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17 17 ifcs2018_poster.out
18 18 ifcs2018_poster.pdf
19 19  
  20 +ifcs2018_journal.aux
  21 +ifcs2018_journal.bbl
  22 +ifcs2018_journal.blg
  23 +ifcs2018_journal.log
  24 +ifcs2018_journal.out
  25 +ifcs2018_journal.pdf
  26 +
20 27 *.bak
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4 4 BIB = bibtex
5 5 TARGET = ifcs2018
6 6  
7   -all: $(TARGET)_abstract $(TARGET)_poster $(TARGET)_proceeding
  7 +all: $(TARGET)_abstract $(TARGET)_poster $(TARGET)_proceeding $(TARGET)_journal
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28 28 $(TEX) $@.tex
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  31 +$(TARGET)_journal: $(TARGET)_journal.tex references.bib biblio.bib
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  38 + rm -f $(TARGET)_abstract.aux $(TARGET)_abstract.log $(TARGET)_abstract.out $(TARGET)_abstract.bbl $(TARGET)_abstract.blg
34 39 rm -f $(TARGET)_poster.aux $(TARGET)_poster.log $(TARGET)_poster.out
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demo_critere_filtre.m
Diff comments   View file @ 17d9a84
  1 +clear all;
  2 +
  3 +global N = 2048;
  4 +
  5 +function rejection = rejection_criteria(log_data, fc)
  6 + N = length(log_data);
  7 +
  8 + % Index of the first point in the tail fir
  9 + index_tail = round((fc + 0.1) * N) + 1;
  10 +
  11 + % Index of th last point on the band
  12 + index_band = round((fc - 0.1) * N);
  13 +
  14 + % Get the worst rejection in stopband
  15 + worst_rejection = -max(log_data(index_tail:end));
  16 +
  17 + % Get the total of deviation in passband
  18 + worst_band = mean(-1 * abs(log_data(1:index_band)));
  19 +
  20 + % Compute the rejection
  21 + passband_malus = 10; % weighted value to penalize the deviation in passband
  22 + rejection = worst_band * passband_malus + worst_rejection;
  23 +endfunction
  24 +
  25 +function [h, log_curve, rejection] = compute_freqz(filename)
  26 + global N;
  27 +
  28 + b = load(filename);
  29 + [h, w] = freqz(b, 1, N/2);
  30 + mag = abs(h);
  31 + mag = mag ./ mag(1);
  32 + log_curve = 20 * log10(mag);
  33 +
  34 + rejection = rejection_criteria(log_curve, 0.5);
  35 +endfunction
  36 +
  37 +# Stages
  38 +hTotal = ones(N/2, 1);
  39 +% [h, curve1, c1] = compute_freqz("filters/fir1/fir1_033_int08"); % 1) -8dB
  40 +% [h, curve1, c1] = compute_freqz("filters/fir1/fir1_037_int08"); % 2) -9dB
  41 +[h, curve1, c1] = compute_freqz("filters/fir1/fir1_037_int08");
  42 +hTotal = hTotal .* h;
  43 +% [h, curve2, c2] = compute_freqz("filters/fir1/fir1_033_int10"); % 1) -8dB
  44 +% [h, curve2, c2] = compute_freqz("filters/fir1/fir1_033_int10"); % 2) -9dB
  45 +[h, curve2, c2] = compute_freqz("filters/fir1/fir1_033_int10");
  46 +hTotal = hTotal .* h;
  47 +% [h, curve3, c3] = compute_freqz("filters/fir1/fir1_033_int08");
  48 +% hTotal = hTotal .* h;
  49 +% [h, curve4, c4] = compute_freqz("filters/fir1/fir1_033_int10");
  50 +% hTotal = hTotal .* h;
  51 +% [h, curve5, c5] = compute_freqz("filters/fir1/fir1_015_int11");
  52 +% hTotal = hTotal .* h;
  53 +
  54 +# Log total
  55 +mag = abs(hTotal);
  56 +mag = mag ./ mag(1);
  57 +log_freqz = 20 * log10(mag);
  58 +cTotal = rejection_criteria(log_freqz, 0.5);
