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% fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee 1 1 % fusionner max rejection a surface donnee v.s minimiser surface a rejection donnee
% demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de 2 2 % demontrer comment la quantification rejette du bruit vers les hautes frequences => 6 dB de
% rejection par bit et perte si moins de bits que rejection/6 3 3 % rejection par bit et perte si moins de bits que rejection/6
% developper programme lineaire en incluant le decalage de bits 4 4 % developper programme lineaire en incluant le decalage de bits
% insister que avant on etait synthetisable mais pas implementable, alors que maintenant on 5 5 % insister que avant on etait synthetisable mais pas implementable, alors que maintenant on
% implemente et on demontre que ca tourne 6 6 % implemente et on demontre que ca tourne
% gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ? 7 7 % gwen : pourquoi le FIR est desormais implementable et ne l'etait pas meme sur zedboard->new FIR ?
% Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer 8 8 % Gwen : peut-on faire un vrai banc de bruit de phase avec ce FIR, ie ajouter ADC, NCO et mixer
% (zedboard ou redpit) 9 9 % (zedboard ou redpit)
10 10
% label schema : verifier que "argumenter de la cascade de FIR" est fait 11 11 % label schema : verifier que "argumenter de la cascade de FIR" est fait
12 12
\documentclass[a4paper,journal]{IEEEtran/IEEEtran} 13 13 \documentclass[a4paper,journal]{IEEEtran/IEEEtran}
\usepackage{graphicx,color,hyperref} 14 14 \usepackage{graphicx,color,hyperref}
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\usepackage[normalem]{ulem} 21 21 \usepackage[normalem]{ulem}
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\usetikzlibrary{positioning,fit} 23 23 \usetikzlibrary{positioning,fit}
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\usepackage{caption} 26 26 \usepackage{caption}
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% correct bad hyphenation here 29 29 % correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor} 30 30 \hyphenation{op-tical net-works semi-conduc-tor}
\textheight=26cm 31 31 \textheight=26cm
\setlength{\footskip}{30pt} 32 32 \setlength{\footskip}{30pt}
\pagenumbering{gobble} 33 33 \pagenumbering{gobble}
\begin{document} 34 34 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 35 35 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 36 36 to oscillator metrology}
37 37
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 38 38 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 39 39 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\ 40 40 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}\\
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\ 41 41 \IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }\\
\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\ 42 42 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 43 43 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 44 44 }
\maketitle 45 45 \maketitle
\thispagestyle{plain} 46 46 \thispagestyle{plain}
\pagestyle{plain} 47 47 \pagestyle{plain}
\newtheorem{definition}{Definition} 48 48 \newtheorem{definition}{Definition}
49 49
\begin{abstract} 50 50 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 51 51 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 52 52 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 53 53 of ultrastable clocks, stringent filtering requirements are defined by spurious signal or
noise rejection needs. Since real time radiofrequency processing must be performed in a 54 54 noise rejection needs. Since real time radiofrequency processing must be performed in a
Field Programmable Array to meet timing constraints, we investigate optimization strategies 55 55 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 56 56 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. The 57 57 required and keeping timing constraints within the targeted measurement bandwidths. The
presented technique is applicable to scheduling any sequence of processing blocks characterized 58 58 presented technique is applicable to scheduling any sequence of processing blocks characterized
by a throughput, resource occupation and performance tabulated as a function of configuration 59 59 by a throughput, resource occupation and performance tabulated as a function of configuration
characateristics, as is the case for filters with their coefficients and resolution yielding 60 60 characateristics, as is the case for filters with their coefficients and resolution yielding
rejection and number of multipliers. 61 61 rejection and number of multipliers.
\end{abstract} 62 62 \end{abstract}
63 63
\begin{IEEEkeywords} 64 64 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 65 65 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 66 66 \end{IEEEkeywords}
67 67
\section{Digital signal processing of ultrastable clock signals} 68 68 \section{Digital signal processing of ultrastable clock signals}
69 69
Analog oscillator phase noise characteristics are classically performed by downconverting 70 70 Analog oscillator phase noise characteristics are classically performed by downconverting
the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband, 71 71 the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,
followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In 72 72 followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In
a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by 73 73 a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by
multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}. 74 74 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
75 75
\begin{figure}[h!tb] 76 76 \begin{figure}[h!tb]
\begin{center} 77 77 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 78 78 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 79 79 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 80 80 \caption{Fully digital oscillator phase noise characterization: the Device Under Test
(DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and 81 81 (DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and
downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals 82 82 downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals
and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite 83 83 and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite
Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays 84 84 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 85 85 the spectral characteristics of the phase fluctuations.}
\label{schema} 86 86 \label{schema}
\end{figure} 87 87 \end{figure}
88 88
As with the analog mixer, 89 89 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 90 90 the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as
well as the generation of the frequency sum signal in addition to the frequency difference. 91 91 well as the generation of the frequency sum signal in addition to the frequency difference.
These unwanted spectral characteristics must be rejected before decimating the data stream 92 92 These unwanted spectral characteristics must be rejected before decimating the data stream
for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the 93 93 for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
downconverter 94 94 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 95 95 and the decimation processing blocks are core characteristics of an oscillator characterization
system, and must reject out-of-band signals below the targeted phase noise -- typically in the 96 96 system, and must reject out-of-band signals below the targeted phase noise -- typically in the
sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will 97 97 sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will
use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency 98 98 use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency
datastream: optimizing the performance of the filter while reducing the needed resources is 99 99 datastream: optimizing the performance of the filter while reducing the needed resources is
hence tackled in a systematic approach using optimization techniques. Most significantly, we 100 100 hence tackled in a systematic approach using optimization techniques. Most significantly, we
tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with 101 101 tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with
tunable number of coefficients and tunable number of bits representing the coefficients and the 102 102 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 103 103 data being processed.
104 104
\section{Finite impulse response filter} 105 105 \section{Finite impulse response filter}
106 106
We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined 107 107 We select FIR filters for their unconditional stability and ease of design. A FIR filter is defined
by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the 108 108 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 109 109 outputs $y_k$
\begin{align} 110 110 \begin{align}
y_n=\sum_{k=0}^N b_k x_{n-k} 111 111 y_n=\sum_{k=0}^N b_k x_{n-k}
\label{eq:fir_equation} 112 112 \label{eq:fir_equation}
\end{align} 113 113 \end{align}
114 114
As opposed to an implementation on a general purpose processor in which word size is defined by the 115 115 As opposed to an implementation on a general purpose processor in which word size is defined by the
processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since 116 116 processor architecture, implementing such a filter on an FPGA offers more degrees of freedom since
not only the coefficient values and number of taps must be defined, but also the number of bits 117 117 not only the coefficient values and number of taps must be defined, but also the number of bits
defining the coefficients and the sample size. For this reason, and because we consider pipeline 118 118 defining the coefficients and the sample size. For this reason, and because we consider pipeline
processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency 119 119 processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency
signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but 120 120 signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but
the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language 121 121 the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language
(VHDL) level. 122 122 (VHDL) level.
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.
Our final criterion to compute the filter rejection considers 298 298 {\color{red} In intermediate criterion considered the minimal rejection within the stopband, to which the sum of the absolute values
% r2.8 et r2.2 r2.3 299
the minimal rejection within the stopband, to which the sum of the absolute values 300
within the passband is subtracted to avoid filters with excessive ripples, normalized to the 301 299 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). With this 302 300 bin width to remain consistent with the passband criterion (dBc/Hz units in all cases).
301 In this case, when we cascaded too filters with a excessive deviation in passband ($>$ 1~dB),
302 the final deviation in passband may be too considerable ($>$ 10~dB). Hence our final
303 criterion always take the minimal rejection in stopband but we substract the maximal
304 amplitude in passband (maximum value minus the minimum value). If this amplitude is
305 greater than 1~dB, we discard the filter.}
306 % Our final criterion to compute the filter rejection considers
307 % % r2.8 et r2.2 r2.3
308 % the minimal rejection within the stopband, to which the sum of the absolute values
309 % within the passband is subtracted to avoid filters with excessive ripples, normalized to the
310 % bin width to remain consistent with the passband criterion (dBc/Hz units in all cases).
311 With this
criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}. 303 312 criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
313 {\color{red} The best filter has a correct rejection estimation and the worst filter
314 is discarded.} % AH 20191609: Utile ?
