From 48d886be993bab8469e9e4f038410adc9e58893d Mon Sep 17 00:00:00 2001
From: Arthur HUGEAT <arthur.hugeat@femto-st.fr>
Date: Mon, 21 May 2018 01:33:13 +0200
Subject: [PATCH] Correction des notations.

---
 ifcs2018_proceeding.tex | 25 ++++++++++++-------------
 1 file changed, 12 insertions(+), 13 deletions(-)

diff --git a/ifcs2018_proceeding.tex b/ifcs2018_proceeding.tex
index da14b26..2021afe 100644
--- a/ifcs2018_proceeding.tex
+++ b/ifcs2018_proceeding.tex
@@ -166,11 +166,10 @@ computational resources: optimizing some criteria within finite, limited
 resources indeed matches the definition of a classical optimization problem.
 
 Specifically the degrees of freedom when addressing the problem of replacing the single monolithic
-FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$,
-the number of bits $C_i$ representing the coefficients and the number of bits $D_i$ representing
-the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage,
+FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$ and
+the number of bits $C_i$ representing the coefficients. Because each FIR in the chain is fed the output of the previous stage,
 the optimization of the complete processing chain within a constrained resource environment is not
-trivial. The resource occupation of a FIR filter is considered as $(D_i+C_i) \times N_i$ which is
+trivial. The resource occupation of a FIR filter is considered as $C_i \times N_i$ which is
 the number of bits needed in a worst case condition to represent the output of the FIR. Such an
 occupied area estimate assumes that the number of gates scales as the number of bits and the number
 of coefficients, but does not account for the detailed implementation of the hardware. Indeed,
@@ -199,7 +198,7 @@ Following these considerations, the model is expressed as:
 \begin{align}
   \begin{cases}
     \mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\
-    \mathcal{A}_i &= N_i * C_i + D_i\\
+    \mathcal{A}_i &= N_i * C_i\\
     \Delta_i &= \Delta _{i-1} + \mathcal{P}_i
   \end{cases}
   \label{model-FIR}
@@ -248,13 +247,13 @@ x_{i,j} \in \lbrace 0,1 \rbrace & \text{ $i$ is a given filter} \\
 \begin{align*}
 \mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\
 & \text{ $p$ is the number of different filters} \\
-C(i) & \text{ % Constant to let the
-number of coefficients %} \\ & \text{
-for filter $i$}\\
-\pi_C(i) & \text{ % Constant to let the
-number of bits of %}\\ & \text{
-each coefficient for filter $i$}\\
-\mathcal{A}_{\max} & \text{ Total space available inside the FPGA}
+% N(i) & \text{ % Constant to let the
+% number of coefficients %} \\ & \text{
+% for filter $i$}\\
+% C(i) & \text{ % Constant to let the
+% number of bits of %}\\ & \text{
+% each coefficient for filter $i$}\\
+\mathcal{S}_{\max} & \text{ Total space available inside the FPGA}
 \end{align*}
 \paragraph{Constraints}
 \begin{align}
@@ -284,7 +283,7 @@ plus the rejection of selected filter.
 
 The MILP solver provides a solution to the problem by selecting a series of small FIR with
 increasing number of bits representing data and coefficients as well as an increasing number
-of coefficients, instead of a single monolithic filter. 
+of coefficients, instead of a single monolithic filter.
 
 \begin{figure}[h!tb]
 % \includegraphics[width=\linewidth]{images/compare-fir.pdf}
-- 
2.16.4