  59 +
  60 +[ c1+c2 cTotal ]
  61 +% [ c1+c2+c3+c4+c5 cTotal ]
  62 +
  63 +clf;
  64 +f_axe = [1:N/2] * 2/N;
  65 +hold on;
  66 +color = [0/255 114/255 189/255];
  67 +plot(f_axe, curve1, "linewidth", 1.5, "color", color);
  68 +plot([0 1], [-c1 -c1], "--", "linewidth", 1.5, "color", color);
  69 +
  70 +color = [217/255 83/255 25/255];
  71 +plot(f_axe, curve2, "linewidth", 1.5, "color", color);
  72 +plot([0 1], [-c2 -c2], "--", "linewidth", 1.5, "color", color);
  73 +% plot(f_axe, curve3, "linewidth", 1.5);
  74 +% plot(f_axe, curve4, "linewidth", 1.5);
  75 +% plot(f_axe, curve5, "linewidth", 1.5);
  76 +
  77 +color = [237/255 177/255 32/255];
  78 +plot(f_axe, log_freqz, "linewidth", 1.5, "color", color);
  79 +plot([0 1], [-cTotal -cTotal], "--", "linewidth", 1.5, "color", color);
  80 +plot([0 1], [-(c1 + c2) -(c1 + c2)], ":", "linewidth", 1.5, "color", color);
  81 +
  82 +plot([0.4 0.4], [-500 50], "k:")
  83 +plot([0.6 0.6], [-500 50], "k:")
  84 +ylim([-200 10])
  85 +hold off;
  86 +
  87 +xlabel("Normalized Frequency (a.u.)")
  88 +ylabel("Rejection (dB)")
  89 +legend("Reponse of 1st filter", "Rejection of 1st filter", "Reponse of 2nd filter", "Rejection of 2nd filter", "Reponse Total", "Actual Rejection", "Expected Rejection", "location", "southwest")
ifcs2018_journal.tex
Diff comments   View file @ 17d9a84
  1 +\documentclass[a4paper,conference]{IEEEtran/IEEEtran}
  2 +\usepackage{graphicx,color,hyperref}
  3 +\usepackage{amsfonts}
  4 +\usepackage{amsthm}
  5 +\usepackage{amssymb}
  6 +\usepackage{amsmath}
  7 +\usepackage{algorithm2e}
  8 +\usepackage{url,balance}
  9 +\usepackage[normalem]{ulem}
  10 +\usepackage{tikz}
  11 +\usetikzlibrary{positioning,fit}
  12 +\usepackage{multirow}
  13 +\usepackage{scalefnt}
  14 +
  15 +% correct bad hyphenation here
  16 +\hyphenation{op-tical net-works semi-conduc-tor}
  17 +\textheight=26cm
  18 +\setlength{\footskip}{30pt}
  19 +\pagenumbering{gobble}
  20 +\begin{document}
  21 +\title{Filter optimization for real time digital processing of radiofrequency signals: application
  22 +to oscillator metrology}
  23 +
  24 +\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
  25 +G. Goavec-M\'erou\IEEEauthorrefmark{1},
  26 +P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}
  27 +\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }
  28 +\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
  29 +Email: \{pyb2,jmfriedt\}@femto-st.fr}
  30 +}
  31 +\maketitle
  32 +\thispagestyle{plain}
  33 +\pagestyle{plain}
  34 +\newtheorem{definition}{Definition}
  35 +
  36 +\begin{abstract}
  37 +Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
  38 +radiofrequency signal processing. Applied to oscillator characterization in the context
  39 +of ultrastable clocks, stringent filtering requirements are defined by spurious signal or
  40 +noise rejection needs. Since real time radiofrequency processing must be performed in a
  41 +Field Programmable Array to meet timing constraints, we investigate optimization strategies
  42 +to design filters meeting rejection characteristics while limiting the hardware resources
  43 +required and keeping timing constraints within the targeted measurement bandwidths.
