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Round-off Effects in IIR Digital Filters 571
x(n)
z −1
y(n)
x(n)
z −1
y(n) ˆ
= y(n) + q 1 (n) + q 2 (n)
−a 1
y(n − 1)
e 1 (n)
−a 1
y(n ˆ − 1)
−a 2
z −1
y(n − 2)
e 2 (n)
−a 2
z −1
y(n ˆ − 2)
(a)
(b)
e(n)
e(n)
x(n)
z −1
y(n) ˆ
= y(n) + q(n)
x(n)
X max
z −1
y(n) ˆ
= y(n) + q(n)
−a 1
y(n ˆ − 1)
−a 1
−a 2
y(n ˆ − 2)
−a 2
z −1
y(n ˆ − 1)
z −1
y(n ˆ − 2)
(c)
(d)
FIGURE 10.22 2nd-order IIR filter: (a) structure, (b) round-off noise model,
(c) simplified model, (d) scaled simplified model
is given by
σ 2 q = σ 2 e
∞∑
n=0
|h(n)| 2 = 2−2B
12
∞∑
|h(n)| 2 (10.54)
n=0
Since x(n) isquantized, we have |x(n)| ≤1. It is then scaled by X max to
avoid overflow in the adder. Hence the output signal power is given by
σ 2 y = X 2 maxσ 2 x
∞∑
n=0
|h(n)| 2 = X2 max
3
∞∑
|h(n)| 2 (10.55)
assuming that x(n) isuniformly distributed over [−1, +1]. Hence the output
SNR is given by
n=0
SNR = σ2 y
σ 2 q
=4 ( 2 2B) X 2 max =2 2(B+1) X 2 max (10.56)
or
SNR dB =6.02+6.02B +20log 10 X max (10.57)
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