Consider the class of FIR filters that have h[n] real, h[n] = 0 for n < 0 and n >...

Consider the class of FIR filters that have h[n] real, h[n] = 0 for n < 0 and n > M, and one of the following symmetry properties:

Symmetric: h[n] = h[M − n]

Antisymmetric: h[n] = −h[M − n]

All filters in this class have generalized linear phase, i.e., have frequency response of the form

H(e ^{j ω} ) = A(e^jω)e^−jαω+jβ ,

where A(e^jω) is a real function of ω, α is a real constant, and β is a real constant.

For the following table, show that A(e^jω) has the indicated form, and find the values of α and β.

Here are several helpful suggestions.

• For type I filters, first show that H(e^jω) can be written in the form

The analysis for type III filters is very similar to that for type I, with the exception

of a sign change and removal of one of the preceding terms.

• For type II filters, first write H(e^jω) in the form

and then pull out a common factor of e^−jω(^M/2) from both sums.

• The analysis for type IV filters is very similar to that for type II filters