  44 +\end{abstract}
  45 +
  46 +\begin{IEEEkeywords}
  47 +Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
  48 +\end{IEEEkeywords}
  49 +
  50 +\section{Digital signal processing of ultrastable clock signals}
  51 +
  52 +Analog oscillator phase noise characteristics are classically performed by downconverting
  53 +the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,
  54 +followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In
  55 +a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by
  56 +multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
  57 +
  58 +\begin{figure}[h!tb]
  59 +\begin{center}
  60 +\includegraphics[width=.8\linewidth]{images/schema}
  61 +\end{center}
  62 +\caption{Fully digital oscillator phase noise characterization: the Device Under Test
  63 +(DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and
  64 +downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals
  65 +and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite
  66 +Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
  67 +the spectral characteristics of the phase fluctuations.}
  68 +\label{schema}
  69 +\end{figure}
  70 +
  71 +As with the analog mixer,
  72 +the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as
  73 +well as the generation of the frequency sum signal in addition to the frequency difference.
  74 +These unwanted spectral characteristics must be rejected before decimating the data stream
  75 +for the phase noise spectral characterization \cite{andrich2018high}. The characteristics introduced between the
  76 +downconverter
  77 +and the decimation processing blocks are core characteristics of an oscillator characterization
  78 +system, and must reject out-of-band signals below the targeted phase noise -- typically in the
  79 +sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will
  80 +use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency
  81 +datastream: optimizing the performance of the filter while reducing the needed resources is
  82 +hence tackled in a systematic approach using optimization techniques. Most significantly, we
  83 +tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with
  84 +tunable number of coefficients and tunable number of bits representing the coefficients and the
  85 +data being processed.
  86 +
  87 +\section{Finite impulse response filter}
  88 +
  89 +We select FIR filter for their unconditional stability and ease of design. A FIR filter is defined
  90 +by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
  91 +outputs $y_k$
  92 +\begin{align}
  93 + y_n=\sum_{k=0}^N b_k x_{n-k}
  94 + \label{eq:fir_equation}
  95 +\end{align}
  96 +
  97 +As opposed to an implementation on a general purpose processor in which word size is defined by the
  98 +processor architecture, implementing such a filter on an FPGA offer more degrees of freedom since
  99 +not only the coefficient values and number of taps must be defined, but also the number of bits
  100 +defining the coefficients and the sample size. For this reason, and because we consider pipeline
  101 +processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency
  102 +signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but
  103 +the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language (VHDL) level.
  104 +Since latency is not an issue in a openloop phase noise characterization instrument, the large
  105 +numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
  106 +is not considered as an issue as would be in a closed loop system.
  107 +
  108 +The coefficients are classically expressed as floating point values. However, this binary
  109 +number representation is not efficient for fast arithmetic computation by an FPGA. Instead,
  110 +we select to quantify these floating point values into integer values. This quantization
  111 +will result in some precision loss.
  112 +
  113 +\begin{figure}[h!tb]
  114 +\includegraphics[width=\linewidth]{images/demo_filtre}
  115 +\caption{Impact of the quantization resolution of the coefficients: the quantization is
  116 +set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting
  117 +the 30~first and 30~last coefficients out of the initial 128~band-pass
  118 +filter coefficients to 0 (red dots).}
  119 +\label{float_vs_int}
  120 +\end{figure}
  121 +
  122 +The tradeoff between quantization resolution and number of coefficients when considering
  123 +integer operations is not trivial. As an illustration of the issue related to the
  124 +relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits
  125 +a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon
  126 +quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the
  127 +taps become null, making the large number of coefficients irrelevant and allowing to save
  128 +processing resource by shrinking the filter length. This tradeoff aimed at minimizing resources
  129 +to reach a given rejection level, or maximizing out of band rejection for a given computational
  130 +resource, will drive the investigation on cascading filters designed with varying tap resolution
  131 +and tap length, as will be shown in the next section. Indeed, our development strategy closely
  132 +follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
  133 +in which basic blocks are defined and characterized before being assembled \cite{hide}
  134 +in a complete processing chain. In our case, assembling the filter blocks is a simpler block
  135 +combination process since we assume a single value to be processed and a single value to be
  136 +generated at each clock cycle. The FIR filters will not be considered to decimate in the
  137 +current implementation: the decimation is assumed to be located after the FIR cascade at the
  138 +moment.
