jfriedt / IFCS2018 article

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Merge branch 'master' of https://lxsd.femto-st.fr/gitlab/jfriedt/ifcs2018-article

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.gitignore
Makefile
demo_critere_filtre.m
ifcs2018_journal.tex
images/custom_criterion.pdf
images/max_rejection/prn_1000.pdf
images/max_rejection/prn_2000.pdf
images/max_rejection/prn_500.pdf
images/mean_criterion.pdf
images/min_area/prn_100.pdf
images/min_area/prn_150.pdf
images/min_area/prn_50.pdf
images/sum_rejection.pdf

.gitignore

Diff comments View file @ 17d9a84

ifcs2018_abstract.aux	1	1	ifcs2018_abstract.aux
ifcs2018_abstract.bbl	2	2	ifcs2018_abstract.bbl
ifcs2018_abstract.blg	3	3	ifcs2018_abstract.blg
ifcs2018_abstract.log	4	4	ifcs2018_abstract.log
ifcs2018_abstract.out	5	5	ifcs2018_abstract.out
ifcs2018_abstract.pdf	6	6	ifcs2018_abstract.pdf
	7	7
ifcs2018_proceeding.aux	8	8	ifcs2018_proceeding.aux
ifcs2018_proceeding.bbl	9	9	ifcs2018_proceeding.bbl
ifcs2018_proceeding.blg	10	10	ifcs2018_proceeding.blg
ifcs2018_proceeding.log	11	11	ifcs2018_proceeding.log
ifcs2018_proceeding.out	12	12	ifcs2018_proceeding.out
ifcs2018_proceeding.pdf	13	13	ifcs2018_proceeding.pdf
	14	14
ifcs2018_poster.aux	15	15	ifcs2018_poster.aux
ifcs2018_poster.log	16	16	ifcs2018_poster.log
ifcs2018_poster.out	17	17	ifcs2018_poster.out
ifcs2018_poster.pdf	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

Makefile

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# source: https://tex.stackexchange.com/questions/40738/how-to-properly-make-a-latex-project	1	1	# source: https://tex.stackexchange.com/questions/40738/how-to-properly-make-a-latex-project
	2	2
TEX = pdflatex -shell-escape -interaction=nonstopmode -file-line-error	3	3	TEX = pdflatex -shell-escape -interaction=nonstopmode -file-line-error
BIB = bibtex	4	4	BIB = bibtex
TARGET = ifcs2018	5	5	TARGET = ifcs2018
	6	6
all: $(TARGET)_abstract $(TARGET)_poster $(TARGET)_proceeding	7	7	all: $(TARGET)_abstract $(TARGET)_poster $(TARGET)_proceeding $(TARGET)_journal
	8	8
view: $(TARGET)	9	9	view: $(TARGET)
evince $(TARGET).pdf	10	10	evince $(TARGET).pdf
	11	11
view_poster: $(TARGET)_poster	12	12	view_poster: $(TARGET)_poster
evince $(TARGET)_poster.pdf	13	13	evince $(TARGET)_poster.pdf
	14	14
$(TARGET)_abstract: $(TARGET)_abstract.tex references.bib biblio.bib	15	15	$(TARGET)_abstract: $(TARGET)_abstract.tex references.bib biblio.bib
$(TEX) $@.tex	16	16	$(TEX) $@.tex
$(BIB) $@	17	17	$(BIB) $@
$(TEX) $@.tex	18	18	$(TEX) $@.tex
$(TEX) $@.tex	19	19	$(TEX) $@.tex
	20	20
$(TARGET)_poster:	21	21	$(TARGET)_poster:
$(TEX) $@.tex	22	22	$(TEX) $@.tex
$(TEX) $@.tex	23	23	$(TEX) $@.tex
	24	24
$(TARGET)_proceeding: $(TARGET)_proceeding.tex references.bib biblio.bib	25	25	$(TARGET)_proceeding: $(TARGET)_proceeding.tex references.bib biblio.bib
$(TEX) $@.tex	26	26	$(TEX) $@.tex
$(BIB) $@	27	27	$(BIB) $@
$(TEX) $@.tex	28	28	$(TEX) $@.tex
$(TEX) $@.tex	29	29	$(TEX) $@.tex
	30	30
		31	$(TARGET)_journal: $(TARGET)_journal.tex references.bib biblio.bib
		32	$(TEX) $@.tex
		33	$(BIB) $@
		34	$(TEX) $@.tex
		35	$(TEX) $@.tex
		36
clean:	31	37	clean:
rm -f $(TARGET).aux $(TARGET).log $(TARGET).out $(TARGET).bbl $(TARGET).blg	32	38	rm -f $(TARGET)_abstract.aux $(TARGET)_abstract.log $(TARGET)_abstract.out $(TARGET)_abstract.bbl $(TARGET)_abstract.blg
rm -f $(TARGET)_proceeding.aux $(TARGET)_proceeding.log $(TARGET)_proceeding.out $(TARGET)_proceeding.bbl $(TARGET)_proceeding.blg	33

demo_critere_filtre.m

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File was created	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);

ifcs2018_journal.tex

Diff comments View file @ 17d9a84

File was created	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 référence 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 schéma
	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 \\