304 315
% \begin{figure} 305 316 % \begin{figure}
% \centering 306 317 % \centering
% \includegraphics[width=\linewidth]{images/colored_mean_criterion} 307 318 % \includegraphics[width=\linewidth]{images/colored_mean_criterion}
% \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters} 308 319 % \caption{Mean stopband rejection criterion comparison between monolithic filter and cascaded filters}
% \label{fig:mean_criterion} 309 320 % \label{fig:mean_criterion}
% \end{figure} 310 321 % \end{figure}
311 322
\begin{figure} 312 323 \begin{figure}
\centering 313 324 \centering
\includegraphics[width=\linewidth]{images/colored_custom_criterion} 314 325 \includegraphics[width=\linewidth]{images/custom_criterion}
\caption{Custom criterion (maximum rejection in the stopband minus the sum of the 315 326 \caption{\color{red}Custom criterion (maximum rejection in the stopband minus the maximal
absolute values of the passband rejection normalized to the bandwidth) 316 327 amplitude in passband (if $>$ 1~dB the filter is discarded) rejection normalized to the bandwidth)
comparison between monolithic filter and cascaded filters} 317 328 comparison between monolithic filter and cascaded filters}
\label{fig:custom_criterion} 318 329 \label{fig:custom_criterion}
\end{figure} 319 330 \end{figure}
320 331
Thanks to the latter criterion which will be used in the remainder of this paper, we are able to automatically generate multiple FIR taps 321 332 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 322 333 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. 323 334 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. 324 335 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. 325 336 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 326 337 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. 327 338 the rejection. Hence the best coefficient set are on the vertex of the pyramid.
328 339
\begin{figure} 329 340 \begin{figure}
\centering 330 341 \centering
\includegraphics[width=\linewidth]{images/rejection_pyramid} 331 342 \includegraphics[width=\linewidth]{images/rejection_pyramid}
\caption{Filter rejection as a function of number of coefficients and number of bits 332 343 \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 333 344 : 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.} 334 345 representing coefficients and number of coefficients -- best match the targeted transfer function.}
\label{fig:rejection_pyramid} 335 346 \label{fig:rejection_pyramid}
\end{figure} 336 347 \end{figure}
337 348
Although we have an efficient criterion to estimate the rejection of one set of coefficients (taps), 338 349 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. 339 350 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: 340 351 If the FIR filter coefficients are the same between the stages, we have:
$$F_{total} = F_1 + F_2$$ 341 352 $$F_{total} = F_1 + F_2$$
But selecting two different sets of coefficient will yield a more complex situation in which 342 353 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 343 354 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. 344 355 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 345 356 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. 346 357 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
% r2.9 347 358 % r2.9
Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection 348 359 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, 349 360 criteria of each filter. However since the individual filter rejection sum underestimates the rejection capability of the cascade,
% r2.10 350 361 % r2.10
this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability 351 362 this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
of the filter cascade to meet design criteria. 352 363 of the filter cascade to meet design criteria.
353 364
\begin{figure} 354 365 \begin{figure}
\centering 355 366 \centering
\includegraphics[width=\linewidth]{images/cascaded_criterion} 356 367 \includegraphics[width=\linewidth]{images/cascaded_criterion}
\caption{Transfer function of individual filters and after cascading the two filters, 357 368 \caption{Transfer function of individual filters and after cascading the two filters,
demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal 358 369 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 359 370 lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop
maximum of each individual filter. 360 371 maximum of each individual filter.
} 361 372 }
\label{fig:sum_rejection} 362 373 \label{fig:sum_rejection}
\end{figure} 363 374 \end{figure}
364 375
Finally in our case, we consider that the input signal are fully known. The 365 376 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 366 377 resolution of the input data stream are fixed and still the same for all experiments
in this paper. 367 378 in this paper.
368 379
Based on this analysis, we address the estimate of resource consumption (called 369 380 Based on this analysis, we address the estimate of resource consumption (called
% r2.11 370 381 % r2.11
silicon area -- in the case of FPGAs this means processing cells) as a function of 371 382 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 372 383 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, 373 384 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 374 385 and will assess after synthesis the adequation of this arbitrary unit with actual
hardware resources provided by FPGA manufacturers. The sum of individual processing 375 386 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. 376 387 unit areas is constrained by a total silicon area representative of FPGA global resources.
Formally, variable $a_i$ is the area taken by filter~$i$ 377 388 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). 378 389 (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: 379 390 Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
380 391
\begin{align} 381 392 \begin{align}
\text{Maximize } & \sum_{i=1}^n r_i \notag \\ 382 393 \text{Maximize } & \sum_{i=1}^n r_i \notag \\
\sum_{i=1}^n a_i & \leq \mathcal{A} & \label{eq:area} \\ 383 394 \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} \\ 384 395 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} \\ 385 396 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} \\ 386 397 \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} \\ 387 398 \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} \\ 388 399 \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} 389 400 \pi_1^- &= \Pi^I \label{eq:init}
\end{align} 390 401 \end{align}
391 402
Equation~\ref{eq:area} states that the total area taken by the filters must be 392 403 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 393 404 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 394 405 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 395 406 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 396 407 $\pi_i^C + \pi_i^-$~bits to store the results of the multiplications of the
input data with the coefficients. Equation~\ref{eq:rejectiondef} gives the 397 408 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 398 409 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, 399 410 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 400 411 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 401 412 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 402 413 $\pi_i^C$ 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 403 414 $\pi_i^S$ 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. 404 415 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 405 416 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 406 417 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 407 418 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 408 419 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 409 420 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 410 421 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: 411 422 shift bit would cause an additional 6~dB rejection rise. A totally equivalent equation is:
$\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. 412 423 $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$.
Finally, equation~\ref{eq:init} gives the number of bits of the global input. 413 424 Finally, equation~\ref{eq:init} gives the number of bits of the global input.
414 425
This model is non-linear since we multiply some variable with another variable 415 426 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 416 427 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. 417 428 linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
% AH: conflit merge 418 429 % AH: conflit merge
% This variable must be defined by the user, it represent the number of different 419 430 % 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} 420 431 % 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} 421 432 % 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 422 433 % 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 423 434 % 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 424 435 % configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant
% and the function $F$ can be estimate for each configurations 425 436 % and the function $F$ can be estimate for each configurations
% thanks our rejection criterion. We also defined binary 426 437 % thanks our rejection criterion. We also defined binary
This variable $p$ is defined by the user, and represents the number of different 427 438 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} 428 439 set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}
functions from GNU Octave) based on the targeted filter characteristics and implementation 429 440 functions from GNU Octave) based on the targeted filter characteristics and implementation
assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and 430 441 assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and
$\pi_{ij}^C$ become constants and 431 442 $\pi_{ij}^C$ become constants and
we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table) 432 443 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 433 444 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~$j$ 434 445 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
and 0 otherwise. The new equations are as follows: 435 446 and 0 otherwise. The new equations are as follows:
436 447
\begin{align} 437 448 \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} \\ 438 449 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} \\ 439 450 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} \\ 440 451 \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} 441 452 \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
\end{align} 442 453 \end{align}
443 454
Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 444 455 Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 445 456 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. 446 457 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
447 458
% JM: conflict merge 448 459 % JM: conflict merge
% However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2} 449 460 % However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
% we multiply 450 461 % we multiply
% $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can 451 462 % $\delta_{ij}$ 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, 452 463 % 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 453 464 % we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is
% assumed on hardware characteristics. 454 465 % assumed on hardware characteristics.
% The Gurobi (\url{www.gurobi.com}) optimization software used to solve this quadratic 455 466 % 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 456 467 % model is able to linearize the model provided as is. This model
% has $O(np)$ variables and $O(n)$ constraints.} 457 468 % has $O(np)$ variables and $O(n)$ constraints.}
The problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2} 458 469 The problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
we multiply 459 470 we multiply
$\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can 460 471 $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
linearize this multiplication. The following formula shows how to linearize 461 472 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 ($0 \leq x \leq X^{max}$): 462 473 this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):
\begin{equation*} 463 474 \begin{equation*}
m = x \times y \implies 464 475 m = x \times y \implies
\left \{ 465 476 \left \{
\begin{split} 466 477 \begin{split}
m & \geq 0 \\ 467 478 m & \geq 0 \\
m & \leq y \times X^{max} \\ 468 479 m & \leq y \times X^{max} \\
m & \leq x \\ 469 480 m & \leq x \\
m & \geq x - (1 - y) \times X^{max} \\ 470 481 m & \geq x - (1 - y) \times X^{max} \\
\end{split} 471 482 \end{split}
\right . 472 483 \right .
\end{equation*} 473 484 \end{equation*}
So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is 474 485 So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is
assumed on hardware characteristics, 475 486 assumed on hardware characteristics,
the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize 476 487 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 477 488 for us the quadratic problem so the model is left as is. This model
has $O(np)$ variables and $O(n)$ constraints. 478 489 has $O(np)$ variables and $O(n)$ constraints.
479 490
% This model is non-linear and even non-quadratic, as $F$ does not have a known 480 491 % 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 481 492 % linear or quadratic expression. We introduce $p$ FIR configurations
% $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 482 493 % $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants.
