EE 643 Project #3 Spring 1997

IIR Filter Design

This project will explore and compare various methods for designing IIR filters.

There are 4 classical IIR filters and their counterparts (1) Butterworth, (2) Chebyshev, (3) Chebyshev II, (4) elliptic function. They represent four different combinations of two error approximation measures. One error measure uses the Taylor series. This method equates as many of the derivatives of the desired response as possible to those of the actual response. The other approximation minimizes the difference between the desired and the actual response over a band of frequencies. The MATLAB analog filter design programs all normalize the band edge to .

1. The analog Butterworth filter is based on a Taylor series approximation in the frequency domain with expansions at w=0 and . This filter is also called maximally flat since it is optimal in the sense that as many derivatives as possible equal zero at w=0 and . The formula for the magnitude squared of the normalized frequency response of an Nth order analog Butterworth lowpass filter is given by

   

   This response is normalized so that the magnitude squared is always 1/2 at for any N. Replacing by would allow an arbitrary band edge at .

2. The analog Chebyshev filter has a minimum maximum error over the passband and Taylor approximation at . The maximum error over a band is called the Chebyshev error.

3. The analog Chebyshev II filter (sometimes called the inverse Chebyshev filter) is a Taylor series approximation at and has minimum Chebyshev error in stop band. This is often a more practical combination of characteristics than the usual Chebyshev filter.

4. The analog elliptic function filter uses a Chebyshev approximation in both the passband and the stopband. Since the elliptic function requires the evaluation of the Jacobian elliptic functions and the complete elliptic integrals the theory is considerably more complicated than the other cases. Fortunately MATLAB does all the work for you so that all are equally easy to design.

These four optimal analog filters can be transformed into optimal digital filters with the bilinear transform that is investigated in the second half of this project. The IIR filter design programs in MATLAB take care of analog filter design and bilinear transformation into the digital form automatically.

Exercises

1. Analyze a 5th order LP IIR filter

   Use the MATLAB command butter to design a 5th-order lowpass IIR filter with sampling frequency of 2 Hz and a band edge of 0.6 Hz. Plot the magnitude and phase frequency responses using freqz. Plot the pole and zero location diagram using zplane. Plot the significant part of the impulse response using filter to give around 20 output values. Discuss briefly how the magnitude response makes sense from the pole-zero plot.

   Use the MATLAB command cheby1 to design a 5th-order lowpass IIR filter with a sampling frequency of 2 Hz, a band edge of 0.6 Hz, and a passband ripple of 0.5 dB. Plot the magnitude and the phase frequency responses. Plot the pole and zero location diagram. Discuss briefly how the magnitude response makes sense from the pole-zero plot.

   Use the MATLAB command cheby2 to design a 5th-order lowpass IIR filter with a sampling frequency of 2 Hz, a band edge of 0.6 Hz, and a maximum stopband ripple 30 dB below the passband response. Plot the magnitude and the phase frequency responses. Plot the pole and zero location diagram. Discuss briefly how the magnitude response makes sense from the pole-zero plot.

   Use the MATLAB command ellip to design a 5th-order lowpass IIR filter with a sampling frequency of 2 Hz, a band edge of 0.6 Hz, a passband ripple of 0.5 dB, and a maximum stopband ripple 30 dB below the passband response. Plot the magnitude and the phase frequency responses. Plot the pole and zero location diagram. Discuss briefly how the magnitude response makes sense from the pole-zero plot.

   In what way(s) can the performance of these filters be compared since their specifications are not the same?

   Do the impulse responses approximate the impulse response of an ideal lowpass filter? Explain.

2. Compare the order of the four designs

   The filtering specifications for a particular job had a sampling rate of 2 Hz, passband ripple of 0.1 dB, passband edge of 0.28 Hz, stopband edge of 0.32 Hz, and a stopband ripple below 30 dB. What order Butterworth, Chebyshev, Chebyshev II, and elliptic filters will meet these requirements? Use the butterord, cheb1ord, cheb2ord, and ellipord commands. Why does the elliptic filter have the lowest order?
   

3. Design a lowpass filter for an audio signal

   Read in the file /opt/demo/audio/doorbell.au using auread. Using the fft function, determine the frequencies of the upper and lower tone of the doorbell. Design a filter such that the lower tone is attenuated no more than 3 dB while the upper tone is attenuated at least 15 dB. Design all four types of filter. Listen to the filtered signal for each case. Compare the results for each of the filters in terms of sound and filter order.

   What happens if you attempt to design and implement a Butterworth filter with passband attenuation less than 1 dB and stopband attenuation greater than 60 dB? Explain.

Write a short report describing your findings. The report should contain a concise description of your results. Include listings of your function. Be sure to answer all questions. You are not expected to include in the report all plots which you were required to do; instead, you should summarize in your report the important features of the unincluded plots.

For further help:

• Matlab Primer

• Matlab Help Desk

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