Computability of the Peak Value of Bandlimited Signals

In this paper we study the peak value problem, i.e., the task of computing the peak value of a bandlimited signal from its samples. The peak value problem is important, for example, in communications, where the peak value of the transmit signal has to be controlled in order that the amplifier is not overloaded, which would generate out-of-band radiation. We prove that the peak value of a computable bandlimited signal is computable on digital hardware if oversampling is used. The computability ensures that the approximation error can be effectively controlled. Further, we provide an algorithm that can be used to perform this computation and prove that oversampling is indeed necessary, because there exist signals for which the peak value problem cannot be algorithmically solved without oversampling. Hence, without oversampling the peak value of such signals cannot be computed on any digital hardware, including DSPs, FPGAs, and CPUs.