jfriedt / IFCS2018 article

Download as
Browse Code »

Commit 8a48cee5806b39457cd429dc2c488a9eaea99b0e

Authored by Arthur HUGEAT
1 parent 365465b1b5
Exists in master

Rajout de la definition de D_i

Showing 1 changed file with 7 additions and 6 deletions Inline Diff

ifcs2018_proceeding.tex
Diff comments   View file @ 8a48cee
% JMF : revoir l'abstract : on y avait mis le Zynq7010 de la redpitaya en montrant 1 1 % JMF : revoir l'abstract : on y avait mis le Zynq7010 de la redpitaya en montrant
% comment optimiser les perfs a surface finie. Ici aussi on tombait dans le cas ou` 2 2 % comment optimiser les perfs a surface finie. Ici aussi on tombait dans le cas ou`
% la solution a 1 seul FIR n'etait simplement pas synthetisable => fusionner les deux 3 3 % la solution a 1 seul FIR n'etait simplement pas synthetisable => fusionner les deux
% contributions pour le papier TUFFC 4 4 % contributions pour le papier TUFFC
5 5
\documentclass[a4paper,conference]{IEEEtran/IEEEtran} 6 6 \documentclass[a4paper,conference]{IEEEtran/IEEEtran}
\usepackage{graphicx,color,hyperref} 7 7 \usepackage{graphicx,color,hyperref}
\usepackage{amsfonts} 8 8 \usepackage{amsfonts}
\usepackage{amsthm} 9 9 \usepackage{amsthm}
\usepackage{amssymb} 10 10 \usepackage{amssymb}
\usepackage{amsmath} 11 11 \usepackage{amsmath}
\usepackage{algorithm2e} 12 12 \usepackage{algorithm2e}
\usepackage{url,balance} 13 13 \usepackage{url,balance}
\usepackage[normalem]{ulem} 14 14 \usepackage[normalem]{ulem}
% correct bad hyphenation here 15 15 % correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor} 16 16 \hyphenation{op-tical net-works semi-conduc-tor}
\textheight=26cm 17 17 \textheight=26cm
\setlength{\footskip}{30pt} 18 18 \setlength{\footskip}{30pt}
\pagenumbering{gobble} 19 19 \pagenumbering{gobble}
\begin{document} 20 20 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 21 21 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 22 22 to oscillator metrology}
23 23
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 24 24 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 25 25 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}} 26 26 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France } 27 27 \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 \\ 28 28 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 29 29 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 30 30 }
\maketitle 31 31 \maketitle
\thispagestyle{plain} 32 32 \thispagestyle{plain}
\pagestyle{plain} 33 33 \pagestyle{plain}
\newtheorem{definition}{Definition} 34 34 \newtheorem{definition}{Definition}
35 35
\begin{abstract} 36 36 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 37 37 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 38 38 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 39 39 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 40 40 noise rejection needs. Since real time radiofrequency processing must be performed in a
Field Programmable Array to meet timing constraints, we investigate optimization strategies 41 41 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 42 42 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. 43 43 required and keeping timing constraints within the targeted measurement bandwidths.
\end{abstract} 44 44 \end{abstract}
45 45
\begin{IEEEkeywords} 46 46 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 47 47 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 48 48 \end{IEEEkeywords}
49 49
\section{Digital signal processing of ultrastable clock signals} 50 50 \section{Digital signal processing of ultrastable clock signals}
51 51
Analog oscillator phase noise characteristics are classically performed by downconverting 52 52 Analog oscillator phase noise characteristics are classically performed by downconverting
the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband, 53 53 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 54 54 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 55 55 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}. 56 56 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
57 57
\begin{figure}[h!tb] 58 58 \begin{figure}[h!tb]
\begin{center} 59 59 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 60 60 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 61 61 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 62 62 \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 63 63 (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 64 64 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 65 65 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 66 66 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 67 67 the spectral characteristics of the phase fluctuations.}
\label{schema} 68 68 \label{schema}
\end{figure} 69 69 \end{figure}
70 70
As with the analog mixer, 71 71 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 72 72 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. 73 73 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 74 74 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 75 75 for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
downconverter 76 76 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 77 77 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 78 78 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 79 79 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 80 80 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 81 81 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 82 82 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 83 83 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 84 84 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 85 85 data being processed.
86 86
\section{Finite impulse response filter} 87 87 \section{Finite impulse response filter}
88 88
We select FIR filter for their unconditional stability and ease of design. A FIR filter is defined 89 89 We select FIR filter 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 90 90 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 91 91 outputs $y_k$
$$y_n=\sum_{k=0}^N b_k x_{n-k}$$ 92 92 $$y_n=\sum_{k=0}^N b_k x_{n-k}$$
93 93
As opposed to an implementation on a general purpose processor in which word size is defined by the 94 94 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 offer more degrees of freedom since 95 95 processor architecture, implementing such a filter on an FPGA offer more degrees of freedom since
not only the coefficient values and number of taps must be defined, but also the number of bits 96 96 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 97 97 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 98 98 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 99 99 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 (VHDL) level. 100 100 the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language (VHDL) level.
Since latency is not an issue in a openloop phase noise characterization instrument, the large 101 101 Since latency is not an issue in a openloop phase noise characterization instrument, the large
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 102 102 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. 103 103 is not considered as an issue as would be in a closed loop system.
104 104
The coefficients are classically expressed as floating point values. However, this binary 105 105 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, 106 106 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 107 107 we select to quantify these floating point values into integer values. This quantization
will result in some precision loss. 108 108 will result in some precision loss.
109 109
%As illustrated in Fig. \ref{float_vs_int}, we see that we aren't 110 110 %As illustrated in Fig. \ref{float_vs_int}, we see that we aren't
%need too coefficients or too sample size. If we have lot of coefficients but a small sample size, 111 111 %need too coefficients or too sample size. If we have lot of coefficients but a small sample size,
%the first and last are equal to zero. But if we have too sample size for few coefficients that not improve the quality. 112 112 %the first and last are equal to zero. But if we have too sample size for few coefficients that not improve the quality.
113 113
% JMF je ne comprends pas la derniere phrase ci-dessus ni la figure ci dessous 114 114 % JMF je ne comprends pas la derniere phrase ci-dessus ni la figure ci dessous
% AH en gros je voulais dire que prendre trop peu de bit avec trop de coeff, ça induit ta figure (bien mieux faite que moi) 115 115 % AH en gros je voulais dire que prendre trop peu de bit avec trop de coeff, ça induit ta figure (bien mieux faite que moi)
% et que l'inverse trop de bit sur pas assez de coeff on ne gagne rien, je vais essayer de la reformuler 116 116 % et que l'inverse trop de bit sur pas assez de coeff on ne gagne rien, je vais essayer de la reformuler
117 117
%\begin{figure}[h!tb] 118 118 %\begin{figure}[h!tb]
%\includegraphics[width=\linewidth]{images/float-vs-integer.pdf} 119 119 %\includegraphics[width=\linewidth]{images/float-vs-integer.pdf}
%\caption{Impact of the quantization resolution of the coefficients} 120 120 %\caption{Impact of the quantization resolution of the coefficients}
%\label{float_vs_int} 121 121 %\label{float_vs_int}
%\end{figure} 122 122 %\end{figure}
123 123
\begin{figure}[h!tb] 124 124 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/demo_filtre} 125 125 \includegraphics[width=\linewidth]{images/demo_filtre}
\caption{Impact of the quantization resolution of the coefficients: the quantization is 126 126 \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 127 127 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 128 128 the 30~first and 30~last coefficients out of the initial 128~band-pass
filter coefficients to 0 (red dots).} 129 129 filter coefficients to 0 (red dots).}
\label{float_vs_int} 130 130 \label{float_vs_int}
\end{figure} 131 131 \end{figure}
132 132
The tradeoff between quantization resolution and number of coefficients when considering 133 133 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 134 134 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 135 135 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 136 136 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 137 137 quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the
taps become null, making the large number of coefficients irrelevant and allowing to save 138 138 taps become null, making the large number of coefficients irrelevant and allowing to save
processing resource by shrinking the filter length. This tradeoff aimed at minimizing resources 139 139 processing resource 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 140 140 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 141 141 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 142 142 and tap length, as will be shown in the next section. Indeed, our development strategy closely
follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards} 143 143 follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
in which basic blocks are defined and characterized before being assembled \cite{hide} 144 144 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 145 145 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 146 146 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 147 147 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 148 148 current implementation: the decimation is assumed to be located after the FIR cascade at the
moment. 149 149 moment.
