Page 225 - DSP Integrated Circuits

210 Chapter 5 Finite Word Length Effects

by iterating the difference equations. The results for the bandpass filter, with dif-
ferent scaling, are shown in Table 5.1.

2
Gj with L,2-Norm
2
Node G; with Loo-Norm Scaling Scaling
u[3] 16.22323 16.05974
u[5] 25.52102 11.12439
u[7] 39.98744 19.32355
u[9] 10.38189 10.38189
Sum G; 2 92.11358 56.88957
2
Go 0.0749669 1.0000
AW 6.29 4.08
Table 5.1 Lp-norms (squared) for the bandpass filter in Example 5.6

The difference in dynamic range between the two scaling strategies is about
2.21 bits. The L2-norm for the whole filter is GQ. The variance of the round-off
noise at the output of the filter is obtained by adding the contribution from each
source. The equivalent noise gain is

The round-off noise, corrected for the filter gain, corresponds to a loss in
dynamic range of AW bits, where

5.6.1 FFT Round-Off Noise
Data word length will affect the cost of butterfly PEs and, more importantly, the
chip area required for the memories. In fact, for large FFTs the chip area required
for the PEs is small compared to the area required for memory. It is therefore
important to determine accurately and, if possible, minimize the required data
word length.
We assume that the computations
inside the butterfly operations are per-
formed with full accuracy using fixed-
point arithmetic. The complex results are
rounded at the outputs of the butterfly as
shown in Figure 5.18.
For simplicity, we neglect the correc-
tion for short coefficient word lengths, Figure 5.18 Butterfly with scaling and
Equation (5.14), and model the round-off quantizations
noise at the two outputs of a butterfly
with uncorrelated complex noise sources
with variance