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584 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
Distribution of Output Error
0.0313
0.0234
0.0156
0.0078
SAMPLE SIZE N = 100000
FILT LENGTH M = 5
SNR(THEORY) = 65.4187
0
−0.5 −0.4 −0.3 −0.2 −0.1
Five Multipliers
ROUNDED TO B = 12 BITS
ERROR MEAN = 1.9514e–007
SNR(COMPUTED) = 65.4241
0 0.1 0.2 0.3 0.4 0.5
Distribution of Output Error
0.0313
0.0234
0.0156
0.0078
SAMPLE SIZE N = 100000
FILT LENGTH M = 5
SNR(THEORY) = 72.4084
0
−0.5 −0.4 −0.3 −0.2 −0.1
One Multiplier
ROUNDED TO B = 12 BITS
ERROR MEAN = 1.9514e–007
SNR(COMPUTED) = 72.4324
0 0.1 0.2 0.3 0.4 0.5
Normalized Error
FIGURE 10.27 Multiplication quantization effects for the direct-form FIR filter
in Example 10.13
in which quantization noise sources, e i (n) 1≤ i ≤ K, ateach section’s
output are incorporated. Let y(n) bethe output due to input x(n), and
let q(n) bethe output due to all noise sources. We make the following
reasonable assumptions:
1. The sections are implemented using the MAC (multiply-accumulate)
architecture so that there is only one independent noise source in each
section that contributes to e i (n). The other possibility of three multipliers
in each section is straightforward.
x(n) H 1 (z) H 2 (z) H K (z)
y(n) ˆ = y(n) + q(n)
e 1 (n) e 2 (n) e K−1 (n) e K (n)
FIGURE 10.28 Cascade form FIR filter structure with noise sources inserted for
multiplication quantization
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