150 150
\section{Filter optimization} 151 151 \section{Filter optimization}
152 152
A basic approach for implementing the FIR filter is to compute the transfer function of 153 153 A basic approach for implementing the FIR filter is to compute the transfer function of
a monolithic filter: this single filter defines all coefficients with the same resolution 154 154 a monolithic filter: this single filter defines all coefficients with the same resolution
(number of bits) and processes data represented with their own resolution. Meeting the 155 155 (number of bits) and processes data represented with their own resolution. Meeting the
filter shape requires a large number of coefficients, limited by resources of the FPGA since 156 156 filter shape requires a large number of coefficients, limited by resources of the FPGA since
this filter must process data stream at the radiofrequency sampling rate after the mixer. 157 157 this filter must process data stream at the radiofrequency sampling rate after the mixer.
158 158
An optimization problem \cite{leung2004handbook} aims at improving one or many 159 159 An optimization problem \cite{leung2004handbook} aims at improving one or many
performance criteria within a constrained resource environment. Amongst the tools 160 160 performance criteria within a constrained resource environment. Amongst the tools
developed to meet this aim, Mixed-Integer Linear Programming (MILP) provides the framework to 161 161 developed to meet this aim, Mixed-Integer Linear Programming (MILP) provides the framework to
formally define the stated problem and search for an optimal use of available 162 162 formally define the stated problem and search for an optimal use of available
resources \cite{yu2007design, kodek1980design}. 163 163 resources \cite{yu2007design, kodek1980design}.
164 164
First we need to ensure that our problem is a real optimization problem. When 165 165 First we need to ensure that our problem is a real optimization problem. When
designing a processing function in the FPGA, we aim at meeting some requirement such as 166 166 designing a processing function in the FPGA, we aim at meeting some requirement such as
the throughput, the computation time or the noise rejection noise. However, due to limited 167 167 the throughput, the computation time or the noise rejection noise. However, due to limited
resources to design the process like BRAM (high performance RAM), DSP (Digital Signal Processor) 168 168 resources to design the process like BRAM (high performance RAM), DSP (Digital Signal Processor)
or LUT (Look Up Table), a tradeoff must be generally searched between performance and available 169 169 or LUT (Look Up Table), a tradeoff must be generally searched between performance and available
computational resources: optimizing some criteria within finite, limited 170 170 computational resources: optimizing some criteria within finite, limited
resources indeed matches the definition of a classical optimization problem. 171 171 resources indeed matches the definition of a classical optimization problem.
172 172
Specifically the degrees of freedom when addressing the problem of replacing the single monolithic 173 173 Specifically the degrees of freedom when addressing the problem of replacing the single monolithic
FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$ and 174 174 FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$,
the number of bits $C_i$ representing the coefficients. Because each FIR in the chain is fed the output of the previous stage, 175 175 the number of bits $C_i$ representing the coefficients and the number of bits $D_i$ representing
176 -the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage,
the optimization of the complete processing chain within a constrained resource environment is not 176 177 the optimization of the complete processing chain within a constrained resource environment is not
trivial. The resource occupation of a FIR filter is considered as $C_i \times N_i$ which aims 177 178 trivial. The resource occupation of a FIR filter is considered as $C_i \times N_i$ which aims
at approximating the number of bits needed in a worst case condition to represent the output of the 178 179 at approximating the number of bits needed in a worst case condition to represent the output of the
FIR. Indeed, the number of bits generated by the FIR is $(C_i+D_i)\times\log_2(N_i)$ with $D_i$ 179 180 FIR. Indeed, the number of bits generated by the FIR is $(C_i+D_i)\times\log_2(N_i)$ with $D_i$
the number of bits needed to represent the data $x_k$ generated by the previous stage, but the 180 181 the number of bits needed to represent the data $x_k$ generated by the previous stage, but the
$\log$ function is avoided for its incompatibility with a linear programming description, and 181 182 $\log$ function is avoided for its incompatibility with a linear programming description, and
the simple product is approximated as the number of gates needed to perform the calculation. Such an 182 183 the simple product is approximated as the number of gates needed to perform the calculation. Such an
occupied area estimate assumes that the number of gates scales as the number of bits and the number 183 184 occupied area estimate assumes that the number of gates scales as the number of bits and the number
of coefficients, but does not account for the detailed implementation of the hardware. Indeed, 184 185 of coefficients, but does not account for the detailed implementation of the hardware. Indeed,
various FPGA implementations will provide different hardware functionalities, and we shall consider 185 186 various FPGA implementations will provide different hardware functionalities, and we shall consider
at the end of the design a synthesis step using vendor software to assess the validity of the solution 186 187 at the end of the design a synthesis step using vendor software to assess the validity of the solution
found. As an example of the limitation linked to the lack of detailed hardware consideration, Block Random 187 188 found. As an example of the limitation linked to the lack of detailed hardware consideration, Block Random
Access Memory (BRAM) used to store filter coefficients are not shared amongst filters, and multiplications 188 189 Access Memory (BRAM) used to store filter coefficients are not shared amongst filters, and multiplications
are most efficiently implemented by using DSP blocks whose input word 189 190 are most efficiently implemented by using DSP blocks whose input word
size is finite. DSPs are a scarce resource to be saved in a practical implementation. Keeping a high 190 191 size is finite. DSPs are a scarce resource to be saved in a practical implementation. Keeping a high
abstraction on the resource occupation is nevertheless selected in the following discussion in order 191 192 abstraction on the resource occupation is nevertheless selected in the following discussion in order
to leave enough degrees of freedom in the problem to try and find original solutions: too many 192 193 to leave enough degrees of freedom in the problem to try and find original solutions: too many
constraints in the initial statement of the problem leave little room for finding an optimal solution. 193 194 constraints in the initial statement of the problem leave little room for finding an optimal solution.
194 195
\begin{figure}[h!tb] 195 196 \begin{figure}[h!tb]
\begin{center} 196 197 \begin{center}
\includegraphics[width=.5\linewidth]{schema2} 197 198 \includegraphics[width=.5\linewidth]{schema2}
\caption{Shape of the filter transmitted power $P$ as a function of frequency: 198 199 \caption{Shape of the filter transmitted power $P$ as a function of frequency:
the bandpass BP is considered to occupy the initial 199 200 the bandpass BP is considered to occupy the initial
40\% of the Nyquist frequency range, the stopband the last 40\%, allowing 20\% transition 200 201 40\% of the Nyquist frequency range, the stopband the last 40\%, allowing 20\% transition
width.} 201 202 width.}
\label{rejection-shape} 202 203 \label{rejection-shape}
\end{center} 203 204 \end{center}
\end{figure} 204 205 \end{figure}
205 206
Following these considerations, the model is expressed as: 206 207 Following these considerations, the model is expressed as:
\begin{align} 207 208 \begin{align}
\begin{cases} 208 209 \begin{cases}
\mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\ 209 210 \mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\
\mathcal{A}_i &= N_i \times C_i\\ 210 211 \mathcal{A}_i &= N_i \times C_i\\
\Delta_i &= \Delta _{i-1} + \mathcal{P}_i 211 212 \Delta_i &= \Delta _{i-1} + \mathcal{P}_i
\end{cases} 212 213 \end{cases}
\label{model-FIR} 213 214 \label{model-FIR}
\end{align} 214 215 \end{align}
To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the stopband rejection dependence with $N_i$ and $C_i$, $\mathcal{A}_i$ 215 216 To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the stopband rejection dependence with $N_i$ and $C_i$, $\mathcal{A}_i$
is a theoretical area occupation of the processing block on the FPGA as discussed earlier, and $\Delta_i$ is the total rejection for the current stage $i$. 216 217 is a theoretical area occupation of the processing block on the FPGA as discussed earlier, and $\Delta_i$ is the total rejection for the current stage $i$.
Since the function $\mathcal{F}$ cannot be explictly expressed, we run simulations to determine the rejection depending 217 218 Since the function $\mathcal{F}$ cannot be explictly expressed, we run simulations to determine the rejection depending
on $N_i$ and $C_i$. However, selecting the right filter requires a clear definition of the rejection criterion. Selecting an 218 219 on $N_i$ and $C_i$. However, selecting the right filter requires a clear definition of the rejection criterion. Selecting an
incorrect criterion will lead the linear program solver to produce a solution which might not meet the user requirements. 219 220 incorrect criterion will lead the linear program solver to produce a solution which might not meet the user requirements.
