Infinite Impulse Response Filter Design

Analog Filter TypePass-Band RippleStop-Band RippleTransition Band

ButterworthMonotonic (Maximally Flat)Monotonic Wide

Chebyshev-IEqui-rippleMonotonicNarrow

Chebyshev-IIMonotonicEqui-rippleNarrow

Elliptic (Cauer)Equi-rippleEqui-rippleVery Narrow

Chapter 6

Infinite Impulse Response Filter Design

Objectives

Describe the general concepts and approaches in IIR filter design.

Demonstrate the design of digital oscillators by pole location.

Demonstrate the design of sharp notch filters by pole-zero location.

Describe the characteristics of the four types of classical prototype analog filters

Demonstrate the design of analog filter prototypes with MATLAB.

Derive and describe the bilinear transformation.

Demonstrate the bilinear transformation method of IIR filter design.

Demonstrate the use of MATLAB functions for IIR design of filters with the response of classical analog filters.

Demonstrate the effect of coefficient quantization on the performance of IIR filters.

Concepts in IIR Filter Design

The frequency response of a DSP filter is the value of the z-domain transfer function on the unit circle

The location of the poles and zeros determines the shape of the transfer function in the complex plane

The poles must be inside the unit circle for stability

Typical IIR Filter Designs

Digital Oscillators

Notch Filters

Digital Equivalents of Classical Analog Prototypes:

Butterworth

Chebyshev I

Chebyshev II

Elliptic or Cauer

Digital Oscillators

Oscillator with frequency 0H(z)

Impulse

Digital Sinusoid

X(z)

Y(z)

Digital Oscillator Transfer Function

0 is the oscillator digital frequency in radians

A is the amplitude of the resulting sinusoid

Often called a two-pole resonator because the transfer function has 2 poles at +/- 0 exactly on the unit circle (meta-stable)

For Hertzian frequencies use = 2f/fs

Oscillator Design Example

Design an oscillator with a frequency of 200 Hz in a system operating with a sampling frequency of 8 kHz. The MATLAB solution is:

>> f=200;>> fs=8000;>> omega=2*pi*f/fs;>> b=[0,sin(omega)];>> a=[1,-2*cos(omega),1];>> fvtool(b,a) % Use fvtool to display various results

Oscillator Design Example Results

At a sampling frequency of 8 kHz each sample is 0.125 ms. 40 samples = 5 ms = the period of a 200 Hz sine function.

Oscillator Design Example Results

Pole locations = 0 = 2 (200/8000) = 0.1571 radians

Notch Filters

Notch filters are designed by pole/zero location

The zeros are located at the notch frequencies

Poles are placed close to the zeros locations, just inside the unit circle, to control the notch width.

A gain factor is included to hold the filter gain to unity at all other frequencies

Notch filters for multiple frequencies can be designed by cascading filters or, equivalently, by convolving the a and b coefficient vectors of individual filters

Notch Filter Transfer Function

The following is the transfer function for a notch filter for a notch frequency 0 and -3 dB width (or quality factor Q). The parameter r is the pole radius. The gain factor is g0. Note the trade-off between pole radius and notch width.

Notch Filter Design Example

Design a notch filter in MATLAB with a notch frequency of 0 = /4 and a Q factor of 20

>> omega=pi/4;>> Q=20;>> delta_omega=omega/Q;>> r=1-delta_omega/2;>> g=abs(1-2*r*cos(omega)+r^2)/(2*abs(1-cos(omega))); % The g0 factor>> bn=g*[1,-2*cos(omega),1]; % The b coefficients of the notch filter>> an=[1,-2*r*cos(omega),r^2]; %The a coefficients of the notch filter>> fvtool(bn,an)

Design Example Results

Design Example Results

Note the zeros on the unit circle and corresponding poles just inside the unit circle at 0 = /4

Analog Filter Prototypes

Filter Specifications

Transfer Functions

Analog Filter:

Digital Filter:

MATLAB Prototype Filter Design Commands

[B,A] = BUTTER(N,Wn)

[B,A] = CHEBY1(N,R,Wn)

[B,A] = CHEBY2(N,R,Wn)

[B,A] = ELLIP(N,Rp,Rs,Wn)

N = filter order

R = pass band ripple (cheby1) or stop-band ripple (cheby2) in dB. (Rp and Rs respectively for the elliptic filter)

Wn = cut-off frequency (radians/sec for analog filters or normalized digital frequencies for digital filters)

[B,A] = filter coefficients, s-domain (analog filter) or z-domain (digital filter)

Analog Design Example

Design an order 4 Elliptic analog filter with a cutoff frequency of 10 Hz, a maximum pass-band ripple of 1 dB, and a minimum stop-band attenuation of -20 dB.

