Question

This problem concerns a discrete-time filter with a real-valued impulse response h[n]. Determine whether the following statement is true or false:

Statement: If the group delay of the filter is a constant for 0<ω<π, then the impulse response must have the property that either
h[n] = h[M - n]
or
h[n] = -h[M - n]
where M is an integer.
If the statement is true, show why it is true. If it is false, provide a counterexample.

Answer 1

The statement is true. If the group delay of the filter is constant for 0<ω<π, then the impulse response must have the property that either h[n] = h[M - n] or h[n] = -h[M - n] where M is an integer.

Step-by-step explanation:

The statement is true. If the group delay of the filter is constant for 0<ω<π, it implies that the filter has linear phase response. For a filter with linear phase response, the impulse response must satisfy the property h[n] = h[M - n] or h[n] = -h[M - n], where M is an integer.

To understand why this is true, let's consider the frequency response of the filter. The group delay is related to the phase response, and a constant group delay means a linear phase response. A linear phase response corresponds to a frequency response that is symmetric around ω=π/2, which leads to the two possible forms of the impulse response mentioned.

For example, if we have a filter with group delay 2, the impulse response will satisfy either h[n] = h[2 - n] or h[n] = -h[2 - n].