Rajout de la definition de D_i

Arthur HUGEAT
1 parent 365465b1b5
Showing 1 changed file with 7 additions and 6 deletions Side-by-side Diff
ifcs2018_proceeding.tex
@@ -171,11 +171,12 @@
 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 $i$ and
-the number of bits $C_i$ representing the coefficients. 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 $i$,
+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 $C_i \times N_i$ which aims
-at approximating the number of bits needed in a worst case condition to represent the output of the 
+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
@@ -231,9 +232,9 @@
 \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 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} 
+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
...	...	@@ -171,11 +171,12 @@
171	171	resources indeed matches the definition of a classical optimization problem.
172	172
173	173	Specifically the degrees of freedom when addressing the problem of replacing the single monolithic
174		-FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$ and
175		-the number of bits $C_i$ representing the coefficients. Because each FIR in the chain is fed the output of the previous stage,
	174	+FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$,
	175	+the number of bits $C_i$ representing the coefficients and the number of bits $D_i$ representing
	176	+-the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage,
176	177	the optimization of the complete processing chain within a constrained resource environment is not
177	178	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	+at approximating the number of bits needed in a worst case condition to represent the output of the
179	180	FIR. Indeed, the number of bits generated by the FIR is $(C_i+D_i)\times\log_2(N_i)$ with $D_i$
180	181	the number of bits needed to represent the data $x_k$ generated by the previous stage, but the
181	182	$\log$ function is avoided for its incompatibility with a linear programming description, and
...	...	@@ -231,9 +232,9 @@
231	232	\label{noise-rejection}
232	233	\end{figure}
233	234
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}
	235	+The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource
	236	+occupation below a user-defined threshold, or as will be discussed here, aims at minimizing the area
	237	+needed to reach a given rejection ($\min(S_q)$ in the forthcoming discussion, Eqs. \ref{cstr_size}
237	238	and \ref{cstr_rejection}). The MILP solver is allowed to choose the number of successive
238	239	filters, within an upper bound. The last problem is to model the noise rejection. Since filter
239	240	noise rejection capability is not modeled with linear equations, a look-up-table is generated