Arthur HUGEAT · Arthur HUGEAT
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ifcs2018_journal.tex
ifcs2018_journal_reponse.tex
images/letter_max_criterion.pdf
images/letter_sum_criterion.pdf
@@ -297,7 +297,7 @@ In the transition band, the behavior of the filter is left free, we only {\color
 % the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients.
 Our criterion to compute the filter rejection considers
 % r2.8 et r2.2 r2.3
-the maximum magnitude within the stopband, to which the {\color{red}sum of the absolute values
+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, 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}.
@@ -312,7 +312,7 @@ criterion, we meet the expected rejection capability of low pass filters as show
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{images/colored_custom_criterion}
-\caption{Custom criterion (maximum rejection in the stopband minus the {\color{red} sum of the 
+\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}
@@ -418,12 +418,21 @@ Finally, equation~\ref{eq:init} gives the number of bits of the global input.
 This model is non-linear since we multiply some variable with another variable
 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.
+% 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 
+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) 
+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:
@@ -441,14 +450,36 @@ respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}
 Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
  
 {\color{red}
-However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2} 
+% 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 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
+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
+  \left \{
+  \begin{split}
+    m & \geq 0 \\
+    m & \leq y \times X^{max} \\
+    m & \leq x \\
+    m & \geq x - (1 - y) \times X^{max} \\
+  \end{split}
+  \right .
+\end{equation*}
+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.}
  
 % This model is non-linear and even non-quadratic, as $F$ does not have a known
@@ -727,7 +758,7 @@ the MAX/1500 problem of maximizing rejection for a given resource allocation (15
   \end{subfigure}
   \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 
+The filter shape constraint (bandpass and bandstop) is shown as thick
 horizontal lines on each chart.}
 \end{figure}
  
@@ -994,9 +1025,9 @@ the MIN/80 problem of minimizing resource allocation for reaching a 80~dB reject
 the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}
     \label{fig:min_100}
   \end{subfigure}
-  \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 
+  \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}
  
@@ -1074,16 +1105,16 @@ previous one.
  
 {\color{red} % r1.4
 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 
+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. 
+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. 
+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  
+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.
  
@@ -74,7 +74,7 @@
 %REVIEWERS' COMMENTS:
  
 \documentclass[a4paper]{article}
-\usepackage{fullpage,graphicx,amsmath}
+\usepackage{fullpage,graphicx,amsmath, subcaption}
 \begin{document}
 {\bf Reviewer: 1}
  
@@ -82,21 +82,21 @@
 %In general, the language/grammar is adequate.
  
 {\bf
-On page 2,  "...allowing to save processing resource..." could be improved.       % r1.1
+On page 2,  "...allowing to save processing resource..." could be improved.       % r1.1 - fait
 }
  
 The sentence was split and now reads ``number of coefficients irrelevant: processing
 resources are hence saved by shrinking the filter length.''
  
 {\bf
-On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what   % r1.2
+On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what   % r1.2 - fait
 the author meant.}
  
 Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s)
 Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
  
 {\bf
-On page 2, the whole paragraph "The first step of our approach is to model..."   % r1.3
+On page 2, the whole paragraph "The first step of our approach is to model..."   % r1.3 - fait
 could be improved.
 }
  
@@ -151,7 +151,7 @@ Reviewer: 2
 %filters are really superior than monolithic filters.
  
 {\bf
-By observing the results presented in fig. 10-16, it is clear that the            % r2.1 - fait
+By observing the results presented in fig. 10-16, it is clear that the            % r2.1
 performances of multi-stage filters are obtained at the expense of their
 selectivity and, in this sense, the filters presented in these figures
 are not equivalent. For example, in Fig. 14, at the limit of the pass band,
@@ -162,23 +162,61 @@ n = 1.
 We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined
 the bandpass and the bandstop of the filter.
  
