biblio en majuscules et description moyenne/median

jfriedt
2 parents 33bcbbbcd8 bee7a1f72b
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ifcs2018_proceeding.tex
@@ -138,7 +138,7 @@
 follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
 in which basic blocks are defined and characterized before being assembled \cite{hide}
 in a complete processing chain. In our case, assembling the filter blocks is a simpler block
-combination process since we assume a single value to be processed and a single value to be 
+combination process since we assume a single value to be processed and a single value to be
 generated at each clock cycle. The FIR filters will not be considered to decimate in the
 current implementation: the decimation is assumed to be located after the FIR cascade at the
 moment.
@@ -170,7 +170,7 @@
 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,
 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 $(D_i+C_i) \times N_i$ 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,
@@ -205,7 +205,7 @@
   \label{model-FIR}
 \end{align}
 To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the rejection of depending on $N_i$ and $C_i$, $\mathcal{A}$
-is a theoretical area occupation of the processing block on the FPGA, and $\Delta_i$ is the total rejection for the current stage $i$. 
+is a theoretical area occupation of the processing block on the FPGA, and $\Delta_i$ is the total rejection for the current stage $i$.
 Since the function $\mathcal{F}$ cannot be explictly expressed, we run simulations to determine the rejection depending
 on $N_i$ and $C_i$. However, selecting the right filter requires a clear definition of the rejection criterion. Selecting an  
 incorrect criterion will lead the linear program solver to produce a solution which might not meet the user requirements. 
  
@@ -248,11 +248,11 @@
 \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{ 
+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{ 
+\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}
 \end{align*}
  
@@ -262,11 +262,11 @@
 1 \leq j \leq q & \text{ $q$ is the max of filter stage} \nonumber \\
 \forall j, \mathlarger{\sum_{i}} x_{i,j} = 1 & \text{ At most one filter by stage} \nonumber\\
 \mathcal{S}_0 = 0 & \text{ initial occupation} \nonumber\\
-\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \forall i, x_{i,j} \times \mathcal{A}_i \label{cstr_size} \
+\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{A}_i)} \label{cstr_size} \
 \mathcal{S} \leq \mathcal{S}_{\max}\nonumber \\
 \mathcal{N}_0 = 0 & \text{ initial rejection}\nonumber\\
-\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \forall i, x_{i,j} \times \mathcal{R}_i \label{cstr_rejection} \\
-\mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\ 
+\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{R}_i)} \label{cstr_rejection} \\
+\mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\
 & \text{ (e.g. 160~dB here)}\nonumber\\\nonumber
 \end{align}
 \paragraph{Goal}
@@ -274,7 +274,6 @@
 \min \mathcal{S}_q
 \end{align*}
  
-% AH j'aimerai mettre deux equations avec un label mais je ne sais pas comment faire
 The constraint \ref{cstr_size} means the occupation for the current stage $j$ depends on
 the previous occupation and the occupation of current selected filter (it is possible
 that no filter is selected for this stage). And the second one \ref{cstr_rejection}
@@ -359,7 +358,7 @@
 The coefficients of a single monolithic filter are computed as the impulse response
 of the filter transfer function, and practically approximated by a multitude of methods
 including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing
-(Matlab's {\tt fir1} function). 
+(Matlab's {\tt fir1} function).
  
