Chapter 8 Design of Infinite Impulse Response (IIR) Digital Filter

Basic properties of IIR filters Recursive equation of IIR filter Transfer function of IIR filter (8-1) where is impulse response of the filter, and are coefficients of the filter (8-2)

Factorized form of transfer function for IIR filter Features of IIR filter IIR filter is a feedback system of some sort IIR filter normally requires fewer coefficients than FIR filter when sharp cutoff and high throughput are the important requirements IIR filter can become unstable or its performance significantly degraded (8-3) where are the zeros of , and are the poles of , and is constant

Design of IIR filter using Analog Filter Most common methods of converting analog filters into equivalent digital filter Impulse invariant method has equal impulse responses between equivalent digital filter and analog filter at sampling point Bilinear z-transform Analog transfer function in s-plane is mapped into the discrete transfer function in z-plane jw axis in the s-plane is mapped onto the unit circle brings frequency warping

Impulse Response Series Impulse invariant method Analog Filter Transfer Function Digital Filter Impulse Response Impulse Response Series Fig. 8-1.

Design a LPF using Impulse invariant method (8-4) Fig. 8-2.

Impulse response using inverse Laplace transform Impulse response of the equivalent digital filter Sampling with period T (substituting t=nT) Transfer function by using z-transform (8-5) (8-6) (8-7)

Analog filter with single poles Impulse response Inverse Laplace transform Impulse response of the equivalent digital filter Sampling with period T(substituting t=nT) (8-8) where and are the real and imaginary part of (8-9) (8-10)

Transfer function of digital filter by using z-transform Using commutative law (8-11) (8-12)

Using an infinite series Designed digital filter by using impulse invariant method (8-13) (8-14)

Analog filter with repeated poles Transfer function with repeated poles of l-th order Using z-transform (8-15) (8-16)

Analog filter with complex conjugate poles (1) Transfer function Using z-transform (8-17) (8-18)

Analog filter with complex conjugate poles (2) Transfer function Using z-transform (8-19) (8-20)

Example 8-1 Transfer function for second-order Butterworth filter Using partial fraction Impulse response of the equivalent digital filter (T=1)

Magnitude response Fig. 8-3.

Summary of impulse invariance method The analog filter is bandlimited before the impulse invariant method is applied To have the same filter gain in digital and analog filters, multiply H(z) by T (8-21)

Bilinear z-transform Replacing s to convert an analog filter into an equivalent digital filter Imaginary axis in the s-plane is mapped onto the unit circle The left-half s-plane is mapped inside the unit circle (8-22) (8-23)

Considering frequency scaling Substituting , (8-24) (8-25) where is analog frequency, and is digital frequency

Relationship between analog and digital frequency Frequency warping (8-26) (8-27)

Frequency warping Fig. 8-4.

Example 8-2 Using bilinear z transform For

Impulse response of analog filter : Magnitude and phase responses of analog filter : Impulse response of digital filter : Magnitude response = Phase response =

Frequency response of digital filter : Relationship between analog and digital frequencies Magnitude response = Phase response =

Example 8-3 Specification of the desired filter Filter response Power gain at 1000Hz : -3dB Power gain at 3000Hz : -10dB Sampling frequency : 10kHz Monotonic decrease in transition region :1000 ~ 3000Hz Digital parameter from specification

Considering Frequency warp Prewarping Determining order of Butterworth filter Desired analog Butterworth filter

Using bilinear z-transform Fig. 8-5.

Comparing the two transform Converting analog filter into equivalent digital filter Impulse invariant method Preserving the impulse response of analog filter Bilinear z-transform Frequency warping (8-28) (8-29)

Example 8-4 Transfer function of analog filter Frequency response

Using impulse invariant method Frequency response substituting

Using bilinear z-transform Frequency response

Analog Impulse Response Bilinear z Transform Fig. 8-6.

Example 8-5 Using partial fraction Inverse Laplace transform

Cutoff frequency(-3dB) of Analog filter : Frequency response Cutoff frequency(-3dB) of Analog filter : (8-30)

Impulse invariant method Using partial fraction Substituting (8-31)

Bilinear z-transform (8-32)

Frequency transformation Design of various filters using frequency transformation Analog low pass filter (normalization filter) Analog frequency transform Low pass Low pass High pass Band pass Band reject Desired digital filter Bilinear z transform or Impulse invariant method Fig. 8-7.

Analog frequency transform Digital frequency transform low pass filter (normalization) Digital Low pass High pass Band pass Band reject Bilinear transform Analog frequency transform Digital frequency transform Fig. 8-8.

Low pass filter Cutoff frequency :

High pass filter Cutoff frequency : Cutoff frequency :

Band pass filter Cutoff frequency : where is passband width is upper passband edge frequency is lower passband edge frequency

Band reject filter Cutoff frequency : where is reject band width is upper band reject frequency is lower band reject frequency

Low pass filter (cutoff frequency ) Table. 8-1 Analog Frequency Transform Low pass filter (cutoff frequency ) Low pass filter (cutoff frequency : ) High pass filter (cutoff frequency : ) Band pass filter (Upper cufoff frequency : , Lower cufoff frequency: , Band pass frequency : ) Band reject filter (Upper cufoff frequency : , Lower cufoff frequency : , Band reject frequency : )

Example 8-6 Specification of filter design Using bilinear z-transform Cutoff frequency : 10Hz Sampling frequency ( ) Transfer function : Using bilinear z-transform Prewarping

Transfer function of analog LPF Substituting

For computational efficiency For accurate frequency

Fig. 8-9.

Example 8-7 Specification of filter design Using bilinear z-transform cutoff frequency : Sampling frequency : Transfer function of low pass filter: Using bilinear z-transform Prewarping

Transfer function of analog HPF from table 8-1 Substituting

Fig. 8-10.

Example 8-8 Specification of filter design Using bilinear z-transform Passband : Sampling frequency : Order of filter : 2 Using bilinear z-transform From table 8-2 For second order bandpass filter Using transfer function of LPF

Calculation of lower and upper passband frequencies

Analog bandpass filter Substituting

Fig. 8-11.