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

Arthur HUGEAT
2 parents 9c253d6d24 db81f7ad9d
Showing 1 changed file Side-by-side Diff
ifcs2018_journal.tex
@@ -174,7 +174,7 @@
 that all solutions found by the solver are synthesized and executed on hardware at the end
 of the analysis.
  
-In this demonstration , we focus on only two operations: filtering and shifting the number of
+In this demonstration, we focus on only two operations: filtering and shifting the number of
 bits needed to represent the data along the processing chain.
 We have chosen these basic operations because shifting and the filtering have already been studied
 in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for
@@ -298,7 +298,8 @@
 Our criterion to compute the filter rejection considers
 % r2.8 et r2.2 r2.3
 the {\color{red}minimal} rejection within the stopband, to which the {\color{red}sum of the absolute values
-within the passband is subtracted to avoid filters with excessive ripples}. With this
+within the passband is subtracted to avoid filters with excessive ripples, normalized to the
+bin width to remain consistent with the passband criterion (dBc/Hz units in all cases)}. With this
 criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
  
 % \begin{figure}
@@ -311,7 +312,8 @@
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{images/colored_custom_criterion}
-\caption{Custom criterion (maximum rejection in the stopband minus the mean of the absolute value of the passband rejection)
+\caption{Custom criterion (maximum rejection in the stopband minus the {\color{red} sum of the
+absolute values of the passband rejection normalized to the bandwidth})
 comparison between monolithic filter and cascaded filters}
 \label{fig:custom_criterion}
 \end{figure}
@@ -327,7 +329,9 @@
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{images/rejection_pyramid}
-\caption{Rejection as a function of number of coefficients and number of bits}
+\caption{{\color{red}{Filter}} rejection as a function of number of coefficients and number of bits
+{\color{red}: this lookup table will be used to identify which filter parameters -- number of bits
+representing coefficients and number of coefficients -- best match the targeted transfer function.}}
 \label{fig:rejection_pyramid}
 \end{figure}
  
@@ -342,7 +346,7 @@
 with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
 % r2.9
 Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection
-criteria of each filter. However since the this sum underestimates the rejection capability of the cascade,
+criteria of each filter. However since the {\color{red}individual filter rejection} sum underestimates the rejection capability of the cascade,
 % r2.10
 this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
 of the filter cascade to meet design criteria.
  
@@ -350,14 +354,19 @@
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{images/cascaded_criterion}
-\caption{Rejection of two cascaded filters}
+\caption{{\color{red}Transfer function of individual filters and after cascading} the two filters,
+{\color{red}demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal
+lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop
+maximum of each individual filter.}
+}
 \label{fig:sum_rejection}
 \end{figure}
  
 % r2.6
-Finally in our case, we consider that the input signal are fully known. So the
-resolution of the data stream are fixed and still the same for all experiments
-in this paper.
+{\color{red}
+Finally in our case, we consider that the input signal are fully known. The
+resolution of the input data stream are fixed and still the same for all experiments
+in this paper.}
  
 Based on this analysis, we address the estimate of resource consumption (called
 % r2.11
  
@@ -407,16 +416,24 @@
  
 {\color{red}
 This model is non-linear since we multiply some variable with another variable
-and it is even non-quadratic, as $F$ does not have a known
+and it is even non-quadratic, as the cost function $F$ does not have a known
 linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
-This variable must be defined by the user, it represent the number of different
-set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
-functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid}
-to restrict the number of configurations. Indeed, it is useless to have too many coefficients or
-too many bits, hence we take the configurations close to edge of pyramid. Thank to theses
-configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant
-and the function $F$ can be estimate for each configurations
-thanks our rejection criterion. We also defined binary
+% AH: conflit merge
+% This variable must be defined by the user, it represent the number of different
+% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
+% functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid}
+% to restrict the number of configurations. Indeed, it is useless to have too many coefficients or
+% too many bits, hence we take the configurations close to edge of pyramid. Thank to theses
+% configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant
+% and the function $F$ can be estimate for each configurations
+% thanks our rejection criterion. We also defined binary
+This variable $p$ is defined by the user, and represents the number of different
+set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}
+functions from GNU Octave) based on the targeted filter characteristics and implementation
+assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and
+$\pi_{ij}^C$ become constants and
+we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table)
+for each configurations thanks to the rejection criterion. We also define the binary
 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
 and 0 otherwise. The new equations are as follows:
 }
@@ -433,9 +450,20 @@
 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
  
