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Notch filters is a family of filters that includes the all-pass filter. For the filter

(a) Determine the values of Î±_{1}, Î±_{2}, and K that would make H(z) an all-pass filter of unit magnitude. Use MATLAB to compute and plot the magnitude response of H(z) using the obtained values for Î± and K. Plot the poles and zeros of this filter.

(b) If we would like the filter H(z) to be a notch filter, of unit gain at at Ï‰ = Ï€/2(rad), and notches at Ï‰ = 0 and Ï€, determine the values of Î± and K to achieve this. Use MATLAB functions to verify that the filter is a notch filter, and to plot the poles and zeros.

(c) Use MATLAB to show that when Î±_{1} = Î±_{2} = Î± and 1 ‰¤ Î± ‰¤ 2, the given H(z) correspond to a family of notch filters with different attenuations. Determine K so that these filters are unity gain at Ï‰ = Ï€/2.

(d) Suppose we use the transformation z^{ˆ’1} = jZ^{ˆ’1} to obtain a filter H(Z) using H(z) obtained in the previous item. Where are the notches of this new filter? What is the difference between the all-filters H(z) and H(Z)?

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