A general solution for the synthesis of binary sequences with desired correlation sequence

Binary sequences are of considerable interest in many applications. In particular, the performance of a radar depends on the ambiguity function. For the case of binary phase modulation, the zero-Doppler performance of a pulse radar is a function of the discrete autocorrelation of the corresponding binary sequence. Alternately, a binary sequence can be expressed as a sequence of runs where the length of a run is equal to the number of contiguous bits of same sign. Given a run pattern the discrete autocorrelation can be computed easily and more rapidly than with the standard technique. The inverse problem, which is the synthesis of a run pattern given the desired discrete autocorrelation sequence, is much more difficult. This paper develops a set of sequential relations which relate the desired autocorrelation sequence to properties of the run pattern. For example, the autocorrelation at lag 1 defines the number of runs for a sequence of specified length. The autocorrelation at lag 2 now determines the number of runs of length 1. More generally the autocorrelation at lag (k+1) establishes a relation between consecutive runs with sum exactly equal to k. Judicious use of these relations greatly facilitates the computer synthesis of binary sequences. While the method is quite general, emphasis is on tracking applications where it is desired to minimize the sidelobes as far as possible.