Notch and all-pass filters-Notch filters is a family of filters that includes th

Question

Notch and all-pass filters-Notch filters is a family of filters that includes the all-pass filter. For the filter H(z) = K (1 - alpha_1 z^-1) (1 + alpha_2 z^-1)/(1 - 0.5 z^-1) (1 + 0.5 z^-1) (a) Determine the values of alpha_1, alpha_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 or 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 omega = pi/2 (rad), and notches at omega = 0 and pi, determine the values of alpha 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 alpha_1 = alpha_2 = alpha and 1 lessthanorequalto alpha lessthanorequalto 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 omega = pi/2. (d) Suppose we use the transformation z^-1 = jZ^-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)?

Explanation / Answer

Folding / Reflection

: It is folding of signal about time origin n=0. In this case replace n by

–

n.

Original signal:

X(n) = { 1,

-

1 , 0 , 4 ,

-

2 , 4 , 0}

n=0

Folded signal:

X(

-

n) = {

0 , 4 ,

-

2 , 4 , 0 ,

-

1 , 1}

n=0

3)

Addition

: Given signals are x1(n) and x2(n), which produces output y(n) where y(n) = x1(n)+ x2(n).

Adder generates the output sequence which is the sum of input sequences.

4)

Scaling

: Amplitude scaling can be

done by multiplying signal with some constant. Suppose original

signal is x(n). Then output signal is A x(n)

5)

Multiplication

: The product of two signals is defined as y(n) = x1(n) * x2(n)

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