-We are aware of this non equivalence but we think that difference is not due to
-the cascaded filters but due to the definition of rejection criterion on the passband.
-Indeed, in this article we have choose to take the summation of absolute values divide
-by the bandwidth but this criterion is maybe too permissive and when we cascade
-some filters this impact is more important.
-
-However if we change the passband
-criterion by the summation of absolute value in passband, weighting given to the
-passband ripples are too strong and the solver are too restricted to provide
-any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in
-passband, the rejection evaluation are too close form the original criterion and
-the result will not be improved.
-
-In this article, we will highlight the methodology instead of the filter conception.
-Even if our rejection criterion is not the best, our methodology was not impacted
-by this. So to improve the results, we can choose another criterion to be more
-selective in passband but it is not the main objective of our article.
+% We are aware of this non equivalence but we think that difference is not due to
+% the cascaded filters but due to the definition of rejection criterion on the passband.
+% Indeed, in this article we have choose to take the summation of absolute values divide
+% by the bandwidth but this criterion is maybe too permissive and when we cascade
+% some filters this impact is more important.
+%
+% However if we change the passband
+% criterion by the summation of absolute value in passband, weighting given to the
+% passband ripples are too strong and the solver are too restricted to provide
+% any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in
+% passband, the rejection evaluation are too close form the original criterion and
+% the result will not be improved.
+%
+% In this article, we will highlight the methodology instead of the filter conception.
+% Even if our rejection criterion is not the best, our methodology was not impacted
+% by this. So to improve the results, we can choose another criterion to be more
+% selective in passband but it is not the main objective of our article.
+
+We are aware of this equivalence but to limit this ripples in passband we need to
+enforce the criterion in passband. If we takes a strong constraint like the sum of
+absolute values in passband. This criterion si too selective because it considers
+all bin on passband while on stopband we consider only the bin with the minimal
+rejection. The figure~\ref{fig:letter_sum_criterion} exhibits the results with this
+criterion for the case MAX/1000. With this criterion, the solver find an optimal
+solution with only two filters in expend of the resource consumption.
+
+
+
+If we relax a little the criterion on passband with taking only the maximum absolute
+value, we will penalize the ripple peak on passband. The figure~\ref{fig:letter_max_criterion}
+shows the results for the case MAX/1000. There as almost no difference with the
+article results. Indeed the only little change are on the case $i = 4$ and $i = 5$
+which they have some minor differences on coefficients choices.
+
+\begin{figure}[h!tb]
+  \centering
+  \begin{subfigure}{0.48\linewidth}
+    \includegraphics[width=\linewidth]{images/letter_sum_criterion}
+    \caption{Results for the case MAX/1000 with as criterion on passband the sum absolute values}
+    \label{fig:letter_sum_criterion}
+  \end{subfigure}
+  \begin{subfigure}{0.48\linewidth}
+    \includegraphics[width=\linewidth]{images/letter_max_criterion}
+    \caption{Results for the case MAX/1000 with as criterion on passband the maximum absolute value}
+    \label{fig:letter_max_criterion}
+  \end{subfigure}
+\end{figure}
+
+Finally, if we ponder the maximum absolute on passband, we should improve the result.
+We have arbitrary pondered by 5 the maximum. Even with this weighting, the solver
+choose the same coefficient set.
+
+To conclude, find a better criterion to avoid the ripples on the passband is difficult.
+In this article we are focused on the methodology so even if our criterion could
+be improved, our methodology still the same and it works independently of rejection criterion.
  
 % %Peut etre refaire une serie de simulation dans lesquelles on impose une coupure
 % %non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte
@@ -197,7 +235,7 @@ selective in passband but it is not the main objective of our article.
 % Dire que la chute n'est pas du à la casacade mais à notre critère de rejection
  
 {\bf
-The reason is in the criterion that considers the average attenuation in          % r2.2 - fait
+The reason is in the criterion that considers the average attenuation in          % r2.2
 the pass band. This criterion does not take into account the maximum attenuation
 in this region, which is a very important parameter for specifying a filter
 and for evaluating its performance. For example, with this criterion, a
@@ -206,8 +244,9 @@ filter with 0.1 dB of ripple is considered equivalent to a filter with
 and in the results that are obtained and has to be reconsidered.
 }
  
-See above: If we choose the maximum absolute value in passband, we penalize the
-case with 10 dB of ripple.
+See above: Choose a criterion is difficult and depending on the context. The main
+contribution on this paper is the methodology not the criterion to quantify the
+rejection.
  