 {\color{red}
-However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply
-$\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can
-linearize this multiplication. The following formula shows how to linearize
+% JM: conflict merge
+% However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
+% we multiply
+% $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
+% linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size,
+% we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is
+% assumed on hardware characteristics.
+% The Gurobi (\url{www.gurobi.com}) optimization software used to solve this quadratic
+% model is able to linearize the model provided as is. This model
+% has $O(np)$ variables and $O(n)$ constraints.}
+However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
+we multiply
+$\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
+linearise linearize this multiplication. The following formula shows how to linearize
 this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):
 \begin{equation*}
   m = x \times y \implies
  
@@ -448,12 +476,11 @@
   \end{split}
   \right .
 \end{equation*}
-
-So if we bound up $\pi_i^-$ by 128~bits to represent the maximum data size tolerated,
+So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is
+assumed on hardware characteristics,
 the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize
-for us the quadratic problem so the model is left as is.
-}
-This model has $O(np)$ variables and $O(n)$ constraints.
+for us the quadratic problem so the model is left as is. This model
+has $O(np)$ variables and $O(n)$ constraints.}
  
 % This model is non-linear and even non-quadratic, as $F$ does not have a known
 % linear or quadratic expression. We introduce $p$ FIR configurations
@@ -532,7 +559,8 @@
     \draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
     \draw[->] (Postproc) -- (Results) ;
   \end{tikzpicture}
-  \caption{Design workflow from the input parameters to the results}
+  \caption{Design workflow from the input parameters to the results {\color{red} allowing for
+a fully automated optimal solution search.}}
   \label{fig:workflow}
 \end{figure}
  
  
  
  
@@ -710,22 +738,28 @@
   \centering
   \begin{subfigure}{\linewidth}
     \includegraphics[width=\linewidth]{images/max_500}
-    \caption{Signal spectrum for MAX/500}
+    \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
+the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).}
     \label{fig:max_500_result}
   \end{subfigure}
  
   \begin{subfigure}{\linewidth}
     \includegraphics[width=\linewidth]{images/max_1000}
-    \caption{Signal spectrum for MAX/1000}
+    \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
+the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).}
     \label{fig:max_1000_result}
   \end{subfigure}
  
   \begin{subfigure}{\linewidth}
     \includegraphics[width=\linewidth]{images/max_1500}
-    \caption{Signal spectrum for MAX/1500}
+    \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
+the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).}
     \label{fig:max_1500_result}
   \end{subfigure}
-  \caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500}
+  \caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing
+rejection for a given resource allocation.
+The filter shape constraint (bandpass and bandstop) is shown as thick
+horizontal lines on each chart.}
 \end{figure}
  
 In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
@@ -747,7 +781,8 @@
 Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
  
 \begin{table}[h!tb]
-  \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
+  \caption{Resource occupation {\color{red}following synthesis of the solutions found for
+the problem of maximizing rejection for a given resource allocation}. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
   \label{tbl:resources_usage}
   \centering
       \begin{tabular}{|c|c|ccc|c|}
  
  
  
  
@@ -964,29 +999,36 @@
 \begin{figure}
   \centering
   \begin{subfigure}{\linewidth}
-    \includegraphics[width=\linewidth]{images/min_40}
-    \caption{Signal spectrum for MIN/40}
+    \includegraphics[width=.91\linewidth]{images/min_40}
+    \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
+the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.}
     \label{fig:min_40}
   \end{subfigure}
  
   \begin{subfigure}{\linewidth}
-    \includegraphics[width=\linewidth]{images/min_60}
-    \caption{Signal spectrum for MIN/60}
+    \includegraphics[width=.91\linewidth]{images/min_60}
+    \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
+the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.}
     \label{fig:min_60}
   \end{subfigure}
  
   \begin{subfigure}{\linewidth}
-    \includegraphics[width=\linewidth]{images/min_80}
-    \caption{Signal spectrum for MIN/80}
+    \includegraphics[width=.91\linewidth]{images/min_80}
+    \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
+the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.}
     \label{fig:min_80}
   \end{subfigure}
  
   \begin{subfigure}{\linewidth}
-    \includegraphics[width=\linewidth]{images/min_100}
-    \caption{Signal spectrum for MIN/100}
+    \includegraphics[width=.91\linewidth]{images/min_100}
+    \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
+the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}
     \label{fig:min_100}
   \end{subfigure}
-  \caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100}
+  \caption{\color{red}Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a
+given rejection while minimizing resource allocation. The filter shape constraint (bandpass and
+bandstop) is shown as thick
+horizontal lines on each chart.}
 \end{figure}
  
 We observe that all rejections given by the quadratic solver are close to the experimentally
@@ -1062,17 +1104,19 @@
 previous one.
  