Hence, amongst various criteria including the mean or median value of the FIR response in the stopband as will 220 221 Hence, amongst various criteria including the mean or median value of the FIR response in the stopband as will
be illustrated lated (section \ref{median}), we have designed 221 222 be illustrated lated (section \ref{median}), we have designed
a criterion aimed at avoiding ripples in the passband and considering the maximum of the FIR spectral response in the stopband 222 223 a criterion aimed at avoiding ripples in the passband and considering the maximum of the FIR spectral response in the stopband
(Fig. \ref{rejection-shape}). The bandpass criterion is defined as the sum of the absolute values of the spectral response 223 224 (Fig. \ref{rejection-shape}). The bandpass criterion is defined as the sum of the absolute values of the spectral response
in the bandpass, reminiscent of a standard deviation of the spectral response: this criterion must be minimized to avoid 224 225 in the bandpass, reminiscent of a standard deviation of the spectral response: this criterion must be minimized to avoid
ripples in the passband. The stopband transfer function maximum must also be minimized in order to improve the filter 225 226 ripples in the passband. The stopband transfer function maximum must also be minimized in order to improve the filter
rejection capability. Weighing these two criteria allows designing the linear program to be solved. 226 227 rejection capability. Weighing these two criteria allows designing the linear program to be solved.
227 228
\begin{figure}[h!tb] 228 229 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/noise-rejection.pdf} 229 230 \includegraphics[width=\linewidth]{images/noise-rejection.pdf}
\caption{Rejection as a function of number of coefficients and number of bits} 230 231 \caption{Rejection as a function of number of coefficients and number of bits}
\label{noise-rejection} 231 232 \label{noise-rejection}
\end{figure} 232 233 \end{figure}
233 234
The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource 234 235 The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource
occupation below a user-defined threshold, or as will be discussed here, aims at minimizing the area 235 236 occupation below a user-defined threshold, or as will be discussed here, aims at minimizing the area
needed to reach a given rejection ($\min(S_q)$ in the forthcoming discussion, Eqs. \ref{cstr_size} 236 237 needed to reach a given rejection ($\min(S_q)$ in the forthcoming discussion, Eqs. \ref{cstr_size}
and \ref{cstr_rejection}). The MILP solver is allowed to choose the number of successive 237 238 and \ref{cstr_rejection}). The MILP solver is allowed to choose the number of successive
filters, within an upper bound. The last problem is to model the noise rejection. Since filter 238 239 filters, within an upper bound. The last problem is to model the noise rejection. Since filter
noise rejection capability is not modeled with linear equations, a look-up-table is generated 239 240 noise rejection capability is not modeled with linear equations, a look-up-table is generated
for multiple filter configurations in which the $C_i$, $D_i$ and $N_i$ parameters are varied: for each 240 241 for multiple filter configurations in which the $C_i$, $D_i$ and $N_i$ parameters are varied: for each
one of these conditions, the low-pass filter rejection is stored as computed by the frequency response 241 242 one of these conditions, the low-pass filter rejection is stored as computed by the frequency response
of the digital filter (Fig. \ref{noise-rejection}). Various rejection criteria have been investigated, 242 243 of the digital filter (Fig. \ref{noise-rejection}). Various rejection criteria have been investigated,
including mean value of the stopband response, median value of the stopband response, or as finally 243 244 including mean value of the stopband response, median value of the stopband response, or as finally
selected, maximum value in the stopband. An intuitive analysis of the chart of Fig. \ref{noise-rejection} 244 245 selected, maximum value in the stopband. An intuitive analysis of the chart of Fig. \ref{noise-rejection}
hints at an optimum 245 246 hints at an optimum
set of tap length and number of bit for representing the coefficients along the line of the pyramidal 246 247 set of tap length and number of bit for representing the coefficients along the line of the pyramidal
shaped rejection capability function. 247 248 shaped rejection capability function.
248 249
Linear program formalism for solving the problem is well documented: an objective function is 249 250 Linear program formalism for solving the problem is well documented: an objective function is
defined which is linearly dependent on the parameters to be optimized. Constraints are expressed 250 251 defined which is linearly dependent on the parameters to be optimized. Constraints are expressed
as linear equation and solved using one of the available solvers, in our case GLPK\cite{glpk}. 251 252 as linear equation and solved using one of the available solvers, in our case GLPK\cite{glpk}.
With the notation explain in system \ref{model-FIR}, we have defined our linear problem like this: 252 253 With the notation explain in system \ref{model-FIR}, we have defined our linear problem like this:
\paragraph{Variables} 253 254 \paragraph{Variables}
\begin{align*} 254 255 \begin{align*}
x_{i,j} \in \lbrace 0,1 \rbrace & \text{ $i$ is a given filter} \\ 255 256 x_{i,j} \in \lbrace 0,1 \rbrace & \text{ $i$ is a given filter} \\
& \text{ $j$ is the stage} \\ 256 257 & \text{ $j$ is the stage} \\
& \text{ If $x_{i,j}$ is equal to 1, the filter is selected} \\ 257 258 & \text{ If $x_{i,j}$ is equal to 1, the filter is selected} \\
\end{align*} 258 259 \end{align*}
\paragraph{Constants} 259 260 \paragraph{Constants}
\begin{align*} 260 261 \begin{align*}
\mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\ 261 262 \mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\
& \text{ $p$ is the number of different filters} \\ 262 263 & \text{ $p$ is the number of different filters} \\
% N(i) & \text{ % Constant to let the 263 264 % N(i) & \text{ % Constant to let the
% number of coefficients %} \\ & \text{ 264 265 % number of coefficients %} \\ & \text{
% for filter $i$}\\ 265 266 % for filter $i$}\\
% C(i) & \text{ % Constant to let the 266 267 % C(i) & \text{ % Constant to let the
% number of bits of %}\\ & \text{ 267 268 % number of bits of %}\\ & \text{
% each coefficient for filter $i$}\\ 268 269 % each coefficient for filter $i$}\\
\mathcal{S}_{\max} & \text{ Total space available inside the FPGA} 269 270 \mathcal{S}_{\max} & \text{ Total space available inside the FPGA}
\end{align*} 270 271 \end{align*}
\paragraph{Constraints} 271 272 \paragraph{Constraints}
\begin{align} 272 273 \begin{align}
1 \leq i \leq p & \nonumber\\ 273 274 1 \leq i \leq p & \nonumber\\
1 \leq j \leq q & \text{ $q$ is the max of filter stage} \nonumber \\ 274 275 1 \leq j \leq q & \text{ $q$ is the max of filter stage} \nonumber \\
\forall j, \mathlarger{\sum_{i}} x_{i,j} = 1 & \text{ At most one filter by stage} \nonumber\\ 275 276 \forall j, \mathlarger{\sum_{i}} x_{i,j} = 1 & \text{ At most one filter by stage} \nonumber\\
\mathcal{S}_0 = 0 & \text{ initial occupation} \nonumber\\ 276 277 \mathcal{S}_0 = 0 & \text{ initial occupation} \nonumber\\
\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{A}_i)} \label{cstr_size} \\ 277 278 \forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{A}_i)} \label{cstr_size} \\
\mathcal{S}_j \leq \mathcal{S}_{\max}\nonumber \\ 278 279 \mathcal{S}_j \leq \mathcal{S}_{\max}\nonumber \\
\mathcal{N}_0 = 0 & \text{ initial rejection}\nonumber\\ 279 280 \mathcal{N}_0 = 0 & \text{ initial rejection}\nonumber\\
\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{R}_i)} \label{cstr_rejection} \\ 280 281 \forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{R}_i)} \label{cstr_rejection} \\
\mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\ 281 282 \mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\
& \text{ (e.g. 160~dB here)}\nonumber\\\nonumber 282 283 & \text{ (e.g. 160~dB here)}\nonumber\\\nonumber
\end{align} 283 284 \end{align}
\paragraph{Goal} 284 285 \paragraph{Goal}
\begin{align*} 285 286 \begin{align*}
\min \mathcal{S}_q 286 287 \min \mathcal{S}_q
\end{align*} 287 288 \end{align*}
288 289
The constraint \ref{cstr_size} means the occupation for the current stage $j$ depends on 289 290 The constraint \ref{cstr_size} means the occupation for the current stage $j$ depends on
the previous occupation and the occupation of current selected filter (it is possible 290 291 the previous occupation and the occupation of current selected filter (it is possible
that no filter is selected for this stage). And the second one \ref{cstr_rejection} 291 292 that no filter is selected for this stage). And the second one \ref{cstr_rejection}
means the same thing but for the rejection, the rejection depends the previous rejection 292 293 means the same thing but for the rejection, the rejection depends the previous rejection
plus the rejection of selected filter. 293 294 plus the rejection of selected filter.
