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Round-off Effects in IIR Digital Filters 569
0.0313
SAMPLE SIZE N = 100000
PARAMETER a = 0.90625
SNR(THEORY) = 22.14
ROUNDED T0 B = 6 BITS
ERROR MEAN = –3.2612e–005
SNR(COMPUTED) = 22.2105
Distribution of Output Error
0.0234
0.0156
0.0078
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 10.20 Multiplication quantization effects in the first-order IIR filter in
Example 10.11, B =6bits
The part of the script not shown above also computes and plots the normalized
histogram of the output error and prints the statistical values in the plot,
as shown in Figure 10.20. The error appears to have a Gaussian distribution,
which is to be expected. The exact value of the output SNR is 22.14 dB, which
agrees with the computed value of 22.21 dB. Similar results done for B =12
bits are shown in Figure 10.21. Again, the simulation results agree with the
model results.
□
2nd-order filter Similar analysis can be done for 2nd-order filters with
poles near the unit circle. Let the two poles be at complex locations re jθ
and re −jθ . Then the system function of the filter is given by
1
H(z) =
(1 − re jθ z −1 )(1 − re −jθ z −1 ) = 1
1 − 2r cos(θ) z −1 + r 2 z −2
with impulse response
(10.51)
h(n) = rn sin{(n +1)θ}
u(n) (10.52)
sin(θ)
The difference equation from (10.51) is given by
y(n) =x(n)−a 1 y(n−1)−a 2 y(n−2); a 1 = −2r cos(θ), a 2 = r 2 (10.53)
which requires two multiplications and two additions, as shown in
Figure 10.22a. Thus, there are two noise sources and two possible locations
for overflow. The round-off noise model for quantization following
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