From 7c78647f195243fda50117a36806d268ded209af Mon Sep 17 00:00:00 2001 From: Arthur HUGEAT Date: Tue, 30 Jul 2019 13:01:23 +0200 Subject: [PATCH] Ajout de correction. --- ifcs2018_journal.tex | 55 +++++++++++++++++++++++++++------ ifcs2018_journal_reponse.tex | 73 +++++++++++++++++++++++++++++++++++--------- 2 files changed, 104 insertions(+), 24 deletions(-) diff --git a/ifcs2018_journal.tex b/ifcs2018_journal.tex index 7e87394..37341f9 100644 --- a/ifcs2018_journal.tex +++ b/ifcs2018_journal.tex @@ -405,16 +405,18 @@ shift bit would cause an additional 6~dB rejection rise. A totally equivalent eq $\pi_i^S \leq \pi_i^- + \pi_i^C - 1 - \sum_{k=1}^{i} \left(1 + \frac{r_j}{6}\right)$. Finally, equation~\ref{eq:init} gives the number of bits of the global input. -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 -$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. -% r2.12 +{\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 +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). -We define binary +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 +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: +} \begin{align} a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\ @@ -427,14 +429,47 @@ Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace 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. -% r2.13 -This modified model is quadratic since we multiply two variables in the -equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary. -The Gurobi +{\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. +} +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 is non-linear and even non-quadratic, as $F$ does not have a known +% linear or quadratic expression. We introduce $p$ FIR configurations +% $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. +% % r2.12 +% 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). +% We define binary +% variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$ +% and 0 otherwise. The new equations are as follows: +% +% \begin{align} +% a_i & = \sum_{j=1}^p \delta_{ij} \times C_{ij} \times (\pi_{ij}^C + \pi_i^-), & \forall i \in [1, n] \label{eq:areadef2} \\ +% r_i & = \sum_{j=1}^p \delta_{ij} \times F(C_{ij}, \pi_{ij}^C), & \forall i \in [1, n] \label{eq:rejectiondef2} \\ +% \pi_i^+ & = \pi_i^- + \left(\sum_{j=1}^p \delta_{ij} \pi_{ij}^C\right) - \pi_i^S, & \forall i \in [1, n] \label{eq:bits2} \\ +% \sum_{j=1}^p \delta_{ij} & \leq 1, & \forall i \in [1, n] \label{eq:config} +% \end{align} +% +% Equations \ref{eq:areadef2}, \ref{eq:rejectiondef2} and \ref{eq:bits2} replace +% 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. +% +% % r2.13 +% This modified model is quadratic since we multiply two variables in the +% equation~\ref{eq:areadef2} ($\delta_{ij}$ by $\pi_{ij}^-$) but it can be linearised if necessary. +% 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. + Two problems will be addressed using the workflow described in the next section: on the one hand maximizing the rejection capability of a set of cascaded filters occupying a fixed arbitrary silcon area (section~\ref{sec:fixed_area}) and on the second hand the dual problem of minimizing the silicon area diff --git a/ifcs2018_journal_reponse.tex b/ifcs2018_journal_reponse.tex index bdc02cd..c914eef 100644 --- a/ifcs2018_journal_reponse.tex +++ b/ifcs2018_journal_reponse.tex @@ -74,7 +74,7 @@ %REVIEWERS' COMMENTS: \documentclass[a4paper]{article} -\usepackage{fullpage,graphicx} +\usepackage{fullpage,graphicx,amsmath} \begin{document} {\bf Reviewer: 1} @@ -85,7 +85,7 @@ On page 2, "...allowing to save processing resource..." could be improved. % r1.1 } -The sentence was split and now reads ``number of coefficients irrelevant: processing +The sentence was split and now reads ``number of coefficients irrelevant: processing resources are hence saved by shrinking the filter length.'' {\bf @@ -95,7 +95,7 @@ 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 +{\bf On page 2, the whole paragraph "The first step of our approach is to model..." % r1.3 could be improved. } @@ -103,7 +103,7 @@ could be improved. Indeed this paragraph has be written again and now reads as\\ ``The first step of our approach is to model the DSP chain. Since we aim at only optimizing the filtering part of the signal processing chain, we have not included the PRN generator or the -ADC in the model: the input data size and rate are considered fixed and defined by the hardware. +ADC in the model: the input data size and rate are considered fixed and defined by the hardware. The filtering can be done in two ways, either by considering a single monolithic FIR filter requiring many coefficients to reach the targeted noise rejection ratio, or by cascading multiple FIR filters, each with fewer coefficients than found in the monolithic filter. @@ -197,8 +197,8 @@ fixing its value to a typical one, as it has been done for the transition bandwidth. } -See above: the absolute value within the passband will reject filters with -excessive ripples, including excessive attenuation, within the passband. +See above: the absolute value within the passband will reject filters with +excessive ripples, including excessive attenuation, within the passband. {\bf In addition, I suggest to address the following points: % r2.4 @@ -211,12 +211,11 @@ Indeed while a typical FIR filter will have 32 to 128~coefficients, few IIR filt have more than 5~coefficients. Hence, while a FIR requires 128 inputs before providing the first output, an IIR will start providing outputs only 5 time steps after the initial input starts feeding the IIR. Hence, the issue we address here is lag and not impulse -response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue +response. We aimed at making this sentence clearer by stating that ``Since latency is not an issue in a openloop phase noise characterization instrument, the large numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, is not considered as an issue as would be in a closed loop system in which lag aims at being -minimized to avoid oscillation conditions. -'' +minimized to avoid oscillation conditions.'' {\bf - Fig. 4: the Author should motivate in the text why it has been chosen % r2.5 @@ -243,8 +242,8 @@ used in the analysis? If not, how is it changed with respect to the coefficient resolution? } -We have now stated in the beginning of the document that ``we have not included the PRN generator -or the ADC in the model: the input data size and rate are considered fixed and defined by the +We have now stated in the beginning of the document that ``we have not included the PRN generator +or the ADC in the model: the input data size and rate are considered fixed and defined by the hardware.'' so indeed the input datastream resolution is considered as a given. {\bf @@ -268,21 +267,67 @@ All typos and grammatical errors have been corrected. - Page 4, line 10: how $p$ is chosen? Which is the criterion used to choose % r2.12 - fait these particular configurations? Are they chosen automatically? } - -JMF : repondre +See below: we have added a better description of $p$ during the transformation explanation. +``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)'' {\bf - Page 4, line 31: how does the delta function transform model from non-linear % r2.13 - fait and non-quadratic to a quadratic?} +The first model is non-quadratic but when we introduce the $p$ configurations, +we can estimate the function $F$ by computing +the rejection for each configuration, so the model become quadratic because we have +some multiplication between variables. With the definition of $\delta_{ij}$ we can +replace the multiplication between variables by multiplication with binary variable and +this one can be linearise as follow:\\ +$y$ is a binary variable \\ +$x$ is a real variable bounded by $X^{max}$ \\ +\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*} +Gurobi does the linearization so we don't explain this step to keep the model more +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. + + {\bf - Captions of figure and tables are too minimal. % r2.14 +} +We have change the captions of fig 10-16. + +{\bf - Figures can be grouped: fig. 10-12 can be grouped as three subplots (a, b, c) % r2.15 - fait of a single figure. Same for fig. 13-16. } +We add two sub figure to group the fig.10-12 and fig. 13-16 {\bf - Please increase the number of averages for the spectrum. Currently the noise % r2.16 - fait @@ -290,7 +335,7 @@ of the curves is about 20 dBpk-pk and it doesn’t allow to appreciate the differences among the curves. I suggest to reduce the noise below 1 dBpk-pk. } -Indeed averaging had been omitted during post-processing and figure generation: we +Indeed averaging had been omitted during post-processing and figure generation: we are grateful to the reviewer for emphasizing this point which has now been corrected. All spectra now exhibit sub-dBpk-pl line thickness. -- 2.16.4