Recursive construction of polynomial with prescribed phase on unit circle

On the phase interpolation problem — A brief review and some new results

“…We present in this section some new generalizations and extensions of the results of the previous section. Merchant's (1988) recursive formulation, though very elegant, is not programmable on a computer. We have extended his technique for computer implementation (Minocha et al 1991).…”

“…In a recent contribution, Merchant (1988) has proposed a recursive technique for the reconstruction, which involves finding a polynomial PN(z), of order N, which satisfies phases at N given frequencies.…”

“…(19e) Merchant (1988) has cited two situations either of which will cause the recursion to terminate, and has also suggested remedial actions. We summarize his findings for the sake of completeness.…”

“…Since the recursion constructs a sequence of polynomials which satisfy a growing subset of the prescribed frequency/phase pairs, each subset also needs to satisfy the same condition, In the present situation where a given subset has a linear phase component but the complete set does not, the procedure will not work unless some remedial action is taken. Merchant (1988) suggested that the frequency/ phase pairs should be reordered by circulating them and then the recursion can be continued.…”

“…Again the frequencies can be circularly shifted as in case 1 and reeursion continued. However, it may happen that even after taking care of these situations, the recursion may terminate at a polynomial of order less than the desired order N (Merchant 1988). This indicates the presence of a linear phase component in the full set of frequency-phase pairs which needs to be removed before the polynomial can be constructed.…”