diff --git a/ifcs2018_journal.tex b/ifcs2018_journal.tex index a44b624..6afeb55 100644 --- a/ifcs2018_journal.tex +++ b/ifcs2018_journal.tex @@ -111,7 +111,7 @@ we select to quantify these floating point values into integer values. This quan 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 @@ A second criterion considers the maximum rejection within the stopband minus the \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 @@ better rejection than expected. \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 @@ r_i & = F(C_i, \pi_i^C), & \forall i \in [1, n] \label{eq:rejectiondef} \\ \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 @@ Finally, equation~\ref{eq:init} gives the global input's number of bits. 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 @@ 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. -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. 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