Strictly complex impulse-equivalent codes and subsets with very uniform amplitude distributions
J De Caprio · IEEE Transactions on Information Theory · 1969
The generation of a code of lengthN + 1with an impulse-equivalent autocorrelation leads to a class of2^{N}possible signals. Examination of the root patterns of the generating polynomials of these signals shows that many of the codes that appear to be complex have real counterparts with identical amplitude distributions. The existence of a certain subclass of complex codes with no real equivalents is demonstrated. It is conjectured that this subclass, owing to its nontrivially complex nature, will contain those impulse-equivalent codes possessing the most uniform amplitudes. A theorem is offered which provides a necessary and sufficient condition for determining this subclass of purely complex signals. To test the conjecture, all impulse-equivalent codes up to the length13of several energy levels are examined under two distinct uniformity-of-amplitude criteria. In every case the most uniform codes are found to be members of the subclass. Finally, "randomly" generated impulse-equivalent codes are investigated. It is shown that a logical interpretation of the notion of random selection leads to a set of impulse-equivalent codes that is generated from maximal shift-register sequences. This set is, in turn, proven to be contained within the nontrivially complex subclass. The amplitude distributions of these shift register-generated codes up to length64are examined.