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1 % JMF : revoir l'abstract : on y avait mis le Zynq7010 de la redpitaya en montrant
2 % comment optimiser les perfs a surface finie. Ici aussi on tombait dans le cas ou`
3 % la solution a 1 seul FIR n'etait simplement pas synthetisable => fusionner les deux
4 % contributions pour le papier TUFFC
5
\documentclass[a4paper,conference]{IEEEtran/IEEEtran} 1 6 \documentclass[a4paper,conference]{IEEEtran/IEEEtran}
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\usepackage[normalem]{ulem} 9 14 \usepackage[normalem]{ulem}
% correct bad hyphenation here 10 15 % correct bad hyphenation here
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\setlength{\footskip}{30pt} 13 18 \setlength{\footskip}{30pt}
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\begin{document} 15 20 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 16 21 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 17 22 to oscillator metrology}
18 23
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 19 24 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 20 25 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}} 21 26 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France } 22 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 \\ 23 28 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 24 29 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 25 30 }
\maketitle 26 31 \maketitle
\thispagestyle{plain} 27 32 \thispagestyle{plain}
\pagestyle{plain} 28 33 \pagestyle{plain}
\newtheorem{definition}{Definition} 29 34 \newtheorem{definition}{Definition}
30 35
\begin{abstract} 31 36 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 32 37 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 33 38 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 34 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 35 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 36 41 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 37 42 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. 38 43 required and keeping timing constraints within the targeted measurement bandwidths.
\end{abstract} 39 44 \end{abstract}
40 45
\begin{IEEEkeywords} 41 46 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 42 47 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 43 48 \end{IEEEkeywords}
44 49
\section{Digital signal processing of ultrastable clock signals} 45 50 \section{Digital signal processing of ultrastable clock signals}
46 51
Analog oscillator phase noise characteristics are classically performed by downconverting 47 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, 48 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 49 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 50 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}. 51 56 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
52 57
\begin{figure}[h!tb] 53 58 \begin{figure}[h!tb]
\begin{center} 54 59 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 55 60 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 56 61 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 57 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 58 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 59 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 60 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 61 66 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 62 67 the spectral characteristics of the phase fluctuations.}
\label{schema} 63 68 \label{schema}
\end{figure} 64 69 \end{figure}
65 70
As with the analog mixer, 66 71 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 67 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. 68 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 69 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 70 75 for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
downconverter 71 76 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 72 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 73 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 74 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 75 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 76 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 77 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 78 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 79 84 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 80 85 data being processed.
81 86
\section{Finite impulse response filter} 82 87 \section{Finite impulse response filter}
83 88
We select FIR filter for their unconditional stability and ease of design. A FIR filter is defined 84 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 85 90 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 86 91 outputs $y_k$
$$y_n=\sum_{k=0}^N b_k x_{n-k}$$ 87 92 $$y_n=\sum_{k=0}^N b_k x_{n-k}$$
88 93
As opposed to an implementation on a general purpose processor in which word size is defined by the 89 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 90 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 91 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 92 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 93 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 94 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). 95 100 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 96 101 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, 97 102 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. 98 103 is not considered as an issue as would be in a closed loop system.
99 104
The coefficients are classically expressed as floating point values. However, this binary 100 105 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, 101 106 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 102 107 we select to quantify these floating point values into integer values. This quantization
will result in some precision loss. 103 108 will result in some precision loss.
104 109
%As illustrated in Fig. \ref{float_vs_int}, we see that we aren't 105 110 %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, 106 111 %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. 107 112 %the first and last are equal to zero. But if we have too sample size for few coefficients that not improve the quality.
