From 3ca9d7dfc351984c982e4c613eeb062715adb85c Mon Sep 17 00:00:00 2001 From: Arthur HUGEAT Date: Sat, 19 May 2018 20:30:43 +0200 Subject: [PATCH] =?UTF-8?q?ajout=20du=20programme=20lin=C3=A9aire=20+=20co?= =?UTF-8?q?rrection=20d'un=20figure=20et=20d'un=20tableau.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ifcs2018_proceeding.tex | 69 +- ...fir-mono-vs-fir-series-noise-fixe-jmf-light.eps | 6775 ++++++++++++++++++++ ...fir-mono-vs-fir-series-noise-fixe-jmf-light.pdf | Bin 0 -> 26342 bytes 3 files changed, 6833 insertions(+), 11 deletions(-) create mode 100644 images/fir-mono-vs-fir-series-noise-fixe-jmf-light.eps create mode 100644 images/fir-mono-vs-fir-series-noise-fixe-jmf-light.pdf diff --git a/ifcs2018_proceeding.tex b/ifcs2018_proceeding.tex index ccd1378..d32acad 100644 --- a/ifcs2018_proceeding.tex +++ b/ifcs2018_proceeding.tex @@ -4,6 +4,7 @@ \usepackage{amsthm} \usepackage{amssymb} \usepackage{amsmath} +\usepackage{algorithm2e} \usepackage{url} \usepackage[normalem]{ulem} \graphicspath{{/home/jmfriedt/gpr/170324_avalanche/}{/home/jmfriedt/gpr/1705_homemade/}} @@ -149,7 +150,14 @@ developed to meet this aim, Mixed-Integer Linear Programming (MILP) provides the provide a formal definition of the stated problem and search for an optimal use of available resources \cite{yu2007design, kodek1980design}. -The degrees of freedom when addressing the problem of replacing the single monolithic +First we need to ensure that our problem is a real optimization problem. When +we design a process inside the FPGA we want reach some requirements by example the +throughput, the computation time or the rejection noise... But we some limited +resources to design the process like BRAM (high performance RAM), DSP (Digital Signal Processor) +or LUT (Look Up Table). Since we want optimize some criteria and we have some +limited resources our problem is a classical optimization problem. + +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 $i$, 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, @@ -158,7 +166,7 @@ trivial. The resource occupation of a FIR filter is considered as $D_i+C_i \time the number of bits needed in a worst case condition to represent the output of the FIR. Unfortunately this representation is not sufficient to represent the real occupation inside FPGA. In fact the FPGA have some BRAM block on which the coefficients are stored and each BRAM are not -share between different filters. Moreover the multiplication need Digital Signal Processor to be +share between different filters. Moreover the multiplication need DSP to be perform. Those DSP are in limited quantity so in the future we shall to consider this. At the moment our model can be express like this : @@ -174,8 +182,8 @@ To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represent the rejection o is just theoretical occupation and $\Delta_i$ is the total rejection for the current stage $i$. At this moment we are not able to express the function $\mathcal{F}$ so we are run some simulations to determine the rejection noise depending on $N_i$ and $C_i$. But to choose the right filter we must define clearly the rejection criterion. If we take incorrect criterion -the linear program will produce a wrong solution. So