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

jfriedt
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ifcs2018_journal.tex
images/cascaded_criterion.pdf
images/colored_custom_criterion.pdf
images/colored_mean_criterion.pdf
images/criterion_cascaded.pdf
images/max_1000.pdf
images/max_1500.pdf
images/max_500.pdf
images/min_40.pdf
images/min_60.pdf
images/min_80.pdf
images/zero_values.pdf
@@ -111,7 +111,7 @@
 will result in some precision loss.
  
 \begin{figure}[h!tb]
-\includegraphics[width=\linewidth]{images/demo_filtre}
+\includegraphics[width=\linewidth]{images/zero_values}
 \caption{Impact of the quantization resolution of the coefficients: the quantization is
 set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting
 the 30~first and 30~last coefficients out of the initial 128~band-pass
  
@@ -251,14 +251,14 @@
  
 \begin{figure}
 \centering
-\includegraphics[width=\linewidth]{images/mean_criterion}
+\includegraphics[width=\linewidth]{images/colored_mean_criterion}
 \caption{Mean criterion comparison between monolithic filter and cascade filters}
 \label{fig:mean_criterion}
 \end{figure}
  
 \begin{figure}
 \centering
-\includegraphics[width=\linewidth]{images/custom_criterion}
+\includegraphics[width=\linewidth]{images/colored_custom_criterion}
 \caption{Custom criterion comparison between monolithic filter and cascade filters}
 \label{fig:custom_criterion}
 \end{figure}
  
@@ -278,11 +278,16 @@
  
 \begin{figure}
 \centering
-\includegraphics[width=\linewidth]{images/sum_rejection}
+\includegraphics[width=\linewidth]{images/cascaded_criterion}
 \caption{Rejection of two cascaded filters}
 \label{fig:sum_rejection}
 \end{figure}
  
+The first problem we address is to maximize the rejection under bounded silicon area
+and feasibility constraints. Variable $a_i$ is the area taken by filter~$i$
+(in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB).
+Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
+
 Finally we can describe our abstract model with following expressions :
 \begin{align}
 \text{Maximize } & \sum_{i=1}^n r_i  \notag \\
@@ -295,10 +300,6 @@
 \pi_1^- &= \Pi^I \label{eq:init}
 \end{align}
  
-{\color{red} Je sais que l'idée est de ne pas parler du programme linéaire mais
-ça me semble quand même indispensable. Au pire, j'essaierai de revoir ça si on
-est vraiment en manque de place.}
-
 Equation~\ref{eq:area} states that the total area taken by the filters must be
 less than the available area. Equation~\ref{eq:areadef} gives the definition of
 the area for a filter. More precisely, it is the area of the FIR as the Shifter
@@ -324,9 +325,9 @@
  
 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. 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:
+$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. 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} \\
@@ -339,7 +340,12 @@
 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.
  
