fix typo

Arthur HUGEAT
1 parent e3580faaed
Showing 1 changed file with 14 additions and 14 deletions Side-by-side Diff
ifcs2018_proceeding.tex
@@ -137,7 +137,7 @@
 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.
@@ -168,7 +168,7 @@
 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,
  
  
@@ -202,12 +202,12 @@
   \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. 
+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.
 Hence, amongst various criteria including the mean or median value of the FIR response in the stopband, we have designed
-a criterion aimed at avoiding ripples on passband and considering the maximum of the FIR spectral response in the stopband 
+a criterion aimed at avoiding ripples on passband and considering the maximum of the FIR spectral response in the stopband
 (Fig. \ref{rejection-shape}). The bandpass criterion is defined as the sum of the absolute values of the spectral response
 in the bandpass, reminiscent of a standard deviation of the spectral response: this criterion must be minimized to avoid
 ripples in the passband. The stopband transfer function maximum must also be minimized in order to improve the filter
  
@@ -242,11 +242,11 @@
 \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*}
  
@@ -256,11 +256,11 @@
 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}
@@ -316,7 +316,7 @@
 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}
...	...	@@ -137,7 +137,7 @@
137	137	follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
138	138	in which basic blocks are defined and characterized before being assembled \cite{hide}
139	139	in a complete processing chain. In our case, assembling the filter blocks is a simpler block
140		-combination process since we assume a single value to be processed and a single value to be
	140	+combination process since we assume a single value to be processed and a single value to be
141	141	generated at each clock cycle. The FIR filters will not be considered to decimate in the
142	142	current implementation: the decimation is assumed to be located after the FIR cascade at the
143	143	moment.
...	...	@@ -168,7 +168,7 @@
168	168	the number of bits $C_i$ representing the coefficients and the number of bits $D_i$ representing
169	169	the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage,
170	170	the optimization of the complete processing chain within a constrained resource environment is not
171		-trivial. The resource occupation of a FIR filter is considered as $D_i+C_i \times N_i)$ which is
	171	+trivial. The resource occupation of a FIR filter is considered as $(D_i+C_i) \times N_i$ which is
172	172	the number of bits needed in a worst case condition to represent the output of the FIR. Such an
173	173	occupied area estimate assumes that the number of gates scales as the number of bits and the number
174	174	of coefficients, but does not account for the detailed implementation of the hardware. Indeed,
175	175
176	176
...	...	@@ -202,12 +202,12 @@
202	202	\label{model-FIR}
203	203	\end{align}
204	204	To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the rejection of depending on $N_i$ and $C_i$, $\mathcal{A}$
205		-is a theoretical area occupation of the processing block on the FPGA, and $\Delta_i$ is the total rejection for the current stage $i$.
	205	+is a theoretical area occupation of the processing block on the FPGA, and $\Delta_i$ is the total rejection for the current stage $i$.
206	206	Since the function $\mathcal{F}$ cannot be explictly expressed, we run simulations to determine the rejection depending
207		-on $N_i$ and $C_i$. However, selecting the right filter requires a clear definition of the rejection criterion. Selecting an
208		-incorrect criterion will lead the linear program solver to produce a solution which might not meet the user requirements.
	207	+on $N_i$ and $C_i$. However, selecting the right filter requires a clear definition of the rejection criterion. Selecting an
	208	+incorrect criterion will lead the linear program solver to produce a solution which might not meet the user requirements.
209	209	Hence, amongst various criteria including the mean or median value of the FIR response in the stopband, we have designed
210		-a criterion aimed at avoiding ripples on passband and considering the maximum of the FIR spectral response in the stopband
	210	+a criterion aimed at avoiding ripples on passband and considering the maximum of the FIR spectral response in the stopband
211	211	(Fig. \ref{rejection-shape}). The bandpass criterion is defined as the sum of the absolute values of the spectral response
212	212	in the bandpass, reminiscent of a standard deviation of the spectral response: this criterion must be minimized to avoid
213	213	ripples in the passband. The stopband transfer function maximum must also be minimized in order to improve the filter
214	214
...	...	@@ -242,11 +242,11 @@
242	242	\begin{align*}
243	243	\mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\
244	244	& \text{ $p$ is the number of different filters} \\
245		-C(i) & \text{ % Constant to let the
246		-number of coefficients %} \\ & \text{
	245	+C(i) & \text{ % Constant to let the
	246	+number of coefficients %} \\ & \text{
247	247	for filter $i$}\\
248		-\pi_C(i) & \text{ % Constant to let the
249		-number of bits of %}\\ & \text{
	248	+\pi_C(i) & \text{ % Constant to let the
	249	+number of bits of %}\\ & \text{
250	250	each coefficient for filter $i$}\\
251	251	\mathcal{A}_{\max} & \text{ Total space available inside the FPGA}
252	252	\end{align*}
253	253
...	...	@@ -256,11 +256,11 @@
256	256	1 \leq j \leq q & \text{ $q$ is the max of filter stage} \nonumber \\
257	257	\forall j, \mathlarger{\sum_{i}} x_{i,j} = 1 & \text{ At most one filter by stage} \nonumber\\
258	258	\mathcal{S}_0 = 0 & \text{ initial occupation} \nonumber\\
259		-\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \forall i, x_{i,j} \times \mathcal{A}_i \label{cstr_size} \
	259	+\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{A}_i)} \label{cstr_size} \
260	260	\mathcal{S} \leq \mathcal{S}_{\max}\nonumber \\
261	261	\mathcal{N}_0 = 0 & \text{ initial rejection}\nonumber\\
262		-\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \forall i, x_{i,j} \times \mathcal{R}_i \label{cstr_rejection} \\
263		-\mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\
	262	+\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{R}_i)} \label{cstr_rejection} \\
	263	+\mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\
264	264	& \text{ (e.g. 160~dB here)}\nonumber\\\nonumber
265	265	\end{align}
266	266	\paragraph{Goal}
...	...	@@ -316,7 +316,7 @@
316	316	The coefficients of a single monolithic filter are computed as the impulse response
317	317	of the filter transfer function, and practically approximated by a multitude of methods
318	318	including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing
319		-(Matlab's {\tt fir1} function).
	319	+(Matlab's {\tt fir1} function).
320	320
321	321	\begin{figure}[h!tb]
322	322	\includegraphics[width=\linewidth]{images/fir1-vs-firls}