 {\color{red} % r1.4
-To conclude we have compared our monolithic filters with the FIR Compiler form
-Xilinx. For each experimentation we use the same coefficient set and we compare the
-resources consumption. The table~\ref{tbl:xilinx_resources} exhibits the results.
-The FIR Compiler never use BRAM while our filter use one block. This difference
-can be explain be our wish to have a reconfigurable FIR filter. In our case, we can
-configure the coefficients set without to have to change the FPGA design. With
-the FIR compiler, the coefficients set are given during the FPGA design conception
-so we have to change the coefficients, we need to regenerate the design. The
-difference with the LUT consumption is also related to the reconfigurability
-logic. However the DSP consumption, the most restricted resource, are the same between the FIR compiler end
-our FIR block. Our solutions are as good as the Xilinx implementation.
+To conclude, we compare our monolithic filters with the FIR Compiler provided by
+Xilinx in the Vivado software suite (v.2018.2). For each experiment we use the
+same coefficient set and we compare the resource consumption, having checked that
+the transfer functions are indeed the same with both implementations.
+Table~\ref{tbl:xilinx_resources} exhibits the results.
+The FIR Compiler never use BRAM while our filter implementation uses one block. This difference
+is explained be our wish to have a dynamically reconfigurable FIR filter whose
+coefficients can be updated from the processing system without having to update the FPGA design.
+With the FIR compiler, the coefficients are defined during the FPGA design so that
+changing coefficients required generating a new design. The difference with the LUT consumption
+is also attributed to the reconfigurability logic. However the DSP consumption, the scarcest
+resource, is the same between the Xilinx FIR Compiler end
+our FIR block: we hence conclude that our solutions are as good as the Xilinx implementation.
  
 \renewcommand{\arraystretch}{1.2}
 \begin{table}
...	...	@@ -174,7 +174,7 @@
174	174	that all solutions found by the solver are synthesized and executed on hardware at the end
175	175	of the analysis.
176	176
177		-In this demonstration , we focus on only two operations: filtering and shifting the number of
	177	+In this demonstration, we focus on only two operations: filtering and shifting the number of
178	178	bits needed to represent the data along the processing chain.
179	179	We have chosen these basic operations because shifting and the filtering have already been studied
180	180	in the literature \cite{lim_1996, lim_1988, young_1992, smith_1998} providing a framework for
...	...	@@ -298,7 +298,8 @@
298	298	Our criterion to compute the filter rejection considers
299	299	% r2.8 et r2.2 r2.3
300	300	the {\color{red}minimal} rejection within the stopband, to which the {\color{red}sum of the absolute values
301		-within the passband is subtracted to avoid filters with excessive ripples}. With this
	301	+within the passband is subtracted to avoid filters with excessive ripples, normalized to the
	302	+bin width to remain consistent with the passband criterion (dBc/Hz units in all cases)}. With this
302	303	criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
303	304
304	305	% \begin{figure}
...	...	@@ -311,7 +312,8 @@
311	312	\begin{figure}
312	313	\centering
313	314	\includegraphics[width=\linewidth]{images/colored_custom_criterion}
314		-\caption{Custom criterion (maximum rejection in the stopband minus the mean of the absolute value of the passband rejection)
	315	+\caption{Custom criterion (maximum rejection in the stopband minus the {\color{red} sum of the
	316	+absolute values of the passband rejection normalized to the bandwidth})
315	317	comparison between monolithic filter and cascaded filters}
316	318	\label{fig:custom_criterion}
317	319	\end{figure}
...	...	@@ -327,7 +329,9 @@
327	329	\begin{figure}
328	330	\centering
329	331	\includegraphics[width=\linewidth]{images/rejection_pyramid}
330		-\caption{Rejection as a function of number of coefficients and number of bits}
	332	+\caption{{\color{red}{Filter}} rejection as a function of number of coefficients and number of bits
	333	+{\color{red}: this lookup table will be used to identify which filter parameters -- number of bits
	334	+representing coefficients and number of coefficients -- best match the targeted transfer function.}}
331	335	\label{fig:rejection_pyramid}
332	336	\end{figure}
333	337
...	...	@@ -342,7 +346,7 @@
342	346	with respect to a basic sum of the rejection criteria shown as a the dotted yellow line.
343	347	% r2.9
344	348	Thus, estimating the rejection of filter cascades is more complex than taking the sum of all the rejection
345		-criteria of each filter. However since the this sum underestimates the rejection capability of the cascade,
	349	+criteria of each filter. However since the {\color{red}individual filter rejection} sum underestimates the rejection capability of the cascade,
346	350	% r2.10
347	351	this upper bound is considered as a conservative and acceptable criterion for deciding on the suitability
348	352	of the filter cascade to meet design criteria.
349	353
...	...	@@ -350,14 +354,19 @@
350	354	\begin{figure}
351	355	\centering
352	356	\includegraphics[width=\linewidth]{images/cascaded_criterion}
353		-\caption{Rejection of two cascaded filters}
	357	+\caption{{\color{red}Transfer function of individual filters and after cascading} the two filters,
	358	+{\color{red}demonstrating that the selected criterion of maximum rejection in the bandstop (horizontal
	359	+lines) is met. Notice that the cascaded filter has better rejection than summing the bandstop
	360	+maximum of each individual filter.}
	361	+}
354	362	\label{fig:sum_rejection}
355	363	\end{figure}
356	364
357	365	% r2.6
358		-Finally in our case, we consider that the input signal are fully known. So the
359		-resolution of the data stream are fixed and still the same for all experiments
360		-in this paper.
	366	+{\color{red}
	367	+Finally in our case, we consider that the input signal are fully known. The
	368	+resolution of the input data stream are fixed and still the same for all experiments
	369	+in this paper.}
361	370
362	371	Based on this analysis, we address the estimate of resource consumption (called
363	372	% r2.11
364	373
...	...	@@ -407,16 +416,24 @@
407	416
408	417	{\color{red}
409	418	This model is non-linear since we multiply some variable with another variable
410		-and it is even non-quadratic, as $F$ does not have a known
	419	+and it is even non-quadratic, as the cost function $F$ does not have a known
411	420	linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
412		-This variable must be defined by the user, it represent the number of different
413		-set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
414		-functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid}
415		-to restrict the number of configurations. Indeed, it is useless to have too many coefficients or
416		-too many bits, hence we take the configurations close to edge of pyramid. Thank to theses
417		-configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant
418		-and the function $F$ can be estimate for each configurations
419		-thanks our rejection criterion. We also defined binary
	421	+% AH: conflit merge