-The next section shows the results for this quadratic program but the section~\ref{sec:fixed_rej}
+This modified model is quadratic, and 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.
+
+The section~\ref{sec:fixed_area} shows the results for the first version of quadratic program but the section~\ref{sec:fixed_rej}
 presents the results for the complementary problem. In this case we want
 minimize the occupied area for a targeted rejection level. Hence we have replace
 the objective function with:
...	...	@@ -111,7 +111,7 @@
111	111	will result in some precision loss.
112	112
113	113	\begin{figure}[h!tb]
114		-\includegraphics[width=\linewidth]{images/demo_filtre}
	114	+\includegraphics[width=\linewidth]{images/zero_values}
115	115	\caption{Impact of the quantization resolution of the coefficients: the quantization is
116	116	set to 6~bits -- with the horizontal black lines indicating $\pm$1 least significant bit -- setting
117	117	the 30~first and 30~last coefficients out of the initial 128~band-pass
118	118
...	...	@@ -251,14 +251,14 @@
251	251
252	252	\begin{figure}
253	253	\centering
254		-\includegraphics[width=\linewidth]{images/mean_criterion}
	254	+\includegraphics[width=\linewidth]{images/colored_mean_criterion}
255	255	\caption{Mean criterion comparison between monolithic filter and cascade filters}
256	256	\label{fig:mean_criterion}
257	257	\end{figure}
258	258
259	259	\begin{figure}
260	260	\centering
261		-\includegraphics[width=\linewidth]{images/custom_criterion}
	261	+\includegraphics[width=\linewidth]{images/colored_custom_criterion}
262	262	\caption{Custom criterion comparison between monolithic filter and cascade filters}
263	263	\label{fig:custom_criterion}
264	264	\end{figure}
265	265
...	...	@@ -278,11 +278,16 @@
278	278
279	279	\begin{figure}
280	280	\centering
281		-\includegraphics[width=\linewidth]{images/sum_rejection}
	281	+\includegraphics[width=\linewidth]{images/cascaded_criterion}
282	282	\caption{Rejection of two cascaded filters}
283	283	\label{fig:sum_rejection}
284	284	\end{figure}
285	285
	286	+The first problem we address is to maximize the rejection under bounded silicon area
	287	+and feasibility constraints. Variable $a_i$ is the area taken by filter~$i$
	288	+(in arbitrary unit). Variable $r_i$ is the rejection of filter~$i$ (in dB).
	289	+Constant $\mathcal{A}$ is the total available area. We model our problem as follows:
	290	+
286	291	Finally we can describe our abstract model with following expressions :
287	292	\begin{align}
288	293	\text{Maximize } & \sum_{i=1}^n r_i \notag \\
...	...	@@ -295,10 +300,6 @@
295	300	\pi_1^- &= \Pi^I \label{eq:init}
296	301	\end{align}
297	302
298		-{\color{red} Je sais que l'idée est de ne pas parler du programme linéaire mais
299		-ça me semble quand même indispensable. Au pire, j'essaierai de revoir ça si on
300		-est vraiment en manque de place.}
301		-
302	303	Equation~\ref{eq:area} states that the total area taken by the filters must be
303	304	less than the available area. Equation~\ref{eq:areadef} gives the definition of
304	305	the area for a filter. More precisely, it is the area of the FIR as the Shifter
...	...	@@ -324,9 +325,9 @@
324	325
325	326	This model is non-linear and even non-quadratic, as $F$ does not have a known
326	327	linear or quadratic expression. We introduce $p$ FIR configurations
327		- $(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. We define binary
328		- variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
329		- and 0 otherwise. The new equations are as follows:
	328	+$(C_{ij}, \pi_{ij}^C), 1 \leq j \leq p$ that are constants. We define binary
	329	+variable $\delta_{ij}$ that has value 1 if stage~$i$ is in configuration~$j$
	330	+and 0 otherwise. The new equations are as follows:
330	331
331	332	\begin{align}
332	333	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} \\
...	...	@@ -339,7 +340,12 @@
339	340	respectively equations \ref{eq:areadef}, \ref{eq:rejectiondef} and \ref{eq:bits}.
340	341	Equation~\ref{eq:config} states that for each stage, a single configuration is chosen at most.
341	342
342		-The next section shows the results for this quadratic program but the section~\ref{sec:fixed_rej}
	343	+This modified model is quadratic, and it can be linearised if necessary. The Gurobi
	344	+(\url{www.gurobi.com}) optimization software is used to solve this quadratic
	345	+model, and since Gurobi is able to linearize, the model is left as is. This model
	346	+has $O(np)$ variables and $O(n)$ constraints.
	347	+
	348	+The section~\ref{sec:fixed_area} shows the results for the first version of quadratic program but the section~\ref{sec:fixed_rej}
343	349	presents the results for the complementary problem. In this case we want
344	350	minimize the occupied area for a targeted rejection level. Hence we have replace
345	351	the objective function with: