Frequency transformation of low-pass to high-pass filters—You have designed an IIR low-pass filter with an input-output relation given by the difference equation where x[n] is the input and y[n] the output. You are told that by changing the difference equation to you obtain a high-pass filter. (a) From the eigenfunction property find the frequency response of the two filters at ω = 0, π/2 and π radians. Use the MATLAB functions freqz and abs to compute the magnitude responses of the two filters. Plot them to verify that the filters are low-pass and high-pass. (b) Call H1(e jω) the frequency response of the first filter and H2(e jω) the frequency response of the second filter. Show that h[n] =    α|n| −2 ≤ n ≤ 2 0 otherwise (i) y[n] = 0.5y[n − 1] + x[n] + x[n − 1] n ≥ 0 (ii) y[n]=−0.5y[n − 1] + x[n] − x[n − 1] n ≥ 0 761 H2(e jω) = H1(e j(π−ω)) and relate the impulse response h2[n] to h1[n]. (c) Use the MATLAB function zplane to find and plot the poles and zeros of the filters and determine the relation between the poles and zeros of the two filters.

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a From the eigenfunction property find the frequency response of the two filters at 0 2 and radians ... View the full answer