	422	+% This variable must be defined by the user, it represent the number of different
	423	+% set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir1}
	424	+% functions from GNU Octave). To choose this value, we consider a subset of the figure~\ref{fig:rejection_pyramid}
	425	+% to restrict the number of configurations. Indeed, it is useless to have too many coefficients or
	426	+% too many bits, hence we take the configurations close to edge of pyramid. Thank to theses
	427	+% configurations $C_{ij}$ and $\pi_{ij}^C$ ($1 \leq j \leq p$) become constant
	428	+% and the function $F$ can be estimate for each configurations
	429	+% thanks our rejection criterion. We also defined binary
	430	+This variable $p$ is defined by the user, and represents the number of different
	431	+set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}
	432	+functions from GNU Octave) based on the targeted filter characteristics and implementation
	433	+assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and
	434	+$\pi_{ij}^C$ become constants and
	435	+we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table)
	436	+for each configurations thanks to the rejection criterion. We also define the binary
420	437	variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
421	438	and 0 otherwise. The new equations are as follows:
422	439	}
...	...	@@ -433,9 +450,20 @@
433	450	Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
434	451
435	452	{\color{red}
436		-However the problem still quadratic since in the constraint~\ref{eq:areadef2} we multiply
437		-$\delta_{ij}$ and $\pi_i^-$. But like $\delta_{ij}$ is a binary variable we can
438		-linearize this multiplication. The following formula shows how to linearize
	453	+% JM: conflict merge
	454	+% However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
	455	+% we multiply
	456	+% $\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
	457	+% linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size,
	458	+% we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is
	459	+% assumed on hardware characteristics.
	460	+% The Gurobi (\url{www.gurobi.com}) optimization software used to solve this quadratic
	461	+% model is able to linearize the model provided as is. This model
	462	+% has $O(np)$ variables and $O(n)$ constraints.}
	463	+However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
	464	+we multiply
	465	+$\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
	466	+linearise linearize this multiplication. The following formula shows how to linearize
439	467	this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):
440	468	\begin{equation*}
441	469	m = x \times y \implies
442	470
...	...	@@ -448,12 +476,11 @@
448	476	\end{split}
449	477	\right .
450	478	\end{equation*}
451		-
452		-So if we bound up $\pi_i^-$ by 128~bits to represent the maximum data size tolerated,
	479	+So if we bound up $\pi_i^-$ by 128~bits which is the maximum data size whose estimation is
	480	+assumed on hardware characteristics,
453	481	the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize
454		-for us the quadratic problem so the model is left as is.
455		-}
456		-This model has $O(np)$ variables and $O(n)$ constraints.
	482	+for us the quadratic problem so the model is left as is. This model
	483	+has $O(np)$ variables and $O(n)$ constraints.}
457	484
458	485	% This model is non-linear and even non-quadratic, as $F$ does not have a known
459	486	% linear or quadratic expression. We introduce $p$ FIR configurations
...	...	@@ -532,7 +559,8 @@
532	559	\draw[->] (Deploy) edge node [left] { (5) } (Postproc) ;
533	560	\draw[->] (Postproc) -- (Results) ;
534	561	\end{tikzpicture}
535		- \caption{Design workflow from the input parameters to the results}
	562	+ \caption{Design workflow from the input parameters to the results {\color{red} allowing for
	563	+a fully automated optimal solution search.}}
536	564	\label{fig:workflow}
537	565	\end{figure}
538	566
539	567
540	568
541	569
...	...	@@ -710,22 +738,28 @@
710	738	\centering
711	739	\begin{subfigure}{\linewidth}
712	740	\includegraphics[width=\linewidth]{images/max_500}
713		- \caption{Signal spectrum for MAX/500}
	741	+ \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
	742	+the MAX/500 problem of maximizing rejection for a given resource allocation (500~arbitrary units).}
714	743	\label{fig:max_500_result}
715	744	\end{subfigure}
716	745
717	746	\begin{subfigure}{\linewidth}
718	747	\includegraphics[width=\linewidth]{images/max_1000}
719		- \caption{Signal spectrum for MAX/1000}
	748	+ \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
	749	+the MAX/1000 problem of maximizing rejection for a given resource allocation (1000~arbitrary units).}
720	750	\label{fig:max_1000_result}
721	751	\end{subfigure}
722	752
723	753	\begin{subfigure}{\linewidth}
724	754	\includegraphics[width=\linewidth]{images/max_1500}
725		- \caption{Signal spectrum for MAX/1500}
	755	+ \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
	756	+the MAX/1500 problem of maximizing rejection for a given resource allocation (1500~arbitrary units).}
726	757	\label{fig:max_1500_result}
727	758	\end{subfigure}
728		- \caption{Signal spectrum of each experimental configurations MAX/500, MAX/1000 and MAX/1500}
	759	+ \caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing
	760	+rejection for a given resource allocation.
	761	+The filter shape constraint (bandpass and bandstop) is shown as thick
