On the autocorrelation of finite-length sequences

lt is well known that all autocorrelation sequences are positive (semi)definite. The correlation sequence is further constrained when, as is often the case, it is obtained from correlating a finite-length data sequence with itself. In this paper we study the effect of the data sequence length on the attainable correlations. We derive a formula for the sequence length needed to match an autocorrelation value at a given lag. The formula indicates that inordinately long sequences are needed to obtain a normalized autocorrelation coefficient near unity. We conclude that all-zero (moving-average) models may be ill suited for highly correlated signals. The formula for the single-lag case is then used to obtain a bound on the sequence length needed to match a set of autocorrelations simultaneously.