A line spectrum approach for the design of optimum compaction FIR filters

In this paper we propose a new design method for optimal compaction gain FIR filters of a finite order. Starting from the odd polyphase component r(1), r(3),…, r(2K − 1) of the input correlation sequence we find a suitable extension to an infinite sequence r
e
(k), such that the corresponding spectrum S
e
(ω) is a line spectrum with K
1
(K
1
≤ (K + l)/2) lines. In quite general conditions it is possible to design the optimal compaction filter H(e
^jω
) of order 2K — 1 such that it has zeros at the line frequencies of S
e
(ω) and |H(e
^jω
)|
²
obeys the Nyquist(2) condition. The analytical method presented in [1] can be easily shown to be a particular case in this framework, but our method finds the optimal solution in some of the cases when [1] fails.