Some criteria for frequency domain design of signals imbedded in Gaussian noise

The design of time-limited binary signals imbedded in colored Gaussian noise resulting from passing white noise through a channel of known transfer function is investigated. It is shown that, for a class of channels whose magnitude characteristic is expressible as the ratio of polynomials with the numerator divisible by and of lower or the same degree as the denominator, the output noise kernel (correlation function) and its inverse will have Fourier transforms which are reciprocals of each other. The above assumption results in an expression for probability of error which is independent of the received waveform and depends solely on the transmitter signal-to-noise ratio. The additional freedom of design afforded by the above enables one to impose further restrictions in the form of energy concentration in the frequency domain on the received signal without deterioration of system performance. Some such bandwidth constraints and their associated optimum signal designs are considered in this paper.