>> cutoff=2*pi*10; % Set the filter parameters>> order=4; >> Rp=1;>> Rs=20;>> [b,a]=ellip(order,Rp,Rs,cutoff,'s'); % Note the s option for an analog filter>> W=linspace(0,2*pi*20); % Create a 100 point linear frequency vector 0 to 20 Hz>> [H,f]=freqs(b,a,W); % The freqs command returns the complex value of the transfer function for the frequency vector W (copied into the vector f) >> plot(f/(2*pi),abs(H)) % Plot the magnitude of H versus the frequency in Hertz>> title('Order 4 Elliptic Filter with 10 Hz Cutoff Frequency')>> xlabel('Frequency, Hz')>> ylabel('Magnitude Response')

Analog Design Example Results

Digital Design of Analog Prototypes
The Bilinear Transformation

The bilinear transformation maps the complex variable s in the analog transfer function to the complex variable z in the digital transfer function

Bilinear Transformation Mapping

Pre-Warping Equation

Design Steps for a DSP Implementation of an Analog Design

Determine the desired cut-off frequency for the digital filter, 0

Compute the equivalent cut-off frequency for the analog filter, 0, using the pre-warping equation.

Design the analog filter (i.e., find its a and b coefficient vectors)

Using the bi-linear transformation (s z), compute the coefficients of the digital filter

MATLAB IIR Design Tools
General Design Approach

The MATLAB method for IIR filter design is a two command process; first, to determine the order and critical frequencies, second to compute the filter coefficients. For the a Butterworth filter:

[N, Wn] = BUTTORD(Wp, Ws, Rp, Rs)

[B,A] = BUTTER(N,Wn,'type') where the option type can be either high or stop if specified.

The command parameters are:

Wp = pass-band edges in units of

Ws = stop-band edges in units of

Rp = pass-band ripple in dB

Rs = stop-band ripple in dB

Design Example

Filter Specifications:

Butterworth response

Pass-band edges = 400 Hz and 600 Hz

Stop-band edges = 300 Hz and 700 Hz

Pass-band ripple = 1 dB

Stop-band attenuation = -20 dB

Sampling Frequency = 2000 Hz

MATLAB Code for Design Example

>> fs=2000;

>> Wp=[2*400/fs,2*600/fs]; % Normalized digital frequencies of pass-band edges>> Ws=[2*300/fs,2*700/fs]; % Normalized digital frequencies of stop-band edges>> [N,Wn]=buttord(Wp,Ws,1,20); % The order command>> [B,A]=butter(N,Wn); % The filter command>> fvtool(B,A)

Design Example Results

Band Edges (-1dB and -20 dB)

Design Example
Chebyshev II High-Pass Filter

Filter specifications:

Chebyshev II response (stop-band ripple)

Pass-band edge = 1000 Hz

Stop-band edge = 900 Hz

Pass-band ripple = 1 dB

Stop-band attenuation = -40 dB

Sampling frequency = 8 kHz

MATLAB Code for Design Example

>> fs=8000;

>> Wp=[2*1000/fs]; % Pass-band edge normalized digital frequency>> Ws=[2*900/fs]; % Stop-band edge normalized digital frequency>> [N,Wn]=cheb2ord(Wp,Ws,1,40); % The order command>> [B,A]=cheby2(N,40,Wn,'high'); % cheby2 is the filter command. In this command % the syntax requires the stop-band attenuation % as the second parameter>> fvtool(B,A)

Design Example Results

Comparison of an Elliptic Filter with a Parks-McClellan Design

Filter Specification:

Low-pass response

Pass-band edge = 475 Hz

Stop-band edge = 525 Hz (i.e., a transition width of 50 Hz)

Pass-band ripple less than 0.01 in absolute terms ( = 20log10(1-.01) = 0.0873 dB)

Stop-band attenuation greater than -40 dB (= 0.01 ripple in absolute terms)

Sampling frequency = 2000 Hz

Finding the Order of a P-M Design

[N,Fo,Ao,W] = FIRPMORD(F,A,DEV,Fs)

B = FIRPM(N,Fo,Ao,W)

N = order

F = band edges, in units of , or in Hz if Fs is specified

A = amplitudes corresponding to the bands defined by the edges in F [length(F) must be 2*length(A)-2]

DEV = deviation (ripple) in each band defined by F in absolute units (not dB)

Fs = sampling frequency in Hz

P-M Design to Specifications

>> F = [475,525];>> A = [1,0];>> DEV = [.01,.01];>> Fs = 2000;>> [N,Fo,Ao,W] = firpmord(F,A,DEV,Fs);>> B = firpm(N,Fo,Ao,W);>> fvtool(B,1)>> NN = 78

P-M Design Results

Elliptic Filter Design to Specifications

>> fs=2000;>> fpass=475;>> fstop=525;>> Wp=2*fpass/fs;>> Ws=2*fstop/fs;>> Rp=.0873;>> Rs=40;>> [N,Wn]=ellipord(Wp,Ws,Rp,Rs);>> [Be,Ae]=ellip(N,Rp,Rs,Wn);>> fvtool(Be,Ae)>> NN = 7

Elliptic Filter Design Results

Coefficient Quantization

The poles of an IIR filter must remain within the unit circle in the complex plane for stability

Quantization and round-off errors can move the poles and create an unusable design

Effect of Coefficient Quantization
Chebyshev II High-pass Filter
Double Precision vs. 16 bits

>> B16=quantize(B,16);>> A16=quantize(A,16);

Summary

IIR filters can be design by pole-zero location

Digital oscillators: poles on the unit circle

Notch filters: zeros on the unit circle with nearby poles to control notch width

Classic analog filters can be designed using the bilinear transformation

IIR filters have the advantage of smaller filter order for a given frequency response.

IIR filters have the disadvantages of possible instability due to coefficient quantization effects and non-linear phase response.