The minimum bandwidth ofMreal equi-energy signals is derived for the cases of either a specified code or a specified correlation matrix. The definition of bandwidth is in terms of the ratio of the energy passed by a linear time-invariant filter to the input energy, and incorporates a great variety of definitions. In one instance, the signals are time limited; in a second, a weighted integral over the infinite time interval is restrained. The extremum of the energy ratio depends only on the nonzero characteristic numbers of the correlation matrix of rankrand on the firstrcharacteristic numbers of a kernel related to the filter used in the bandwidth definition. The optimum signal waveforms also depend on the characteristic vectors of the correlation matrix and are given by linear sums of the firstrcharacteristic functions of the kernel. Bounds on the extrema are given, approximations in narrow-band situations are developed, and several useful examples are presented and solved.