ 100 LUT) 1094 1101 If we keep the previous estimation of cost of one DSP in terms of LUT (1 DSP $\approx$ 100 LUT)
the real resource consumption decreases as a function of the number of stages in the cascaded 1095 1102 the real resource consumption decreases as a function of the number of stages in the cascaded
filter according 1096 1103 filter according
ifcs2018_journal_reponse2.tex
Diff comments   View file @ a45e29d
% MANUSCRIPT NO. TUFFC-09469-2019.R1 1 1 % MANUSCRIPT NO. TUFFC-09469-2019.R1
% MANUSCRIPT TYPE: Papers 2 2 % MANUSCRIPT TYPE: Papers
% TITLE: Filter optimization for real time digital processing of radiofrequency signals: application to oscillator metrology 3 3 % TITLE: Filter optimization for real time digital processing of radiofrequency signals: application to oscillator metrology
% AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, Pierre-Yves; Friedt, Jean-Michel 4 4 % AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, Pierre-Yves; Friedt, Jean-Michel
5 5
\documentclass[a4paper]{article} 6 6 \documentclass[a4paper,8pt]{article}
\usepackage[english]{babel} 7 7 \usepackage[english]{babel}
\usepackage{fullpage,graphicx,amsmath, subcaption} 8 8 \usepackage{fullpage,graphicx,amsmath, subcaption}
9 \pagestyle{empty}
\begin{document} 9 10 \begin{document}
\begin{center} 10 11 \begin{center}
{\bf\Large 11 12 {\bf\Large
Rebuttal letter to the review #2 of the manuscript entitled 12 13 Rebuttal letter to the review \#2 of the manuscript entitled\\
13 14
``Filter optimization for real time digital processing of radiofrequency 14 15 ``Filter optimization for real time digital processing of radiofrequency
signals: application to oscillator metrology'' 15 16 signals: application to oscillator metrology''
} 16 17 }
17 18
by A. Hugeat \& al. 18 19 by A. Hugeat \& al.
\end{center} 19 20 \end{center}
20 21
% 21 22 %
% REVIEWERS' COMMENTS: 22 23 % REVIEWERS' COMMENTS:
% Reviewer: 1 23 24 % Reviewer: 1
% 24 25 %
% Comments to the Author 25 26 \vfill
% The Authors have implemented all Reviewers’ remarks except the one related to the criterion that, in my opinion, is the most important one. By considering ``the minimal rejection within the stopband, to which the sum of the absolute values within the passband is subtracted to avoid filters with excessive ripples, normalized to the bin width to remain consistent with the passband criterion (dBc/Hz units in all cases)'' (please, find a way to state criterions more clearly), the Authors get filters with very different behaviors in pass band and, consequently, their comparison loses its meaning. 26
% In practice, the Authors use a good method based on a bad criterion, and this point weakens a lot the results they present. 27
% In phase noise metrology, the target is an uncertainty of 1 dB, even less. In this regard, I would personally use a maximum ripple in pass band of 1 dB (or less), while, in some cases, the filters presented in the Manuscript exceed 10 dB of ripple, which is definitely too much. 28
% The Authors seem to be reactive in redoing the measures and it does not seem a big problem for them to re-run the analysis with a better criterion. The article would gain a lot, because, in addition to the methodology, the reader could understand if it is actually better to put a cascade of small filters rather than a single large filter that is an interesting point. 29
% To help the Authors in finding a better criterion (``…finding a better criterion to avoid the ripples in the passband is challenging...''), in addition to the minimum rejection in stop band, I suggest to specify also the maximum ripple in pass band as it is done, for example, in fig. 4.10, pg. 146 of Crochierie R. E. and Rabiner L. R. (1983) ``Multirate Digital Signal Processing'', Prentice-Hall (see attach). This suggestion, in practice, specify the maximum allowed deviation from the transfer function modulus of an ideal filter: 1 in pass band and 0 in stop band. As a result, it should solve one of the Authors’ concerns: ``Selecting a strong constraint such as the sum of absolute values in the passband is too selective because it considers all frequency bins in the passband while the stopband criterion is limited to a single bin at which rejection is poorest…'' since both pass and stop bands are considered in the same way. 30
% 31
% I understand that the Manuscript is devoted to present a methodology (``In this article we focus on the methodology, so even if our criterion could be improved, our methodology still remains and works independently of rejection criterion.''). Please, remember that a methodology is a solution to a class of problems and the example chosen to present the methodology plays a key role in showing to the reader if the method is valid or not. Here the example problem is represented by the synthesis of a decimation filter to be used in phase noise metrology. Many of the filters presented by the Authors in figures 9 and 10 as the output of this methodology are not suitable to be used in this context, since, for example, some of them have an attenuation as high as 50 dB in DC (!) that poses severe problems in interpreting the phase noise power spectral densities. What is the cause of this fail? The methodology or the criterion? 32
33
{\bf 34 27 {\bf
28 \noindent
29 Comments to the Author
30
31 The Authors have implemented all Reviewers’ remarks except the one related to the criterion that, in my opinion, is the most important one. By considering ``the minimal rejection within the stopband, to which the sum of the absolute values within the passband is subtracted to avoid filters with excessive ripples, normalized to the bin width to remain consistent with the passband criterion (dBc/Hz units in all cases)'' (please, find a way to state criterions more clearly), the Authors get filters with very different behaviors in pass band and, consequently, their comparison loses its meaning.
