The theory and application of reciprocal pairs of periodic sequences

Abstract

Recently, there has been a substantial interest shown in the development of the concept of inverse filtering in which the inverse $\tilde s$(t), of a given sequence s(t) is such that the convolution between s(t) and $\tilde s$(t) is a perfect impulse. In the context of this paper, the sequences s(t) and $\tilde s$(t) will be termed a reciprocal pair of sequences. The authors develop the theory of reciprocal pairs of periodic sequences via the problem of linear time-invariant (LTI) system identification. Initially, the classical correlation-based method of system identification is considered which makes use of test signals with impulsive autocorrelation functions. Analysis of the effects of a non-impulsive autocorrelation function is addressed and a method for its compensation is presented. A new more general formulation of the identification problem is then proposed; it is demonstrated that the classical correlation-based approach is a special case of this more general formulation. Employing the generalised approach and inverse filtering, we show that it is possible to obtain accurate estimates of LTI impulse responses using test signals with non-impulsive autocorrelation functions. The performance is demonstrated under noise-free conditions.