294 295
\subsection{Low bandpass ripple and maximum rejection criteria} 295 296 \subsection{Low bandpass ripple and maximum rejection criteria}
296 297
The MILP solver provides a solution to the problem by selecting a series of small FIR with 297 298 The MILP solver provides a solution to the problem by selecting a series of small FIR with
increasing number of bits representing data and coefficients as well as an increasing number 298 299 increasing number of bits representing data and coefficients as well as an increasing number
of coefficients, instead of a single monolithic filter. 299 300 of coefficients, instead of a single monolithic filter.
300 301
\begin{figure}[h!tb] 301 302 \begin{figure}[h!tb]
% \includegraphics[width=\linewidth]{images/compare-fir.pdf} 302 303 % \includegraphics[width=\linewidth]{images/compare-fir.pdf}
\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-jmf-light.pdf} 303 304 \includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-jmf-light.pdf}
\caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR 304 305 \caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR
with a cutoff frequency set at half the Nyquist frequency.} 305 306 with a cutoff frequency set at half the Nyquist frequency.}
\label{compare-fir} 306 307 \label{compare-fir}
\end{figure} 307 308 \end{figure}
308 309
Fig. \ref{compare-fir} exhibits the 309 310 Fig. \ref{compare-fir} exhibits the
performance comparison between one solution and a monolithic FIR when selecting a cutoff 310 311 performance comparison between one solution and a monolithic FIR when selecting a cutoff
frequency of half the Nyquist frequency: a series of 5 FIR and a series of 10 FIR with the 311 312 frequency of half the Nyquist frequency: a series of 5 FIR and a series of 10 FIR with the
same space usage are provided as selected by the MILP solver. The FIR cascade provides improved 312 313 same space usage are provided as selected by the MILP solver. The FIR cascade provides improved
rejection than the monolithic FIR at the expense of a lower cutoff frequency which remains to 313 314 rejection than the monolithic FIR at the expense of a lower cutoff frequency which remains to
be tuned or compensated for. 314 315 be tuned or compensated for.
315 316
316 317
The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarized as Tab. \ref{t1}. 317 318 The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarized as Tab. \ref{t1}.
We have considered a set of resources representative of the hardware platform we work on, 318 319 We have considered a set of resources representative of the hardware platform we work on,
Avnet's Zedboard featuring a Xilinx XC7Z020-CLG484-1 Zynq System on Chip (SoC). The results reported in 319 320 Avnet's Zedboard featuring a Xilinx XC7Z020-CLG484-1 Zynq System on Chip (SoC). The results reported in
Tab. \ref{t1} emphasize that implementing the monolithic single FIR is impossible due to 320 321 Tab. \ref{t1} emphasize that implementing the monolithic single FIR is impossible due to
the insufficient hardware resources (exhausted LUT resources), while the FIR cascading 5 or 10 321 322 the insufficient hardware resources (exhausted LUT resources), while the FIR cascading 5 or 10
filters fit in the available resources. However, in all cases the DSP resources are fully 322 323 filters fit in the available resources. However, in all cases the DSP resources are fully
used: while the design can be synthesized using Xilinx proprietary Vivado 2016.2 software, 323 324 used: while the design can be synthesized using Xilinx proprietary Vivado 2016.2 software,
implementing the design fails due to the excessive resource usage preventing routing the signals 324 325 implementing the design fails due to the excessive resource usage preventing routing the signals
on the FPGA. Such results emphasize on the one hand the improvement prospect of the optimization 325 326 on the FPGA. Such results emphasize on the one hand the improvement prospect of the optimization
procedure by finding non-trivial solutions matching resource constraints, but on the other 326 327 procedure by finding non-trivial solutions matching resource constraints, but on the other
hand also illustrates the limitation of a model with an abstraction layer that does not account 327 328 hand also illustrates the limitation of a model with an abstraction layer that does not account
for the detailed architecture of the hardware. 328 329 for the detailed architecture of the hardware.
329 330
\begin{table}[h!tb] 330 331 \begin{table}[h!tb]
\caption{Resource occupation on a Xilinx Zynq-7000 series FPGA when synthesizing the FIR cascade 331 332 \caption{Resource occupation on a Xilinx Zynq-7000 series FPGA when synthesizing the FIR cascade
identified as optimal by the MILP solver within a finite resource criterion. The last line refers 332 333 identified as optimal by the MILP solver within a finite resource criterion. The last line refers
to available resources on a Zynq-7020 as found on the Zedboard.} 333 334 to available resources on a Zynq-7020 as found on the Zedboard.}
\begin{center} 334 335 \begin{center}
\begin{tabular}{|c|cccc|}\hline 335 336 \begin{tabular}{|c|cccc|}\hline
FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline 336 337 FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline
1 (monolithic) & 1 & 76183 & 220 & -162 \\ 337 338 1 (monolithic) & 1 & 76183 & 220 & -162 \\
5 & 5 & 18597 & 220 & -160 \\ 338 339 5 & 5 & 18597 & 220 & -160 \\
10 & 8 & 24729 & 220 & -161 \\\hline\hline 339 340 10 & 8 & 24729 & 220 & -161 \\\hline\hline
\textbf{Zynq 7020} & \textbf{420} & \textbf{53200} & \textbf{220} & \\\hline 340 341 \textbf{Zynq 7020} & \textbf{420} & \textbf{53200} & \textbf{220} & \\\hline
%\begin{tabular}{|c|ccccc|}\hline 341 342 %\begin{tabular}{|c|ccccc|}\hline
%FIR & BRAM36 & BRAM18 & LUT & DSP & rejection (dB)\\\hline\hline 342 343 %FIR & BRAM36 & BRAM18 & LUT & DSP & rejection (dB)\\\hline\hline
%1 (monolithic) & 1 & 0 & {\color{Red}76183} & 220 & -162 \\ 343 344 %1 (monolithic) & 1 & 0 & {\color{Red}76183} & 220 & -162 \\
%5 & 0 & 5 & {\color{Green}18597} & 220 & -160 \\ 344 345 %5 & 0 & 5 & {\color{Green}18597} & 220 & -160 \\
%10 & 0 & 8 & {\color{Green}24729} & 220 & -161 \\\hline\hline 345 346 %10 & 0 & 8 & {\color{Green}24729} & 220 & -161 \\\hline\hline
%\textbf{Zynq 7020} & \textbf{140} & \textbf{280} & \textbf{53200} & \textbf{220} & \\\hline 346 347 %\textbf{Zynq 7020} & \textbf{140} & \textbf{280} & \textbf{53200} & \textbf{220} & \\\hline
\end{tabular} 347 348 \end{tabular}
\end{center} 348 349 \end{center}
%\vspace{-0.7cm} 349 350 %\vspace{-0.7cm}
\label{t1} 350 351 \label{t1}
\end{table} 351 352 \end{table}
352 353
\subsection{Alternate criteria}\label{median} 353 354 \subsection{Alternate criteria}\label{median}
354 355
Fig. \ref{compare-fir} provides FIR solutions matching well the targeted transfer 355 356 Fig. \ref{compare-fir} provides FIR solutions matching well the targeted transfer
function, namely low ripple in the bandpass defined as the first 40\% of the frequency 356 357 function, namely low ripple in the bandpass defined as the first 40\% of the frequency
range and maximum rejection of 160~dB in the last 40\% stopband. We illustrate now, for 357 358 range and maximum rejection of 160~dB in the last 40\% stopband. We illustrate now, for
demonstrating the need to properly select the optimization criterion, two cases of poor 358 359 demonstrating the need to properly select the optimization criterion, two cases of poor
filter shapes obtained by selecting the mean value and median value of the rejection, 359 360 filter shapes obtained by selecting the mean value and median value of the rejection,
with no consideration for the ripples in the bandpass. The results of the optimizations, 360 361 with no consideration for the ripples in the bandpass. The results of the optimizations,
in these cases, are shown in Figs. \ref{compare-mean} and \ref{compare-median}. 361 362 in these cases, are shown in Figs. \ref{compare-mean} and \ref{compare-median}.