% % r2.12 483 494 % % r2.12
% This variable must be defined by the user, it represent the number of different 484 495 % 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} 485 496 % set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
% functions from GNU Octave). 486 497 % functions from GNU Octave).
% We define binary 487 498 % We define binary
% variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ 488 499 % variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
% and 0 otherwise. The new equations are as follows: 489 500 % and 0 otherwise. The new equations are as follows:
% 490 501 %
% \begin{align} 491 502 % \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} \\ 492 503 % 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} \\ 493 504 % 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} \\ 494 505 % \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} 495 506 % \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config}
% \end{align} 496 507 % \end{align}
% 497 508 %
% Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 498 509 % Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace
% respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}. 499 510 % 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. 500 511 % Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
% 501 512 %
% % r2.13 502 513 % % r2.13
% This modified model is quadratic since we multiply two variables in the 503 514 % 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. 504 515 % equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary.
% The Gurobi 505 516 % The Gurobi
% (\url{www.gurobi.com}) optimization software is used to solve this quadratic 506 517 % (\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 507 518 % model, and since Gurobi is able to linearize, the model is left as is. This model
% has $O(np)$ variables and $O(n)$ constraints. 508 519 % has $O(np)$ variables and $O(n)$ constraints.
509 520
Two problems will be addressed using the workflow described in the next section: on the one 510 521 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 511 522 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 512 523 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 513 524 for a fixed rejection criterion (section~\ref{sec:fixed_rej}). In the latter case, the
objective function is replaced with: 514 525 objective function is replaced with:
\begin{align} 515 526 \begin{align}
\text{Minimize } & \sum_{i=1}^n a_i \notag 516 527 \text{Minimize } & \sum_{i=1}^n a_i \notag
\end{align} 517 528 \end{align}
We adapt our constraints of quadratic program to replace equation \ref{eq:area} 518 529 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 519 530 with equation \ref{eq:rejection_min} where $\mathcal{R}$ is the minimal
rejection required. 520 531 rejection required.
521 532
\begin{align} 522 533 \begin{align}
\sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min} 523 534 \sum_{i=1}^n r_i & \geq \mathcal{R} & \label{eq:rejection_min}
\end{align} 524 535 \end{align}
525 536
\section{Design workflow} 526 537 \section{Design workflow}
\label{sec:workflow} 527 538 \label{sec:workflow}
528 539
In this section, we describe the workflow to compute all the results presented in sections~\ref{sec:fixed_area} 529 540 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 530 541 and \ref{sec:fixed_rej}. Figure~\ref{fig:workflow} shows the global workflow and the different steps involved
in the computation of the results. 531 542 in the computation of the results.
532 543
\begin{figure} 533 544 \begin{figure}
\centering 534 545 \centering
\begin{tikzpicture}[node distance=0.75cm and 2cm] 535 546 \begin{tikzpicture}[node distance=0.75cm and 2cm]
\node[draw,minimum size=1cm] (Solver) { Filter Solver } ; 536 547 \node[draw,minimum size=1cm] (Solver) { Filter Solver } ;
\node (Start) [left= 3cm of Solver] { } ; 537 548 \node (Start) [left= 3cm of Solver] { } ;
\node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ; 538 549 \node[draw,minimum size=1cm] (TCL) [right= of Solver] { TCL Script } ;
\node (Input) [above= of TCL] { } ; 539 550 \node (Input) [above= of TCL] { } ;
\node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ; 540 551 \node[draw,minimum size=1cm] (Deploy) [below= of Solver] { Deploy Script } ;
\node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ; 541 552 \node[draw,minimum size=1cm] (Bitstream) [below= of TCL] { Bitstream } ;
\node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ; 542 553 \node[draw,minimum size=1cm,rounded corners] (Board) [below right= of Deploy] { Board } ;
\node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ; 543 554 \node[draw,minimum size=1cm] (Postproc) [below= of Deploy] { Post-Processing } ;
\node (Results) [left= of Postproc] { } ; 544 555 \node (Results) [left= of Postproc] { } ;
545 556
\draw[->] (Start) edge node [above] { $\mathcal{A}, n, \Pi^I$ } node [below] { $(C_{ij}, \pi_{ij}^C), F$ } (Solver) ; 546 557 \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) ; 547 558 \draw[->] (Input) edge node [left] { ADC or PRN } (TCL) ;
\draw[->] (Solver) edge node [below] { (1a) } (TCL) ; 548 559 \draw[->] (Solver) edge node [below] { (1a) } (TCL) ;
\draw[->] (Solver) edge node [right] { (1b) } (Deploy) ; 549 560 \draw[->] (Solver) edge node [right] { (1b) } (Deploy) ;
\draw[->] (TCL) edge node [left] { (2) } (Bitstream) ; 550 561 \draw[->] (TCL) edge node [left] { (2) } (Bitstream) ;
\draw[->,dashed] (Bitstream) -- (Deploy) ; 551 562 \draw[->,dashed] (Bitstream) -- (Deploy) ;
\draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ; 552 563 \draw[->] (Deploy) to[out=-30,in=120] node [above] { (3) } (Board) ;
\draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ; 553 564 \draw[->] (Board) to[out=150,in=-60] node [below] { (4) } (Deploy) ;
\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ; 554 565 \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
\draw[->] (Postproc) -- (Results) ; 555 566 \draw[->] (Postproc) -- (Results) ;
\end{tikzpicture} 556 567 \end{tikzpicture}
\caption{Design workflow from the input parameters to the results allowing for 557 568 \caption{Design workflow from the input parameters to the results allowing for
a fully automated optimal solution search.} 558 569 a fully automated optimal solution search.}
\label{fig:workflow} 559 570 \label{fig:workflow}
\end{figure} 560 571 \end{figure}
561 572
The filter solver is a C++ program that takes as input the maximum area 562 573 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 $\Pi^I$, 563 574 $\mathcal{A}$, 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 564 575 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. 565 576 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}) 566 577 Then it produces two scripts: a TCL script ((1a) on figure~\ref{fig:workflow})
and a deploy script ((1b) on figure~\ref{fig:workflow}). 567 578 and a deploy script ((1b) on figure~\ref{fig:workflow}).
568 579
The TCL script describes the whole digital processing chain from the beginning 569 580 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 570 581 (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 571 582 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) 572 583 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. 573 584 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 574 585 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 575 586 comes from a skeleton library. The generic FIR is highly configurable
with the number of coefficients and the size of the coefficients. The coefficients 576 587 with the number of coefficients and the size of the coefficients. The coefficients
themselves are not stored in the script. 577 588 themselves are not stored in the script.
As the signal is processed in real-time, the output signal is stored as 578 589 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 579 590 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 580 591 implemented FIR cascade transfer function with the design criteria and the expected
transfer function. 581 592 transfer function.
582 593
The TCL script is used by Vivado to produce the FPGA bitstream ((2) on figure~\ref{fig:workflow}). 583 594 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 584 595 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 585 596 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 586 597 FPGA (xc7z010clg400-1) and two LTC2145 14-bit 125~MS/s ADC, loaded with 50~$\Omega$ resistors to
provide a broadband noise source. 587 598 provide a broadband noise source.
The board runs the Linux kernel and surrounding environment produced from the 588 599 The board runs the Linux kernel and surrounding environment produced from the
Buildroot framework available at \url{https://github.com/trabucayre/redpitaya/}: configuring 589 600 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 590 601 the Zynq FPGA, feeding the FIR with the set of coefficients, executing the simulation and
fetching the results is automated. 591 602 fetching the results is automated.
592 603
The deploy script uploads the bitstream to the board ((3) on 593 604 The deploy script uploads the bitstream to the board ((3) on
figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers, 594 605 figure~\ref{fig:workflow}), flashes the FPGA, loads the different drivers,
configures the coefficients of the FIR filters. It then waits for the results 595 606 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}). 596 607 and retrieves the data to the main computer ((4) on figure~\ref{fig:workflow}).
597 608
Finally, an Octave post-processing script computes the final results thanks to 598 609 Finally, an Octave post-processing script computes the final results thanks to
the output data ((5) on figure~\ref{fig:workflow}). 599 610 the output data ((5) on figure~\ref{fig:workflow}).
The results are normalized so that the Power Spectrum Density (PSD) starts at zero 600 611 The results are normalized so that the Power Spectrum Density (PSD) starts at zero
and the different configurations can be compared. 601 612 and the different configurations can be compared.
602 613
\section{Maximizing the rejection at fixed silicon area} 603 614 \section{Maximizing the rejection at fixed silicon area}
\label{sec:fixed_area} 604 615 \label{sec:fixed_area}
This section presents the output of the filter solver {\em i.e.} the computed 605 616 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. 606 617 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. 607 618 Such results allow for understanding the choices made by the solver to compute its solutions.