108 113
% JMF je ne comprends pas la derniere phrase ci-dessus ni la figure ci dessous 109 114 % 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) 110 115 % 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 111 116 % et que l'inverse trop de bit sur pas assez de coeff on ne gagne rien, je vais essayer de la reformuler
112 117
%\begin{figure}[h!tb] 113 118 %\begin{figure}[h!tb]
%\includegraphics[width=\linewidth]{images/float-vs-integer.pdf} 114 119 %\includegraphics[width=\linewidth]{images/float-vs-integer.pdf}
%\caption{Impact of the quantization resolution of the coefficients} 115 120 %\caption{Impact of the quantization resolution of the coefficients}
%\label{float_vs_int} 116 121 %\label{float_vs_int}
%\end{figure} 117 122 %\end{figure}
118 123
\begin{figure}[h!tb] 119 124 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/demo_filtre} 120 125 \includegraphics[width=\linewidth]{images/demo_filtre}
\caption{Impact of the quantization resolution of the coefficients: the quantization is 121 126 \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 122 127 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 123 128 the 30~first and 30~last coefficients out of the initial 128~band-pass
filter coefficients to 0 (red dots).} 124 129 filter coefficients to 0 (red dots).}
\label{float_vs_int} 125 130 \label{float_vs_int}
\end{figure} 126 131 \end{figure}
127 132
The tradeoff between quantization resolution and number of coefficients when considering 128 133 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 129 134 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 130 135 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 131 136 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 132 137 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 133 138 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 134 139 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 135 140 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 136 141 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 137 142 and tap length, as will be shown in the next section. Indeed, our development strategy closely
follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards} 138 143 follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
in which basic blocks are defined and characterized before being assembled \cite{hide} 139 144 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 140 145 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 141 146 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 142 147 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 143 148 current implementation: the decimation is assumed to be located after the FIR cascade at the
moment. 144 149 moment.
145 150
\section{Filter optimization} 146 151 \section{Filter optimization}
147 152
A basic approach for implementing the FIR filter is to compute the transfer function of 148 153 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 149 154 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 150 155 (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 151 156 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. 152 157 this filter must process data stream at the radiofrequency sampling rate after the mixer.
153 158
An optimization problem \cite{leung2004handbook} aims at improving one or many 154 159 An optimization problem \cite{leung2004handbook} aims at improving one or many
performance criteria within a constrained resource environment. Amongst the tools 155 160 performance criteria within a constrained resource environment. Amongst the tools
developed to meet this aim, Mixed-Integer Linear Programming (MILP) provides the framework to 156 161 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 157 162 formally define the stated problem and search for an optimal use of available
resources \cite{yu2007design, kodek1980design}. 158 163 resources \cite{yu2007design, kodek1980design}.
159 164
First we need to ensure that our problem is a real optimization problem. When 160 165 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 161 166 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 162 167 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) 163 168 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 164 169 or LUT (Look Up Table), a tradeoff must be generally searched between performance and available
computational resources: optimizing some criteria within finite, limited 165 170 computational resources: optimizing some criteria within finite, limited
resources indeed matches the definition of a classical optimization problem. 166 171 resources indeed matches the definition of a classical optimization problem.
167 172
Specifically the degrees of freedom when addressing the problem of replacing the single monolithic 168 173 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 $i$, 169 174 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 and the number of bits $D_i$ representing 170 175 the number of bits $C_i$ representing the coefficients and the number of bits $D_i$ representing
the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage, 171 176 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 172 177 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 $(D_i+C_i) \times N_i$ which is 173 178 trivial. The resource occupation of a FIR filter is considered as $(D_i+C_i) \times N_i$ which is
the number of bits needed in a worst case condition to represent the output of the FIR. Such an 174 179 the number of bits needed in a worst case condition to represent the output of the FIR. Such an
occupied area estimate assumes that the number of gates scales as the number of bits and the number 175 180 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, 176 181 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 177 182 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 178 183 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 179 184 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 180 185 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 181 186 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 182 187 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 183 188 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 184 189 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. 185 190 constraints in the initial statement of the problem leave little room for finding an optimal solution.