we define a criterion to avoid ripple on baseband and just keep -the maximum of rejection (see the figure \ref{rejection-shape}). Thank to this system, we can able to design our linear program. +the linear program will produce a wrong solution. So we define a criterion to avoid ripple on passband and just keep +the maximum of rejection on the stopband (see the figure \ref{rejection-shape}). Thank to this system, we can able to design our linear program. \begin{figure}[h!tb] \begin{center} @@ -203,6 +211,46 @@ of the digital filter (Fig. \ref{noise-rejection}). 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 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: +\paragraph{Variables} +\begin{align*} +x_{i,j} \in \lbrace 0,1 \rbrace & \text{ $i$ is a specific filter} \\ +& \text{ $j$ is the stage} \\ +& \text{ If $x_{i,j}$ is equal to 1, the filter is selected} \\ +\end{align*} +\paragraph{Constants} +\begin{align*} +\mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\ +& \text{ $p$ is the number of different filters} \\ +C(i) & \text{ Constant to let the number of coefficients}\\ +& \text{ for the filter $i$}\\ +\pi_C(i) & \text{ Constant to let the number of bits of}\\ +& \text{ each coefficient for the filter $i$}\\ +\mathcal{A}_{\max} & \text{ Max space available inside the FPGA} +\end{align*} +\paragraph{Constraints} +\begin{align*} +1 \leq i \leq p & \\ +1 \leq j \leq q & \text{ $q$ is the max of filter stage} \\ +\forall j, \mathlarger{\sum_{i}} x_{i,j} = 1 & \text{ At most one filter by stage} \\ +\mathcal{S}_0 = 0 & \text{ initial occupation}\\ +\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \forall i, x_{i,j} \times \mathcal{A}_i \\%\label{cstr_size} +\mathcal{S} \leq \mathcal{S}_{\max} \\ +\mathcal{N}_0 = 0 & \text{ initial rejection}\\ +\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \forall i, x_{i,j} \times \mathcal{R}_i \\%\label{cstr_rejection} +\mathcal{N}_q \geqslant 160 & \text{ an user's bound}\\ +\end{align*} +\paragraph{Goal} +\begin{align*} +\min \mathcal{S}_q +\end{align*} + +% AH j'aimerai mettre deux equations avec un label mais je ne sais pas comment faire +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 +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 +plus the rejection of selected filter. 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 @@ -215,7 +263,7 @@ be tuned or compensated for. \begin{figure}[h!tb] % \includegraphics[width=\linewidth]{images/compare-fir.pdf} -\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-200dB.pdf} +\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 with a cutoff frequency set at half the Nyquist frequency.} \label{compare-fir} @@ -226,15 +274,14 @@ The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarize \begin{table}[h!tb] \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 -to available resources on a Zynq-7010 as found on the Redpitaya board. The rejection is the mean -value from 0.6 to 1 Nyquist frequency.} +to available resources on a Zynq-7020 as found on the Zedboard.