0.6 Digital filtering and the dft  (Page 13/13)

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Designing IIR filters is also straightforward. For example,the code [b,a]=cheby1(10,5,0.25); designs a 10th order lowpass filter with 5 dB of ripple in the passband, which extends from DC to 0.25 times the Nyquist rate.

While commands such as firpm and cheby1 make filter design easy, be warned—strange things can happen,even to nice filters. Whenever using any of the filter design tools, always check freqz(b,a) to verify that the output of the design behaves as anticipated.There are many other ways to design LTI filters, and M atlab includes several commands that design filter coefficients: cfirpm , firls , fir1 , fir2 , butter , cheby2 , and ellip . The subject of filter design is vast, and each of theseis useful in certain applications. For simplicity, we have chosen to present most examples throughout Software Receiver Design by using firpm .

Rerun bandex.m with very narrow transition regions, for instance fbe =[0, 0.24, 0.2401, 0.6, 0.601, 1]. What happens to theripple in the passband? Compare the minimum magnitude in the passband with the maximum value in the stopband.

Returning to the filter specified in [link] , try using different numbers of terms in the impulse response, fl= 5, 10, 100, 500, 1000. Comment on the resulting designs in terms of flatness of the frequency response in the passband,attenuation from the passband to the stopband, and the width of the transition band.

Specify and design a lowpass filter with cutoff at $0.15$ . What values of fl , fbe , and damps work best?

Specify and design a filter that has two passbands, one between [0.2, 0.3] and another between [0.5 0.6]. What values of fl , fbe , and damps work best?

Use the filter designed in bandex.m to filter a white noise sequence (generated by randn ) using the time domain method from waystofilt.m . Verify that the output has a bandpass spectrum.

The preceding filter designs do not explicitly require the sampling rate of the signal. However, since the sampling rate determinesthe Nyquist rate, it is used implicitly. The next exercise asks that you familiarize yourself with “real” units of frequencyin the filter design task.

In Exercise  [link] , the program specgong.m was used to analyze the sound of an Indonesian gong. The three most prominent partials (or narrowband components)were found to be at about 520, 630, and 660 Hz.

1. Design a filter using firpm that will remove the two highest partials from this sound without affecting the lowest partial.
2. Use the filter command to process the gong.wav file with your filter.
3. Take the FFT of the resulting signal (the output of your filter) and verify that the partial at 520 remains while the others areremoved.
4. If a sound card is attached to your computer, compare the sound of the raw and the filtered gong sound by using M atlab s sound command. Comment on what you hear.

The next set of problems examines how accurate digital filters really are.

With a sampling rate of 44100 Hz, let $x\left[k\right]$ be a sinusoid of frequency 3000 Hz. Design a lowpass filter with a cutofffrequency fl of 1500 Hz, and let $y\left[k\right]=LPF\left\{x\left[k\right]\right\}$ be the output of the filter.

1. How much does the filter attenuate the signal? (Express your answer as the ratio of the power in the output $y\left[k\right]$ to the power in the input $x\left[k\right]$ .)
2. Now use a LPF with a cutoff of 2500 Hz. How much does the filter attenuate the signal?
3. Now use a LPF with a cutoff of 2900 Hz. How much does the filter attenuate the signal?

without using the filter command (implement the filtering, using the time domain method in waystofilt.m ).

With the same setup as in Exercise  [link] , generate $x\left[k\right]$ as a bandlimited noise signal containing frequencies between 3000 Hz and the Nyquist rate.

1. Use a LPF with cutoff frequency fl of 1500 Hz. How much does the filter attenuate the signal?
2. Now use a LPF with a cutoff of 2500 Hz. How much does the filter attenuate the signal?
3. Now use a LPF with a cutoff of 3100 Hz. How much does the filter attenuate the signal?
4. Now use a LPF with a cutoff of 4000 Hz. How much does the filter attenuate the signal?

Let ${f}_{1}<{f}_{2}<{f}_{3}$ . Suppose $x\left[k\right]$ has no frequencies above ${f}_{1}$ Hz, while $z\left[k\right]$ has no frequencies below ${f}_{3}$ . If a LPF has cutoff frequency ${f}_{2}$ . In principle,

$\begin{array}{ccc}\hfill \text{LPF}\left\{x\left[k\right]+z\left[k\right]\right\}& =& \text{LPF}\left\{x\left[k\right]\right\}+\text{LPF}\left\{z\left[k\right]\right\}\hfill \\ & =& x\left[k\right]+0=x\left[k\right].\hfill \end{array}$

Explain how this is (and is not) consistent with the results of Exercises  [link] and  [link] .

Let the output $y\left[k\right]$ of a LTI system be created from the input $x\left[k\right]$ according to the formula

$y\left[k+1\right]=y\left[k\right]+\mu x\left[k\right],$

where $\mu$ is a small constant. This is drawn in [link] .

1. What is the impulse response of this filter?
2. What is the frequency response of this filter?
3. Would you call this filter lowpass, highpass, or bandpass?

Using one of the alternative filter design routines ( cfirpm , firls , fir1 , fir2 , butter , cheby1 , cheby2 , or ellip` ), repeat Exercises  [link][link] . Comment on the subtle (and the not-so-subtle)differences in the resulting designs.

The effect of bandpass filtering can be accomplished by

1. modulating to DC,
2. lowpass filtering, and
3. modulating back.

Repeat the task given in Exercise  [link] (the Indonesian gong filter design problem) by modulating with a 520 Hz cosine, lowpass filtering,and then remodulating. Compare the final output of this method with the direct bandpass filter design.

Here are some of our favorite books on signal processing:

• K. Steiglitz, A Digital Signal Processing Primer , Addison-Wesley, Pubs, 1996.
• J. H. McClellan, R. W. Schafer, M. A. Yoder, DSP First: A Multimedia Approach , Prentice Hall, 1998.
• C. S. Burrus and T. W. Parks, DFT/FFT and Convolution Algorithms: Theory and Implementation, Wiley-Interscience, 1985.

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