Some optimal signals for time measurement

Some optimal transmitted signals are given for communications systems in which the time of occurrence of a signal is desired in the presence of additive Gaussian noise. These signals are of two classes; those in which bipolar signal excursions are permitted and those in which only unipolar signal excursions are permissible. Some bipolar codes used by R. H. Barker are supplemented when conditions permit unlimited negative excursions of the optimally correlated signal output. Other constraints, such as limiting the positive and negative excursions of the optimally correlated signal output, except at the proper phase, will lead to different codes. When only unipolar codes are permitted, optimum repetitive and nonrepetitive codes (embedded in zeros) are given for various code lengths. The minimum length of such codes is given. A mathematical resemblance is indicated to a frequency allocation problem in which third-order intermodulation products must be avoided. The closely related concepts of error-detecting and correcting codes and optimally correlated signals are illustrated in the derivation of these codes. The problem of generating such codes by other than trial and error is not solved and remains as a provocative problem.