= k) of Problem 3 Design a causal, linear phase, low pass filter (cutoff = 0c...

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= k) of Problem 3 Design a causal, linear phase, low pass filter (cutoff = 0c and gain length 3 using the optimization (CAD) method discussed in lecture 20 by following the steps outlined below. h(n) = bod(n) + b₁d(n − 1) + bod(n − 2) (a) Show that the mean square error (bo, b₁; K, 0c) is given by k²0c 2kb1 + (262 +6²) -- sinec -sin20c ㅠ πT (b) Show that the values of bo and b₁ that minimize the mean square error e(bo, b₁; K, 0c) are given by €(bo, b₁; k, 0c) = k k 4π -sinec π Make sure that you verify that the Jacobian J for the above values of bo and b₁ gives a minimum, i.e., J> 0. bo = -sin20 and b₁ kbo π = = (c) Plot the normalized magnitude H(0)| and phase ZH(0) for cutoff c π/4. (To plot the normalized magnitude, divide |H(0)| by |H(0)|max.) On the |H(0)| plot, show also the normalized ideal low pass filter magnitude with cutoff at c. Comment on the magnitude and phase response of the designed filter. = (d) Plot the normalized magnitude |H(0)| and phase ZH(0) for cutoff Oc π/2. On the |H(0)| plot, show also the normalized ideal low pass filter magnitude with cutoff at 0c. Com- ment on the magnitude and phase response of the designed filter. (e) Plot the normalized magnitude | H(0)| and phase ZH(0) for cutoff 0e = 3π/4. On the H(0) plot, show also the normalized ideal low pass filter magnitude with cutoff at 0c. Comment on the magnitude and phase response of the designed filter. = k) of Problem 3 Design a causal, linear phase, low pass filter (cutoff = 0c and gain length 3 using the optimization (CAD) method discussed in lecture 20 by following the steps outlined below. h(n) = bod(n) + b₁d(n − 1) + bod(n − 2) (a) Show that the mean square error (bo, b₁; K, 0c) is given by k²0c 2kb1 + (262 +6²) -- sinec -sin20c ㅠ πT (b) Show that the values of bo and b₁ that minimize the mean square error e(bo, b₁; K, 0c) are given by €(bo, b₁; k, 0c) = k k 4π -sinec π Make sure that you verify that the Jacobian J for the above values of bo and b₁ gives a minimum, i.e., J> 0. bo = -sin20 and b₁ kbo π = = (c) Plot the normalized magnitude H(0)| and phase ZH(0) for cutoff c π/4. (To plot the normalized magnitude, divide |H(0)| by |H(0)|max.) On the |H(0)| plot, show also the normalized ideal low pass filter magnitude with cutoff at c. Comment on the magnitude and phase response of the designed filter. = (d) Plot the normalized magnitude |H(0)| and phase ZH(0) for cutoff Oc π/2. On the |H(0)| plot, show also the normalized ideal low pass filter magnitude with cutoff at 0c. Com- ment on the magnitude and phase response of the designed filter. (e) Plot the normalized magnitude | H(0)| and phase ZH(0) for cutoff 0e = 3π/4. On the H(0) plot, show also the normalized ideal low pass filter magnitude with cutoff at 0c. Comment on the magnitude and phase response of the designed filter.

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To design the causal linear phase low pass filter of length 3 using the optimization CAD method lets ... View the full answer