jfriedt / IFCS2018 article

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ifcs2018_proceeding.tex
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... ... @@ -171,11 +171,10 @@ computational resources: optimizing some criteria within finite, limited
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 $i$,
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,
  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,
177 176 the optimization of the complete processing chain within a constrained resource environment is not
178   -trivial. The resource occupation of a FIR filter is considered as $(D_i+C_i) \times N_i$ which is
  177 +trivial. The resource occupation of a FIR filter is considered as $C_i \times N_i$ which is
179 178 the number of bits needed in a worst case condition to represent the output of the FIR. Such an
180 179 occupied area estimate assumes that the number of gates scales as the number of bits and the number
181 180 of coefficients, but does not account for the detailed implementation of the hardware. Indeed,
... ... @@ -204,7 +203,7 @@ Following these considerations, the model is expressed as:
204 203 \begin{align}
205 204 \begin{cases}
206 205 \mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\
207   - \mathcal{A}_i &= N_i \times (C_i + D_i)\
  206 + \mathcal{A}_i &= N_i * C_i\
208 207 \Delta_i &= \Delta _{i-1} + \mathcal{P}_i
209 208 \end{cases}
210 209 \label{model-FIR}
... ... @@ -230,7 +229,7 @@ rejection capability. Weighing these two criteria allows designing the linear pr
230 229  
231 230 The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource occupation below
232 231 a user-defined threshold, or aims at minimizing the area needed to reach a given rejection ($\min(S_q)$ in
233   -the forthcoming discussion, Eqs. \ref{cstr_size} and \ref{cstr_rejection}).
  232 +the forthcoming discussion, Eqs. \ref{cstr_size} and \ref{cstr_rejection}).
234 233 The MILP solver is allowed to choose the number of successive
235 234 filters, within an upper bound. The last problem is to model the noise rejection. Since filter
236 235 noise rejection capability is not modeled with linear equations, a look-up-table is generated
... ... @@ -238,7 +237,7 @@ for multiple filter configurations in which the $C_i$, $D_i$ and $N_i$ parameter
238 237 one of these conditions, the low-pass filter rejection is stored as computed by the frequency response
239 238 of the digital filter (Fig. \ref{noise-rejection}). Various rejection criteria have been investigated,
240 239 including mean value of the stopband response, median value of the stopband response, or as finally
241   -selected, maximum value in the stopband. An intuitive analysis of the chart of Fig. \ref{noise-rejection}
  240 +selected, maximum value in the stopband. An intuitive analysis of the chart of Fig. \ref{noise-rejection}
242 241 hints at an optimum
243 242 set of tap length and number of bit for representing the coefficients along the line of the pyramidal
244 243 shaped rejection capability function.
... ... @@ -257,13 +256,13 @@ x_{i,j} \in \lbrace 0,1 \rbrace & \text{ $i$ is a given filter} \\
257 256 \begin{align*}
258 257 \mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\
259 258 & \text{ $p$ is the number of different filters} \\
260   -C(i) & \text{ % Constant to let the
261   -number of coefficients %} \\ & \text{
262   -for filter $i$}\\
263   -\pi_C(i) & \text{ % Constant to let the
264   -number of bits of %}\\ & \text{
265   -each coefficient for filter $i$}\\
266   -\mathcal{A}_{\max} & \text{ Total space available inside the FPGA}
  259 +% N(i) & \text{ % Constant to let the
  260 +% number of coefficients %} \\ & \text{
  261 +% for filter $i$}\\
  262 +% C(i) & \text{ % Constant to let the
  263 +% number of bits of %}\\ & \text{
  264 +% each coefficient for filter $i$}\\
  265 +\mathcal{S}_{\max} & \text{ Total space available inside the FPGA}
267 266 \end{align*}
268 267 \paragraph{Constraints}
269 268 \begin{align}
... ... @@ -293,7 +292,7 @@ plus the rejection of selected filter.
293 292  
294 293 The MILP solver provides a solution to the problem by selecting a series of small FIR with
295 294 increasing number of bits representing data and coefficients as well as an increasing number
296   -of coefficients, instead of a single monolithic filter.
  295 +of coefficients, instead of a single monolithic filter.
297 296  
298 297 \begin{figure}[h!tb]
299 298 % \includegraphics[width=\linewidth]{images/compare-fir.pdf}