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The outputs of the command are n = filter length - 1 , and the vectors fo , mo , and w which are intermediate filter parameters.

Once the filter length, n , is obtained, the Matlab command for designing a Parks-McClellan filteris b = firpm(n,fo,mo,w) . The inputs n , fo , mo , and w are the corresponding outputs of firpmord , and the output b is a vector of FIR filter coefficients such that

H ( z ) = b ( 1 ) + b ( 2 ) z - 1 + + b ( n + 1 ) z - n

(What is the impulse response of this filter?)

For further information, read the help document on using Matlab to implement the Parks-McClellan algorithm.

Now design a symmetric FIR filter using firpmord and firpm in Matlab to meet the design specifications given in the "Filter Design Using the Kaiser Window" section. Compute the DTFT of the filter's response for at least 512 points,and use this result to compute the passband and stopband ripple of the filter that was designed.Adjust the filter length until the minimum order which meets the design constraints is found.Plot the magnitude of the DTFT in dB of the final filter design.

Inlab report

Do the following:
  1. Submit the final measured values of filter length, passband ripple, and stopband ripple.How accurate was the filter order computation using Matlab's firpmord ? How does the length of this filter compare to the filterdesigned using a Kaiser window?
  2. Submit the plot of the filter's DTFT. How does the frequency response of the Parks-McClellan filtercompare to the filter designed using the Kaiser window? Comment on the shape of both the passband and stopband.

Use the filter you have designed to remove the noise from the signal nspeech2.mat . Play the noisy and filtered speech signals back using sound and listen to them carefully. Compute theDTFT of 400 samples of the filtered signal starting at time n = 20 , 001 (i.e. 20001:20400 ).Plot the magnitude of the DTFT in decibels versus frequency in radians for | ω | < π . Compare this with the spectrum of the noisy speech signal shownin [link] , and also with the magnitude of the DTFT of the Kaiser filtered signal.

Submit the plot of the DTFT magnitude for the filtered signal. Comment on how the audio qualityof the signal changes after filtering. Also comment on any differences in audio quality between the Parks-McClellan filteredspeech and the Kaiser filtered speech.

Design of discrete-time iir filters using numerical optimization

In this section, we consider the design of discrete-time IIR filters through the direct search of filter parametersthat will minimize a specific design criterion. Such “brute force” approaches to filter design have becomeincreasingly more popular due to the wide availability of high speed computers and robust numerical optimization methods.

Typically, numerical approaches to filter design have two parts. First, they design a cost , or error criterion. This criterion is a measure of the difference betweenthe ideal filter response and the response of the computed or “approximate” filter. The goal is to findthe approximate filter with the lowest cost (error). Mean square error isa popular cost criterion. The second part is to minimize the cost with respect to the filter parameters.We will perform the required numerical optimization with the fminsearch function in Matlab's Optimization Toolbox .

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