	762	+horizontal lines on each chart.}
729	763	\end{figure}
730	764
731	765	In all cases, we observe that the actual rejection is close to the rejection computed by the solver.
...	...	@@ -747,7 +781,8 @@
747	781	Logic (PL -- FPGA) to Processing System (PS -- general purpose processor) communication.
748	782
749	783	\begin{table}[h!tb]
750		- \caption{Resource occupation. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
	784	+ \caption{Resource occupation {\color{red}following synthesis of the solutions found for
	785	+the problem of maximizing rejection for a given resource allocation}. The last column refers to available resources on a Zynq-7010 as found on the Redpitaya.}
751	786	\label{tbl:resources_usage}
752	787	\centering
753	788	\begin{tabular}{\|c\|c\|ccc\|c\|}
754	789
755	790
756	791
757	792
...	...	@@ -964,29 +999,36 @@
964	999	\begin{figure}
965	1000	\centering
966	1001	\begin{subfigure}{\linewidth}
967		- \includegraphics[width=\linewidth]{images/min_40}
968		- \caption{Signal spectrum for MIN/40}
	1002	+ \includegraphics[width=.91\linewidth]{images/min_40}
	1003	+ \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
	1004	+the MIN/40 problem of minimizing resource allocation for reaching a 40~dB rejection.}
969	1005	\label{fig:min_40}
970	1006	\end{subfigure}
971	1007
972	1008	\begin{subfigure}{\linewidth}
973		- \includegraphics[width=\linewidth]{images/min_60}
974		- \caption{Signal spectrum for MIN/60}
	1009	+ \includegraphics[width=.91\linewidth]{images/min_60}
	1010	+ \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
	1011	+the MIN/60 problem of minimizing resource allocation for reaching a 60~dB rejection.}
975	1012	\label{fig:min_60}
976	1013	\end{subfigure}
977	1014
978	1015	\begin{subfigure}{\linewidth}
979		- \includegraphics[width=\linewidth]{images/min_80}
980		- \caption{Signal spectrum for MIN/80}
	1016	+ \includegraphics[width=.91\linewidth]{images/min_80}
	1017	+ \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
	1018	+the MIN/80 problem of minimizing resource allocation for reaching a 80~dB rejection.}
981	1019	\label{fig:min_80}
982	1020	\end{subfigure}
983	1021
984	1022	\begin{subfigure}{\linewidth}
985		- \includegraphics[width=\linewidth]{images/min_100}
986		- \caption{Signal spectrum for MIN/100}
	1023	+ \includegraphics[width=.91\linewidth]{images/min_100}
	1024	+ \caption{\color{red}Filter transfer functions for varying number of cascaded filters solving
	1025	+the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}
987	1026	\label{fig:min_100}
988	1027	\end{subfigure}
989		- \caption{Signal spectrum of each experimental configurations MIN/40, MIN/60, MIN/80 and MIN/100}
	1028	+ \caption{\color{red}Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a
	1029	+given rejection while minimizing resource allocation. The filter shape constraint (bandpass and
	1030	+bandstop) is shown as thick
	1031	+horizontal lines on each chart.}
990	1032	\end{figure}
991	1033
992	1034	We observe that all rejections given by the quadratic solver are close to the experimentally
...	...	@@ -1062,17 +1104,19 @@
1062	1104	previous one.
1063	1105
1064	1106	{\color{red} % r1.4
1065		-To conclude we have compared our monolithic filters with the FIR Compiler form
1066		-Xilinx. For each experimentation we use the same coefficient set and we compare the
1067		-resources consumption. The table~\ref{tbl:xilinx_resources} exhibits the results.
1068		-The FIR Compiler never use BRAM while our filter use one block. This difference
1069		-can be explain be our wish to have a reconfigurable FIR filter. In our case, we can
1070		-configure the coefficients set without to have to change the FPGA design. With
1071		-the FIR compiler, the coefficients set are given during the FPGA design conception
1072		-so we have to change the coefficients, we need to regenerate the design. The
1073		-difference with the LUT consumption is also related to the reconfigurability
1074		-logic. However the DSP consumption, the most restricted resource, are the same between the FIR compiler end
1075		-our FIR block. Our solutions are as good as the Xilinx implementation.
	1107	+To conclude, we compare our monolithic filters with the FIR Compiler provided by
	1108	+Xilinx in the Vivado software suite (v.2018.2). For each experiment we use the
	1109	+same coefficient set and we compare the resource consumption, having checked that
	1110	+the transfer functions are indeed the same with both implementations.
	1111	+Table~\ref{tbl:xilinx_resources} exhibits the results.
	1112	+The FIR Compiler never use BRAM while our filter implementation uses one block. This difference
	1113	+is explained be our wish to have a dynamically reconfigurable FIR filter whose
	1114	+coefficients can be updated from the processing system without having to update the FPGA design.
	1115	+With the FIR compiler, the coefficients are defined during the FPGA design so that
	1116	+changing coefficients required generating a new design. The difference with the LUT consumption
	1117	+is also attributed to the reconfigurability logic. However the DSP consumption, the scarcest
	1118	+resource, is the same between the Xilinx FIR Compiler end
	1119	+our FIR block: we hence conclude that our solutions are as good as the Xilinx implementation.
1076	1120
1077	1121	\renewcommand{\arraystretch}{1.2}
1078	1122	\begin{table}