A maximum-phase sequence is a stable sequence whose z-transform has all its poles and ze...

A maximum-phase sequence is a stable sequence whose z-transform has all its poles and zeros outside the unit circle.

(a) Show that maximum-phase sequences are necessarily anti-causal, i.e., that they are zero for n > 0.

FIR maximum-phase sequences can be made causal by including a finite amount of delay. A finite-duration causal maximum-phase sequence having a Fourier transform of a given magnitude can be obtained by reflecting all the zeros of the z-transform of a minimum-phase sequence to conjugate-reciprocal positions outside the unit circle. That is, we can express the z-transform of a maximum-phase causal finite-duration sequence as

Obviously, this process ensures that |Hmax(e^jω)| = |Hmin(e^jω)|. Now, the z-transform of a finite-duration minimum-phase sequence can be expressed as

(b) Obtain an expression for the all-pass system function required to reflect all the zeros

of H_min(z) to positions outside the unit circle.

(c) Show that H_max(z) can be expressed as

(d) Using the result of part (c), express the maximum-phase sequence h_max[n] in terms of h_min[n].