186 191
\begin{figure}[h!tb] 187 192 \begin{figure}[h!tb]
\begin{center} 188 193 \begin{center}
\includegraphics[width=.5\linewidth]{schema2} 189 194 \includegraphics[width=.5\linewidth]{schema2}
\caption{Shape of the filter transmitted power $P$ as a function of frequency: 190 195 \caption{Shape of the filter transmitted power $P$ as a function of frequency:
the bandpass BP is considered to occupy the initial 191 196 the bandpass BP is considered to occupy the initial
40\% of the Nyquist frequency range, the stopband the last 40\%, allowing 20\% transition 192 197 40\% of the Nyquist frequency range, the stopband the last 40\%, allowing 20\% transition
width.} 193 198 width.}
\label{rejection-shape} 194 199 \label{rejection-shape}
\end{center} 195 200 \end{center}
\end{figure} 196 201 \end{figure}
197 202
Following these considerations, the model is expressed as: 198 203 Following these considerations, the model is expressed as:
\begin{align} 199 204 \begin{align}
\begin{cases} 200 205 \begin{cases}
\mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\ 201 206 \mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\
\mathcal{A}_i &= N_i * C_i + D_i\\ 202 207 \mathcal{A}_i &= N_i \times (C_i + D_i)\\
\Delta_i &= \Delta _{i-1} + \mathcal{P}_i 203 208 \Delta_i &= \Delta _{i-1} + \mathcal{P}_i
\end{cases} 204 209 \end{cases}
\label{model-FIR} 205 210 \label{model-FIR}
\end{align} 206 211 \end{align}
To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the rejection of depending on $N_i$ and $C_i$, $\mathcal{A}$ 207 212 To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the stopband rejection dependence with $N_i$ and $C_i$, $\mathcal{A}$
is a theoretical area occupation of the processing block on the FPGA, and $\Delta_i$ is the total rejection for the current stage $i$. 208 213 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 209 214 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 210 215 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. 211 216 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 212 217 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 213 218 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 214 219 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 215 220 (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 216 221 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 217 222 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. 218 223 rejection capability. Weighing these two criteria allows designing the linear program to be solved.
219 224
\begin{figure}[h!tb] 220 225 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/noise-rejection.pdf} 221 226 \includegraphics[width=\linewidth]{images/noise-rejection.pdf}
\caption{Rejection as a function of number of coefficients and number of bits} 222 227 \caption{Rejection as a function of number of coefficients and number of bits}
\label{noise-rejection} 223 228 \label{noise-rejection}
\end{figure} 224 229 \end{figure}
225 230
The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource occupation below 226 231 The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource occupation below
a user-defined threshold. The MILP solver is allowed to choose the number of successive 227 232 a user-defined threshold, or aims at minimizing the area needed to reach a given rejection ($\min(S_q)$ in
233 the forthcoming discussion, Eqs. \ref{cstr_size} and \ref{cstr_rejection}).
234 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 228 235 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 229 236 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 230 237 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 defined as the mean power between 231 238 one of these conditions, the low-pass filter rejection is stored as computed by the frequency response
half the Nyquist frequency and the Nyquist frequency is stored as computed by the frequency response 232 239 of the digital filter (Fig. \ref{noise-rejection}). Various rejection criteria have been investigated,
of the digital filter (Fig. \ref{noise-rejection}). An intuitive analysis of this chart hints at an optimum 233 240 including mean value of the stopband response, median value of the stopband response, or as finally
241 selected, maximum value in the stopband. An intuitive analysis of the chart of Fig. \ref{noise-rejection}
242 hints at an optimum
set of tap length and number of bit for representing the coefficients along the line of the pyramidal 234 243 set of tap length and number of bit for representing the coefficients along the line of the pyramidal
shaped rejection capability function. 235 244 shaped rejection capability function.
236 245
Linear program formalism for solving the problem is well documented: an objective function is 237 246 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 238 247 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}. 239 248 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: 240 249 With the notation explain in system \ref{model-FIR}, we have defined our linear problem like this:
\paragraph{Variables} 241 250 \paragraph{Variables}
\begin{align*} 242 251 \begin{align*}
x_{i,j} \in \lbrace 0,1 \rbrace & \text{ $i$ is a given filter} \\ 243 252 x_{i,j} \in \lbrace 0,1 \rbrace & \text{ $i$ is a given filter} \\