  139 +
  140 +\section{Methodology description}
  141 +We want create a new methodology to develop any Digital Signal Processing (DSP) chain
  142 +and for any hardware platform (Altera, Xilinx...). To do this we have defined an
  143 +abstract model to represent some basic operations of DSP.
  144 +
  145 +For the moment, we are focused on only two operations: the filtering and the shift of data.
  146 +We have chosen this basic operation because the shifting and the filtering have already be studied in
  147 +lot of works {\color{red} mettre les nouvelles rรฉfรฉrence ici} hence it will be easier
  148 +to check and validate our results.
  149 +
  150 +However having only two operations is insufficient to work with complex DSP but
  151 +in this paper we only want demonstrate the relevance and the efficiency of our approach.
  152 +In future work it will be possible to add more operations and we are able to
  153 +model any DSP chain.
  154 +
  155 +We will apply our methodology on very simple DSP chain. We generate a digital signal
  156 +thanks at generator of Pseudo-Random Number (PRN) or thanks at an Analog to Digital
  157 +Converter (ADC). Once we have a digital signal, we filter it to decrease the noise level.
  158 +Finally we stored some burst of filtered samples before post-processing it.
  159 +% TODO: faire un schรฉma
  160 +In this particular case, we want optimize the filtering step to have the best noise
  161 +rejection for constrain number of resource or to have the minimal resources
  162 +consumption for a given rejection objective.
  163 +
  164 +The first step of our approach is to model the DSP chain and since we just optimize
  165 +the filtering, we have not modeling the PRN generator or the ADC. The filtering can be
  166 +done by two ways. The first one we use only one FIR filter with lot of coefficients
  167 +to rejection the noise, we called this approach a monolithic approach. And the second one
  168 +we select different FIR filters with less coefficients the monolithic filter and we cascaded
  169 +it to filtering the signal.
  170 +
  171 +After each filter we leave the possibility of shifting the filtered data to consume
  172 +less resources. Hence in the case of cascaded filter, we define a stage as a filter
  173 +and a shifter (the shift could be omitted if we do not need to divide the filtered data).
  174 +
  175 +\subsection{Model of a FIR filter}
  176 +A cascade of filter are composed of $n$ stage. In stage $i$ ($1 \leq i \leq n$)
  177 +the FIR has $C_i$ coefficients and each coefficients are integer values with $\pi^C_i$
  178 +bits and the filtered data are shifted of $\pi^S_i$ bits. We define also $\pi^-_i$ as
  179 +the size of input data and $\pi^+_i$ as the size of output data. The figure~\ref{fig:fir_stage}
  180 +shows a filtering stage.
  181 +
  182 +\begin{figure}
  183 + \centering
  184 + \begin{tikzpicture}[node distance=2cm]
  185 + \node[draw,minimum size=1.3cm] (FIR) { $C_i, \pi_i^C$ } ;
  186 + \node[draw,minimum size=1.3cm] (Shift) [right of=FIR, ] { $\pi_i^S$ } ;
  187 + \node (Start) [left of=FIR] { } ;
  188 + \node (End) [right of=Shift] { } ;
  189 +
  190 + \node[draw,fit=(FIR) (Shift)] (Filter) { } ;
  191 +
  192 + \draw[->] (Start) edge node [above] { $\pi_i^-$ } (FIR) ;
  193 + \draw[->] (FIR) -- (Shift) ;
  194 + \draw[->] (Shift) edge node [above] { $\pi_i^+$ } (End) ;
  195 + \end{tikzpicture}
  196 + \caption{A single filter is composed of a FIR (on the left) and a Shifter (on the right)}
  197 + \label{fig:fir_stage}
  198 +\end{figure}
  199 +
  200 +FIR $i$ can reject $F(C_i, \pi_i^C)$ dB. $F$ is determined numerically.
