notations

jfriedt
1 parent 9db1d56ab9
Showing 1 changed file with 13 additions and 9 deletions Side-by-side Diff
ifcs2018_proceeding.tex
@@ -174,8 +174,12 @@
 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 $C_i \times N_i$ which is
-the number of bits needed in a worst case condition to represent the output of the FIR. Such an
+trivial. The resource occupation of a FIR filter is considered as $C_i \times N_i$ which aims
+at approximating the number of bits needed in a worst case condition to represent the output of the 
+FIR. Indeed, the number of bits generated by the FIR is $(C_i+D_i)\times\log_2(N_i)$ with $D_i$
+the number of bits needed to represent the data $x_k$ generated by the previous stage, but the
+$\log$ function is avoided for its incompatibility with a linear programming description, and
+the simple product is approximated as the number of gates needed to perform the calculation. 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,
 various FPGA implementations will provide different hardware functionalities, and we shall consider
  
@@ -203,12 +207,12 @@
 \begin{align}
   \begin{cases}
     \mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\
-    \mathcal{A}_i &= N_i * C_i\
+    \mathcal{A}_i &= N_i \times C_i\
     \Delta_i &= \Delta _{i-1} + \mathcal{P}_i
   \end{cases}
   \label{model-FIR}
 \end{align}
-To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the stopband rejection dependence with $N_i$ and $C_i$, $\mathcal{A}$
+To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the stopband rejection dependence with $N_i$ and $C_i$, $\mathcal{A}_i$
 is a theoretical area occupation of the processing block on the FPGA as discussed earlier, 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
@@ -227,10 +231,10 @@
 \label{noise-rejection}
 \end{figure}
  
-The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource occupation below
-a user-defined threshold, or aims at minimizing the area needed to reach a given rejection ($\min(S_q)$ in
-the forthcoming discussion, Eqs. \ref{cstr_size} and \ref{cstr_rejection}).
-The MILP solver is allowed to choose the number of successive
+The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource 
+occupation below a user-defined threshold, or as will be discussed here, aims at minimizing the area 
+needed to reach a given rejection ($\min(S_q)$ in the forthcoming discussion, Eqs. \ref{cstr_size} 
+and \ref{cstr_rejection}). The MILP solver is allowed to choose the number of successive
 filters, within an upper bound. The last problem is to model the noise rejection. Since filter
 noise rejection capability is not modeled with linear equations, a look-up-table is generated
 for multiple filter configurations in which the $C_i$, $D_i$ and $N_i$ parameters are varied: for each
@@ -271,7 +275,7 @@
 \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} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{A}_i)} \label{cstr_size} \\
-\mathcal{S} \leq \mathcal{S}_{\max}\nonumber \
+\mathcal{S}_j \leq \mathcal{S}_{\max}\nonumber \
 \mathcal{N}_0 = 0 & \text{ initial rejection}\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\\
...	...	@@ -174,8 +174,12 @@
174	174	FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$ and
175	175	the number of bits $C_i$ representing the coefficients. Because each FIR in the chain is fed the output of the previous stage,
176	176	the optimization of the complete processing chain within a constrained resource environment is not
177		-trivial. The resource occupation of a FIR filter is considered as $C_i \times N_i$ which is
178		-the number of bits needed in a worst case condition to represent the output of the FIR. Such an
	177	+trivial. The resource occupation of a FIR filter is considered as $C_i \times N_i$ which aims
	178	+at approximating the number of bits needed in a worst case condition to represent the output of the
	179	+FIR. Indeed, the number of bits generated by the FIR is $(C_i+D_i)\times\log_2(N_i)$ with $D_i$
	180	+the number of bits needed to represent the data $x_k$ generated by the previous stage, but the
	181	+$\log$ function is avoided for its incompatibility with a linear programming description, and
	182	+the simple product is approximated as the number of gates needed to perform the calculation. Such an
179	183	occupied area estimate assumes that the number of gates scales as the number of bits and the number
180	184	of coefficients, but does not account for the detailed implementation of the hardware. Indeed,
181	185	various FPGA implementations will provide different hardware functionalities, and we shall consider
182	186
...	...	@@ -203,12 +207,12 @@
203	207	\begin{align}
204	208	\begin{cases}
205	209	\mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\
206		- \mathcal{A}_i &= N_i * C_i\
	210	+ \mathcal{A}_i &= N_i \times C_i\
207	211	\Delta_i &= \Delta _{i-1} + \mathcal{P}_i
208	212	\end{cases}
209	213	\label{model-FIR}
210	214	\end{align}
211		-To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the stopband rejection dependence with $N_i$ and $C_i$, $\mathcal{A}$
	215	+To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the stopband rejection dependence with $N_i$ and $C_i$, $\mathcal{A}_i$
212	216	is a theoretical area occupation of the processing block on the FPGA as discussed earlier, and $\Delta_i$ is the total rejection for the current stage $i$.
213	217	Since the function $\mathcal{F}$ cannot be explictly expressed, we run simulations to determine the rejection depending
214	218	on $N_i$ and $C_i$. However, selecting the right filter requires a clear definition of the rejection criterion. Selecting an
...	...	@@ -227,10 +231,10 @@
227	231	\label{noise-rejection}
228	232	\end{figure}
229	233
230		-The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource occupation below
231		-a user-defined threshold, or aims at minimizing the area needed to reach a given rejection ($\min(S_q)$ in
232		-the forthcoming discussion, Eqs. \ref{cstr_size} and \ref{cstr_rejection}).
233		-The MILP solver is allowed to choose the number of successive
	234	+The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource
	235	+occupation below a user-defined threshold, or as will be discussed here, aims at minimizing the area
	236	+needed to reach a given rejection ($\min(S_q)$ in the forthcoming discussion, Eqs. \ref{cstr_size}
	237	+and \ref{cstr_rejection}). The MILP solver is allowed to choose the number of successive
234	238	filters, within an upper bound. The last problem is to model the noise rejection. Since filter
235	239	noise rejection capability is not modeled with linear equations, a look-up-table is generated
236	240	for multiple filter configurations in which the $C_i$, $D_i$ and $N_i$ parameters are varied: for each
...	...	@@ -271,7 +275,7 @@
271	275	\forall j, \mathlarger{\sum_{i}} x_{i,j} = 1 & \text{ At most one filter by stage} \nonumber\\
272	276	\mathcal{S}_0 = 0 & \text{ initial occupation} \nonumber\\
273	277	\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{A}_i)} \label{cstr_size} \\
274		-\mathcal{S} \leq \mathcal{S}_{\max}\nonumber \
	278	+\mathcal{S}_j \leq \mathcal{S}_{\max}\nonumber \
275	279	\mathcal{N}_0 = 0 & \text{ initial rejection}\nonumber\\
276	280	\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{R}_i)} \label{cstr_rejection} \\
277	281	\mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\