& \text{ $j$ is the stage} \\ 244 253 & \text{ $j$ is the stage} \\
& \text{ If $x_{i,j}$ is equal to 1, the filter is selected} \\ 245 254 & \text{ If $x_{i,j}$ is equal to 1, the filter is selected} \\
\end{align*} 246 255 \end{align*}
\paragraph{Constants} 247 256 \paragraph{Constants}
\begin{align*} 248 257 \begin{align*}
\mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\ 249 258 \mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\
& \text{ $p$ is the number of different filters} \\ 250 259 & \text{ $p$ is the number of different filters} \\
C(i) & \text{ % Constant to let the 251 260 C(i) & \text{ % Constant to let the
number of coefficients %} \\ & \text{ 252 261 number of coefficients %} \\ & \text{
for filter $i$}\\ 253 262 for filter $i$}\\
\pi_C(i) & \text{ % Constant to let the 254 263 \pi_C(i) & \text{ % Constant to let the
number of bits of %}\\ & \text{ 255 264 number of bits of %}\\ & \text{
each coefficient for filter $i$}\\ 256 265 each coefficient for filter $i$}\\
\mathcal{A}_{\max} & \text{ Total space available inside the FPGA} 257 266 \mathcal{A}_{\max} & \text{ Total space available inside the FPGA}
\end{align*} 258 267 \end{align*}
\paragraph{Constraints} 259 268 \paragraph{Constraints}
\begin{align} 260 269 \begin{align}
1 \leq i \leq p & \nonumber\\ 261 270 1 \leq i \leq p & \nonumber\\
1 \leq j \leq q & \text{ $q$ is the max of filter stage} \nonumber \\ 262 271 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\\ 263 272 \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\\ 264 273 \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} \\ 265 274 \forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{A}_i)} \label{cstr_size} \\
\mathcal{S} \leq \mathcal{S}_{\max}\nonumber \\ 266 275 \mathcal{S} \leq \mathcal{S}_{\max}\nonumber \\
\mathcal{N}_0 = 0 & \text{ initial rejection}\nonumber\\ 267 276 \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} \\ 268 277 \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\\ 269 278 \mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\
& \text{ (e.g. 160~dB here)}\nonumber\\\nonumber 270 279 & \text{ (e.g. 160~dB here)}\nonumber\\\nonumber
\end{align} 271 280 \end{align}
\paragraph{Goal} 272 281 \paragraph{Goal}
\begin{align*} 273 282 \begin{align*}
\min \mathcal{S}_q 274 283 \min \mathcal{S}_q
\end{align*} 275 284 \end{align*}
276 285
The constraint \ref{cstr_size} means the occupation for the current stage $j$ depends on 277 286 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 278 287 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} 279 288 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 280 289 means the same thing but for the rejection, the rejection depends the previous rejection
plus the rejection of selected filter. 281 290 plus the rejection of selected filter.
282 291
\subsection{Low bandpass ripple and maximum rejection criteria} 283 292 \subsection{Low bandpass ripple and maximum rejection criteria}
284 293
The MILP solver provides a solution to the problem by selecting a series of small FIR with 285 294 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 286 295 increasing number of bits representing data and coefficients as well as an increasing number
of coefficients, instead of a single monolithic filter. 287 296 of coefficients, instead of a single monolithic filter.
288 297
\begin{figure}[h!tb] 289 298 \begin{figure}[h!tb]
% \includegraphics[width=\linewidth]{images/compare-fir.pdf} 290 299 % \includegraphics[width=\linewidth]{images/compare-fir.pdf}
\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-jmf-light.pdf} 291 300 \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 292 301 \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.} 293 302 with a cutoff frequency set at half the Nyquist frequency.}
\label{compare-fir} 294 303 \label{compare-fir}
\end{figure} 295 304 \end{figure}
296 305
Fig. \ref{compare-fir} exhibits the 297 306 Fig. \ref{compare-fir} exhibits the
performance comparison between one solution and a monolithic FIR when selecting a cutoff 298 307 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 299 308 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 300 309 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 301 310 rejection than the monolithic FIR at the expense of a lower cutoff frequency which remains to
be tuned or compensated for. 302 311 be tuned or compensated for.
303 312
304 313
The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarized as Tab. \ref{t1}. 305 314 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, 306 315 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 on 307 316 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 308 317 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 10 309 318 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 310 319 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, 311 320 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 312 321 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 313 322 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 314 323 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 315 324 hand also illustrates the limitation of a model with an abstraction layer that does not account
for the detailed architecture of the hardware. 316 325 for the detailed architecture of the hardware.