 % The manuscript erroneously stated that we considered the mean of the absolute
 % value within the bandpass: the manuscript has now been corrected to properly state
@@ -231,7 +270,7 @@ excessive ripples, including excessive attenuation, within the passband.
 % TODO: test max(stopband) - max(abs(passband))
  
 {\bf
-In addition, I suggest to address the following points:                           % r2.4
+In addition, I suggest to address the following points:                           % r2.4 - fait
 - Page 1, line 50: the Authors state that IIR have shorter impulse response
 than FIR. This is not true in general. The sentence should be reconsidered.
 }
@@ -248,7 +287,7 @@ is not considered as an issue as would be in a closed loop system in which lag a
 minimized to avoid oscillation conditions.''
  
 {\bf
-- Fig. 4: the Author should motivate in the text why it has been chosen           % r2.5
+- Fig. 4: the Author should motivate in the text why it has been chosen           % r2.5 - fait
 this transition bandwidth and if it is a typical requirement for phase-noise
 metrology.
 }
@@ -279,13 +318,16 @@ hardware.&#39;&#39; so indeed the input datastream resolution is considered as a given.
 {\bf
 - Page 3, line 47: the initial criterion can be omitted and, consequently,        % r2.7  - fait
 Fig. 5 can be removed.
+}
+
+Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats
+
+{\bf
 - Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4.             % r2.8  - fait
 It should be ``minimum''
 }
-AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max
-de la magnitude. J'ai corrigé en ce sens.
  
-Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats
+This typo has been corrected.
  
 {\bf
 - Page e, line 55, second column: ``takin''                                       % r2.9  - fait
@@ -309,7 +351,7 @@ set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir
 functions from GNU Octave)''
  
 {\bf
-- Page 4, line 31: how does the delta function transform model from non-linear    % r2.13 - fait
+- Page 4, line 31: how does the delta function transform model from non-linear    % r2.13
 and non-quadratic to a quadratic?}
  
 The first model is non-quadratic but when we introduce the $p$ configurations,
@@ -335,22 +377,22 @@ Gurobi does the linearization so we don&#39;t explain this step to keep the model mo
 simple. However, to improve the transformation explanation we have rewrote the
 paragraph ``This model is non-linear and even non-quadratic...''.
  
-JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
-je ne comprends pas comment ca repond a la question
-
-AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
-
-Le problème n'est pas linéaire car nous multiplions des variables
-entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
-des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
-quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
-quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
-$\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
-de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
-cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
-nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
-qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
-linéarisation pour nous.
+% JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
+% je ne comprends pas comment ca repond a la question
+%
+% AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
+%
+% Le problème n'est pas linéaire car nous multiplions des variables
+% entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
+% des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
+% quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
+% quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
+% $\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
+% de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
+% cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
+% nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
+% qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
+% linéarisation pour nous.
  
