Amplitude- and Time-Limited Functions With Minimum Out-of-Band Energy

The theory of time-limited functions with minimum out-of-band energy is extended to apply to functions whose amplitude is held constant and hence, are limited to phase modulation. In the absence of the amplitude constraint, the desired functions are prolate spheroidal functions of time, of order zero, with time-bandwidth product as a parameter. With the amplitude constraint, the desired functions can be expressed in terms of sums of even-order prolate spheroidal functions. An algorithm suitable for computations of the amplitude constrained functions is derived.