317 326
\begin{table}[h!tb] 318 327 \begin{table}[h!tb]
\caption{Resource occupation on a Xilinx Zynq-7000 series FPGA when synthesizing the FIR cascade 319 328 \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 320 329 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.} 321 330 to available resources on a Zynq-7020 as found on the Zedboard.}
\begin{center} 322 331 \begin{center}
\begin{tabular}{|c|cccc|}\hline 323 332 \begin{tabular}{|c|cccc|}\hline
FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline 324 333 FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline
1 (monolithic) & 1 & 76183 & 220 & -162 \\ 325 334 1 (monolithic) & 1 & 76183 & 220 & -162 \\
5 & 5 & 18597 & 220 & -160 \\ 326 335 5 & 5 & 18597 & 220 & -160 \\
10 & 8 & 24729 & 220 & -161 \\\hline\hline 327 336 10 & 8 & 24729 & 220 & -161 \\\hline\hline
\textbf{Zynq 7020} & \textbf{420} & \textbf{53200} & \textbf{220} & \\\hline 328 337 \textbf{Zynq 7020} & \textbf{420} & \textbf{53200} & \textbf{220} & \\\hline
%\begin{tabular}{|c|ccccc|}\hline 329 338 %\begin{tabular}{|c|ccccc|}\hline
%FIR & BRAM36 & BRAM18 & LUT & DSP & rejection (dB)\\\hline\hline 330 339 %FIR & BRAM36 & BRAM18 & LUT & DSP & rejection (dB)\\\hline\hline
%1 (monolithic) & 1 & 0 & {\color{Red}76183} & 220 & -162 \\ 331 340 %1 (monolithic) & 1 & 0 & {\color{Red}76183} & 220 & -162 \\
%5 & 0 & 5 & {\color{Green}18597} & 220 & -160 \\ 332 341 %5 & 0 & 5 & {\color{Green}18597} & 220 & -160 \\
%10 & 0 & 8 & {\color{Green}24729} & 220 & -161 \\\hline\hline 333 342 %10 & 0 & 8 & {\color{Green}24729} & 220 & -161 \\\hline\hline
%\textbf{Zynq 7020} & \textbf{140} & \textbf{280} & \textbf{53200} & \textbf{220} & \\\hline 334 343 %\textbf{Zynq 7020} & \textbf{140} & \textbf{280} & \textbf{53200} & \textbf{220} & \\\hline
\end{tabular} 335 344 \end{tabular}
\end{center} 336 345 \end{center}
%\vspace{-0.7cm} 337 346 %\vspace{-0.7cm}
\label{t1} 338 347 \label{t1}
\end{table} 339 348 \end{table}
340 349
\subsection{Alternate criteria}\label{median} 341 350 \subsection{Alternate criteria}\label{median}
342 351
Fig. \ref{compare-fir} provides FIR solutions matching well the targeted transfer 343 352 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 344 353 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 345 354 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 346 355 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, 347 356 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, 348 357 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}. 349 358 in these cases, are shown in Figs. \ref{compare-mean} and \ref{compare-median}.
350 359
\begin{figure}[h!tb] 351 360 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-mean-light.pdf} 352 361 \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 353 362 \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.} 354 363 with a cutoff frequency set at half the Nyquist frequency.}
\label{compare-mean} 355 364 \label{compare-mean}
\end{figure} 356 365 \end{figure}
357 366
In the case of the mean value criterion (Fig. \ref{compare-mean}), the solution is not 358 367 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 359 368 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 360 369 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. 361 370 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 362 371 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 363 372 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 364 373 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 365 374 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 366 375 the reasons stated previously and much poorer than those found with the maximum rejection criterion
selected earlier (Fig. \ref{compare-fir}). 367 376 selected earlier (Fig. \ref{compare-fir}).
368 377
\begin{figure}[h!tb] 369 378 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-median-light.pdf} 370 379 \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 371 380 \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.} 372 381 with a cutoff frequency set at half the Nyquist frequency.}
\label{compare-median} 373 382 \label{compare-median}
\end{figure} 374 383 \end{figure}
375 384
\section{Filter coefficient selection} 376 385 \section{Filter coefficient selection}
377 386
The coefficients of a single monolithic filter are computed as the impulse response 378 387 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 379 388 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 380 389 including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing
(Matlab's {\tt fir1} function). 381 390 (Matlab's {\tt fir1} function).
382 391
\begin{figure}[h!tb] 383 392 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/fir1-vs-firls} 384 393 \includegraphics[width=\linewidth]{images/fir1-vs-firls}
\caption{Evolution of the rejection capability of least-square optimized filters and Hamming 385 394 \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 386 395 FIR filters as a function of the number of coefficients, for floating point numbers and 8-bit
encoded integers.} 387 396 encoded integers.}
\label{2} 388 397 \label{2}
\end{figure} 389 398 \end{figure}
390 399
Cascading filters opens a new optimization opportunity by 391 400 Cascading filters opens a new optimization opportunity by
selecting various coefficient sets depending on the number of coefficients. Fig. \ref{2} 392 401 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 393 402 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 394 403 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 395 404 the processing chain, the type of selected filter also changes depending on the number of coefficients
and evolves along the processing chain. 396 405 and evolves along the processing chain.
