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570 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
0.0313
SAMPLE SIZE N = 100000
PARAMETER a = 0.8999
SNR(THEORY) = 58.26
ROUNDED TO B = 12 BITS
ERROR MEAN = –1.282e–007
SNR(COMPUTED) = 58.134
Distribution of Output Error
0.0234
0.0156
0.0078
0.0
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized Error
FIGURE 10.21 Multiplication quantization effects in the 1st-order IIR filter in
Example 10.11, B =12bits
the two multipliers is shown in Figure 10.22b, where the responses q 1 (n)
and q 2 (n) are due to noise sources e 1 (n) and e 2 (n), respectively. We can
combine two noise sources into one. However, to avoid overflow we have
to scale signals at the input of each adder, which can complicate this
consolidation of sources.
In modern DSP chips, the intermediate results of multiply-add operations
are stored in a multiply-accumulate or MAC unit that has
a double precision register to accumulate sums. The final sum [which
for Figure 10.22b is at the output of the top adder] is quantized to
obtain ŷ(n). This implementation not only reduces the total multiplication
quantization noise but also makes the resulting analysis easier.
Assuming this modern implementation, the resulting simplified model is
shown in Figure 10.22c, where e(n) isthe single noise source that is uniformly
distributed between [−2 −(B+1) , 2 −(B+1) ] and q(n) isthe response
due to e(n). Note that e(n) ≠ e 1 (n)+e 2 (n) and that q(n) ≠ q 1 (n)+q 2 (n).
The only overflow that we have to worry about is at the output of the
top adder, which can be controlled by scaling the input sequence x(n)
as shown in Figure 10.22d. Now the round-off noise analysis can be carried
out in a fashion similar to that of the 1st-order filter. The details,
however, are more involved due to the impulse response in (10.52).
Signal-to-noise ratio Referring to Figure 10.22d, the noise impulse
response h e (n) isequal to h(n). Hence the output round-off noise power
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