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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 $ :
For each case use at least 512 points in the plot of the DTFT.
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.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:
The low pass filter designed with the Kaiser method will automatically have a cut-offfrequency centered between ${\omega}_{p}$ and ${\omega}_{s}$ .
Plot the magnitude of the DTFT of $h\left(n\right)$ for $\left|\omega \right|<\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.
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
$\left|\omega \right|<\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.
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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