} \begin{center} \begin{tabular}{|c|cccc|}\hline FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline -1 (monolithic) & 1 & 4064 & 40 & -72 \\ -5 & 5 & 12332 & 0 & -217 \\ -10 & 10 & 12717 & 0 & -251 \\\hline\hline -Zynq 7010 & 60 & 17600 & 80 & \\\hline +1 (monolithic) & 1 & 76183 & 220 & -162 \\ +5 & 5 & 18597 & 220 & -160 \\ +10 & 8 & 24729 & 220 & -161 \\\hline\hline +\textbf{Zynq 7020} & \textbf{420} & \textbf{53200} & \textbf{220} & \\\hline \end{tabular} \end{center} %\vspace{-0.7cm} diff --git a/images/fir-mono-vs-fir-series-noise-fixe-jmf-light.eps b/images/fir-mono-vs-fir-series-noise-fixe-jmf-light.eps new file mode 100644 index 0000000..8bb66fe --- /dev/null +++ b/images/fir-mono-vs-fir-series-noise-fixe-jmf-light.eps @@ -0,0 +1,6775 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: /tmp/figure.eps +%%Creator: gnuplot 5.2 patchlevel 2 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{currentpoint stroke M 0 vshift R {ExtendTextBox} forall} def +/GRwidth {currentpoint stroke M dup Gwidth vshift R {ExtendTextBox} forall} def +/GCwidth {currentpoint stroke M dup Gwidth 2 div vshift R {ExtendTextBox} forall} def +/GLwidth2 {0 Gwidth AddGlyphWidth} def +/GRwidth2 {Gwidth -1 mul 0 AddGlyphWidth} def +/GCwidth2 {Gwidth 2 div dup -1 mul AddGlyphWidth} def +/AddGlyphWidth { dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse + dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse } def +/MFshow { + { dup 5 get 3 ge + { 5 get 3 eq {gsave} {grestore} ifelse } + {dup dup 0 get findfont exch 1 get scalefont setfont + [ currentpoint ] exch dup 2 get 0 exch R dup 5 get 2 ne {dup dup 6 + get exch 4 get {textshow} {Metrics pop 0 R} ifelse }if dup 5 get 0 eq + {dup 3 get {2 get neg 0 exch R pop} {pop aload pop M} ifelse} {dup 5 + get 1 eq {dup 2 get exch dup 3 get exch 6 get Gswidth pop -2 div + dup 0 R} {dup 6 get Gswidth pop -2 div 0 R 6 get + textshow 2 index {aload pop M neg 3 -1 roll neg R pop pop} {pop pop pop + pop aload pop M} ifelse }ifelse }ifelse } + ifelse } + forall} def +/Gswidth {dup type /stringtype eq {stringwidth} {pop (n) stringwidth} ifelse} def +/MFwidth {0 exch { dup 5 get 3 ge { 5 get 3 eq { 0 } { pop } ifelse } + {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont + 6 get Gswidth pop add} {pop} ifelse} ifelse} forall} def +/MLshow { currentpoint stroke M + 0 exch R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/MRshow { currentpoint stroke M + exch dup MFwidth neg 3 -1 roll R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/MCshow { currentpoint stroke M + exch dup MFwidth -2 div 3 -1 roll R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/XYsave { [( ) 1 2 true false 3 ()] } bind def +/XYrestore { [( ) 1 2 true false 4 ()] } bind def +Level1 SuppressPDFMark or +{} { +/SDict 10 dict def +systemdict /pdfmark known not { + userdict /pdfmark systemdict /cleartomark get put +} if +SDict begin [ + /Title (/tmp/figure.eps) + /Subject (gnuplot plot) + /Creator (gnuplot 5.2 patchlevel 2) +% /Producer (gnuplot) +% /Keywords () + /CreationDate (Sat May 19 20:12:47 2018) + /DOCINFO pdfmark +end +} ifelse +% +% Support for boxed text - Ethan A Merritt Sep 2016 +% +/InitTextBox { userdict /TBy2 3 -1 roll put userdict /TBx2 3 -1 roll put + userdict /TBy1 3 -1 roll put userdict /TBx1 3 -1 roll put + /Boxing true def } def +/ExtendTextBox { dup type /stringtype eq + { Boxing { gsave dup false charpath pathbbox + dup TBy2 gt {userdict /TBy2 