397 406
\section{Conclusion} 398 407 \section{Conclusion}
399 408
We address the optimization problem of designing a low-pass filter chain in a Field Programmable Gate 400 409 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 401 410 Array for improved noise rejection within constrained resource occupation, as needed for
real time processing of radiofrequency signal when characterizing spectral phase noise 402 411 real time processing of radiofrequency signal when characterizing spectral phase noise
characteristics of stable oscillators. The flexibility of the digital approach makes the result 403 412 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 404 413 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 405 414 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 406 415 is controlled by non-piezoelectric resonator (sapphire resonator, microwave or optical
atomic transition). 407 416 atomic transition).
408 417
\section*{Acknowledgement} 409 418 \section*{Acknowledgement}
410 419
This work is supported by the ANR Programme d'Investissement d'Avenir in 411 420 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 412 421 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. 413 422 (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 414 423 The authors would like to thank E. Rubiola, F. Vernotte, G. Cabodevila for support and
fruitful discussions. 415 424 fruitful discussions.
416 425
\bibliographystyle{IEEEtran} 417 426 \bibliographystyle{IEEEtran}
\balance 418 427 \balance
\bibliography{references,biblio} 419 428 \bibliography{references,biblio}
\end{document} 420 429 \end{document}
421 430
\section{Contexte d'ordonnancement} 422 431 \section{Contexte d'ordonnancement}
Dans cette partie, nous donnerons des d\'efinitions de termes rattach\'es au domaine de l'ordonnancement 423 432 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 424 433 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 425 434 nous pourrons aller plus loin que les travaux vus pr\'ec\'edemment et nous tenterons des approches d'ordonnancement
et d'optimisation. 426 435 et d'optimisation.
427 436
\subsection{D\'efinition du vocabulaire} 428 437 \subsection{D\'efinition du vocabulaire}
Avant tout, il faut d\'efinir ce qu'est un problème d'optimisation. Il y a deux d\'efinitions 429 438 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} : 430 439 importantes à donner. La première est propos\'ee par Legrand et Robert dans leur livre \cite{def1-ordo} :
\begin{definition} 431 440 \begin{definition}
\label{def-ordo1} 432 441 \label{def-ordo1}
Un ordonnancement d'un système de t\^aches $G\ =\ (V,\ E,\ w)$ est une fonction $\sigma$ : 433 442 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 $(u,\ v) \in E$. 434 443 $V \rightarrow \mathbb{N}$ telle que $\sigma(u) + w(u) \leq \sigma(v)$ pour toute arête $(u,\ v) \in E$.
\end{definition} 435 444 \end{definition}
436 445
Dit plus simplement, l'ensemble $V$ repr\'esente les t\^aches à ex\'ecuter, l'ensemble $E$ repr\'esente les d\'ependances 437 446 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 438 447 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 439 448 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 440 449 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. 441 450 temps d'ex\'ecution.
442 451
Une autre d\'efinition importante qui est propos\'ee par Leung et al. \cite{def2-ordo} est : 443 452 Une autre d\'efinition importante qui est propos\'ee par Leung et al. \cite{def2-ordo} est :
\begin{definition} 444 453 \begin{definition}
\label{def-ordo2} 445 454 \label{def-ordo2}
L'ordonnancement traite de l'allocation de ressources rares à des activit\'es avec 446 455 L'ordonnancement traite de l'allocation de ressources rares à des activit\'es avec
l'objectif d'optimiser un ou plusieurs critères de performance. 447 456 l'objectif d'optimiser un ou plusieurs critères de performance.
\end{definition} 448 457 \end{definition}
449 458
Cette d\'efinition est plus g\'en\'erique mais elle nous int\'eresse d'avantage que la d\'efinition \ref{def-ordo1}. 450 459 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. 451 460 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. 452 461 Dans les faits les dates de d\'ebut ne nous int\'eressent pas r\'eellement.
453 462
En revanche la d\'efinition \ref{def-ordo2} sera au c\oe{}ur du projet. Pour se convaincre de cela, 454 463 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 455 464 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. 456 465 sont les m\'ethodes qu'on peut appliquer.