362 363
\begin{figure}[h!tb] 363 364 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-mean-light.pdf} 364 365 \includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-mean-light.pdf}
\caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR 365 366 \caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR
with a cutoff frequency set at half the Nyquist frequency.} 366 367 with a cutoff frequency set at half the Nyquist frequency.}
\label{compare-mean} 367 368 \label{compare-mean}
\end{figure} 368 369 \end{figure}
369 370
In the case of the mean value criterion (Fig. \ref{compare-mean}), the solution is not 370 371 In the case of the mean value criterion (Fig. \ref{compare-mean}), the solution is not
acceptable since the notch at the end of the transition band compensates for some unacceptable 371 372 acceptable since the notch at the end of the transition band compensates for some unacceptable
rise in the rejection close to the Nyquist frequency. Applying such a filter might yield excessive 372 373 rise in the rejection close to the Nyquist frequency. Applying such a filter might yield excessive
high frequency spurious components to be aliased at low frequency when decimating the signal. 373 374 high frequency spurious components to be aliased at low frequency when decimating the signal.
Similarly, the lack of criterion on the bandpass shape induces a shape with poor flatness and 374 375 Similarly, the lack of criterion on the bandpass shape induces a shape with poor flatness and
and slowly decaying transfer function starting to attenuate spectral components well before the 375 376 and slowly decaying transfer function starting to attenuate spectral components well before the
transition band starts. Such issues are partly aleviated by replacing a mean rejection value with 376 377 transition band starts. Such issues are partly aleviated by replacing a mean rejection value with
a median rejection value (Fig. \ref{compare-median}) but solutions remain unacceptable for 377 378 a median rejection value (Fig. \ref{compare-median}) but solutions remain unacceptable for
the reasons stated previously and much poorer than those found with the maximum rejection criterion 378 379 the reasons stated previously and much poorer than those found with the maximum rejection criterion
selected earlier (Fig. \ref{compare-fir}). 379 380 selected earlier (Fig. \ref{compare-fir}).
380 381
\begin{figure}[h!tb] 381 382 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-median-light.pdf} 382 383 \includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-median-light.pdf}
\caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR 383 384 \caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR
with a cutoff frequency set at half the Nyquist frequency.} 384 385 with a cutoff frequency set at half the Nyquist frequency.}
\label{compare-median} 385 386 \label{compare-median}
\end{figure} 386 387 \end{figure}
387 388
\section{Filter coefficient selection} 388 389 \section{Filter coefficient selection}
389 390
The coefficients of a single monolithic filter are computed as the impulse response 390 391 The coefficients of a single monolithic filter are computed as the impulse response
of the filter transfer function, and practically approximated by a multitude of methods 391 392 of the filter transfer function, and practically approximated by a multitude of methods
including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing 392 393 including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing
(Matlab's {\tt fir1} function). 393 394 (Matlab's {\tt fir1} function).
394 395
\begin{figure}[h!tb] 395 396 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/fir1-vs-firls} 396 397 \includegraphics[width=\linewidth]{images/fir1-vs-firls}
\caption{Evolution of the rejection capability of least-square optimized filters and Hamming 397 398 \caption{Evolution of the rejection capability of least-square optimized filters and Hamming
FIR filters as a function of the number of coefficients, for floating point numbers and 8-bit 398 399 FIR filters as a function of the number of coefficients, for floating point numbers and 8-bit
encoded integers.} 399 400 encoded integers.}
\label{2} 400 401 \label{2}
\end{figure} 401 402 \end{figure}
402 403
Cascading filters opens a new optimization opportunity by 403 404 Cascading filters opens a new optimization opportunity by
selecting various coefficient sets depending on the number of coefficients. Fig. \ref{2} 404 405 selecting various coefficient sets depending on the number of coefficients. Fig. \ref{2}
illustrates that for a number of coefficients ranging from 8 to 47, {\tt fir1} provides a better 405 406 illustrates that for a number of coefficients ranging from 8 to 47, {\tt fir1} provides a better
rejection than {\tt firls}: since the linear solver increases the number of coefficients along 406 407 rejection than {\tt firls}: since the linear solver increases the number of coefficients along
the processing chain, the type of selected filter also changes depending on the number of coefficients 407 408 the processing chain, the type of selected filter also changes depending on the number of coefficients
and evolves along the processing chain. 408 409 and evolves along the processing chain.
409 410
\section{Conclusion} 410 411 \section{Conclusion}
411 412
We address the optimization problem of designing a low-pass filter chain in a Field Programmable Gate 412 413 We address the optimization problem of designing a low-pass filter chain in a Field Programmable Gate
Array for improved noise rejection within constrained resource occupation, as needed for 413 414 Array for improved noise rejection within constrained resource occupation, as needed for
real time processing of radiofrequency signal when characterizing spectral phase noise 414 415 real time processing of radiofrequency signal when characterizing spectral phase noise
characteristics of stable oscillators. The flexibility of the digital approach makes the result 415 416 characteristics of stable oscillators. The flexibility of the digital approach makes the result
best suited for closing the loop and using the measurement output in a feedback loop for 416 417 best suited for closing the loop and using the measurement output in a feedback loop for
controlling clocks, e.g. in a quartz-stabilized high performance clock whose long term behavior 417 418 controlling clocks, e.g. in a quartz-stabilized high performance clock whose long term behavior
is controlled by non-piezoelectric resonator (sapphire resonator, microwave or optical 418 419 is controlled by non-piezoelectric resonator (sapphire resonator, microwave or optical
atomic transition). 419 420 atomic transition).
420 421
\section*{Acknowledgement} 421 422 \section*{Acknowledgement}
422 423
This work is supported by the ANR Programme d'Investissement d'Avenir in 423 424 This work is supported by the ANR Programme d'Investissement d'Avenir in
progress at the Time and Frequency Departments of the FEMTO-ST Institute 424 425 progress at the Time and Frequency Departments of the FEMTO-ST Institute
(Oscillator IMP, First-TF and Refimeve+), and by R\'egion de Franche-Comt\'e. 425 426 (Oscillator IMP, First-TF and Refimeve+), and by R\'egion de Franche-Comt\'e.
The authors would like to thank E. Rubiola, F. Vernotte, G. Cabodevila for support and 426 427 The authors would like to thank E. Rubiola, F. Vernotte, G. Cabodevila for support and
fruitful discussions. 427 428 fruitful discussions.
428 429
\bibliographystyle{IEEEtran} 429 430 \bibliographystyle{IEEEtran}
\balance 430 431 \balance
\bibliography{references,biblio} 431 432 \bibliography{references,biblio}
\end{document} 432 433 \end{document}
433 434
\section{Contexte d'ordonnancement} 434 435 \section{Contexte d'ordonnancement}
Dans cette partie, nous donnerons des d\'efinitions de termes rattach\'es au domaine de l'ordonnancement 435 436 Dans cette partie, nous donnerons des d\'efinitions de termes rattach\'es au domaine de l'ordonnancement
et nous verrons que le sujet trait\'e se rapproche beaucoup d'un problème d'ordonnancement. De ce fait 436 437 et nous verrons que le sujet trait\'e se rapproche beaucoup d'un problème d'ordonnancement. De ce fait
nous pourrons aller plus loin que les travaux vus pr\'ec\'edemment et nous tenterons des approches d'ordonnancement 437 438 nous pourrons aller plus loin que les travaux vus pr\'ec\'edemment et nous tenterons des approches d'ordonnancement
et d'optimisation. 438 439 et d'optimisation.
439 440
\subsection{D\'efinition du vocabulaire} 440 441 \subsection{D\'efinition du vocabulaire}
Avant tout, il faut d\'efinir ce qu'est un problème d'optimisation. Il y a deux d\'efinitions 441 442 Avant tout, il faut d\'efinir ce qu'est un problème d'optimisation. Il y a deux d\'efinitions
importantes à donner. La première est propos\'ee par Legrand et Robert dans leur livre \cite{def1-ordo} : 442 443 importantes à donner. La première est propos\'ee par Legrand et Robert dans leur livre \cite{def1-ordo} :
\begin{definition} 443 444 \begin{definition}
\label{def-ordo1} 444 445 \label{def-ordo1}
Un ordonnancement d'un système de t\^aches $G\ =\ (V,\ E,\ w)$ est une fonction $\sigma$ : 445 446 Un ordonnancement d'un système de t\^aches $G\ =\ (V,\ E,\ w)$ est une fonction $\sigma$ :
$V \rightarrow \mathbb{N}$ telle que $\sigma(u) + w(u) \leq \sigma(v)$ pour toute arête $(u,\ v) \in E$. 446 447 $V \rightarrow \mathbb{N}$ telle que $\sigma(u) + w(u) \leq \sigma(v)$ pour toute arête $(u,\ v) \in E$.
