Minimum rms bandwidth ofMtime-limited signals with specified code or correlation matrix

The minimum rms (root-mean-square) bandwidth of
M
real equi-energy time-limited signals is derived for the case of either a specified code or a specified correlation matrix. The minimum bandwidth depends only on the nonzero characteristic roots of the correlation matrix, or equivalently, of a matrix derived from column properties of the code. The optimum waveforms depend additionally on the characteristic vectors of the respective matrix and are given by linear combinations of a time-limited fundamental sine wave and its harmonics. An upper bound on the minimum rms bandwidth is approximately proportional to the rank of either matrix. Applications to several codes and correlation matrices are given, including some best binary group codes, which indicate a worthwhile saving of bandwidth for the optimum waveforms as compared with conventional waveforms. The energy concentration of these optimum rms waveforms compares very favorably with that of the time-truncated prolate spheroidal wave functions, which concentrate most energy in an assigned frequency band. Furthermore, the optimum rms waveforms are easier to generate and process.