  201 +To measure this rejection, we use GNU Octave software to design FIR filter coefficients thanks to two
  202 +algorithms (\texttt{firls} and \texttt{fir1}).
  203 +For each configuration $(C_i, \pi_i^C)$, we first create a FIR with floating point coefficients and a given $C_i$ number of coefficients.
  204 +Then, the floating point coefficients are discretized into integers. In order to ensure that the coefficients are coded on $\pi_i^C$~bits effectively,
  205 +the coefficients are normalized by their absolute maximum before being scaled to integer coefficients.
  206 +At least one coefficient is coded on $\pi_i^C$~bits, and in practice only $b_{C_i/2}$ is coded on $\pi_i^C$~bits while the other are coded on very fewer bits.
  207 +
  208 +With these coefficients, the \texttt{freqz} function is used to estimate the magnitude of the filter.
  209 +Comparing the performance between FIRs requires however a unique criterion. As shown in figure~\ref{fig:fir_mag},
  210 +the FIR magnitude exhibits two parts.
  211 +
  212 +\begin{figure}
  213 + \centering
  214 + \begin{tikzpicture}[scale=0.3]
  215 + \draw[<->] (0,15) -- (0,0) -- (21,0) ;
  216 + \draw[thick] (0,12) -- (8,12) -- (20,0) ;
  217 +
  218 + \draw (0,14) node [left] { $P$ } ;
  219 + \draw (20,0) node [below] { $f$ } ;
  220 +
  221 + \draw[>=latex,<->] (0,14) -- (8,14) ;
  222 + \draw (4,14) node [above] { passband } node [below] { $40\%$ } ;
  223 +
  224 + \draw[>=latex,<->] (8,14) -- (12,14) ;
  225 + \draw (10,14) node [above] { transition } node [below] { $20\%$ } ;
  226 +
  227 + \draw[>=latex,<->] (12,14) -- (20,14) ;
  228 + \draw (16,14) node [above] { stopband } node [below] { $40\%$ } ;
  229 +
  230 + \draw[>=latex,<->] (16,12) -- (16,8) ;
  231 + \draw (16,10) node [right] { rejection } ;
  232 +
  233 + \draw[dashed] (8,-1) -- (8,14) ;
  234 + \draw[dashed] (12,-1) -- (12,14) ;
  235 +
  236 + \draw[dashed] (8,12) -- (16,12) ;
  237 + \draw[dashed] (12,8) -- (16,8) ;
  238 +
  239 + \end{tikzpicture}
  240 +
  241 +% \includegraphics[width=.5\linewidth]{images/fir_magnitude}
  242 +\caption{Shape of the filter transmitted power $P$ as a function of frequency $f$:
  243 +the passband is considered to occupy the initial 40\% of the Nyquist frequency range,
  244 +the stopband the last 40\%, allowing 20\% transition width.}
  245 +\label{fig:fir_mag}
  246 +\end{figure}
  247 +
  248 +In the transition band, the behavior of the filter is left free, we only care about the passband and the stopband.
  249 +Our first criterion considers the mean value of the stopband rejection, as shown in figure~\ref{fig:mean_criterion}. This criterion does not work because we do not consider the shape of the passband.
  250 +A second criterion considers the maximum rejection within the stopband minus the mean of the absolute value of passband rejection. With this criterion, the results are significantly improved as shown in figure~\ref{fig:custom_criterion}.
  251 +
  252 +\begin{figure}
  253 +\centering
  254 +\includegraphics[width=\linewidth]{images/mean_criterion}
  255 +\caption{Mean criterion comparison between monolithic filter and cascade filters}
  256 +\label{fig:mean_criterion}
  257 +\end{figure}
  258 +
  259 +\begin{figure}
  260 +\centering
  261 +\includegraphics[width=\linewidth]{images/custom_criterion}
  262 +\caption{Custom criterion comparison between monolithic filter and cascade filters}
  263 +\label{fig:custom_criterion}
  264 +\end{figure}
  265 +
  266 +Although we have a efficient criterion to estimate the rejection of one set of coefficient
  267 +we have a problem when we sum two or more criterion. If the FIR filter coefficients are the same
  268 +between the stage, we have:
  269 +$$F_{total} = F_1 + F_2$$
  270 +But when we choose two different set of coefficient, the previous equality are not
  271 +true. The figure~\ref{fig:sum_rejection} illustrates the problem. The red and blue curves
  272 +are two different filter coefficient and we can see that their maximum on the stopband
  273 +are not at the same frequency. So when we sum the rejection criteria (the dotted yellow line)
  274 +we do not meet the dashed yellow line. Define the rejection of cascaded filters
  275 +is more difficult than just take the summation between all the rejection criteria of each filter.
