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588 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
Distribution of output error
0.1563
0.1172
0.0781
0.0391
SAMPLE SIZE N = 100000
FILT LENGTH M = 5
SNR(THEORY) = 70.9563
Cascade Structure
ROUNDED TO B = 12 BITS
ERROR MEAN = 1.9514e–007
SNR(COMPUTED) = 70.8517
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 10.29 Multiplication quantization effects for the cascade-form FIR
filter in Example 10.14
the relative errors in adders and {ε i (n)}, 0≤ i ≤ M − 1 are the relative
errors in multipliers, with |η i |≤2 −2B and |ε i |≤2 −2B .
Let A(n, k) bethe gain from the kth multiplier to the output node,
which is given by
⎧
⎨
A(n, k) =
⎩
(1 + ε k (n)) ∏ M−1
r=k
(1 + η r(n)) ,k≠0;
(1 + ε 0 (n)) ∏ M−1
r=k
(1 + η r(n)) ,k=0.
(10.98)
Let ŷ(n) △ = y(n) +q(n) bethe overall output where y(n) isthe output
due to the input x(n) and q(n) isthe output due to noise sources. Then
ŷ(n) =
M−1
∑
k=0
A(n, k) h(k) x(n − k) (10.99)
x(n)
z −1 z −1 z −1 z −1
h(0) h(1) h(2) h(3) h(M − 2) h(M − 1)
1 + ε 0 (n) 1 + ε 1 (n)
1 + ε 2 (n) 1 + ε 3 (n) 1 + ε M−2 (n) 1 + ε M−1 (n)
1
1 + η 1 (n) 1 + η 2 (n)
1 + η M−2 (n) 1 + η M−1 (n)
y(n) + q(n)
FIGURE 10.30 Multiplication quantization model for direct-form floating-point
implementation of an FIR filter
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