\end{definition} 447 448 \end{definition}
448 449
Dit plus simplement, l'ensemble $V$ repr\'esente les t\^aches à ex\'ecuter, l'ensemble $E$ repr\'esente les d\'ependances 449 450 Dit plus simplement, l'ensemble $V$ repr\'esente les t\^aches à ex\'ecuter, l'ensemble $E$ repr\'esente les d\'ependances
des t\^aches et $w$ les temps d'ex\'ecution de la t\^ache. La fonction $\sigma$ donne donc l'heure de d\'ebut de 450 451 des t\^aches et $w$ les temps d'ex\'ecution de la t\^ache. La fonction $\sigma$ donne donc l'heure de d\'ebut de
chacune des t\^aches. La d\'efinition dit que si une t\^ache $v$ d\'epend d'une t\^ache $u$ alors 451 452 chacune des t\^aches. La d\'efinition dit que si une t\^ache $v$ d\'epend d'une t\^ache $u$ alors
la date de d\'ebut de $v$ sera plus grande ou \'egale au d\'ebut de l'ex\'ecution de la t\^ache $u$ plus son 452 453 la date de d\'ebut de $v$ sera plus grande ou \'egale au d\'ebut de l'ex\'ecution de la t\^ache $u$ plus son
temps d'ex\'ecution. 453 454 temps d'ex\'ecution.
454 455
Une autre d\'efinition importante qui est propos\'ee par Leung et al. \cite{def2-ordo} est : 455 456 Une autre d\'efinition importante qui est propos\'ee par Leung et al. \cite{def2-ordo} est :
\begin{definition} 456 457 \begin{definition}
\label{def-ordo2} 457 458 \label{def-ordo2}
L'ordonnancement traite de l'allocation de ressources rares à des activit\'es avec 458 459 L'ordonnancement traite de l'allocation de ressources rares à des activit\'es avec
l'objectif d'optimiser un ou plusieurs critères de performance. 459 460 l'objectif d'optimiser un ou plusieurs critères de performance.
\end{definition} 460 461 \end{definition}
461 462
Cette d\'efinition est plus g\'en\'erique mais elle nous int\'eresse d'avantage que la d\'efinition \ref{def-ordo1}. 462 463 Cette d\'efinition est plus g\'en\'erique mais elle nous int\'eresse d'avantage que la d\'efinition \ref{def-ordo1}.
En effet, la partie qui nous int\'eresse dans cette première d\'efinition est le respect de la pr\'ec\'edance des t\^aches. 463 464 En effet, la partie qui nous int\'eresse dans cette première d\'efinition est le respect de la pr\'ec\'edance des t\^aches.
Dans les faits les dates de d\'ebut ne nous int\'eressent pas r\'eellement. 464 465 Dans les faits les dates de d\'ebut ne nous int\'eressent pas r\'eellement.
465 466
En revanche la d\'efinition \ref{def-ordo2} sera au c\oe{}ur du projet. Pour se convaincre de cela, 466 467 En revanche la d\'efinition \ref{def-ordo2} sera au c\oe{}ur du projet. Pour se convaincre de cela,
il nous faut d'abord d\'efinir quel est le type de problème d'ordonnancement qu'on traite et quelles 467 468 il nous faut d'abord d\'efinir quel est le type de problème d'ordonnancement qu'on traite et quelles
sont les m\'ethodes qu'on peut appliquer. 468 469 sont les m\'ethodes qu'on peut appliquer.
469 470
Les problèmes d'ordonnancement peuvent être class\'es en diff\'erentes cat\'egories : 470 471 Les problèmes d'ordonnancement peuvent être class\'es en diff\'erentes cat\'egories :
\begin{itemize} 471 472 \begin{itemize}
\item T\^aches ind\'ependantes : dans cette cat\'egorie de problèmes, les t\^aches sont complètement ind\'ependantes 472 473 \item T\^aches ind\'ependantes : dans cette cat\'egorie de problèmes, les t\^aches sont complètement ind\'ependantes
les unes des autres. Dans notre cas, ce n'est pas le plus adapt\'e. 473 474 les unes des autres. Dans notre cas, ce n'est pas le plus adapt\'e.
\item Graphe de t\^aches : la d\'efinition \ref{def-ordo1} d\'ecrit cette cat\'egorie. La plupart du temps, 474 475 \item Graphe de t\^aches : la d\'efinition \ref{def-ordo1} d\'ecrit cette cat\'egorie. La plupart du temps,
les t\^aches sont repr\'esent\'ees par une DAG. Cette cat\'egorie est très proche de notre cas puisque nous devons \'egalement ex\'ecuter 475 476 les t\^aches sont repr\'esent\'ees par une DAG. Cette cat\'egorie est très proche de notre cas puisque nous devons \'egalement ex\'ecuter
des t\^aches qui ont un certain nombre de d\'ependances. On pourra même dire que dans certain cas, 476 477 des t\^aches qui ont un certain nombre de d\'ependances. On pourra même dire que dans certain cas,
on a des anti-arbres, c'est à dire que nous avons une multitude de t\^aches d'entr\'ees qui convergent vers une 477 478 on a des anti-arbres, c'est à dire que nous avons une multitude de t\^aches d'entr\'ees qui convergent vers une
t\^ache de fin. 478 479 t\^ache de fin.
\item Workflow : cette cat\'egorie est une sous cat\'egorie des graphes de t\^aches dans le sens où 479 480 \item Workflow : cette cat\'egorie est une sous cat\'egorie des graphes de t\^aches dans le sens où
il s'agit d'un graphe de t\^aches r\'ep\'et\'e de nombreuses de fois. C'est exactement ce type de problème 480 481 il s'agit d'un graphe de t\^aches r\'ep\'et\'e de nombreuses de fois. C'est exactement ce type de problème
que nous traitons ici. 481 482 que nous traitons ici.
\end{itemize} 482 483 \end{itemize}
483 484
Bien entendu, cette liste n'est pas exhaustive et il existe de nombreuses autres classifications et sous-classifications 484 485 Bien entendu, cette liste n'est pas exhaustive et il existe de nombreuses autres classifications et sous-classifications
de ces problèmes. Nous n'avons parl\'e ici que des cat\'egories les plus communes. 485 486 de ces problèmes. Nous n'avons parl\'e ici que des cat\'egories les plus communes.
486 487
Un autre point à d\'efinir, est le critère d'optimisation. Il y a là encore un grand nombre de 487 488 Un autre point à d\'efinir, est le critère d'optimisation. Il y a là encore un grand nombre de
critères possibles. Nous allons donc parler des principaux : 488 489 critères possibles. Nous allons donc parler des principaux :
\begin{itemize} 489 490 \begin{itemize}
\item Temps de compl\'etion total (ou Makespan en anglais) : ce critère est l'un des critères d'optimisation 490 491 \item Temps de compl\'etion total (ou Makespan en anglais) : ce critère est l'un des critères d'optimisation
les plus courant. Il s'agit donc de minimiser la date de fin de la dernière t\^ache de l'ensemble des 491 492 les plus courant. Il s'agit donc de minimiser la date de fin de la dernière t\^ache de l'ensemble des
t\^aches à ex\'ecuter. L'enjeu de cette optimisation est donc de trouver l'ordonnancement optimal permettant 492 493 t\^aches à ex\'ecuter. L'enjeu de cette optimisation est donc de trouver l'ordonnancement optimal permettant
la fin d'ex\'ecution au plus tôt. 493 494 la fin d'ex\'ecution au plus tôt.
\item Somme des temps d'ex\'ecution (Flowtime en anglais) : il s'agit de faire la somme des temps d'ex\'ecution de toutes les t\^aches 494 495 \item Somme des temps d'ex\'ecution (Flowtime en anglais) : il s'agit de faire la somme des temps d'ex\'ecution de toutes les t\^aches
et d'optimiser ce r\'esultat. 495 496 et d'optimiser ce r\'esultat.
\item Le d\'ebit : ce critère quant à lui, vise à augmenter au maximum le d\'ebit de traitement des donn\'ees. 496 497 \item Le d\'ebit : ce critère quant à lui, vise à augmenter au maximum le d\'ebit de traitement des donn\'ees.
\end{itemize} 497 498 \end{itemize}
498 499
En plus de cela, on peut avoir besoin de plusieurs critères d'optimisation. Il s'agit dans ce cas d'une optimisation 499 500 En plus de cela, on peut avoir besoin de plusieurs critères d'optimisation. Il s'agit dans ce cas d'une optimisation
multi-critères. Bien entendu, cela complexifie d'autant plus le problème car la solution la plus optimale pour un 500 501 multi-critères. Bien entendu, cela complexifie d'autant plus le problème car la solution la plus optimale pour un
des critères peut être très mauvaise pour un autre critère. De ce cas, il s'agira de trouver une solution qui permet 501 502 des critères peut être très mauvaise pour un autre critère. De ce cas, il s'agira de trouver une solution qui permet
de faire le meilleur compromis entre tous les critères. 502 503 de faire le meilleur compromis entre tous les critères.
