Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 302

OBJECTIVES:
This chapter investigates a bilinear transformation method for infinite impulse response (IIR) filter design
and develops a procedure to design digital Butterworth and Chebyshev filters. The chapter also investigates
other IIR filter design methods, such as impulse-invariant design and pole-zero placement design. Finally,
the chapter illustrates how to apply the designed IIR filters to solve real-world problems such as digital audio
equalization, 60-Hz interference cancellation in audio and electrocardiography signals, dual-tone multifre-
quency tone generation, and detection using the Goertzel algorithm.

8.1 INFINITE IMPULSE RESPONSE FILTER FORMAT
In this chapter, we will study several methods for infinite impulse response (IIR) filter design. An IIR
filter is described using the difference equation, as discussed in Chapter 6:

yðnÞ¼b 0 xðnÞþb 1 xðn 1 Þþ / þ b M xðn MÞ
a 1 yðn 1 Þ / a N yðn NÞ
Chapter 6 also gives the IIR filter transfer function as

XðzÞ ¼ b 0 þ b 1 z 1

þ / þ b M z M

HðzÞ¼ YðzÞ

þ / þ a N z N
where b i and a i are the ðM þ 1 Þ numerator and N denominator coefficients, respectively. YðzÞ and XðzÞ
are the z-transform functions of the filter input xðnÞ and filter output yðnÞ . To become familiar with the
form of the IIR filter, let us look at the following example.

1 þ a 1 z 1

EXAMPLE 8.1
Given the IIR filter

yðnÞ¼0:2xðnÞþ0:4xðn 1Þþ0:5yðn 1Þ

determine the transfer function, nonzero coefficients, and impulse response.
Solution:
Applying the z-transform and solving for a ratio of the z-transform output over input, we have

X ðzÞ ¼ 0:2 þ 0:4 z 1

Y ð z Þ

HðzÞ¼

1 0:5z 1

We also identify the nonzero numerator coefficients and denominator coefficient as

b 0 ¼ 0:2; b 1 ¼ 0:4; and a 1 ¼0:5

To determine the impulse response, we rewrite the transfer function as

0:4z 1
1 0:5z 1

0:2
1 0:5z 1 þ

HðzÞ¼