457 466
Les problèmes d'ordonnancement peuvent être class\'es en diff\'erentes cat\'egories : 458 467 Les problèmes d'ordonnancement peuvent être class\'es en diff\'erentes cat\'egories :
\begin{itemize} 459 468 \begin{itemize}
\item T\^aches ind\'ependantes : dans cette cat\'egorie de problèmes, les t\^aches sont complètement ind\'ependantes 460 469 \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. 461 470 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, 462 471 \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 463 472 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, 464 473 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 465 474 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. 466 475 t\^ache de fin.
\item Workflow : cette cat\'egorie est une sous cat\'egorie des graphes de t\^aches dans le sens où 467 476 \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 468 477 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. 469 478 que nous traitons ici.
\end{itemize} 470 479 \end{itemize}
471 480
Bien entendu, cette liste n'est pas exhaustive et il existe de nombreuses autres classifications et sous-classifications 472 481 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. 473 482 de ces problèmes. Nous n'avons parl\'e ici que des cat\'egories les plus communes.
474 483
Un autre point à d\'efinir, est le critère d'optimisation. Il y a là encore un grand nombre de 475 484 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 : 476 485 critères possibles. Nous allons donc parler des principaux :
\begin{itemize} 477 486 \begin{itemize}
\item Temps de compl\'etion total (ou Makespan en anglais) : ce critère est l'un des critères d'optimisation 478 487 \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 479 488 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 480 489 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. 481 490 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 482 491 \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. 483 492 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. 484 493 \item Le d\'ebit : ce critère quant à lui, vise à augmenter au maximum le d\'ebit de traitement des donn\'ees.
\end{itemize} 485 494 \end{itemize}
486 495
En plus de cela, on peut avoir besoin de plusieurs critères d'optimisation. Il s'agit dans ce cas d'une optimisation 487 496 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 488 497 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 489 498 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. 490 499 de faire le meilleur compromis entre tous les critères.
491 500
\subsection{Formalisation du problème} 492 501 \subsection{Formalisation du problème}
\label{formalisation} 493 502 \label{formalisation}
Maintenant que nous avons donn\'e le vocabulaire li\'e à l'ordonnancement, nous allons pouvoir essayer caract\'eriser 494 503 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} 495 504 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. 496 505 et nous essayerons de les formaliser le plus finement possible.
497 506
Comme nous l'avons dit, une t\^ache est un bloc de traitement. Chaque t\^ache $i$ dispose d'un ensemble de paramètres 498 507 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 499 508 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. 500 509 t\^ache à l'autre. Nous reviendrons plus tard sur les paramètres qui peuvent composer cet ensemble.
501 510
Outre cet ensemble $\mathcal{P}_i$, chaque t\^ache dispose de paramètres communs : 502 511 Outre cet ensemble $\mathcal{P}_i$, chaque t\^ache dispose de paramètres communs :
\begin{itemize} 503 512 \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. 504 513 \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. 505 514 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 506 515 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 507 516 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 508 517 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 509 518 \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 $\pi_i^+$. 510 519 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). 511 520 \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 512 521 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 $f_i^+$. 513 522 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. 514 523 \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 515 524 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 $q_i^-$ 516 525 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 $i$. 517 526 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 518 527 \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 519 528 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 $d_i^+\ =\ q_i^+\ *\ f_i^+$. 520 529 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. 521 530 \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. 522 531 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 523 532 \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 ^+$ \\ 524 533 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 ? 525 534 %TODO Est-ce vraiment un paramètre ?
\end{itemize} 526 535 \end{itemize}
527 536
Ces diff\'erents paramètres communs sont fortement li\'es aux \'el\'ements de $\mathcal{P}_i$. Voici quelques exemples de relations 528 537 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 : 529 538 que nous avons identifi\'ees :
\begin{itemize} 530 539 \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 531 540 \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. 532 541 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 533 542 \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. 534 543 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 535 544 \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. 536 545 sortant et des variables internes de la t\^ache.
\item $A_i^+\ =\ \mathcal{F}_A(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+, \mathcal{P}_i)$ 537 546 \item $A_i^+\ =\ \mathcal{F}_A(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+, \mathcal{P}_i)$
\end{itemize} 538 547 \end{itemize}
Pour le moment, nous ne sommes pas capables de donner une d\'efinition g\'en\'erale de ces fonctions. Mais en revanche, 539 548 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. 540 549 sur quelques exemples simples (cf. \ref{def-contraintes}), nous parvenons à donner une \'evaluation de ces fonctions.
