From 9c253d6d247e44f3801c0463afe289f421ab5133 Mon Sep 17 00:00:00 2001 From: Arthur HUGEAT Date: Fri, 2 Aug 2019 12:30:22 +0200 Subject: [PATCH] =?UTF-8?q?Correction=20sur=20le=20crit=C3=A8re=20de=20sel?= =?UTF-8?q?ection.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ifcs2018_journal.tex | 33 +++++++--- ifcs2018_journal_reponse.tex | 136 ++++++++++++++++++++++++++-------------- images/letter_max_criterion.pdf | Bin 0 -> 35402 bytes images/letter_sum_criterion.pdf | Bin 0 -> 17125 bytes 4 files changed, 113 insertions(+), 56 deletions(-) create mode 100644 images/letter_max_criterion.pdf create mode 100644 images/letter_sum_criterion.pdf diff --git a/ifcs2018_journal.tex b/ifcs2018_journal.tex index bcd8d3d..204c082 100644 --- a/ifcs2018_journal.tex +++ b/ifcs2018_journal.tex @@ -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}. With this criterion, we meet the expected rejection capability of low pass filters as shown in figure~\ref{fig:custom_criterion}. @@ -411,8 +411,11 @@ and it is even non-quadratic, as $F$ 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). So $C_{ij}$ and $\pi_{ij}^C$ become constant and -we defined $1 \leq j \leq p$ and the function $F$ can be estimate for each configurations +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 variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ and 0 otherwise. The new equations are as follows: @@ -432,13 +435,25 @@ Equation~\ref{eq:config} states that for each stage, a single configuration is c {\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 -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 maximal data size that we can process. +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 to represent the maximum data size tolerated, +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. } -Moreover the Gurobi -(\url{www.gurobi.com}) optimization software is used to solve this quadratic -model, and since Gurobi is able to linearize, the model is left as is. This model -has $O(np)$ variables and $O(n)$ constraints. +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 diff --git a/ifcs2018_journal_reponse.tex b/ifcs2018_journal_reponse.tex index bb55f51..77fbc18 100644 --- a/ifcs2018_journal_reponse.tex +++ b/ifcs2018_journal_reponse.tex @@ -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.'' 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'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 diff --git a/images/letter_max_criterion.pdf b/images/letter_max_criterion.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ed87a4917a169fcae68f2556a3e5a76da3c1a5c4 GIT binary patch literal 35402 zcmXVWbzIZm_rHOFfaFN0G-Gu08YvPZhcFOOLOK;j2}ryJsZkP24Un94ml6U)dPqqr zF*+2G?(aUo$M2tW?>(=bdfn$a=k9L2Pt-Jo#YJQ(czd=J@+hP^#5mk-TqqP2M0M?7 zIeI&B$PhV(q8iR_-u9jxq8e`2-u7ztw(fTJN=g)7-k$c>FDd*mv6ni-J1q^oM~BB) z)1lV;j_RNti`x-5Z{MXozu`Rfip57Y)JNd6b}Yx~!D5Twhf_ zUly5JuyViNy54W}5`MXpd{%kAr+o4HI=r?w#W(T{U6XV_vn93&LtVo?99l% z`nLabez+H8{-sgia=Lx~YPugfUp{y2+&=JH5PLD@^5^p0dA>X)wEQZ;VNIz04g1n0 zru{x%re;6q%0N)`U`O%l8X|huB7A<_?7T9md_9}$$_L;Nk1K;o75+(;nr+`JJ-0 z$5%Pm{KGqQGZ*I}!~nA(j^oo;SEAQ|Z14xOtE=|akgNR!-|R!<{`u>j)5`04f*?Dc7- zRrBxbi{m#IvbTv_1^gj;*tZA6pSI<|!)Us_9-{wkUGK?`Q#w|z{VH2Kc3TT~GI*E$ z`l!5pE_tzCU)(GEosWnkCgoV+gvJuzif3pV9zZ_747xflz;G3{oEL0+Jt-=iudUn% 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