 \begin{figure}[h!tb]
 \includegraphics[width=\linewidth]{images/fir1-vs-firls}
...	...	@@ -138,7 +138,7 @@
138	138	follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
139	139	in which basic blocks are defined and characterized before being assembled \cite{hide}
140	140	in a complete processing chain. In our case, assembling the filter blocks is a simpler block
141		-combination process since we assume a single value to be processed and a single value to be
	141	+combination process since we assume a single value to be processed and a single value to be
142	142	generated at each clock cycle. The FIR filters will not be considered to decimate in the
143	143	current implementation: the decimation is assumed to be located after the FIR cascade at the
144	144	moment.
...	...	@@ -170,7 +170,7 @@
170	170	the number of bits $C_i$ representing the coefficients and the number of bits $D_i$ representing
171	171	the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage,
172	172	the optimization of the complete processing chain within a constrained resource environment is not
173		-trivial. The resource occupation of a FIR filter is considered as $D_i+C_i \times N_i)$ which is
	173	+trivial. The resource occupation of a FIR filter is considered as $(D_i+C_i) \times N_i$ which is
174	174	the number of bits needed in a worst case condition to represent the output of the FIR. Such an
175	175	occupied area estimate assumes that the number of gates scales as the number of bits and the number
176	176	of coefficients, but does not account for the detailed implementation of the hardware. Indeed,
...	...	@@ -205,7 +205,7 @@
205	205	\label{model-FIR}
206	206	\end{align}
207	207	To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the rejection of depending on $N_i$ and $C_i$, $\mathcal{A}$
208		-is a theoretical area occupation of the processing block on the FPGA, and $\Delta_i$ is the total rejection for the current stage $i$.
	208	+is a theoretical area occupation of the processing block on the FPGA, and $\Delta_i$ is the total rejection for the current stage $i$.
209	209	Since the function $\mathcal{F}$ cannot be explictly expressed, we run simulations to determine the rejection depending
210	210	on $N_i$ and $C_i$. However, selecting the right filter requires a clear definition of the rejection criterion. Selecting an
211	211	incorrect criterion will lead the linear program solver to produce a solution which might not meet the user requirements.
212	212
...	...	@@ -248,11 +248,11 @@
248	248	\begin{align*}
249	249	\mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\
250	250	& \text{ $p$ is the number of different filters} \\
251		-C(i) & \text{ % Constant to let the
252		-number of coefficients %} \\ & \text{
	251	+C(i) & \text{ % Constant to let the
	252	+number of coefficients %} \\ & \text{
253	253	for filter $i$}\\
254		-\pi_C(i) & \text{ % Constant to let the
255		-number of bits of %}\\ & \text{
	254	+\pi_C(i) & \text{ % Constant to let the
	255	+number of bits of %}\\ & \text{
256	256	each coefficient for filter $i$}\\
257	257	\mathcal{A}_{\max} & \text{ Total space available inside the FPGA}
258	258	\end{align*}
259	259
...	...	@@ -262,11 +262,11 @@
262	262	1 \leq j \leq q & \text{ $q$ is the max of filter stage} \nonumber \\
263	263	\forall j, \mathlarger{\sum_{i}} x_{i,j} = 1 & \text{ At most one filter by stage} \nonumber\\
264	264	\mathcal{S}_0 = 0 & \text{ initial occupation} \nonumber\\
265		-\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \forall i, x_{i,j} \times \mathcal{A}_i \label{cstr_size} \
	265	+\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{A}_i)} \label{cstr_size} \
266	266	\mathcal{S} \leq \mathcal{S}_{\max}\nonumber \\
267	267	\mathcal{N}_0 = 0 & \text{ initial rejection}\nonumber\\
268		-\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \forall i, x_{i,j} \times \mathcal{R}_i \label{cstr_rejection} \\
269		-\mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\
	268	+\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{R}_i)} \label{cstr_rejection} \\
	269	+\mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\
270	270	& \text{ (e.g. 160~dB here)}\nonumber\\\nonumber
271	271	\end{align}
272	272	\paragraph{Goal}
...	...	@@ -274,7 +274,6 @@
274	274	\min \mathcal{S}_q
275	275	\end{align*}
276	276
277		-% AH j'aimerai mettre deux equations avec un label mais je ne sais pas comment faire
278	277	The constraint \ref{cstr_size} means the occupation for the current stage $j$ depends on
279	278	the previous occupation and the occupation of current selected filter (it is possible
280	279	that no filter is selected for this stage). And the second one \ref{cstr_rejection}
...	...	@@ -359,7 +358,7 @@
359	358	The coefficients of a single monolithic filter are computed as the impulse response
360	359	of the filter transfer function, and practically approximated by a multitude of methods
361	360	including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing
362		-(Matlab's {\tt fir1} function).
	361	+(Matlab's {\tt fir1} function).
363	362
364	363	\begin{figure}[h!tb]
365	364	\includegraphics[width=\linewidth]{images/fir1-vs-firls}