  276 +However this summation gives us an upper bound for rejection although in fact we obtain
  277 +better rejection than expected.
  278 +
  279 +\begin{figure}
  280 +\centering
  281 +\includegraphics[width=\linewidth]{images/sum_rejection}
  282 +\caption{Rejection of two cascaded filters}
  283 +\label{fig:sum_rejection}
  284 +\end{figure}
  285 +
  286 +\section{Experiments with fixed area space}
  287 +
  288 +\begin{figure}
  289 +\centering
  290 +\includegraphics[width=\linewidth]{images/max_rejection/prn_500}
  291 +\caption{Experimental results for design with PRN as data input and 500 a.u. as max arbitrary space}
  292 +\label{fig:prn_500}
  293 +\end{figure}
  294 +
  295 +\begin{figure}
  296 +\centering
  297 +\includegraphics[width=\linewidth]{images/max_rejection/prn_1000}
  298 +\caption{Experimental results for design with PRN as data input and 1000 a.u. as max arbitrary space}
  299 +\label{fig:prn_1000}
  300 +\end{figure}
  301 +
  302 +\begin{figure}
  303 +\centering
  304 +\includegraphics[width=\linewidth]{images/max_rejection/prn_2000}
  305 +\caption{Experimental results for design with PRN as data input and 2000 a.u. as max arbitrary space}
  306 +\label{fig:prn_2000}
  307 +\end{figure}
  308 +
  309 +\begin{table}
  310 +\centering
  311 +\begin{tabular}{|c|c|ccc|c|c|}
  312 +\hline
  313 +\multicolumn{2}{|c|}{\multirow{2}{*}{Stage}} & \multicolumn{3}{c|}{Stage} & \multirow{2}{*}{Rejection} & \multirow{2}{*}{Area} \\ \cline{3-5}
  314 +\multicolumn{2}{|c|}{} & i = 1 & i = 2 & i = 3 & & \\ \hline
  315 + & C & 19 & - & - & & \\
  316 +n = 1 & $pi^C$ & 7 & - & - & 33 dB & 437 a.u. \\
  317 + & $pi^S$ & 0 & - & - & & \\ \hline
  318 + & C & 11 & 19 & - & & \\
  319 +n = 2 & $pi^C$ & 5 & 7 & - & 53 dB & 478 a.u. \\
  320 + & $pi^S$ & 16 & 0 & - & & \\ \hline
  321 + & C & 9 & 15 & 11 & & \\
  322 +n = 3 & $pi^C$ & 4 & 6 & 5 & 57 dB & 499 a.u. \\
  323 + & $pi^S$ & 16 & 3 & 0 & & \\ \hline
  324 +\end{tabular}
  325 +\caption{Solver results for design with PRN as data input and 500 a.u. as max arbitrary space}
  326 +\label{tbl:prn_500}
  327 +\end{table}
  328 +
  329 +\begin{table}
  330 +\centering
  331 +{\scalefont{0.85}
  332 +\begin{tabular}{|c|c|ccccc|c|c|}
  333 +\hline
  334 +\multicolumn{2}{|c|}{\multirow{2}{*}{Stage}} & \multicolumn{5}{c|}{Stage} & \multirow{2}{*}{Rejection} & \multirow{2}{*}{Area} \\ \cline{3-7}
  335 +\multicolumn{2}{|c|}{} & i = 1 & i = 2 & i = 3 & i = 4 & i = 5 & & \\ \hline
  336 + & C & 37 & - & - & - & - & & \\
  337 +n = 1 & $pi^C$ & 11 & - & - & - & - & 56 dB & 999 a.u. \\
  338 + & $pi^S$ & 0 & - & - & - & - & & \\ \hline
  339 + & C & 11 & 39 & - & - & - & & \\
