# 0.6 Lab 5b - digital filter design (part 2)  (Page 3/6)

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To further investigate the Kaiser window, plot the Kaiser windows and their DTFT magnitudes (in dB)for $N=21$ and the following values of $\beta$ :

• $\beta =0$
• $\beta =1$
• $\beta =5$

For each case use at least 512 points in the plot of the DTFT.

To create theKaiser windows, use the Matlab command kaiser(N,beta) command where N is the length of the filter and betais the shape parameter $\beta$ . To insure at least 512 points in the plot use the command DTFT(window,512) when computing the DTFT.
Submit the plots of the 3 Kaiser windows and the magnitude of their DTFT's in decibels.Comment on how the value $\beta$ affects the shape of the window and the sidelobes of the DTFT.

Next use a Kaiser window to design a low pass filter, $h\left(n\right)$ , to remove the noise from the signal in nspeech2.mat using equations [link] and [link] . To do this, use equations [link] and [link] to compute the values of $\beta$ and $N$ that will yield the following design specifications:

$\begin{array}{ccc}\hfill {\omega }_{p}& =& 1.8\hfill \\ \hfill {\omega }_{c}& =& 2.0\hfill \\ \hfill {\omega }_{s}& =& 2.2\hfill \\ \hfill {\delta }_{p}& =& 0.05\hfill \\ \hfill {\delta }_{s}& =& 0.005\hfill \end{array}$

The low pass filter designed with the Kaiser method will automatically have a cut-offfrequency centered between ${\omega }_{p}$ and ${\omega }_{s}$ .

${\omega }_{c}=\frac{{\omega }_{p}+{\omega }_{s}}{2}$

Plot the magnitude of the DTFT of $h\left(n\right)$ for $|\omega |<\pi$ . Create three plots in the same figure:one that shows the entire frequency response, and ones that zoom in on the passband and stopband ripple, respectively.Mark ${\omega }_{p}$ , ${\omega }_{s}$ , ${\delta }_{p}$ , and ${\delta }_{s}$ on these plots where appropriate. Note:Since the ripple is measured on a magnitude scale, DO NOT use a decibel scale on this set of plots.

From the Matlab prompt, compute the stopband and passband ripple (do not do this graphically). Record the stopband and passband ripple tothree decimal places.

To compute the passband ripple, find the value of the DTFT at frequencies corresponding to the passband using the command H(abs(w)<=1.8) where H is the DTFT of $h\left(n\right)$ and w is the corresponding vector of frequencies. Then use this vector to compute the passband ripple.Use a similar procedure for the stopband ripple.

Filter the noisy speech signal in nspeech2.mat using the filter you have designed. Then compute theDTFT of 400 samples of the filtered signal starting at time $n=20000$ (i.e. 20001:20400 ).Plot the magnitude of the DTFT samples in decibels versus frequency in radians for $|\omega |<\pi$ . Compare this with the spectrum of the noisy speech signal shownin [link] . Play the noisy and filtered speech signals back using sound and listen to them carefully.

## Inlab report

Do the following:
1. Submit the values of $\beta$ and $N$ that you computed.
2. Submit the three plots of the filter's magnitude response. Make sure the plots are labeled.
3. Submit the values of the passband and stopband ripple. Does this filter meet the design specifications?
4. Submit the magnitude plot of the DTFT in dB for the filtered signal.Compare this plot to the plot of [link] .
5. Comment on how the frequency content and the audio quality of the filtered signal have changed afterfiltering.

## Fir filter design using parks-mcclellan algorithm

Click here for help on the firpm function for Parks-McClellan filter design. Download the data file nspeech2.mat for the following section.

Kaiser windows are versatile since they allow the design of arbitrary filters which meet specific design constraints.However, filters designed with Kaiser windows still have a number of disadvantages. For example,

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