608 619
The experimental setup is composed of three cases. The raw input is generated 609 620 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 $\Pi^I$. 610 621 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 611 622 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. 612 623 arbitrary units. Hence, the three cases have been named: MAX/500, MAX/1000, MAX/1500.
The number of configurations $p$ is 1827, with $C_i$ ranging from 3 to 60 and $\pi^C$ 613 624 The number of configurations $p$ is \color{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 614 625 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. 615 626 result up to five stages ($n = 5$) in the cascaded filter.
616 627
Table~\ref{tbl:gurobi_max_500} shows the results obtained by the filter solver for MAX/500. 617 628 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. 618 629 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. 619 630 Table~\ref{tbl:gurobi_max_1500} shows the results obtained by the filter solver for MAX/1500.
620 631
\renewcommand{\arraystretch}{1.4} 621 632 \renewcommand{\arraystretch}{1.4}
622 633
\begin{table} 623 634 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500} 624 635 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/500}
\label{tbl:gurobi_max_500} 625 636 \label{tbl:gurobi_max_500}
\centering 626 637 \centering
{\scalefont{0.77} 627 638 {\color{red}
639 \scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 628 640 \begin{tabular}{|c|ccccc|c|c|}
\hline 629 641 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 630 642 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 631 643 \hline
1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\ 632 644 1 & (21, 7, 0) & - & - & - & - & 32~dB & 483 \\
2 & (3, 3, 15) & (31, 9, 0) & - & - & - & 58~dB & 460 \\ 633 645 2 & (3, 5, 18) & (33, 10, 0) & - & - & - & 48~dB & 492 \\
3 & (3, 3, 15) & (27, 9, 0) & (5, 3, 0) & - & - & 66~dB & 488 \\ 634 646 3 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\
4 & (3, 3, 15) & (19, 7, 0) & (11, 5, 0) & (3, 3, 0) & - & 74~dB & 499 \\ 635 647 4 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\
5 & (3, 3, 15) & (23, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 78~dB & 489 \\ 636 648 5 & (3, 5, 18) & (19, 7, 1) & (15, 7, 0) & - & - & 56~dB & 493 \\
\hline 637 649 \hline
\end{tabular} 638 650 \end{tabular}
} 639 651 }
\end{table} 640 652 \end{table}
641 653
\begin{table} 642 654 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000} 643 655 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1000}
\label{tbl:gurobi_max_1000} 644 656 \label{tbl:gurobi_max_1000}
\centering 645 657 \centering
{\scalefont{0.77} 646 658 {\color{red}\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 647 659 \begin{tabular}{|c|ccccc|c|c|}
\hline 648 660 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 649 661 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 650 662 \hline
1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\ 651 663 1 & (37, 11, 0) & - & - & - & - & 56~dB & 999 \\
2 & (3, 3, 15) & (51, 14, 0) & - & - & - & 87~dB & 975 \\ 652 664 2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\
3 & (3, 3, 15) & (35, 11, 0) & (19, 7, 0) & - & - & 99~dB & 1000 \\ 653 665 3 & (3, 13, 26) & (31, 9, 1) & (27, 9, 0) & - & - & 92~dB & 999 \\
4 & (3, 4, 16) & (27, 8, 0) & (19, 7, 1) & (11, 5, 0) & - & 103~dB & 998 \\ 654 666 4 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\
5 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 1) & (3, 3, 0) & 111~dB & 984 \\ 655 667 5 & (3, 5, 18) & (19, 7, 1) & (19, 7, 0) & (19, 7, 0) & - & 98~dB & 994 \\
\hline 656 668 \hline
\end{tabular} 657 669 \end{tabular}
} 658 670 }
\end{table} 659 671 \end{table}
660 672
\begin{table} 661 673 \begin{table}
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500} 662 674 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MAX/1500}
\label{tbl:gurobi_max_1500} 663 675 \label{tbl:gurobi_max_1500}
\centering 664 676 \centering
{\scalefont{0.77} 665 677 {\color{red}\scalefont{0.77}
\begin{tabular}{|c|ccccc|c|c|} 666 678 \begin{tabular}{|c|ccccc|c|c|}
\hline 667 679 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 668 680 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 669 681 \hline
1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\ 670 682 1 & (47, 15, 0) & - & - & - & - & 71~dB & 1457 \\
2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 103~dB & 1489 \\ 671 683 2 & (19, 6, 15) & (51, 14, 0) & - & - & - & 102~dB & 1489 \\
3 & (3, 3, 15) & (35, 11, 0) & (35, 11, 0) & - & - & 122~dB & 1492 \\ 672 684 3 & (15, 9, 18) & (31, 8, 0) & (27, 9, 0) & - & - & 116~dB & 1488 \\
4 & (3, 3, 15) & (27, 8, 0) & (19, 7, 0) & (27, 9, 0) & - & 129~dB & 1498 \\ 673 685 4 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\
5 & (3, 3, 15) & (23, 9, 2) & (27, 9, 0) & (19, 7, 0) & (3, 3, 0) & 136~dB & 1499 \\ 674 686 5 & (3, 9, 22) & (31, 9, 1) & (27, 9, 0) & (19, 7, 0) & - & 125~dB & 1500 \\
\hline 675 687 \hline
\end{tabular} 676 688 \end{tabular}
} 677 689 }
\end{table} 678 690 \end{table}
679 691
\renewcommand{\arraystretch}{1} 680 692 \renewcommand{\arraystretch}{1}
681 693
From these tables, we can first state that the more stages are used to define 682 694 % From these tables, we can first state that the more stages are used to define
the cascaded FIR filters, the better the rejection. It was an expected result as it has 683 695 % the cascaded FIR filters, the better the rejection.
696 {\color{red} From these tables, we can first state that we reach an optimal solution
697 for each case : $n = 3$ for MAX/500 and $n = 4$ for MAX/1000 and MAX/1500. Moreover
698 the cascade filters always are better than monolithic solution.}
699 It was an expected result as it has
been previously observed that many small filters are better than 684 700 been previously observed that many small filters are better than
a single large filter \cite{lim_1988, lim_1996, young_1992}, despite such conclusions 685 701 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 686 702 being hardly used in practice due to the lack of tools for identifying individual filter
coefficients in the cascaded approach. 687 703 coefficients in the cascaded approach.
688 704
Second, the larger the silicon area, the better the rejection. This was also an 689 705 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 690 706 expected result as more area means a filter of better quality with more coefficients
or more bits per coefficient. 691 707 or more bits per coefficient.
692 708
Then, we also observe that the first stage can have a larger shift than the other 693 709 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 694 710 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 695 711 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} 696 712 balance between a strong rejection with a low number of bits is targeted. Equation~\ref{eq:maxshift}
gives the relation between both values. 697 713 gives the relation between both values.
698 714
Finally, we note that the solver consumes all the given silicon area. 699 715 Finally, we note that the solver consumes all the given silicon area.
700 716
The following graphs present the rejection for real data on the FPGA. In all the following 701 717 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 702 718 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 703 719 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. 704 720 given by the quadratic solver. The configurations are those computed in the previous section.
705 721
Figure~\ref{fig:max_500_result} shows the rejection of the different configurations in the case of MAX/500. 706 722 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. 707 723 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. 708 724 Figure~\ref{fig:max_1500_result} shows the rejection of the different configurations in the case of MAX/1500.
