Exact distribution of the levels of periodic correlation functions of complete and truncated binary codes
The selection of signal systems for M-ary data-transmission channels is frequently based on orthogonalization of the system (see ). At the same time, it is expedient in a number of applications to use random binary codes, while a system of M signals may be formed by cyclical shifts of the code sequence formed by a M-digit random-number generator. In the first approximation, the estimate of the correlation properties of such codes has been given in , while in  an analysis was performed of the energy losses of channel interference immunity for conversion from orthogonal codes to random codes. These losses turned out to be comparatively low (about 1.5 dB for an error probability pe = 10–3).
The Gaussian approximation derived in  for the levels of the periodic correlation function (PCF) regrettably yields a very high error on the distribution boundaries; this leads to incorrect estimates of interference immunity in the region of high signal/noise ratios. The present work presents a procedure for determining the exact distribution of PCF levels for complete and truncated binary codes, and the distributions are likewise obtained for the PCF levels in the case of codes having a length N = 2p (p = 2 to 5).
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