  
 {\bf
...	...	@@ -297,7 +297,7 @@ In the transition band, the behavior of the filter is left free, we only {\color
297	297	% the passband are not considered and might be excessive for excessively wide transitions widths introduced for filters with few coefficients.
298	298	Our criterion to compute the filter rejection considers
299	299	% r2.8 et r2.2 r2.3
300		-the maximum magnitude within the stopband, to which the {\color{red}sum of the absolute values
	300	+the {\color{red}minimal} rejection within the stopband, to which the {\color{red}sum of the absolute values
301	301	within the passband is subtracted to avoid filters with excessive ripples, normalized to the
302	302	bin width to remain consistent with the passband criterion (dBc/Hz units in all cases)}. With this
303	303	criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}.
...	...	@@ -312,7 +312,7 @@ criterion, we meet the expected rejection capability of low pass filters as show
312	312	\begin{figure}
313	313	\centering
314	314	\includegraphics[width=\linewidth]{images/colored_custom_criterion}
315		-\caption{Custom criterion (maximum rejection in the stopband minus the {\color{red} sum of the
	315	+\caption{Custom criterion (maximum rejection in the stopband minus the {\color{red} sum of the
316	316	absolute values of the passband rejection normalized to the bandwidth})
317	317	comparison between monolithic filter and cascaded filters}
318	318	\label{fig:custom_criterion}
...	...	@@ -418,12 +418,21 @@ Finally, equation~\ref{eq:init} gives the number of bits of the global input.
418	418	This model is non-linear since we multiply some variable with another variable
419	419	and it is even non-quadratic, as the cost function $F$ does not have a known
420	420	linear or quadratic expression. To linearize this problem, we introduce $p$ FIR configurations.
	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
421	430	This variable $p$ is defined by the user, and represents the number of different
422	431	set of coefficients generated (remember, we use \texttt{firls} and \texttt{fir1}
423	432	functions from GNU Octave) based on the targeted filter characteristics and implementation
424		-assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and
	433	+assumptions (estimated number of bits defining the coefficients). Hence, $C_{ij}$ and
425	434	$\pi_{ij}^C$ become constants and
426		-we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table)
	435	+we define $1 \leq j \leq p$ so that the function $F$ can be estimated (Look Up Table)
427	436	for each configurations thanks to the rejection criterion. We also define the binary
428	437	variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
429	438	and 0 otherwise. The new equations are as follows:
...	...	@@ -441,14 +450,36 @@ respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}
441	450	Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
442	451
443	452	{\color{red}
444		-However the problem remains quadratic at this stage since in the constraint~\ref{eq:areadef2}
	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}
445	464	we multiply
446	465	$\delta_{ij}$ and $\pi_i^-$. However, since $\delta_{ij}$ is a binary variable we can
447		-linearise this multiplication if we can bound $\pi_i^-$. As $\pi_i^-$ is the data size,
448		-we define $0 < \pi_i^- \leq 128$ which is the maximum data size whose estimation is
449		-assumed on hardware characteristics.
450		-The Gurobi (\url{www.gurobi.com}) optimization software used to solve this quadratic
451		-model is able to linearize the model provided as is. This model
	466	+linearise linearize this multiplication. The following formula shows how to linearize
	467	+this situation in general case with $y$ a binary variable and $x$ a real variable ($0 \leq x \leq X^{max}$):
	468	+\begin{equation*}
	469	+ m = x \times y \implies
	470	+ \left \{
	471	+ \begin{split}
	472	+ m & \geq 0 \\
	473	+ m & \leq y \times X^{max} \\
	474	+ m & \leq x \\
	475	+ m & \geq x - (1 - y) \times X^{max} \\
	476	+ \end{split}
	477	+ \right .
	478	+\end{equation*}
	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,
	481	+the Gurobi (\url{www.gurobi.com}) optimization software will be able to linearize
	482	+for us the quadratic problem so the model is left as is. This model
452	483	has $O(np)$ variables and $O(n)$ constraints.}
453	484
454	485	% This model is non-linear and even non-quadratic, as $F$ does not have a known
...	...	@@ -727,7 +758,7 @@ the MAX/1500 problem of maximizing rejection for a given resource allocation (15
727	758	\end{subfigure}
728	759	\caption{\color{red}Solutions for the MAX/500, MAX/1000 and MAX/1500 problems of maximizing
729	760	rejection for a given resource allocation.
730		-The filter shape constraint (bandpass and bandstop) is shown as thick
	761	+The filter shape constraint (bandpass and bandstop) is shown as thick
731	762	horizontal lines on each chart.}
732	763	\end{figure}
733	764
