9.1

Use the window design method to design a linear phase FIR filter of order N = 24 to approximate the following ideal frequency response magnitude:

The ideal filter that we would like to approximate is a low-pass filter with a cutoff frequency ? _p = 0.2 ?. With N = 24, the frequency response of the filter that is to be designed has the form

Therefore, the delay of h( n) is ? = N/2 = 12, and the ideal unit sample response that is to be windowed is

All that is left to do in the design is to select a window. With the length of the window fixed, there is a trade-off between the width of the transition band and the amplitude of the passband and stopband ripple. With a rectangular window, which provides the smallest transition band,

and the filter is

However, the stopband attenuation is only 21 dB, which is equivalent to a ripple of ? _s = 0.089. With a Hamming window, on the other hand,

and the stopband attenuation is 53 dB, or ? _s = 0.0022. However, the width of the transition band increases to

which, for most designs, would be too wide.

9.2

Use the window design method to design a minimum-order high-pass filter with a stopband cutoff frequency ? _s = 0.22 ?, a passband cutoff frequency ? _p = 0.28