0.5 Lab 5a - digital filter design (part 1)  (Page 4/5)

Consider the following transfer function for a second order IIR filter with complex-conjugate poles:

[link] shows the locations of the two poles of this filter. The poles have the form

where $r$ is the distance from the origin, and $\theta$ is the angle of $p_1$ relative to the positive real axis. From the theory of Z-transforms,we know that a causal filter is stable if and only if its poles are located within the unit circle.This implies that this filter is stable if and only if $\left|r\right|<1$ . However, we will see that by locating the poles close to theunit circle, the filter's bandwidth may be made extremely narrow around $\theta$ .

This two-pole system is an example of an IIR filter because its impulse response has infinite duration.Any filter with nontrivial poles (not at z = 0 or $\pm \infty$ ) is an IIR filter unless the poles are canceled by zeros.

Calculate the magnitude of the filter's frequency response $|H_i\left(e^{jw}\right)|$ on $\left|\omega \right|<\pi$ for $\theta =\pi /3$ and the following three values of $r$ .

• $r=0.99$
• $r=0.9$
• $r=0.7$

Put all three plots on the same figure using the subplot command.

In the following experiment, we will use the filter $H_i\left(z\right)$ to separate a modulated sinusoid from background noise.To run the experiment, first download the file pcm.mat and load it into the Matlab workspace using the command load pcm . Play pcm using the sound command. Plot 101 samples of the signal for indices (100:200),and then compute the magnitude of the DTFT of 1001 samples of pcm using the time indices (100:1100). Plot the magnitude of the DTFT samples versus radial frequencyfor $\left|\omega \right|<\pi$ . The two peaks in the spectrum correspond to the center frequency of the modulated signal.The low amplitude wideband content is the background noise. In this exercise, you will use the IIR filter described above toamplify the desired signal, relative to the background noise.

The pcm signal is modulated at 3146Hz and sampled at 8kHz.Use these values to calculate the value of $\theta$ for the filter $H_i\left(z\right)$ . Remember from the sampling theorem that a radial frequency of $2\pi$ corresponds to the sampling frequency.

Write a Matlab function IIRfilter(x) that implements the filter $H_i\left(z\right)$ . In this case, you need to use a for loop to implement the recursive difference equation. Use your calculated value of $\theta$ and $r=0.995$ . You can assume that y(n)is equal to 0 for negative values of $n$ . Apply the new command IIRfilter to the signal pcm to separate the desired signal from the background noise, and listen to the filtered signal to hear the effects.Plot the filtered signal for indices (100:200), and then compute the DTFT of 1001 samples of the filtered signal using the timeindices (100:1100). Plot the magnitude of this DTFT. In order to see the DTFT around $\omega =\theta$ more clearly, plot also the portion of this DTFT for the values of $\omega$ in the range $[\theta -0.02,\theta +0.02]$ . Use your calculated value of $\theta$ .