541 550
Maintenant que nous avons donn\'e toutes les notations utiles, nous allons \'enoncer des contraintes relatives à notre problème. Soit 542 551 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 : 543 552 un DGA $G(V,\ E)$, on a pour toutes arêtes $(i, j)\ \in\ E$ les in\'equations suivantes :
544 553
\paragraph{Contrainte de pr\'ecision :} 545 554 \paragraph{Contrainte de pr\'ecision :}
Cette in\'equation traduit la contrainte de pr\'ecision d'une t\^ache à l'autre : 546 555 Cette in\'equation traduit la contrainte de pr\'ecision d'une t\^ache à l'autre :
\begin{align*} 547 556 \begin{align*}
\pi _i ^+ \geq \pi _j ^- 548 557 \pi _i ^+ \geq \pi _j ^-
\end{align*} 549 558 \end{align*}
550 559
\paragraph{Contrainte de d\'ebit :} 551 560 \paragraph{Contrainte de d\'ebit :}
Cette in\'equation traduit la contrainte de d\'ebit d'une t\^ache à l'autre : 552 561 Cette in\'equation traduit la contrainte de d\'ebit d'une t\^ache à l'autre :
\begin{align*} 553 562 \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} 554 563 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*} 555 564 \end{align*}
556 565
\paragraph{Contrainte de synchronisation :} 557 566 \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 558 567 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. 559 568 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 : 560 569 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*} 561 570 \begin{align*}
\forall \text{ chemin } \mathcal{C}1(s, .., f), 562 571 \forall \text{ chemin } \mathcal{C}1(s, .., f),
\forall \text{ chemin } \mathcal{C}2(s, .., f) 563 572 \forall \text{ chemin } \mathcal{C}2(s, .., f)
\text{ tel que } \mathcal{C}1 \neq \mathcal{C}2 564 573 \text{ tel que } \mathcal{C}1 \neq \mathcal{C}2
\Rightarrow 565 574 \Rightarrow
\sum _{i} ^{i \in \mathcal{C}1} \delta_i = \sum _{i} ^{i \in \mathcal{C}2} \delta_i 566 575 \sum _{i} ^{i \in \mathcal{C}1} \delta_i = \sum _{i} ^{i \in \mathcal{C}2} \delta_i
\end{align*} 567 576 \end{align*}
568 577
\paragraph{Contrainte de place :} 569 578 \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 $\mathcal{A}_{FPGA}$ : 570 579 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*} 571 580 \begin{align*}
\sum ^{\text{t\^ache } i} \mathcal{A}_i \leq \mathcal{A}_{FPGA} 572 581 \sum ^{\text{t\^ache } i} \mathcal{A}_i \leq \mathcal{A}_{FPGA}
\end{align*} 573 582 \end{align*}
574 583
\subsection{Exemples de mod\'elisation} 575 584 \subsection{Exemples de mod\'elisation}
\label{exemples-modeles} 576 585 \label{exemples-modeles}
Nous allons maintenant prendre quelques blocs de traitement simples afin d'illustrer au mieux notre modèle. 577 586 Nous allons maintenant prendre quelques blocs de traitement simples afin d'illustrer au mieux notre modèle.
Pour tous nos exemple, nous prendrons un d\'ebit en entr\'ee de 200 Mo/s avec une pr\'ecision de 16 bit. 578 587 Pour tous nos exemple, nous prendrons un d\'ebit en entr\'ee de 200 Mo/s avec une pr\'ecision de 16 bit.
579 588
Prenons tout d'abord l'exemple d'un bloc de d\'ecimation. Le but de ce bloc est de ralentir le flux en ne gardant 580 589 Prenons tout d'abord l'exemple d'un bloc de d\'ecimation. Le but de ce bloc est de ralentir le flux en ne gardant
que certaines donn\'ees à intervalle r\'egulier. Cet intervalle est appel\'e le facteur de d\'ecimation, on le notera $N$. 581 590 que certaines donn\'ees à intervalle r\'egulier. Cet intervalle est appel\'e le facteur de d\'ecimation, on le notera $N$.