One of the zeros of a third order Linear Phase filter lies at z = -0.5. Findtransfer function and Impulse Response of the filter

To find the transfer function and impulse response of the filter, we can start by considering the zero at z = -0.5.

1. Transfer Function: The transfer function of a third-order linear phase filter can be written as H(z) = (z - z1)(z - z2)(z - z3), where z1, z2, and z3 are the zeros of the filter. In this case, we have one zero at z = -0.5.

Therefore, the transfer function can be written as H(z) = (z - (-0.5))(z - z2)(z - z3).

2. Impulse Response: To find the impulse response of the filter, we can take the inverse Z-transform of the transfer function.

Using partial fraction decomposition, we can express the transfer function as H(z) = A/(z - z1) + B/(z - z2) + C/(z - z3), where A, B, and C are constants.

Substituting z = 1 into the transfer function, we can find the value of A. Similarly, substituting z = 2 and z = 3, we can find the values of B and C, respectively.

Once we have the values of A, B, and C, we can write the impulse response as h[n] = A*(z1^n) + B*(z2^n) + C*(z3^n), where n is the sample index.

By substituting the value of z = -0.5 into the impulse response equation, we can find the coefficients A, B, and C, and thus determine the impulse response of the filter.