32
33 In practice, the Authors use a good method based on a bad criterion, and this point weakens a lot the results they present.
34
35 In phase noise metrology, the target is an uncertainty of 1 dB, even less. In this regard, I would personally use a maximum ripple in pass band of 1 dB (or less), while, in some cases, the filters presented in the Manuscript exceed 10 dB of ripple, which is definitely too much.
36
37 The Authors seem to be reactive in redoing the measures and it does not seem a big problem for them to re-run the analysis with a better criterion. The article would gain a lot, because, in addition to the methodology, the reader could understand if it is actually better to put a cascade of small filters rather than a single large filter that is an interesting point.
38
39 To help the Authors in finding a better criterion (``...finding a better criterion to avoid the ripples in the passband is challenging...''), in addition to the minimum rejection in stop band, I suggest to specify also the maximum ripple in pass band as it is done, for example, in fig. 4.10, pg. 146 of Crochierie R. E. and Rabiner L. R. (1983) ``Multirate Digital Signal Processing'', Prentice-Hall (see attach). This suggestion, in practice, specify the maximum allowed deviation from the transfer function modulus of an ideal filter: 1 in pass band and 0 in stop band. As a result, it should solve one of the Authors’ concerns: ``Selecting a strong constraint such as the sum of absolute values in the passband is too selective because it considers all frequency bins in the passband while the stopband criterion is limited to a single bin at which rejection is poorest…'' since both pass and stop bands are considered in the same way.
40
41 I understand that the Manuscript is devoted to present a methodology (``In this article we focus on the methodology, so even if our criterion could be improved, our methodology still remains and works independently of rejection criterion.''). Please, remember that a methodology is a solution to a class of problems and the example chosen to present the methodology plays a key role in showing to the reader if the method is valid or not. Here the example problem is represented by the synthesis of a decimation filter to be used in phase noise metrology. Many of the filters presented by the Authors in figures 9 and 10 as the output of this methodology are not suitable to be used in this context, since, for example, some of them have an attenuation as high as 50 dB in DC (!) that poses severe problems in interpreting the phase noise power spectral densities. What is the cause of this fail? The methodology or the criterion?
42
In my opinion, it is mandatory to correct the criterion and to re-run the analysis for checking if the methodology works properly or not. 35 43 In my opinion, it is mandatory to correct the criterion and to re-run the analysis for checking if the methodology works properly or not.
In the end, I suggest to publish the Manuscript After Minor Revisions. 36 44 In the end, I suggest to publish the Manuscript After Minor Revisions.
} 37 45 }
38 46
We have change our criterion to be more selective in passband. Now, when the filter response 39 47 \noindent
exceed 1~dB in the passband, we discard the filter. We have re-run all experimentation 40 48 Our answer:
and we have updated the dataset and our conclusion. The methodology provide the 41
same results but since we have less filters we found the optimal solution earlier. 42
Our argumentation about the needed time to compute the optimal solution is not so 43
valid anymore since we need less time but we can also see that for biggest cases 44
we need more time. 45
46 49
50 We are grateful for the opportunity provided by the reviewer to implement the constructive
51 comment provided in the review. Indeed we have thoroughly reviewed our investigation by implementing
52 a threshold criterion on the ripple level in the passband of the filters considered in the analysis.
53 By selecting a 1~dB maximum ripple level in the passband, the transfer functions indeed
54 closely match the targeted shape, as expected from an optimization analysis. The conclusion
55 of the initial paper are not changed but the numerical values of all tables as well as all figures
56 have been updated accordingly, as highlighted in red in the submitted manuscript. All
57 datasets have been re-computed with the updated criterion.
58
59 While the methodology provides the same results as proposed in the initial manuscript,
60 a significant fraction of the initial filter set is discarded by the hard threshold rejection
61 criterion, thus reducing the search space and hence reducing the search time. The conclusion
62 about the computation duration has been updated accordingly since on the one hand all possible
63 filter combinations could now be analyzed, and a maximum number of four cascaded filters meets
64 the optimum solution with no additional improvement when adding a fifth stage. Hence, the search
65 is now exhaustive for solving the considered problem. Such a statement has been added in the
66 conclusion
67
68 The DC component cancellation was an erroneous analysis of the filter transfer function when normalizing
69 the output power to the input power: removing the DC component erroneously led to this unexpected
70 drop of the filter transfer function close to 0~Hz. All charts have been updated accordingly
71 after correcting for this mistake.
72 \vfill
73 \hfill{\mbox{Yours sincerely, A. Hugeat}