709 725
% \begin{figure} 710 726 % \begin{figure}
% \centering 711 727 % \centering
% \includegraphics[width=\linewidth]{images/max_500} 712 728 % \includegraphics[width=\linewidth]{images/max_500}
% \caption{Signal spectrum for MAX/500} 713 729 % \caption{Signal spectrum for MAX/500}
% \label{fig:max_500_result} 714 730 % \label{fig:max_500_result}
% \end{figure} 715 731 % \end{figure}
% 716 732 %
% \begin{figure} 717 733 % \begin{figure}
% \centering 718 734 % \centering
% \includegraphics[width=\linewidth]{images/max_1000} 719 735 % \includegraphics[width=\linewidth]{images/max_1000}
% \caption{Signal spectrum for MAX/1000} 720 736 % \caption{Signal spectrum for MAX/1000}
% \label{fig:max_1000_result} 721 737 % \label{fig:max_1000_result}
% \end{figure} 722 738 % \end{figure}
% 723 739 %
% \begin{figure} 724 740 % \begin{figure}
% \centering 725 741 % \centering
% \includegraphics[width=\linewidth]{images/max_1500} 726 742 % \includegraphics[width=\linewidth]{images/max_1500}
% \caption{Signal spectrum for MAX/1500} 727 743 % \caption{Signal spectrum for MAX/1500}
% \label{fig:max_1500_result} 728 744 % \label{fig:max_1500_result}
% \end{figure} 729 745 % \end{figure}
730 746
% r2.14 et r2.15 et r2.16 731 747 % r2.14 et r2.15 et r2.16
\begin{figure} 732 748 \begin{figure}
\centering 733 749 \centering
\begin{subfigure}{\linewidth} 734 750 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_500} 735 751 \includegraphics[width=\linewidth]{images/max_500}
\caption{Filter transfer functions for varying number of cascaded filters solving 736 752 \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).} 737 753 the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).}
\label{fig:max_500_result} 738 754 \label{fig:max_500_result}
\end{subfigure} 739 755 \end{subfigure}
740 756
\begin{subfigure}{\linewidth} 741 757 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1000} 742 758 \includegraphics[width=\linewidth]{images/max_1000}
\caption{Filter transfer functions for varying number of cascaded filters solving 743 759 \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).} 744 760 the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).}
\label{fig:max_1000_result} 745 761 \label{fig:max_1000_result}
\end{subfigure} 746 762 \end{subfigure}
747 763
\begin{subfigure}{\linewidth} 748 764 \begin{subfigure}{\linewidth}
\includegraphics[width=\linewidth]{images/max_1500} 749 765 \includegraphics[width=\linewidth]{images/max_1500}
\caption{Filter transfer functions for varying number of cascaded filters solving 750 766 \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).} 751 767 the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).}
\label{fig:max_1500_result} 752 768 \label{fig:max_1500_result}
\end{subfigure} 753 769 \end{subfigure}
\caption{Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing 754 770 \caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing
rejection for a given resource allocation. 755 771 rejection for a given resource allocation.
The filter shape constraint (bandpass and bandstop) is shown as thick 756 772 The filter shape constraint (bandpass and bandstop) is shown as thick
horizontal lines on each chart.} 757 773 horizontal lines on each chart.}
\end{figure} 758 774 \end{figure}
759 775
In all cases, we observe that the actual rejection is close to the rejection computed by the solver. 760 776 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
761 777
We compare the actual silicon resources given by Vivado to the 762 778 We compare the actual silicon resources given by Vivado to the
resources in arbitrary units. 763 779 resources in arbitrary units.
The goal is to check that our arbitrary units of silicon area models well enough 764 780 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 765 781 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 766 782 number of arbitrary units, the actual silicon resources do not depend on the
number of stages $n$. Most significantly, our approach aims 767 783 number of stages $n$. Most significantly, our approach aims
at remaining far enough from the practical logic gate implementation used by 768 784 at remaining far enough from the practical logic gate implementation used by
various vendors to remain platform independent and be portable from one 769 785 various vendors to remain platform independent and be portable from one
architecture to another. 770 786 architecture to another.
771 787
Table~\ref{tbl:resources_usage} shows the resources usage in the case of MAX/500, MAX/1000 and 772 788 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 773 789 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 774 790 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 775 791 the FIR filters and remove additional processing blocks including FIFO and Programmable
Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication. 776 792 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
777 793
\begin{table}[h!tb] 778 794 \begin{table}[h!tb]
\caption{Resource occupation following synthesis of the solutions found for 779 795 \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.} 780 796 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} 781 797 \label{tbl:resources_usage}
798 \color{red}
\centering 782 799 \centering
\begin{tabular}{|c|c|ccc|c|} 783 800 \begin{tabular}{|c|c|ccc|c|}
\hline 784 801 \hline
$n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline 785 802 $n$ & & MAX/500 & MAX/1000 & MAX/1500 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 249 & 453 & 627 & \emph{17600} \\ 786 803 & LUT & 249 & 453 & 627 & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & \emph{120} \\ 787 804 1 & BRAM & 1 & 1 & 1 & \emph{120} \\
& DSP & 21 & 37 & 47 & \emph{80} \\ \hline 788 805 & DSP & 21 & 37 & 47 & \emph{80} \\ \hline
& LUT & 2374 & 5494 & 691 & \emph{17600} \\ 789 806 & LUT & 2253 & 474 & 691 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & \emph{120} \\ 790 807 2 & BRAM & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 70 & \emph{80} \\ \hline 791 808 & DSP & 0 & 50 & 70 & \emph{80} \\ \hline
& LUT & 2443 & 3304 & 3521 & \emph{17600} \\ 792 809 & LUT & 1329 & 2006 & 3158 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & \emph{120} \\ 793 810 3 & BRAM & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 19 & 35 & \emph{80} \\ \hline 794 811 & DSP & 15 & 30 & 42 & \emph{80} \\ \hline
& LUT & 2634 & 3753 & 2557 & \emph{17600} \\ 795 812 & LUT & 1329 & 1600 & 2260 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & \emph{120} \\ 796 813 4 & BRAM & 3 & 4 & 4 & \emph{120} \\
& DPS & 0 & 19 & 46 & \emph{80} \\ \hline 797 814 & DPS & 15 & 38 & 49 & \emph{80} \\ \hline
& LUT & 2423 & 3047 & 2847 & \emph{17600} \\ 798 815 & LUT & 1329 & 1600 & 2260 & \emph{17600} \\
5 & BRAM & 5 & 5 & 5 & \emph{120} \\ 799 816 5 & BRAM & 3 & 4 & 4 & \emph{120} \\
& DPS & 0 & 22 & 46 & \emph{80} \\ \hline 800 817 & DPS & 15 & 38 & 49 & \emph{80} \\ \hline
\end{tabular} 801 818 \end{tabular}
\end{table} 802 819 \end{table}
803 820
In some cases, Vivado replaces the DSPs by Look Up Tables (LUTs). We assume that, 804 821 {\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 805 822 when the filter coefficients are small enough, or when the input size is small
enough, Vivado optimizes resource consumption by selecting multiplexers to 806 823 enough, Vivado optimizes resource consumption by selecting multiplexers to
implement the multiplications instead of a DSP. In this case, it is quite difficult 807 824 implement the multiplications instead of a DSP. In this case, it is quite difficult
to compare the whole silicon budget. 808 825 to compare the whole silicon budget.
809 826
However, a rough estimation can be made with a simple equivalence: looking at 810 827 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 $n \geq 2$, 811 828 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 812 829 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, 813 830 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 814 831 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 815 832 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 816 833 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. 817 834 by the optimizations done by Vivado based on the detailed map of available processing resources.
818 835
We now present the computation time needed to solve the quadratic problem. 819 836 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 820 837 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 821 838 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 822 839 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. 823 840 problem when the maximal area is fixed to 500, 1000 and 1500 arbitrary units.
824 841
\begin{table}[h!tb] 825 842 \begin{table}[h!tb]
\caption{Time needed to solve the quadratic program with Gurobi} 826 843 \caption{Time needed to solve the quadratic program with Gurobi}
\label{tbl:area_time} 827 844 \label{tbl:area_time}
\centering 828 845 \centering
846 \color{red}
\begin{tabular}{|c|c|c|c|}\hline 829 847 \begin{tabular}{|c|c|c|c|}\hline
$n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline 830 848 $n$ & Time (MAX/500) & Time (MAX/1000) & Time (MAX/1500) \\\hline\hline
1 & 0.1~s & 0.1~s & 0.3~s \\ 831 849 1 & 0.01~s & 0.02~s & 0.03~s \\
2 & 1.1~s & 2.2~s & 12~s \\ 832 850 2 & 0.1~s & 1~s & 2~s \\
3 & 17~s & 137~s ($\approx$ 2~min) & 275~s ($\approx$ 4~min) \\ 833 851 3 & 5~s & 27~s & 351~s ($\approx$ 6~min) \\
4 & 52~s & 5448~s ($\approx$ 90~min) & 5505~s ($\approx$ 17~h) \\ 834 852 4 & 4~s & 141~s ($\approx$ 3~min) & 1134~s ($\approx$ 18~min) \\
5 & 286~s ($\approx$ 4~min) & 4119~s ($\approx$ 68~min) & 235479~s ($\approx$ 3~days) \\\hline 835 853 5 & 6~s & 630~s ($\approx$ 10~min) & 49400~s ($\approx$ 13~h) \\\hline
\end{tabular} 836 854 \end{tabular}
\end{table} 837 855 \end{table}
838 856
As expected, the computation time seems to rise exponentially with the number of stages. % TODO: exponentiel ? 839 857 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 840 858 When the area is limited, the design exploration space is more limited and the solver is able to
find an optimal solution faster. 841 859 find an optimal solution faster.
860 {\color{red} We can also notice that the solution with $n$ greater than the optimal $n$
861 take more time than the optimal one. This can be explain since the search space is
862 more important and we need more time to ensure that the previous solution (from the
863 smaller value of $n$) still the optimal solution.}
842 864
\subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej} 843 865 \subsection{Minimizing resource occupation at fixed rejection}\label{sec:fixed_rej}
844 866
This section presents the results of the complementary quadratic program aimed at 845 867 This section presents the results of the complementary quadratic program aimed at
minimizing the area occupation for a targeted rejection level. 846 868 minimizing the area occupation for a targeted rejection level.