3 -1 roll put} {pop} ifelse + dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse + dup TBy1 lt {userdict /TBy1 3 -1 roll put} {pop} ifelse + dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse + grestore } if } + {} ifelse} def +/PopTextBox { newpath TBx1 TBxmargin sub TBy1 TBymargin sub M + TBx1 TBxmargin sub TBy2 TBymargin add L + TBx2 TBxmargin add TBy2 TBymargin add L + TBx2 TBxmargin add TBy1 TBymargin sub L closepath } def +/DrawTextBox { PopTextBox stroke /Boxing false def} def +/FillTextBox { gsave PopTextBox fill grestore /Boxing false def} def +0 0 0 0 InitTextBox +/TBxmargin 20 def +/TBymargin 20 def +/Boxing false def +/textshow { ExtendTextBox Gshow } def +% +end +%%EndProlog +%%Page: 1 1 +gnudict begin +gsave +doclip +50 50 translate +0.050 0.050 scale +0 setgray +newpath +(Helvetica) findfont 200 scalefont setfont +BackgroundColor 0 lt 3 1 roll 0 lt exch 0 lt or or not {BackgroundColor C 1.000 0 0 11520.00 8640.00 BoxColFill} if +gsave % colour palette begin +/maxcolors 64 def +/HSV2RGB { exch dup 0.0 eq {pop exch pop dup dup} % achromatic gray + { /HSVs exch def /HSVv exch def 6.0 mul dup floor dup 3 1 roll sub + /HSVf exch def /HSVi exch cvi def /HSVp HSVv 1.0 HSVs sub mul def + /HSVq HSVv 1.0 HSVs HSVf mul sub mul def + /HSVt HSVv 1.0 HSVs 1.0 HSVf sub mul sub mul def + /HSVi HSVi 6 mod def 0 HSVi eq {HSVv HSVt HSVp} + {1 HSVi eq {HSVq HSVv HSVp}{2 HSVi eq {HSVp HSVv HSVt} + {3 HSVi eq {HSVp HSVq HSVv}{4 HSVi eq {HSVt HSVp HSVv} + {HSVv HSVp HSVq} ifelse} ifelse} ifelse} ifelse} ifelse + } ifelse} def +/Constrain { + dup 0 lt {0 exch pop}{dup 1 gt {1 exch pop} if} ifelse} def +/YIQ2RGB { + 3 copy -1.702 mul exch -1.105 mul add add Constrain 4 1 roll + 3 copy -0.647 mul exch -0.272 mul add add Constrain 5 1 roll + 0.621 mul exch -0.956 mul add add Constrain 3 1 roll } def +/CMY2RGB { 1 exch sub exch 1 exch sub 3 2 roll 1 exch sub 3 1 roll exch } def +/XYZ2RGB { 3 copy -0.9017 mul exch -0.1187 mul add exch 0.0585 mul exch add + Constrain 4 1 roll 3 copy -0.0279 mul exch 1.999 mul add exch + -0.9844 mul add Constrain 5 1 roll -0.2891 mul exch -0.5338 mul add + exch 1.91 mul exch add Constrain 3 1 roll} def +/SelectSpace {ColorSpace (HSV) eq {HSV2RGB}{ColorSpace (XYZ) eq { + XYZ2RGB}{ColorSpace (CMY) eq {CMY2RGB}{ColorSpace (YIQ) eq {YIQ2RGB} + if} ifelse} ifelse} ifelse} def +/InterpolatedColor true def +/grayindex {/gidx 0 def + {GrayA gidx get grayv ge {exit} if /gidx gidx 1 add def} loop} def +/dgdx {grayv GrayA gidx get sub GrayA gidx 1 sub get + GrayA gidx get sub div} def +/redvalue {RedA gidx get RedA gidx 1 sub get + RedA gidx get sub dgdxval mul add} def +/greenvalue {GreenA gidx get GreenA gidx 1 sub get + GreenA gidx get sub dgdxval mul add} def +/bluevalue {BlueA gidx get BlueA gidx 1 sub get + BlueA gidx get sub dgdxval mul add} def +/interpolate { + grayindex grayv GrayA gidx get sub abs 1e-5 le + {RedA gidx get GreenA gidx get BlueA gidx get} + {/dgdxval dgdx def redvalue greenvalue bluevalue} ifelse} def +/GrayA [0 .0159 .0317 .0476 .0635 .0794 .0952 .1111 .127 .1429 .1587 .1746 + .1905 .2063 .2222 .2381 .254 .2698 .2857 .3016 .3175 .3333 .3492 .3651 + .381 .3968 .4127 .4286 .4444 .4603 .4762 .4921 .5079 .5238 .5397 .5556 + .5714 .5873 .6032 .619 .6349 .6508 .6667 .6825 .6984 .7143 .7302 .746 + .7619 .7778 .7937 .8095 .8254 .8413 .8571 .873 .8889 .9048 .9206 .9365 + .9524 .9683 .9841 1 ] def +/RedA [.267 .2727 .2771 .2804 .2824 .2832 .2828 .2812 .2785 .2747 .27 .2644 + .258 .2511 .2437 .2361 .2283 .2204 .2127 .2051 .1977 .1906 .1838 .1773 + .171 .1648 .1588 .153 .1471 .1414 .1358 .1306 .1259 .1222 .1199 .1196 .122 + .1277 .1368 .1496 .166 .1855 .208 .2331 .2605 .29 .3213 .3544 .3889 .4249 + .4622 .5008 .5403 .5809 .6222 .6641 .7064 .7489 .7913 .8333 .8747 .9153 + .9548 .9932 ] def +/GreenA [.0049 .0258 .0509 .0742 .096 .1169 .1374 .1575 .1773 .197 .2163 + .2354 .2542 .2726 .2906 .3083 .3256 .3425 .3591 .3754 .3913 .4071 .4226 + .4379 .4531 .4681 .4831 .4981 .513 .5279 .5428 .5577 .5726 .5875 .6024 + .6173 .6321 .6469 .6616 .6761 .6905 .7047 .7187 .7324 .7458 .7588 .7715 + .7837 .7955 .8067 .8173 .8274 .8369 .8457 .8538 .8613 .8682 .8745 .8803 + .8858 .8909 .896 .901 .9062 ] def +/BlueA [.3294 .3534 .3762 .3979 .4183 .4372 .4546 .4704 .4847 .4973 .5083 + .5177 .5258 .5325 .5381 .5427 .5463 .5493 .5516 .5535 .555 .5561 .557 + .5576 .558 .5581 .5581 .5577 .557 .5559 .5543 .5522 .5494 .546 .5418 .5368 + .5308 .5239 .516 .5069 .4968 .4854 .4729 .4591 .4441 .4278 .4103 .3915 + .3714 .3501 .3275 .3038 .2789 .253 .2262 .1989 .1715 .145 .1213 .1033 + .0954 .1005 .1179 .1439 ] def +/pm3dround {maxcolors 0 gt {dup 1 ge + {pop 1} {maxcolors mul floor maxcolors 1 sub div} ifelse} if} def +/pm3dGamma 1.0 1.5 Gamma mul div def +/ColorSpace (RGB) def +Color InterpolatedColor or { % COLOUR vs. GRAY map + InterpolatedColor { %% Interpolation vs. RGB-Formula + /g {stroke pm3dround /grayv exch def interpolate + SelectSpace setrgbcolor} bind def + }{ + /g {stroke pm3dround dup cF7 Constrain exch dup cF5 Constrain exch cF15 Constrain + SelectSpace setrgbcolor} bind def + } ifelse +}{ + /g {stroke pm3dround pm3dGamma exp setgray} bind def +} ifelse +1.000 UL +LTb +1.00 1.00 1.00 C 1.000 1497 5067 8926 2923 BoxColFill +1.000 UL +LTb +0.00 0.00 0.00 C 1497 5067 M +88 0 V +stroke +0.15 0.15 0.15 C 1377 5067 M +[ [(Helvetica) 200.0 0.0 true true 0 (-300)] +] -66.7 MRshow 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4867 M +[ [(Helvetica) 200.0 0.0 true true 0 (0)] +] -66.7 MCshow +1.000 UL +LTb +0.00 0.00 0.00 C 3282 5067 M +0 88 V +stroke +0.15 0.15 0.15 C 3282 4867 M +[ [(Helvetica) 200.0 0.0 true true 0 (0.2)] +] -66.7 MCshow +1.000 UL +LTb +0.00 0.00 0.00 C 5068 5067 M +0 88 V +stroke +0.15 0.15 0.15 C 5068 4867 M +[ [(Helvetica) 200.0 0.0 true true 0 (0.4)] +] -66.7 MCshow +1.000 UL +LTb +0.00 0.00 0.00 C 6853 5067 M +0 88 V +stroke +0.15 0.15 0.15 C 6853 4867 M +[ [(Helvetica) 200.0 0.0 true true 0 (0.6)] +] -66.7 MCshow +1.000 UL +LTb +0.00 0.00 0.00 C 8639 5067 M +0 88 V +stroke +0.15 0.15 0.15 C 8639 4867 M +[ [(Helvetica) 200.0 0.0 true true 0 (0.8)] +] -66.7 MCshow +1.000 UL +LTb +0.00 0.00 0.00 C 10424 5067 M +0 88 V +stroke +0.15 0.15 0.15 C 10424 4867 M +[ [(Helvetica) 200.0 0.0 true true 0 (1)] +] -66.7 MCshow +1.000 UL +LTb +0.00 0.00 0.00 C 1.000 UP +1.000 UL +LTb +0.00 0.00 0.00 C 0.15 0.15 0.15 C 697 6529 M +currentpoint gsave translate 90 rotate 0 0 moveto +[ [(Helvetica) 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