...	...	@@ -994,9 +1025,9 @@ the MIN/80 problem of minimizing resource allocation for reaching a 80~dB reject
994	1025	the MIN/100 problem of minimizing resource allocation for reaching a 100~dB rejection.}
995	1026	\label{fig:min_100}
996	1027	\end{subfigure}
997		- \caption{\color{red}Solutions for the MIN/40, MIN/60, MIN/80 and MIN/100 problems of reaching a
998		-given rejection while minimizing resource allocation. The filter shape constraint (bandpass and
999		-bandstop) is shown as thick
	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
1000	1031	horizontal lines on each chart.}
1001	1032	\end{figure}
1002	1033
...	...	@@ -1074,16 +1105,16 @@ previous one.
1074	1105
1075	1106	{\color{red} % r1.4
1076	1107	To conclude, we compare our monolithic filters with the FIR Compiler provided by
1077		-Xilinx in the Vivado software suite (v.2018.2). For each experiment we use the
	1108	+Xilinx in the Vivado software suite (v.2018.2). For each experiment we use the
1078	1109	same coefficient set and we compare the resource consumption, having checked that
1079		-the transfer functions are indeed the same with both implementations.
	1110	+the transfer functions are indeed the same with both implementations.
1080	1111	Table~\ref{tbl:xilinx_resources} exhibits the results.
1081	1112	The FIR Compiler never use BRAM while our filter implementation uses one block. This difference
1082	1113	is explained be our wish to have a dynamically reconfigurable FIR filter whose
1083		-coefficients can be updated from the processing system without having to update the FPGA design.
	1114	+coefficients can be updated from the processing system without having to update the FPGA design.
1084	1115	With the FIR compiler, the coefficients are defined during the FPGA design so that
1085		-changing coefficients required generating a new design. The difference with the LUT consumption
1086		-is also attributed to the reconfigurability logic. However the DSP consumption, the scarcest
	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
1087	1118	resource, is the same between the Xilinx FIR Compiler end
1088	1119	our FIR block: we hence conclude that our solutions are as good as the Xilinx implementation.
1089	1120
...	...	@@ -74,7 +74,7 @@
74	74	%REVIEWERS' COMMENTS:
75	75
76	76	\documentclass[a4paper]{article}
77		-\usepackage{fullpage,graphicx,amsmath}
	77	+\usepackage{fullpage,graphicx,amsmath, subcaption}
78	78	\begin{document}
79	79	{\bf Reviewer: 1}
80	80
...	...	@@ -82,21 +82,21 @@
82	82	%In general, the language/grammar is adequate.
83	83
84	84	{\bf
85		-On page 2, "...allowing to save processing resource..." could be improved. % r1.1
	85	+On page 2, "...allowing to save processing resource..." could be improved. % r1.1 - fait
86	86	}
87	87
88	88	The sentence was split and now reads ``number of coefficients irrelevant: processing
89	89	resources are hence saved by shrinking the filter length.''
90	90
91	91	{\bf
92		-On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2
	92	+On page 2, "... or thanks at a radiofrequency-grade..." isn't at all clear what % r1.2 - fait
93	93	the author meant.}
94	94
95	95	Grammatical error: this sentence now reads ``or by sampling a wideband (125~MS/s)
96	96	Analog to Digital Converter (ADC) loaded by a 50~$\Omega$ resistor.''
97	97
98	98	{\bf
99		-On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3
	99	+On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 - fait
100	100	could be improved.
101	101	}
102	102
...	...	@@ -151,7 +151,7 @@ Reviewer: 2
151	151	%filters are really superior than monolithic filters.
152	152
153	153	{\bf
154		-By observing the results presented in fig. 10-16, it is clear that the % r2.1 - fait
	154	+By observing the results presented in fig. 10-16, it is clear that the % r2.1
155	155	performances of multi-stage filters are obtained at the expense of their
156	156	selectivity and, in this sense, the filters presented in these figures
157	157	are not equivalent. For example, in Fig. 14, at the limit of the pass band,
...	...	@@ -162,23 +162,61 @@ n = 1.
162	162	We have added on Figs 10--16 (now Fig 9(a)--(c)) the templates used to defined
163	163	the bandpass and the bandstop of the filter.
164	164
165		-We are aware of this non equivalence but we think that difference is not due to
166		-the cascaded filters but due to the definition of rejection criterion on the passband.
167		-Indeed, in this article we have choose to take the summation of absolute values divide
168		-by the bandwidth but this criterion is maybe too permissive and when we cascade
169		-some filters this impact is more important.
170		-
171		-However if we change the passband
172		-criterion by the summation of absolute value in passband, weighting given to the
173		-passband ripples are too strong and the solver are too restricted to provide
174		-any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in
175		-passband, the rejection evaluation are too close form the original criterion and