503 504
\subsection{Formalisation du problème} 504 505 \subsection{Formalisation du problème}
\label{formalisation} 505 506 \label{formalisation}
Maintenant que nous avons donn\'e le vocabulaire li\'e à l'ordonnancement, nous allons pouvoir essayer caract\'eriser 506 507 Maintenant que nous avons donn\'e le vocabulaire li\'e à l'ordonnancement, nous allons pouvoir essayer caract\'eriser
formellement notre problème. En effet, nous allons reprendre les contraintes \'enonc\'ees dans la sections \ref{def-contraintes} 507 508 formellement notre problème. En effet, nous allons reprendre les contraintes \'enonc\'ees dans la sections \ref{def-contraintes}
et nous essayerons de les formaliser le plus finement possible. 508 509 et nous essayerons de les formaliser le plus finement possible.
509 510
Comme nous l'avons dit, une t\^ache est un bloc de traitement. Chaque t\^ache $i$ dispose d'un ensemble de paramètres 510 511 Comme nous l'avons dit, une t\^ache est un bloc de traitement. Chaque t\^ache $i$ dispose d'un ensemble de paramètres
que nous nommerons $\mathcal{P}_{i}$. Cet ensemble $\mathcal{P}_i$ est propre à chaque t\^ache et il variera d'une 511 512 que nous nommerons $\mathcal{P}_{i}$. Cet ensemble $\mathcal{P}_i$ est propre à chaque t\^ache et il variera d'une
t\^ache à l'autre. Nous reviendrons plus tard sur les paramètres qui peuvent composer cet ensemble. 512 513 t\^ache à l'autre. Nous reviendrons plus tard sur les paramètres qui peuvent composer cet ensemble.
513 514
Outre cet ensemble $\mathcal{P}_i$, chaque t\^ache dispose de paramètres communs : 514 515 Outre cet ensemble $\mathcal{P}_i$, chaque t\^ache dispose de paramètres communs :
\begin{itemize} 515 516 \begin{itemize}
\item Dur\'ee de la t\^ache : Comme nous l'avons dit auparavant, dans le cadre d'un FPGA le temps est compt\'e en nombre de coup d'horloge. 516 517 \item Dur\'ee de la t\^ache : Comme nous l'avons dit auparavant, dans le cadre d'un FPGA le temps est compt\'e en nombre de coup d'horloge.
En outre, les blocs sont toujours sollicit\'es, certains même sont capables de lire et de renvoyer une r\'esultat à chaque coups d'horloge. 517 518 En outre, les blocs sont toujours sollicit\'es, certains même sont capables de lire et de renvoyer une r\'esultat à chaque coups d'horloge.
Donc la dur\'ee d'une t\^ache ne peut être le laps de temps entre l'entr\'ee d'une donn\'ee et la sortie d'une autre. Nous d\'efinirons la 518 519 Donc la dur\'ee d'une t\^ache ne peut être le laps de temps entre l'entr\'ee d'une donn\'ee et la sortie d'une autre. Nous d\'efinirons la
dur\'ee comme le temps de traitement d'une donn\'ee, c'est à dire la diff\'erence de temps entre la date de sortie d'une donn\'ee 519 520 dur\'ee comme le temps de traitement d'une donn\'ee, c'est à dire la diff\'erence de temps entre la date de sortie d'une donn\'ee
et de sa date d'entr\'ee. Nous nommerons cette dur\'ee $\delta_i$. % Je devrais la nomm\'ee w comme dans la def2 520 521 et de sa date d'entr\'ee. Nous nommerons cette dur\'ee $\delta_i$. % Je devrais la nomm\'ee w comme dans la def2
\item La pr\'ecision : La pr\'ecision d'une donn\'ee est le nombre de bits significatifs qu'elle compte. En effet, au fil des traitements 521 522 \item La pr\'ecision : La pr\'ecision d'une donn\'ee est le nombre de bits significatifs qu'elle compte. En effet, au fil des traitements
les pr\'ecisions peuvent varier. On nomme donc la pr\'ecision d'entr\'ee d'une t\^ache $i$ comme $\pi_i^-$ et la pr\'ecision en sortie $\pi_i^+$. 522 523 les pr\'ecisions peuvent varier. On nomme donc la pr\'ecision d'entr\'ee d'une t\^ache $i$ comme $\pi_i^-$ et la pr\'ecision en sortie $\pi_i^+$.
\item La fr\'equence du flux en entr\'ee (ou sortie) : Cette fr\'equence repr\'esente la fr\'equence des donn\'ees qui arrivent (resp. sortent). 523 524 \item La fr\'equence du flux en entr\'ee (ou sortie) : Cette fr\'equence repr\'esente la fr\'equence des donn\'ees qui arrivent (resp. sortent).
Selon les t\^aches, les fr\'equences varieront. En effet, certains blocs ralentissent le flux c'est pourquoi on distingue la fr\'equence du 524 525 Selon les t\^aches, les fr\'equences varieront. En effet, certains blocs ralentissent le flux c'est pourquoi on distingue la fr\'equence du
flux en entr\'ee et la fr\'equence en sortie. Nous nommerons donc la fr\'equence du flux en entr\'ee $f_i^-$ et la fr\'equence en sortie $f_i^+$. 525 526 flux en entr\'ee et la fr\'equence en sortie. Nous nommerons donc la fr\'equence du flux en entr\'ee $f_i^-$ et la fr\'equence en sortie $f_i^+$.
\item La quantit\'e de donn\'ees en entr\'ee (ou en sortie) : Il s'agit de la quantit\'e de donn\'ees que le bloc s'attend à traiter (resp. 526 527 \item La quantit\'e de donn\'ees en entr\'ee (ou en sortie) : Il s'agit de la quantit\'e de donn\'ees que le bloc s'attend à traiter (resp.
est capable de produire). Les t\^aches peuvent avoir à traiter des gros volumes de donn\'ees et n'en ressortir qu'une partie. Cette 527 528 est capable de produire). Les t\^aches peuvent avoir à traiter des gros volumes de donn\'ees et n'en ressortir qu'une partie. Cette
fois encore, il nous faut donc diff\'erencier l'entr\'ee et la sortie. Nous nommerons donc la quantit\'e de donn\'ees entrantes $q_i^-$ 528 529 fois encore, il nous faut donc diff\'erencier l'entr\'ee et la sortie. Nous nommerons donc la quantit\'e de donn\'ees entrantes $q_i^-$
et la quantit\'e de donn\'ees sortantes $q_i^+$ pour une t\^ache $i$. 529 530 et la quantit\'e de donn\'ees sortantes $q_i^+$ pour une t\^ache $i$.
\item Le d\'ebit d'entr\'ee (ou de sortie) : Ce paramètre correspond au d\'ebit de donn\'ees que la t\^ache est capable de traiter ou qu'elle 530 531 \item Le d\'ebit d'entr\'ee (ou de sortie) : Ce paramètre correspond au d\'ebit de donn\'ees que la t\^ache est capable de traiter ou qu'elle
fournit en sortie. Il s'agit simplement de l'expression des deux pr\'ec\'edents paramètres. Nous d\'efinirons donc la d\'ebit entrant de la 531 532 fournit en sortie. Il s'agit simplement de l'expression des deux pr\'ec\'edents paramètres. Nous d\'efinirons donc la d\'ebit entrant de la
t\^ache $i$ comme $d_i^-\ =\ q_i^-\ *\ f_i^-$ et le d\'ebit sortant comme $d_i^+\ =\ q_i^+\ *\ f_i^+$. 532 533 t\^ache $i$ comme $d_i^-\ =\ q_i^-\ *\ f_i^-$ et le d\'ebit sortant comme $d_i^+\ =\ q_i^+\ *\ f_i^+$.
\item La taille de la t\^ache : La taille dans les FPGA \'etant limit\'ee, ce paramètre exprime donc la place qu'occupe la t\^ache au sein du bloc. 533 534 \item La taille de la t\^ache : La taille dans les FPGA \'etant limit\'ee, ce paramètre exprime donc la place qu'occupe la t\^ache au sein du bloc.
Nous nommerons $\mathcal{A}_i$ cette taille. 534 535 Nous nommerons $\mathcal{A}_i$ cette taille.