847 869
The experimental setup is composed of four cases. The raw input is the same 848 870 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 $\Pi^I$. 849 871 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. 850 872 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. 851 873 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. 852 874 The number of configurations $p$ is the same as previous section.
853 875
Table~\ref{tbl:gurobi_min_40} shows the results obtained by the filter solver for MIN/40. 854 876 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. 855 877 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. 856 878 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. 857 879 Table~\ref{tbl:gurobi_min_100} shows the results obtained by the filter solver for MIN/100.
858 880
\renewcommand{\arraystretch}{1.4} 859 881 \renewcommand{\arraystretch}{1.4}
860 882
\begin{table}[h!tb] 861 883 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40} 862 884 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/40}
\label{tbl:gurobi_min_40} 863 885 \label{tbl:gurobi_min_40}
\centering 864 886 \centering
{\scalefont{0.77} 865 887 {\scalefont{0.77}\color{red}
\begin{tabular}{|c|ccccc|c|c|} 866 888 \begin{tabular}{|c|ccccc|c|c|}
\hline 867 889 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 868 890 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 869 891 \hline
1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\ 870 892 1 & (27, 8, 0) & - & - & - & - & 41~dB & 648 \\
2 & (3, 2, 14) & (19, 7, 0) & - & - & - & 40~dB & 263 \\ 871 893 2 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\
3 & (3, 3, 15) & (11, 5, 0) & (3, 3, 0) & - & - & 41~dB & 192 \\ 872 894 3 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\
4 & (3, 3, 15) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & - & 42~dB & 147 \\ 873 895 4 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\
896 5 & (3, 5, 18) & (27, 8, 0) & - & - & - & 42~dB & 360 \\
\hline 874 897 \hline
\end{tabular} 875 898 \end{tabular}
} 876 899 }
\end{table} 877 900 \end{table}
878 901
\begin{table}[h!tb] 879 902 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60} 880 903 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/60}
\label{tbl:gurobi_min_60} 881 904 \label{tbl:gurobi_min_60}
\centering 882 905 \centering
{\scalefont{0.77} 883 906 {\scalefont{0.77}\color{red}
\begin{tabular}{|c|ccccc|c|c|} 884 907 \begin{tabular}{|c|ccccc|c|c|}
\hline 885 908 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 886 909 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 887 910 \hline
1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\ 888 911 1 & (39, 13, 0) & - & - & - & - & 60~dB & 1131 \\
2 & (3, 3, 15) & (35, 10, 0) & - & - & - & 60~dB & 547 \\ 889 912 2 & (15, 6, 16) & (23, 9, 0) & - & - & - & 60~dB & 675 \\
3 & (3, 3, 15) & (27, 8, 0) & (3, 3, 0) & - & - & 62~dB & 426 \\ 890 913 3 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\
4 & (3, 2, 14) & (11, 5, 1) & (11, 5, 0) & (3, 3, 0) & - & 60~dB & 344 \\ 891 914 4 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\
5 & (3, 2, 14) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & (3, 3, 0) & 60~dB & 279 \\ 892 915 5 & (3, 5, 18) & (15, 6, 2) & (23, 8, 0) & - & - & 60~dB & 543 \\
\hline 893 916 \hline
\end{tabular} 894 917 \end{tabular}
} 895 918 }
\end{table} 896 919 \end{table}
897 920
\begin{table}[h!tb] 898 921 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80} 899 922 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/80}
\label{tbl:gurobi_min_80} 900 923 \label{tbl:gurobi_min_80}
\centering 901 924 \centering
{\scalefont{0.77} 902 925 {\scalefont{0.77}\color{red}
\begin{tabular}{|c|ccccc|c|c|} 903 926 \begin{tabular}{|c|ccccc|c|c|}
\hline 904 927 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 905 928 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 906 929 \hline
1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\ 907 930 1 & (55, 16, 0) & - & - & - & - & 81~dB & 1760 \\
2 & (3, 3, 15) & (47, 14, 0) & - & - & - & 80~dB & 903 \\ 908 931 2 & (15, 8, 17) & (35, 11, 0) & - & - & - & 80~dB & 990 \\
3 & (3, 3, 15) & (23, 9, 0) & (19, 7, 0) & - & - & 80~dB & 698 \\ 909 932 3 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\
4 & (3, 3, 15) & (27, 9, 0) & (7, 7, 4) & (3, 3, 0) & - & 80~dB & 605 \\ 910 933 4 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\
5 & (3, 2, 14) & (27, 8, 0) & (3, 3, 1) & (3, 3, 0) & (3, 3, 0) & 81~dB & 534 \\ 911 934 5 & (3, 7, 20) & (31, 9, 1) & (19, 7, 0) & - & - & 80~dB & 783 \\
\hline 912 935 \hline
\end{tabular} 913 936 \end{tabular}
} 914 937 }
\end{table} 915 938 \end{table}
916 939
\begin{table}[h!tb] 917 940 \begin{table}[h!tb]
\caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100} 918 941 \caption{Configurations $(C_i, \pi_i^C, \pi_i^S)$, rejections and areas (in arbitrary units) for MIN/100}
\label{tbl:gurobi_min_100} 919 942 \label{tbl:gurobi_min_100}
\centering 920 943 \centering
{\scalefont{0.77} 921 944 {\scalefont{0.77}\color{red}
\begin{tabular}{|c|ccccc|c|c|} 922 945 \begin{tabular}{|c|ccccc|c|c|}
\hline 923 946 \hline
$n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\ 924 947 $n$ & $i = 1$ & $i = 2$ & $i = 3$ & $i = 4$ & $i = 5$ & Rejection & Area \\
\hline 925 948 \hline
1 & - & - & - & - & - & - & - \\ 926 949 1 & - & - & - & - & - & - & - \\
2 & (15, 7, 17) & (51, 14, 0) & - & - & - & 100~dB & 1365 \\ 927 950 2 & (27, 9, 15) & (35, 11, 0) & - & - & - & 100~dB & 1410 \\
3 & (3, 3, 15) & (27, 9, 0) & (27, 9, 0) & - & - & 100~dB & 1002 \\ 928 951 3 & (3, 5, 18) & (35, 11, 1) & (27, 9, 0) & - & - & 100~dB & 1147 \\
4 & (3, 3, 15) & (31, 9, 0) & (19, 7, 0) & (3, 3, 0) & - & 101~dB & 909 \\ 929 952 4 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\
5 & (3, 3, 15) & (23, 8, 1) & (19, 7, 0) & (3, 3, 0) & (3, 3, 0) & 101~dB & 810 \\ 930 953 5 & (3, 5, 18) & (15, 6, 2) & (27, 9, 0) & (19, 7, 0) & - & 100~dB & 1067 \\
\hline 931 954 \hline
\end{tabular} 932 955 \end{tabular}
} 933 956 }
\end{table} 934 957 \end{table}
\renewcommand{\arraystretch}{1} 935 958 \renewcommand{\arraystretch}{1}
936 959
From these tables, we can first state that almost all configurations reach the targeted rejection 937 960 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 938 961 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 939 962 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. 940 963 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 941 964 Furthermore, the area of the monolithic filter is twice as big as the two cascaded filters
(1131 and 1760 arbitrary units v.s 547 and 903 arbitrary units for 60 and 80~dB rejection 942 965 {\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. 943 966 respectively). More generally, the more filters are cascaded, the lower the occupied area.
944 967
Like in previous section, the solver chooses always a little filter as first 945 968 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 946 969 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 947 970 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 948 971 signal and in the second filter selecting a better filter to improve rejection without
having too many bits in the output data. 949 972 having too many bits in the output data.
950 973
For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal 951 974 {\color{red} For each case, we found an optimal solution with $n < 5$: for MIN/40 $n=2$,
number of filters is 4 so it did not chose any configuration for the last filter. Hence this 952 975 for MIN/60 and MIN/80 $n = 3$ and for MIN/100 $n = 4$. In all cases, the solutions
solution is equivalent to the result for $n = 4$. 953 976 when $n$ is greater than the optimal $n$ they remain identical to the optimal one.}
977 % For the specific case of MIN/40 for $n = 5$ the solver has determined that the optimal
978 % number of filters is 4 so it did not chose any configuration for the last filter. Hence this
979 % solution is equivalent to the result for $n = 4$.
954 980
The following graphs present the rejection for real data on the FPGA. In all the following 955 981 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 956 982 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 957 983 data on the FPGA as measured experimentally and the dashed line is the noise level
given by the quadratic solver. 958 984 given by the quadratic solver.
