A method of digital signalling in the presence of additive Gaussian noise

This paper considers the basic problem of transmitting digital information through a noisy channel with minimum probability of error in finite time. The transmitted signals are average-power limited, and the noise is assumed to be additive Gaussian with a power spectrum which may be nonwhite. A theory of so-called efficient codes (minimax, equal separation, and nearly equal separation) is developed. Efficient codes are formed from weighted sums of eigenfunctions generated by an integral equation with its kernel corresponding to the inverse Fourier transform of the Gaussian noise power spectrum. In addition, the theory of equidistant and nearly equidistant codes [1] is extended to the case of nonwhite Gaussian noise. It is shown that efficient codes perform better than equidistant codes if the noise is nonwhite; i.e., properly chosen waveforms are more efficient than binary coding. Performance results are given for several different codes when the interference is white Gaussian noise and when the noise power density increases with increasing frequency. The detection scheme used does not require estimation of the signal or the noise levels at the receiver and is thus independent of fading.