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552 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
There are two types of limit cycles. The granular limit cycles are due
to nonlinearities in multiplication quantization and are of low amplitude.
The overflow limit cycles are a result of overflow in addition and can have
large amplitudes.
10.2.2 GRANULAR LIMIT CYCLES
This type of limit cycle can easily be demonstrated with a simple rounding
quantizer following a multiplication. We illustrate with the following
example.
□ EXAMPLE 10.5 Consider a simple 1st-order IIR filter given by
y(n) =αy(n − 1) + x(n); y(−1) = 0, n ≥ 0 (10.24)
Let α = − 1 ; then this is a highpass filter, since its pole is near z = −1.
2
Determine the output y(n) when x(n) = 7 δ(n), assuming a 3-bit quantizer in
8
the multiplier.
Solution
After multiplication by α, wehave toquantize the result. Let the output due
to this quantization be ŷ(n). Then the actual implementable digital filter is
ŷ(n) =Q
[− 1 ]
2 ŷ(n − 1) + x(n); ŷ(−1) = 0, n ≥ 0 (10.25)
We assume that the input in (10.24) is quantized and that there is no overflow
due to addition. Let B =3(that is, we have 3 fraction bits and 1 sign bit) and
let x(n) = 7 δ(n). Now α = − 1 is represented by 1110 in two’s-complement
8 2
format. Hence the output sequence is obtained as:
ŷ(0) = x(0) = + 7 8 :0111
[
ŷ(1) = Q [α ŷ(0)] = Q − 1 (
+ 7 )] [
= Q − 7 ]
= − 1 2 8
16 2 :1100
[
ŷ(2) = Q [α ŷ(1)] = Q − 1 (
− 1 )] [
= Q + 1 ]
=+ 1 2 2
4 4 :0010
[
ŷ(3) = Q [α ŷ(2)] = Q − 1 (
+ 1 )] [
= Q − 1 ]
= − 1 2 4
8 8 :1111
[
ŷ(4) = Q [α ŷ(3)] = Q − 1 (
− 1 )] [
= Q + 1 ]
=+ 1 2 8
16 8 :0001
[
ŷ(5) = Q [α ŷ(4)] = Q − 1 (
+ 1 )] [
= Q − 1
2 8
16
.
.
.
.
]
= − 1 8 :1111
.
(10.26)
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