  340 +n = 2 & $pi^C$ & 5 & 13 & - & - & - & 82 dB & 972 a.u. \\
  341 + & $pi^S$ & 16 & 0 & - & - & - & & \\ \hline
  342 + & C & 9 & 31 & 19 & - & - & & \\
  343 +n = 3 & $pi^C$ & 7 & 8 & 7 & - & - & 93 dB & 990 a.u. \\
  344 + & $pi^S$ & 19 & 2 & 0 & - & - & & \\ \hline
  345 + & C & 9 & 19 & 17 & 11 & - & & \\
  346 +n = 4 & $pi^C$ & 4 & 7 & 7 & 5 & - & 99 dB & 992 a.u. \\
  347 + & $pi^S$ & 16 & 3 & 3 & 0 & - & & \\ \hline
  348 + & C & 9 & 15 & 11 & 11 & 11 & & \\
  349 +n = 5 & $pi^C$ & 4 & 7 & 5 & 5 & 5 & 99 dB & 998 a.u. \\
  350 + & $pi^S$ & 16 & 3 & 2 & 1 & 1 & & \\ \hline
  351 +\end{tabular}
  352 +}
  353 +\caption{Solver results for design with PRN as data input and 1000 a.u. as max arbitrary space}
  354 +\label{tbl:prn_1000}
  355 +\end{table}
  356 +
  357 +\begin{table}
  358 +\centering
  359 +{\scalefont{0.85}
  360 +\begin{tabular}{|c|c|ccccc|c|c|}
  361 +\hline
  362 +\multicolumn{2}{|c|}{\multirow{2}{*}{Stage}} & \multicolumn{5}{c|}{Stage} & \multirow{2}{*}{Rejection} & \multirow{2}{*}{Area} \\ \cline{3-7}
  363 +\multicolumn{2}{|c|}{} & i = 1 & i = 2 & i = 3 & i = 4 & i = 5 & & \\ \hline
  364 + & C & 39 & - & - & - & - & & \\
  365 +n = 1 & $pi^C$ & 13 & - & - & - & - & 61 dB & 1131 a.u. \\
  366 + & $pi^S$ & 0 & - & - & - & - & & \\ \hline
  367 + & C & 37 & 39 & - & - & - & & \\
  368 +n = 2 & $pi^C$ & 11 & 13 & - & - & - & 117 dB & 1974 a.u. \\
  369 + & $pi^S$ & 17 & 0 & - & - & - & & \\ \hline
  370 + & C & 15 & 35 & 35 & - & - & & \\
  371 +n = 3 & $pi^C$ & 9 & 11 & 11 & - & - & 138 dB & 1985 a.u. \\
  372 + & $pi^S$ & 19 & 3 & 0 & - & - & & \\ \hline
  373 + & C & 11 & 27 & 27 & 23 & - & & \\
  374 +n = 4 & $pi^C$ & 5 & 9 & 9 & 9 & - & 148 dB & 1993 a.u. \\
  375 + & $pi^S$ & 16 & 3 & 2 & 0 & - & & \\ \hline
  376 + & C & 11 & 27 & 31 & 11 & 11 & & \\
  377 +n = 5 & $pi^C$ & 5 & 9 & 8 & 5 & 5 & 153 dB & 2000 a.u. \\
  378 + & $pi^S$ & 16 & 3 & 1 & 0 & 1 & & \\ \hline
  379 +\end{tabular}
  380 +}
  381 +\caption{Solver results for design with PRN as data input and 2000 a.u. as max arbitrary space}
  382 +\label{tbl:prn_2000}
  383 +\end{table}
  384 +
  385 +\begin{table}
  386 +\centering
  387 +\begin{tabular}{|c|c|c|c|c|}\hline
  388 +Input & Stages & Computation time & Vivado time & Redpitaya time \\\hline\hline
  389 + & 1 & 0.02~s & $\approx$ 20 min & $\approx$ 1 min \\
  390 +PRN & 2 & 1.70~s & $\approx$ 20 min & $\approx$ 1 min \\
  391 + & 3 & 19~s & $\approx$ 20 min & $\approx$ 1 min \\\hline
  392 +\end{tabular}
  393 +\caption{Time to compute and deploy the designs for PRN 500}
  394 +\label{tbl:time_prn_500}
  395 +\end{table}
  396 +
  397 +\begin{table}
  398 +\centering
  399 +\begin{tabular}{|c|c|c|c|c|}\hline
