Notch filters are a family of filters that include the all-pass filter. For the

Question

Notch filters are a family of filters that include the all-pass filter. For the filter H(z) = K(1 - alpha_1 z^-1)(1 + alpha_2 z^-1)/(1 - 0.5z^-1)(1 + 0.5z^-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 a and K. Plot the poles and the zeros of this filter. (b) If we would like the filter H(z) to be a notch filter of unit gain at omega = pi/2 rad, determine the values of alpha and K to achieve this and then determine where the notch(es) are Use MATLAB functions to verify that the filter is a notch filter, and to plot the poles and the zeros. (c) Place the zeros of H(z) at positions between the zeros for the all-pass and the notch filters, and use MATLAB to plot the magnitude responses. Each of these filters must have unit gain at omega = pi/2 rad. Explain the connection between the all-pass and the notch filters. (d) Suppose we use the transformation z^-1 = jZ^-1 to obtain a filter H(Z). Repeat the above part of the problem for H(Z). Where are the notches of this new filter. What would be the difference between the all-pass 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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