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566 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
X max yˆ
1 (n)
X max yˆ
1 (n)
x(n) y(n) ˆ x(n) y(n) + q(n)
z −1
z −1
α
α
Q
e(n)
(a)
(b)
FIGURE 10.19 Scaled first-order IIR filter: (a) structure with quantizer, (b)
round-off noise model
However, we also have to prevent a possible overflow following the
adder. Let y 1 (n) bethe signal at the output of the adder in Figure 10.18a,
which in this case is equal to y(n). Now the upper bound on y 1 (n) is
∞∑
∞ |y 1 (n)| = |y(n)| =
h(k) x(n − k)
∣
∣ ≤ ∑
|h(k)| |x(n − k)| (10.42)
0
Let the input sequence be bounded by X max (i.e., |x(n)| ≤X max ). Then
0
∑ ∞
|y 1 (n)| ≤X max |h(k)| (10.43)
Since y 1 (n) isrepresented by B fraction bits, we have |y 1 (n)| ≤1. The
condition (10.43) can be satisfied by requiring
X max =
Thus, to prevent overflow x(n) must satisfy
0
1
∑ ∞
0 |h(k)| = 1
=1−|α| (10.44)
1/ (1 −|α|)
− (1 −|α|) ≤ x(n) ≤ (1 −|α|) (10.45)
Thus, the input must be scaled before it is applied to the filter as shown
in Figure 10.19.
Signal-to-noise ratio We will now compute the finite word-length
effect in terms of the output signal-to-noise ratio (SNR). We assume
that there is no overflow at the output by properly scaling x(n). Let
x(n) be a stationary white sequence, uniformly distributed between
[− (1 −|α|) , (1 −|α|)]. Then
m x =0 and σx 2 (1 −|α|)2
= (10.46)
3
Therefore, y(n) isalso a stationary random sequence with mean m y =0
and
∞∑
σy 2 = σx
2 |h(n)| 2 (1 −|α|)2 1 (1 −|α|)2
= =
3 1 −|α|
2
3(1−|α| 2 (10.47)
)
0
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