176		-the result will not be improved.
177		-
178		-In this article, we will highlight the methodology instead of the filter conception.
179		-Even if our rejection criterion is not the best, our methodology was not impacted
180		-by this. So to improve the results, we can choose another criterion to be more
181		-selective in passband but it is not the main objective of our article.
	165	+% We are aware of this non equivalence but we think that difference is not due to
	166	+% the cascaded filters but due to the definition of rejection criterion on the passband.
	167	+% Indeed, in this article we have choose to take the summation of absolute values divide
	168	+% by the bandwidth but this criterion is maybe too permissive and when we cascade
	169	+% some filters this impact is more important.
	170	+%
	171	+% However if we change the passband
	172	+% criterion by the summation of absolute value in passband, weighting given to the
	173	+% passband ripples are too strong and the solver are too restricted to provide
	174	+% any interesting solution but the ripples in passband will be minimal. And if we take the maximum absolute value in
	175	+% passband, the rejection evaluation are too close form the original criterion and
	176	+% the result will not be improved.
	177	+%
	178	+% In this article, we will highlight the methodology instead of the filter conception.
	179	+% Even if our rejection criterion is not the best, our methodology was not impacted
	180	+% by this. So to improve the results, we can choose another criterion to be more
	181	+% selective in passband but it is not the main objective of our article.
	182	+
	183	+We are aware of this equivalence but to limit this ripples in passband we need to
	184	+enforce the criterion in passband. If we takes a strong constraint like the sum of
	185	+absolute values in passband. This criterion si too selective because it considers
	186	+all bin on passband while on stopband we consider only the bin with the minimal
	187	+rejection. The figure~\ref{fig:letter_sum_criterion} exhibits the results with this
	188	+criterion for the case MAX/1000. With this criterion, the solver find an optimal
	189	+solution with only two filters in expend of the resource consumption.
	190	+
	191	+
	192	+
	193	+If we relax a little the criterion on passband with taking only the maximum absolute
	194	+value, we will penalize the ripple peak on passband. The figure~\ref{fig:letter_max_criterion}
	195	+shows the results for the case MAX/1000. There as almost no difference with the
	196	+article results. Indeed the only little change are on the case $i = 4$ and $i = 5$
	197	+which they have some minor differences on coefficients choices.
	198	+
	199	+\begin{figure}[h!tb]
	200	+ \centering
	201	+ \begin{subfigure}{0.48\linewidth}
	202	+ \includegraphics[width=\linewidth]{images/letter_sum_criterion}
	203	+ \caption{Results for the case MAX/1000 with as criterion on passband the sum absolute values}
	204	+ \label{fig:letter_sum_criterion}
	205	+ \end{subfigure}
	206	+ \begin{subfigure}{0.48\linewidth}
	207	+ \includegraphics[width=\linewidth]{images/letter_max_criterion}
	208	+ \caption{Results for the case MAX/1000 with as criterion on passband the maximum absolute value}
	209	+ \label{fig:letter_max_criterion}
	210	+ \end{subfigure}
	211	+\end{figure}
	212	+
	213	+Finally, if we ponder the maximum absolute on passband, we should improve the result.
	214	+We have arbitrary pondered by 5 the maximum. Even with this weighting, the solver
	215	+choose the same coefficient set.
	216	+
	217	+To conclude, find a better criterion to avoid the ripples on the passband is difficult.
	218	+In this article we are focused on the methodology so even if our criterion could
	219	+be improved, our methodology still the same and it works independently of rejection criterion.
182	220
183	221	% %Peut etre refaire une serie de simulation dans lesquelles on impose une coupure
184	222	% %non pas entre 40 et 60\% mais entre 50 et 60\% pour demontrer que l'outil s'adapte
...	...	@@ -197,7 +235,7 @@ selective in passband but it is not the main objective of our article.
197	235	% Dire que la chute n'est pas du à la casacade mais à notre critère de rejection
198	236
199	237	{\bf
200		-The reason is in the criterion that considers the average attenuation in % r2.2 - fait
	238	+The reason is in the criterion that considers the average attenuation in % r2.2
201	239	the pass band. This criterion does not take into account the maximum attenuation
202	240	in this region, which is a very important parameter for specifying a filter
203	241	and for evaluating its performance. For example, with this criterion, a
...	...	@@ -206,8 +244,9 @@ filter with 0.1 dB of ripple is considered equivalent to a filter with
206	244	and in the results that are obtained and has to be reconsidered.
207	245	}
208	246
209		-See above: If we choose the maximum absolute value in passband, we penalize the
210		-case with 10 dB of ripple.
