function [a,b] = cascadeIIRNotch(notchFrequencies, alpha, beta)
% y = IIRNotchFilter(soundfilename, zeroCoefficients, poleCoefficients, alpha)
% Computes the coefficients for a cascaded IIR digital notch filter. Each
% cascaded IIR notch filter is a second order filter. Thus, for each notch
% frequency specified a second order filter is added to the cascaded system.
% notchFrequencies A vector specifying the digital frequencies to be notched.
% alpha The magnitude of the poles
% beta The magnitude of the zeros
% a Coefficients of the numerator polynomial
% b Coefficient of the denominator polynomial
% Cascaded Digital Notch Filter Implementation
% Dr. Ratliff
% ECE505

%Calculate locations of each pole and zero, noting that each pole-zero
%also has a matching complex conjugates. All I am doing here is using the
%convolution command to multiply out the transfer function polynomials to
%obtain the coefficients.
w = notchFrequencies;
for k=1:length(w)
Z = conv([ 1 -beta*exp(1i*w(k))], [ 1 -beta*exp(-1i*w(k))]);
P = conv([ 1 -alpha*exp(1i*w(k))], [ 1 -alpha*exp(-1i*w(k))]);
Z = conv(Z, [ 1 -beta*exp(1i*w(k))]);
Z = conv(Z, [ 1 -beta*exp(-1i*w(k))]);
P = conv(P, [ 1 -alpha*exp(1i*w(k))]);
P = conv(P, [ 1 -alpha*exp(-1i*w(k))]);

%Take the real part of the coefficients for use in the difference equation and display them
a = real(Z); b = real(P);
disp(sprintf(‘Cascaded Notch Filter Numerator (Zero) Coefficients’));
disp(sprintf(‘Cascaded Notch Filter Denominator (Pole) Coefficients’));
disp(b);ECE505 Digital Signal Processing
Laboratory Project – IIR Notch Filter Design

Dr. Ratliff

1 Project Definition:

In this project, you are given a digital audio clip that has been corrupted by sinusoidal tones resulting from
pressing the numeric keys on a touch-tone phone. The goal is to remove the tones from the signal by designing
cascaded digital notch filters. This project is designed to make use of the mathematical tools we have learned
this semester by applying them to a practical DSP problem.

2 Problem Description:

The Touch-Tone phone system, developed during the 1960’s by AT&T, was created to eliminate the need for
rotary-dialing telephone systems. This is accomplished using Dual-Tone Multi-Frequency (DTMF) signaling
and has resulted in the familiar tones we hear when dialing a telephone. As the name implies, each tone is
generated by introducing a sinusoidal wave at two distinct frequencies. As such, the DTMF telephone keypad
is laid out in a 4 × 4 matrix of push buttons in which each row represents the low frequency component and
each column represents the high frequency component of the DTMF signal. Pressing a key sends a combination
of the row and column frequencies, as described in the figure below:

For example, pressing the ‘1’ key produces a superposition of tones at 697 and 1209 hertz (Hz). When these
keys are pressed, they are mixed with the audio signal from the phone microphone. When such an audio stream
is digitized and recorded, it may be desirable to later remove these tones from the audio stream. One way to
do this is to design a group of filters such that each respective filter removes the content at a single frequency.
Such filters are called digital notch filters. A block diagram for such a system can be represented by:

In this case each hk[n], for k = 1, 2, . . . , 8, represents a digital notch filter for one of the numeric tone frequencies
from the DTMF keypad. Thus, if such an ideal filter bank is employed, it would indeed remove the tones resulting
from any key-press. Notice that the total impulse response of such a filter bank can be obtained as

h[n] = h1[n] ∗ h2[n] ∗ . . . ∗ h8[n]. (1)


In this project we are going to design second order IIR notch filters to remove each sinusoidal frequency term.
Thus, we expect to have two poles and two zeros for each filter. For notch filters of this type the design is straight-
forward. For each frequency we want to filter, we simply place a pole and zero along a line corresponding to
the digital frequency we wish to notch, along with their respective complex conjugate pairs, as shown in the
pole-zero diagram below.

Notice that the zero is placed upon the unit circle with a magnitude of 1, i.e., znk
= e±jθk , and each pole is

placed at the same angle, θk, only at a magnitude of α, i.e., zdk = αe±jθk , where 0 < α < 1, znk denotes system zeros and zdk denote system poles. Here θk corresponds directly to the digital frequency of the sinusoid to be notched. Later we

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