\item Les pr\'ed\'ecesseurs et successeurs d'une t\^ache : cela nous permet de connaître les t\^aches requises pour pouvoir traiter 535 536 \item Les pr\'ed\'ecesseurs et successeurs d'une t\^ache : cela nous permet de connaître les t\^aches requises pour pouvoir traiter
la t\^ache $i$ ainsi que les t\^aches qui en d\'ependent. Ces ensemble sont not\'es $\Gamma _i ^-$ et $ \Gamma _i ^+$ \\ 536 537 la t\^ache $i$ ainsi que les t\^aches qui en d\'ependent. Ces ensemble sont not\'es $\Gamma _i ^-$ et $ \Gamma _i ^+$ \\
%TODO Est-ce vraiment un paramètre ? 537 538 %TODO Est-ce vraiment un paramètre ?
\end{itemize} 538 539 \end{itemize}
539 540
Ces diff\'erents paramètres communs sont fortement li\'es aux \'el\'ements de $\mathcal{P}_i$. Voici quelques exemples de relations 540 541 Ces diff\'erents paramètres communs sont fortement li\'es aux \'el\'ements de $\mathcal{P}_i$. Voici quelques exemples de relations
que nous avons identifi\'ees : 541 542 que nous avons identifi\'ees :
\begin{itemize} 542 543 \begin{itemize}
\item $ \delta _i ^+ \ = \ \mathcal{F}_{\delta}(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+,\ \mathcal{P}_i) $ donne le temps d'ex\'ecution 543 544 \item $ \delta _i ^+ \ = \ \mathcal{F}_{\delta}(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+,\ \mathcal{P}_i) $ donne le temps d'ex\'ecution
de la t\^ache en fonction de la pr\'ecision voulue, du d\'ebit et des paramètres internes. 544 545 de la t\^ache en fonction de la pr\'ecision voulue, du d\'ebit et des paramètres internes.
\item $ \pi _i ^+ \ = \ \mathcal{F}_{p}(\pi_i^-,\ \mathcal{P}_i) $, la fonction $F_p$ donne la pr\'ecision en sortie selon la pr\'ecision de d\'epart 545 546 \item $ \pi _i ^+ \ = \ \mathcal{F}_{p}(\pi_i^-,\ \mathcal{P}_i) $, la fonction $F_p$ donne la pr\'ecision en sortie selon la pr\'ecision de d\'epart
et les paramètres internes de la t\^ache. 546 547 et les paramètres internes de la t\^ache.
\item $d_i^+\ =\ \mathcal{F}_d(d_i^-, \mathcal{P}_i)$, la fonction $F_d$ donne le d\'ebit sortant de la t\^ache en fonction du d\'ebit 547 548 \item $d_i^+\ =\ \mathcal{F}_d(d_i^-, \mathcal{P}_i)$, la fonction $F_d$ donne le d\'ebit sortant de la t\^ache en fonction du d\'ebit
sortant et des variables internes de la t\^ache. 548 549 sortant et des variables internes de la t\^ache.
\item $A_i^+\ =\ \mathcal{F}_A(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+, \mathcal{P}_i)$ 549 550 \item $A_i^+\ =\ \mathcal{F}_A(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+, \mathcal{P}_i)$
\end{itemize} 550 551 \end{itemize}
Pour le moment, nous ne sommes pas capables de donner une d\'efinition g\'en\'erale de ces fonctions. Mais en revanche, 551 552 Pour le moment, nous ne sommes pas capables de donner une d\'efinition g\'en\'erale de ces fonctions. Mais en revanche,
sur quelques exemples simples (cf. \ref{def-contraintes}), nous parvenons à donner une \'evaluation de ces fonctions. 552 553 sur quelques exemples simples (cf. \ref{def-contraintes}), nous parvenons à donner une \'evaluation de ces fonctions.
553 554
Maintenant que nous avons donn\'e toutes les notations utiles, nous allons \'enoncer des contraintes relatives à notre problème. Soit 554 555 Maintenant que nous avons donn\'e toutes les notations utiles, nous allons \'enoncer des contraintes relatives à notre problème. Soit
un DGA $G(V,\ E)$, on a pour toutes arêtes $(i, j)\ \in\ E$ les in\'equations suivantes : 555 556 un DGA $G(V,\ E)$, on a pour toutes arêtes $(i, j)\ \in\ E$ les in\'equations suivantes :
556 557
\paragraph{Contrainte de pr\'ecision :} 557 558 \paragraph{Contrainte de pr\'ecision :}
Cette in\'equation traduit la contrainte de pr\'ecision d'une t\^ache à l'autre : 558 559 Cette in\'equation traduit la contrainte de pr\'ecision d'une t\^ache à l'autre :
\begin{align*} 559 560 \begin{align*}
\pi _i ^+ \geq \pi _j ^- 560 561 \pi _i ^+ \geq \pi _j ^-
\end{align*} 561 562 \end{align*}
562 563
\paragraph{Contrainte de d\'ebit :} 563 564 \paragraph{Contrainte de d\'ebit :}
Cette in\'equation traduit la contrainte de d\'ebit d'une t\^ache à l'autre : 564 565 Cette in\'equation traduit la contrainte de d\'ebit d'une t\^ache à l'autre :
\begin{align*} 565 566 \begin{align*}
d _i ^+ = q _j ^- * (f_i + (1 / s_j) ) & \text{ où } s_j \text{ est une valeur positive de temporisation de la t\^ache} 566 567 d _i ^+ = q _j ^- * (f_i + (1 / s_j) ) & \text{ où } s_j \text{ est une valeur positive de temporisation de la t\^ache}
\end{align*} 567 568 \end{align*}
568 569
\paragraph{Contrainte de synchronisation :} 569 570 \paragraph{Contrainte de synchronisation :}
Il s'agit de la contrainte qui impose que si à un moment du traitement, le DAG se s\'epare en plusieurs branches parallèles 570 571 Il s'agit de la contrainte qui impose que si à un moment du traitement, le DAG se s\'epare en plusieurs branches parallèles
et qu'elles se rejoignent plus tard, la somme des latences sur chacune des branches soit la même. 571 572 et qu'elles se rejoignent plus tard, la somme des latences sur chacune des branches soit la même.
Plus formellement, s'il existe plusieurs chemins disjoints, partant de la t\^ache $s$ et allant à la t\^ache de $f$ alors : 572 573 Plus formellement, s'il existe plusieurs chemins disjoints, partant de la t\^ache $s$ et allant à la t\^ache de $f$ alors :
\begin{align*} 573 574 \begin{align*}
\forall \text{ chemin } \mathcal{C}1(s, .., f), 574 575 \forall \text{ chemin } \mathcal{C}1(s, .., f),
\forall \text{ chemin } \mathcal{C}2(s, .., f) 575 576 \forall \text{ chemin } \mathcal{C}2(s, .., f)
\text{ tel que } \mathcal{C}1 \neq \mathcal{C}2 576 577 \text{ tel que } \mathcal{C}1 \neq \mathcal{C}2
\Rightarrow 577 578 \Rightarrow
\sum _{i} ^{i \in \mathcal{C}1} \delta_i = \sum _{i} ^{i \in \mathcal{C}2} \delta_i 578 579 \sum _{i} ^{i \in \mathcal{C}1} \delta_i = \sum _{i} ^{i \in \mathcal{C}2} \delta_i
\end{align*} 579 580 \end{align*}
580 581
\paragraph{Contrainte de place :} 581 582 \paragraph{Contrainte de place :}
Cette in\'equation traduit la contrainte de place dans le FPGA. La taille max de la puce FPGA est nomm\'e $\mathcal{A}_{FPGA}$ : 582 583 Cette in\'equation traduit la contrainte de place dans le FPGA. La taille max de la puce FPGA est nomm\'e $\mathcal{A}_{FPGA}$ :
\begin{align*} 583 584 \begin{align*}
\sum ^{\text{t\^ache } i} \mathcal{A}_i \leq \mathcal{A}_{FPGA} 584 585 \sum ^{\text{t\^ache } i} \mathcal{A}_i \leq \mathcal{A}_{FPGA}
\end{align*} 585 586 \end{align*}
586 587
\subsection{Exemples de mod\'elisation} 587 588 \subsection{Exemples de mod\'elisation}
\label{exemples-modeles} 588 589 \label{exemples-modeles}
Nous allons maintenant prendre quelques blocs de traitement simples afin d'illustrer au mieux notre modèle. 589 590 Nous allons maintenant prendre quelques blocs de traitement simples afin d'illustrer au mieux notre modèle.