  400 +Input & Stages & Computation time & Vivado time & Redpitaya time \\\hline\hline
  401 + & 1 & 0.07~s & $\approx$ 20 min & $\approx$ 1 min \\
  402 + & 2 & 1.31~s & $\approx$ 20 min & $\approx$ 1 min \\
  403 +PRN & 3 & 119~s ($\approx$ 2~min) & $\approx$ 20 min & $\approx$ 1 min \\
  404 + & 4 & 270~s ($\approx$ 5~min) & $\approx$ 20 min & $\approx$ 1 min \\
  405 + & 5 & 5998~s ($\approx$ 2~h) & $\approx$ 20 min & $\approx$ 1 min \\\hline
  406 +\end{tabular}
  407 +\caption{Time to compute and deploy the designs for PRN 1000}
  408 +\label{tbl:time_prn_1000}
  409 +\end{table}
  410 +
  411 +\begin{table}
  412 +\centering
  413 +\begin{tabular}{|c|c|c|c|c|}\hline
  414 +Input & Stages & Computation time & Vivado time & Redpitaya time \\\hline\hline
  415 + & 1 & 0.07~s & $\approx$ 20 min & $\approx$ 1 min \\
  416 + & 2 & 0.75~s & $\approx$ 20 min & $\approx$ 1 min \\
  417 +PRN & 3 & 36~s & - & - \\
  418 + & 4 & 14500~s ($\approx$ 4~h) & $\approx$ 20 min & $\approx$ 1 min \\
  419 + & 5 & 74237~s ($\approx$ 20~h) & $\approx$ 20 min & $\approx$ 1 min \\\hline
  420 +\end{tabular}
  421 +\caption{Time to compute and deploy the designs for PRN 2000}
  422 +\label{tbl:time_prn_2000}
  423 +\end{table}
  424 +
  425 +\section{Experiments with fixed rejection target}
  426 +
  427 +\begin{figure}
  428 +\centering
  429 +\includegraphics[width=\linewidth]{images/min_area/prn_50}
  430 +\caption{Results for design with PRN as data input and 50 dB as aimed rejection level}
  431 +\label{fig:prn_500}
  432 +\end{figure}
  433 +
  434 +\begin{figure}
  435 +\centering
  436 +\includegraphics[width=\linewidth]{images/min_area/prn_100}
  437 +\caption{Results for design with PRN as data input and 50 dB as aimed rejection level}
  438 +\label{fig:prn_100}
  439 +\end{figure}
  440 +
  441 +\begin{figure}
  442 +\centering
  443 +\includegraphics[width=\linewidth]{images/min_area/prn_150}
  444 +\caption{Results for design with PRN as data input and 2000 a.u. as max arbitrary space}
  445 +\label{fig:prn_150}
  446 +\end{figure}
  447 +
  448 +\section{Conclusion}
  449 +
  450 +\section*{Acknowledgement}
  451 +
  452 +This work is supported by the ANR Programme d'Investissement d'Avenir in
  453 +progress at the Time and Frequency Departments of the FEMTO-ST Institute
  454 +(Oscillator IMP, First-TF and Refimeve+), and by R\'egion de Franche-Comt\'e.
  455 +The authors would like to thank E. Rubiola, F. Vernotte, and G. Cabodevila
  456 +for support and fruitful discussions.
  457 +
  458 +\bibliographystyle{IEEEtran}
  459 +\balance
  460 +\bibliography{references,biblio}
  461 +\end{document}
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