959 985
Figure~\ref{fig:min_40} shows the rejection of the different configurations in the case of MIN/40. 960 986 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. 961 987 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. 962 988 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. 963 989 Figure~\ref{fig:min_100} shows the rejection of the different configurations in the case of MIN/100.
964 990
% \begin{figure} 965 991 % \begin{figure}
% \centering 966 992 % \centering
% \includegraphics[width=\linewidth]{images/min_40} 967 993 % \includegraphics[width=\linewidth]{images/min_40}
% \caption{Signal spectrum for MIN/40} 968 994 % \caption{Signal spectrum for MIN/40}
% \label{fig:min_40} 969 995 % \label{fig:min_40}
% \end{figure} 970 996 % \end{figure}
% 971 997 %
% \begin{figure} 972 998 % \begin{figure}
% \centering 973 999 % \centering
% \includegraphics[width=\linewidth]{images/min_60} 974 1000 % \includegraphics[width=\linewidth]{images/min_60}
% \caption{Signal spectrum for MIN/60} 975 1001 % \caption{Signal spectrum for MIN/60}
% \label{fig:min_60} 976 1002 % \label{fig:min_60}
% \end{figure} 977 1003 % \end{figure}
% 978 1004 %
% \begin{figure} 979 1005 % \begin{figure}
% \centering 980 1006 % \centering
% \includegraphics[width=\linewidth]{images/min_80} 981 1007 % \includegraphics[width=\linewidth]{images/min_80}
% \caption{Signal spectrum for MIN/80} 982 1008 % \caption{Signal spectrum for MIN/80}
% \label{fig:min_80} 983 1009 % \label{fig:min_80}
% \end{figure} 984 1010 % \end{figure}
% 985 1011 %
% \begin{figure} 986 1012 % \begin{figure}
% \centering 987 1013 % \centering
% \includegraphics[width=\linewidth]{images/min_100} 988 1014 % \includegraphics[width=\linewidth]{images/min_100}
% \caption{Signal spectrum for MIN/100} 989 1015 % \caption{Signal spectrum for MIN/100}
% \label{fig:min_100} 990 1016 % \label{fig:min_100}
% \end{figure} 991 1017 % \end{figure}
992 1018
% r2.14 et r2.15 et r2.16 993 1019 % r2.14 et r2.15 et r2.16
\begin{figure} 994 1020 \begin{figure}
\centering 995 1021 \centering
\begin{subfigure}{\linewidth} 996 1022 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_40} 997 1023 \includegraphics[width=.91\linewidth]{images/min_40}
\caption{Filter transfer functions for varying number of cascaded filters solving 998 1024 \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.} 999 1025 the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.}
\label{fig:min_40} 1000 1026 \label{fig:min_40}
\end{subfigure} 1001 1027 \end{subfigure}
1002 1028
\begin{subfigure}{\linewidth} 1003 1029 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_60} 1004 1030 \includegraphics[width=.91\linewidth]{images/min_60}
\caption{Filter transfer functions for varying number of cascaded filters solving 1005 1031 \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.} 1006 1032 the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.}
\label{fig:min_60} 1007 1033 \label{fig:min_60}
\end{subfigure} 1008 1034 \end{subfigure}
1009 1035
\begin{subfigure}{\linewidth} 1010 1036 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_80} 1011 1037 \includegraphics[width=.91\linewidth]{images/min_80}
\caption{Filter transfer functions for varying number of cascaded filters solving 1012 1038 \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.} 1013 1039 the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.}
\label{fig:min_80} 1014 1040 \label{fig:min_80}
\end{subfigure} 1015 1041 \end{subfigure}
1016 1042
\begin{subfigure}{\linewidth} 1017 1043 \begin{subfigure}{\linewidth}
\includegraphics[width=.91\linewidth]{images/min_100} 1018 1044 \includegraphics[width=.91\linewidth]{images/min_100}
\caption{Filter transfer functions for varying number of cascaded filters solving 1019 1045 \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.} 1020 1046 the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}
\label{fig:min_100} 1021 1047 \label{fig:min_100}
\end{subfigure} 1022 1048 \end{subfigure}
\caption{Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a 1023 1049 \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 1024 1050 given rejection while minimizing resource allocation. The filter shape constraint (bandpass and
bandstop) is shown as thick 1025 1051 bandstop) is shown as thick
horizontal lines on each chart.} 1026 1052 horizontal lines on each chart.}
\end{figure} 1027 1053 \end{figure}
1028 1054
We observe that all rejections given by the quadratic solver are close to the experimentally 1029 1055 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 1030 1056 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. 1031 1057 respected with both monolithic (except in MIN/100 which has no monolithic solution) or cascaded filters.
1032 1058
Table~\ref{tbl:resources_usage} shows the resource usage in the case of MIN/40, MIN/60; 1033 1059 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 1034 1060 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 1035 1061 have taken care to extract solely the resources used by
the FIR filters and remove additional processing blocks including FIFO and PL to 1036 1062 the FIR filters and remove additional processing blocks including FIFO and PL to
PS communication. 1037 1063 PS communication.
1038 1064
\renewcommand{\arraystretch}{1.2} 1039 1065 \renewcommand{\arraystretch}{1.2}
\begin{table} 1040 1066 \begin{table}
\caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.} 1041 1067 \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
\label{tbl:resources_usage_comp} 1042 1068 \label{tbl:resources_usage_comp}
\centering 1043 1069 \centering
{\scalefont{0.90} 1044 1070 {\scalefont{0.90}\color{red}
\begin{tabular}{|c|c|cccc|c|} 1045 1071 \begin{tabular}{|c|c|cccc|c|}
\hline 1046 1072 \hline
$n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline 1047 1073 $n$ & & MIN/40 & MIN/60 & MIN/80 & MIN/100 & \emph{Zynq 7010} \\ \hline\hline
& LUT & 343 & 334 & 772 & - & \emph{17600} \\ 1048 1074 & LUT & 343 & 334 & 772 & - & \emph{17600} \\
1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\ 1049 1075 1 & BRAM & 1 & 1 & 1 & - & \emph{120} \\
& DSP & 27 & 39 & 55 & - & \emph{80} \\ \hline 1050 1076 & DSP & 27 & 39 & 55 & - & \emph{80} \\ \hline
& LUT & 1252 & 2862 & 5099 & 640 & \emph{17600} \\ 1051 1077 & LUT & 1664 & 2329 & 474 & 620 & \emph{17600} \\
2 & BRAM & 2 & 2 & 2 & 2 & \emph{120} \\ 1052 1078 2 & BRAM & 2 & 2 & 2 & 2 & \emph{120} \\
& DSP & 0 & 0 & 0 & 66 & \emph{80} \\ \hline 1053 1079 & DSP & 0 & 15 & 50 & 62 & \emph{80} \\ \hline
& LUT & 891 & 2148 & 2023 & 2448 & \emph{17600} \\ 1054 1080 & LUT & 1664 & 3114 & 1884 & 2873 & \emph{17600} \\
3 & BRAM & 3 & 3 & 3 & 3 & \emph{120} \\ 1055 1081 3 & BRAM & 2 & 3 & 3 & 3 & \emph{120} \\
& DSP & 0 & 0 & 19 & 27 & \emph{80} \\ \hline 1056 1082 & DSP & 0 & 0 & 22 & 27 & \emph{80} \\ \hline
& LUT & 662 & 1729 & 2451 & 2893 & \emph{17600} \\ 1057 1083 & LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\
4 & BRAM & 4 & 4 & 4 & 4 & \emph{120} \\ 1058 1084 4 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\
& DPS & 0 & 0 & 7 & 19 & \emph{80} \\ \hline 1059 1085 & DPS & 0 & 15 & 19 & 19 & \emph{80} \\ \hline
& LUT & - & 1259 & 2602 & 2505 & \emph{17600} \\ 1060 1086 & LUT & 1664 & 3114 & 2570 & 4318 & \emph{17600} \\
5 & BRAM & - & 5 & 5 & 5 & \emph{120} \\ 1061 1087 5 & BRAM & 2 & 3 & 4 & 4 & \emph{120} \\
& DPS & - & 0 & 0 & 19 & \emph{80} \\ \hline 1062 1088 & DPS & 0 & 0 & 19 & 19 & \emph{80} \\ \hline
\end{tabular} 1063 1089 \end{tabular}
} 1064 1090 }
\end{table} 1065 1091 \end{table}
\renewcommand{\arraystretch}{1} 1066 1092 \renewcommand{\arraystretch}{1}
1067 1093
If we keep the previous estimation of cost of one DSP in terms of LUT (1 DSP $\approx$ 100 LUT) 1068 1094 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 1069 1095 the real resource consumption decreases as a function of the number of stages in the cascaded
filter according 1070 1096 filter according
to the solution given by the quadratic solver. Indeed, we have always a decreasing 1071 1097 to the solution given by the quadratic solver. Indeed, we have always a decreasing
consumption even if the difference between the monolithic and the two cascaded 1072 1098 consumption even if the difference between the monolithic and the two cascaded
filters is less than expected. 1073 1099 filters is less than expected.
