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Round-off Effects in IIR Digital Filters 565
x(n) y 1 (n) y(n) x(n) yˆ
1 (n)
Y x(n) yˆ
1 (n) y(n) + q(n)
z −1 z −1 z −1
α α α
Q
e(n)
(a) (b) (c)
FIGURE 10.18 First-order IIR filter: (a) structure, (b) structure with quantizer,
(c) round-off noise model
previous section. Since we emphasize the rounding operation, this model
is also known as a round-off noise model. We will limit ourselves to the 1stand
2nd-order filters since practical realizations involve 1st- or 2nd-order
sections.
1st-order filter Consider the 1st-order filter shown in Figure 10.18a.
When a quantizer Q[·] isintroduced after the multiplier, the resulting
filter model is shown in Figure 10.18b, which is a nonlinear system. When
Q [·] isaquantizer based on the round-off characteristics, then its effect is
to add a zero-mean, stationary white noise sequence e(n) atthe multiplier
output as shown in Figure 10.18c.
Let q(n)bethe response due to e(n) and let h e (n)bethe noise impulse
response (i.e., between e(n) and q(n)). For the system in Figure 10.18c
Using (10.12) and (10.7), the mean of q(n) is
h e (n) =h(n) =α n u(n) (10.38)
m q = m e
∞ ∑
0
h e (n) =0 (10.39)
Similarly, using (10.15), the variance of q(n) is
( ∞
)
∑
σq 2 = σe
2 |h e (n)| 2
0
(10.40)
Substituting σe
2 = 2 −2B /12 for rounding and h e (n) from (10.38), we
obtain
( ∞
)
σq 2 = 2−2B ∑
∞∑
|α n | 2 = 2−2B (
|α|
2 ) n 2 −2B
=
12
12
12 (1 −|α| 2 (10.41)
)
0
which is the output noise power due to rounding following the multiplication.
0
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