Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 370

After filter coefficient quantization, we have the quantized digital IIR filter transfer function:
H q ðzÞ¼ b 0 þ b 1 z 1
1 þ a 1 z 1

(8.58)

Solving for the pole and zero, we get

z 1 ¼ b 1 (8.59)

b 0

p 1 ¼a 1

(8.60)

Now considering a second-order IIR filter transfer function as
HðzÞ¼ b 0 þ b 1 z 1

þ b 2 z 2

(8.61)

1 þ a 1 z 1 þ a 2 z 2

and its quantized IIR filter transfer function
H q ðzÞ¼ b 0 þ b 1 z 1

þ b 2 z 2

(8.62)

1 þ a 1 z 1 þ a 2 z 2

solving for poles and zeros yields
z 1 ; 2 ¼ 0 : 5 $ b 1

b 2 b 1

b 0

2

1
2

b 0 j b 0 0 : 25 $

(8.63)

a 2 0 : 25 $ a 1 2

1
2

p 1 ; 2 ¼ 0 : 5 $a 1 j

(8.64)

With Equations (8.59) and (8.60) for the first-order IIR filter, and Equations (8.63) and (8.64) for the
second-order IIR filter, we can study the effects of location changes of the poles and zeros, and the
frequency responses due to filter coefficient quantization.

EXAMPLE 8.24
Given the first-order IIR filter

HðzÞ¼ 1:2341 þ 0:2126 z 1
1 0:5126z 1

and assuming that we use 1 sign bit and 6 bits for encoding the magnitude of the filter coefficients, find the
quantized transfer function and pole-zero locations.
Solution:
Let us find the pole and zero for infinite precision filter coefficients. Solving 1:2341z þ 0:2126 ¼ 0 leads to
a zero location z 1 ¼0:17227. Solving z 0:5126 ¼ 0 gives a pole location p 1 ¼ 0:5126.
Now let us quantize the filter coefficients. Quantizing 1.2341 can be illustrated as
1:2341 2 5 ¼ 39:4912 ¼ 39 ðrounded to integerÞ