1074 1100
Finally, table~\ref{tbl:area_time_comp} shows the computation time to solve 1075 1101 Finally, table~\ref{tbl:area_time_comp} shows the computation time to solve
the quadratic program. 1076 1102 the quadratic program.
1077 1103
\renewcommand{\arraystretch}{1.2} 1078 1104 \renewcommand{\arraystretch}{1.2}
\begin{table}[h!tb] 1079 1105 \begin{table}[h!tb]
\caption{Time to solve the quadratic program with Gurobi} 1080 1106 \caption{Time to solve the quadratic program with Gurobi}
\label{tbl:area_time_comp} 1081 1107 \label{tbl:area_time_comp}
\centering 1082 1108 \centering
{\scalefont{0.90} 1083 1109 {\scalefont{0.90}\color{red}
\begin{tabular}{|c|c|c|c|c|}\hline 1084 1110 \begin{tabular}{|c|c|c|c|c|}\hline
$n$ & Time (MIN/40) & Time (MIN/60) & Time (MIN/80) & Time (MIN/100) \\\hline\hline 1085 1111 $n$ & Time (MIN/40) & Time (MIN/60) & Time (MIN/80) & Time (MIN/100) \\\hline\hline
1 & 0.07~s & 0.02~s & 0.01~s & - \\ 1086 1112 1 & 0.04~s & 0.01~s & 0.01~s & - \\
2 & 7.8~s & 16~s & 14~s & 1.8~s \\ 1087 1113 2 & 2.7~s & 2.4~s & 2.4~s & 0.8~s \\
3 & 4.7~s & 14~s & 28~s & 39~s \\ 1088 1114 3 & 4.6~s & 7~s & 7~s & 18~s \\
4 & 39~s & 20~s & 193~s & 522~s ($\approx$ 9~min) \\ 1089 1115 4 & 3~s & 22~s & 70~s & 220~s ($\approx$ 3~min) \\
5 & - & 12~s & 170~s & 1048~s ($\approx$ 17~min) \\\hline 1090 1116 5 & 5~s & 122~s & 200~s & 384~s ($\approx$ 5~min) \\\hline
\end{tabular} 1091 1117 \end{tabular}
} 1092 1118 }
\end{table} 1093 1119 \end{table}
\renewcommand{\arraystretch}{1} 1094 1120 \renewcommand{\arraystretch}{1}
1095 1121
The time needed to solve this configuration is significantly shorter than the time 1096 1122 The time needed to solve this configuration is significantly shorter than the time
needed in the previous section. Indeed the worst time in this case is only 17~minutes, 1097 1123 needed in the previous section. Indeed the worst time in this case is only {\color{red}5~minutes,
compared to 3~days in the previous section: this problem is more easily solved than the 1098 1124 compared to 13~hours} in the previous section: this problem is more easily solved than the
previous one. 1099 1125 previous one.
1100 1126
To conclude, we compare our monolithic filters with the FIR Compiler provided by 1101 1127 To conclude, we compare our monolithic filters with the FIR Compiler provided by
Xilinx in the Vivado software suite (v.2018.2). For each experiment we use the 1102 1128 Xilinx in the Vivado software suite (v.2018.2). For each experiment we use the
same coefficient set and we compare the resource consumption, having checked that 1103 1129 same coefficient set and we compare the resource consumption, having checked that
the transfer functions are indeed the same with both implementations. 1104 1130 the transfer functions are indeed the same with both implementations.
Table~\ref{tbl:xilinx_resources} exhibits the results. 1105 1131 Table~\ref{tbl:xilinx_resources} exhibits the results.
The FIR Compiler never uses BRAM while our filter implementation uses one block. This difference 1106 1132 The FIR Compiler never uses BRAM while our filter implementation uses one block. This difference
is explained be our wish to have a dynamically reconfigurable FIR filter whose 1107 1133 is explained be our wish to have a dynamically reconfigurable FIR filter whose
coefficients can be updated from the processing system without having to update the FPGA design. 1108 1134 coefficients can be updated from the processing system without having to update the FPGA design.
With the FIR compiler, the coefficients are defined during the FPGA design so that 1109 1135 With the FIR compiler, the coefficients are defined during the FPGA design so that
changing coefficients required generating a new design. The difference with the LUT consumption 1110 1136 changing coefficients required generating a new design. The difference with the LUT consumption
is also attributed to the reconfigurability logic. However the DSP consumption, the scarcest 1111 1137 is also attributed to the reconfigurability logic. However the DSP consumption, the scarcest
resource, is the same between the Xilinx FIR Compiler end 1112 1138 resource, is the same between the Xilinx FIR Compiler end
our FIR block: we hence conclude that our solutions are as good as the Xilinx implementation. 1113 1139 our FIR block: we hence conclude that our solutions are as good as the Xilinx implementation.
1114 1140
\renewcommand{\arraystretch}{1.2} 1115 1141 \renewcommand{\arraystretch}{1.2}
\begin{table} 1116 1142 \begin{table}
\centering 1117 1143 \centering
\caption{Resource consumption compared between the FIR Compiler from Xilinx and our FIR block} 1118 1144 \caption{Resource consumption compared between the FIR Compiler from Xilinx and our FIR block}
\label{tbl:xilinx_resources} 1119 1145 \label{tbl:xilinx_resources}
\begin{tabular}{|c|c|c|c|c|c|c|} 1120 1146 \begin{tabular}{|c|c|c|c|c|c|c|}
\hline 1121 1147 \hline
\multirow{2}{*}{} & \multicolumn{3}{c|}{Xilinx} & \multicolumn{3}{c|}{Our FIR block} \\ \cline{2-7} 1122 1148 \multirow{2}{*}{} & \multicolumn{3}{c|}{Xilinx} & \multicolumn{3}{c|}{Our FIR block} \\ \cline{2-7}
& LUT & BRAM & DSP & LUT & BRAM & DSP \\ \hline 1123 1149 & LUT & BRAM & DSP & LUT & BRAM & DSP \\ \hline
MAX/500 & 177 & 0 & 21 & 249 & 1 & 21 \\ \hline 1124 1150 MAX/500 & 177 & 0 & 21 & 249 & 1 & 21 \\ \hline
MAX/1000 & 306 & 0 & 37 & 453 & 1 & 37 \\ \hline 1125 1151 MAX/1000 & 306 & 0 & 37 & 453 & 1 & 37 \\ \hline
MAX/1500 & 418 & 0 & 47 & 627 & 1 & 47 \\ \hline 1126 1152 MAX/1500 & 418 & 0 & 47 & 627 & 1 & 47 \\ \hline
MIN/40 & 225 & 0 & 27 & 347 & 1 & 27 \\ \hline 1127 1153 MIN/40 & 225 & 0 & 27 & 347 & 1 & 27 \\ \hline
MIN/60 & 322 & 0 & 39 & 334 & 1 & 39 \\ \hline 1128 1154 MIN/60 & 322 & 0 & 39 & 334 & 1 & 39 \\ \hline
MIN/80 & 482 & 0 & 55 & 772 & 1 & 55 \\ \hline 1129 1155 MIN/80 & 482 & 0 & 55 & 772 & 1 & 55 \\ \hline
\end{tabular} 1130 1156 \end{tabular}
\end{table} 1131 1157 \end{table}
\renewcommand{\arraystretch}{1} 1132 1158 \renewcommand{\arraystretch}{1}
1133 1159
\section{Conclusion} 1134 1160 \section{Conclusion}
1135 1161
We have proposed a new approach to schedule a set of signal processing blocks whose performances 1136 1162 We have proposed a new approach to optimize a set of signal processing blocks whose performances
ifcs2018_journal_reponse2.tex
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File was created 1 % MANUSCRIPT NO. TUFFC-09469-2019.R1
2 % MANUSCRIPT TYPE: Papers
3 % TITLE: Filter optimization for real time digital processing of radiofrequency signals: application to oscillator metrology
4 % AUTHOR(S): HUGEAT, Arthur; BERNARD, Julien; Goavec-Mérou, Gwenhaël; Bourgeois, Pierre-Yves; Friedt, Jean-Michel
5 %
6 % REVIEWERS' COMMENTS:
7 % Reviewer: 1
8 %
9 % Comments to the Author
10 % 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.
11 % In practice, the Authors use a good method based on a bad criterion, and this point weakens a lot the results they present.
12 % 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.
13 % 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.
14 % 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.
15 % 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?
16 % 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.
17 % In the end, I suggest to publish the Manuscript After Minor Revisions.
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