	247	+See above: Choose a criterion is difficult and depending on the context. The main
	248	+contribution on this paper is the methodology not the criterion to quantify the
	249	+rejection.
211	250
212	251	% The manuscript erroneously stated that we considered the mean of the absolute
213	252	% value within the bandpass: the manuscript has now been corrected to properly state
...	...	@@ -231,7 +270,7 @@ excessive ripples, including excessive attenuation, within the passband.
231	270	% TODO: test max(stopband) - max(abs(passband))
232	271
233	272	{\bf
234		-In addition, I suggest to address the following points: % r2.4
	273	+In addition, I suggest to address the following points: % r2.4 - fait
235	274	- Page 1, line 50: the Authors state that IIR have shorter impulse response
236	275	than FIR. This is not true in general. The sentence should be reconsidered.
237	276	}
...	...	@@ -248,7 +287,7 @@ is not considered as an issue as would be in a closed loop system in which lag a
248	287	minimized to avoid oscillation conditions.''
249	288
250	289	{\bf
251		-- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5
	290	+- Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 - fait
252	291	this transition bandwidth and if it is a typical requirement for phase-noise
253	292	metrology.
254	293	}
...	...	@@ -279,13 +318,16 @@ hardware.'' so indeed the input datastream resolution is considered as a given.
279	318	{\bf
280	319	- Page 3, line 47: the initial criterion can be omitted and, consequently, % r2.7 - fait
281	320	Fig. 5 can be removed.
	321	+}
	322	+
	323	+Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats
	324	+
	325	+{\bf
282	326	- Page 3, line 55: ``maximum rejection'' is not compatible with fig. 4. % r2.8 - fait
283	327	It should be ``minimum''
284	328	}
285		-AH: Je ne suis pas d'accord, le critère n'est pas le min de la rejection mais le max
286		-de la magnitude. J'ai corrigé en ce sens.
287	329
288		-Juste mettre une phrase pour dire que la mean ne donnait pas de bons résultats
	330	+This typo has been corrected.
289	331
290	332	{\bf
291	333	- Page e, line 55, second column: ``takin'' % r2.9 - fait
...	...	@@ -309,7 +351,7 @@ set of coefficients generated (for memory, we use \texttt{firls} and \texttt{fir
309	351	functions from GNU Octave)''
310	352
311	353	{\bf
312		-- Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait
	354	+- Page 4, line 31: how does the delta function transform model from non-linear % r2.13
313	355	and non-quadratic to a quadratic?}
314	356
315	357	The first model is non-quadratic but when we introduce the $p$ configurations,
...	...	@@ -335,22 +377,22 @@ Gurobi does the linearization so we don't explain this step to keep the model mo
335	377	simple. However, to improve the transformation explanation we have rewrote the
336	378	paragraph ``This model is non-linear and even non-quadratic...''.
337	379
338		-JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
339		-je ne comprends pas comment ca repond a la question
340		-
341		-AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
342		-
343		-Le problème n'est pas linéaire car nous multiplions des variables
344		-entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
345		-des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
346		-quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
347		-quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
348		-$\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
349		-de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
350		-cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
351		-nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
352		-qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
353		-linéarisation pour nous.
	380	+% JMF : il faudra mettre une phrase qui explique, ca en lisant cette reponse dans l'article
	381	+% je ne comprends pas comment ca repond a la question
	382	+%
	383	+% AH: Je mets l'idée en français, je vais essayer de traduire ça au mieux.
	384	+%
	385	+% Le problème n'est pas linéaire car nous multiplions des variables
	386	+% entre elles. Pour y remédier, on considère que $\pi_{ij}^C$ et que $C_{ij}$ deviennent
	387	+% des constantes. On introduit donc la variable binaire $\delta_{ij}$ qui nous indique
	388	+% quel filtre est sélectionné étage par étage. Malgré cela, notre programme est encore
	389	+% quadratique car pour la contrainte~\ref{eq:areadef2}, il reste une multiplication entre
	390	+% $\delta_{ij}$ et $\pi_i^-$. Mais comme $\delta_{ij}$ est binaire, il est possible
	391	+% de linéariser cette multiplication pour peu qu'on puisse borner $\pi_i^-$. Dans notre
	392	+% cas définir la borne est facile car $\pi_i^-$ représente une taille de donnée,
	393	+% nous définission donc $0 < \pi_i^- \leq 128$ car il s'agit de la plus grande valeur
	394	+% qu'on puisse traiter. De plus nous utiliserons Gurobi qui se chargera de faire la
	395	+% linéarisation pour nous.
354	396
355	397
356	398	{\bf