For all parts of this problem, H(ejω) is the frequency response of a DT filter and can b...

For all parts of this problem, H(ejω) is the frequency response of a DT filter and can be expressed in polar coordinates as

where A(ω) is even and real-valued and θ(ω) is a continuous, odd function of ω for −π < ω < π, i.e., θ(ω) is what we have referred to as the unwrapped phase. Recall:

• The group delay τ(ω) associated with the filter is defined as

• An LTI filter is called minimum phase if it is stable and causal and has a stable and causal inverse.

For each of the following statements, state whether it is TRUE or FALSE. If you state

that it is TRUE, give a clear, brief justification. If you state that it is FALSE, give a simple

counterexample with a clear, brief explanation of why it is a counterexample.

(a) “If the filter is causal, its group delay must be nonnegative at all frequencies in the range |ω| < π.”

(b) “If the group delay of the filter is a positive constant integer for |ω| < π the filter must be a simple integer delay.”

(c) “If the filter is minimum phase and all the poles and zeros are on the real axis then