A computer-simulation algorithm is described for calculating the performance of a maximum-likelihood (ML) receiver in the nonlinear satellite communications channel. It is assumed that performance is dominated by isolated single errors and that the channel can be approximated by one of finite memory,M \leq 10bits. The algorithm employs the brute force method of comparing the wave forms of all sequence pairs of length2M - 1bits that differ in theirM'th bit, and in spite of the large number of sequences involved requires only modest computer time. Examples for a rudimentary channel employing a traveling wave tube at saturation with modulation formats QPSK, O-QPSK, and MSK are computed and ML performance is compared to that of suboptimum receivers using bit-by-bit decision or discrete ML sequence estimation. Results indicate that the inherent channel degradation that cannot be corrected by appropriate receiver design is not great even for transmission at the Nyquist rate limit as defined relative to the 3 dB bandwidth of a link employing 6-pole Butterworth filters.

In most digital satellite communications links the message is determined on the basis of periodically sampled voltages of the received signal. Assuming that in the absence of noise each sample voltage depends significantly only on a finite number of message bits, one can define a discrete channel in the form of a power series in these bits. From a complete set of sample voltages computed by simulation the coefficients of the series are readily obtained by means of the fast Hadamard transform. Examples computed for rudimentary nonlinear channels that include limiters, amplitude-to-phase converting elements, or traveling wave tubes show that the series contains only a small number of significant terms, and that each specific type of nonlinearity is associated with specific nonlinear terms. The degradation in performance in the presence of noise that results from the largest nonlinear terms in the expansion is of an inherent nature, and can only be corrected in small part by receiver equalization. On the other hand, a simple minded predistortion scheme appears to remove most of this degradation.

The detection of light signals using photomultipliers is presented in terms of an idealized model in which the received signal consists of a series of pulses of random height with arrival times that are Poisson distributed with an instantaneous rate proportional to transmitted signal intensity. Linear detection, effected through multiplication by locally generated functions followed by integration, is considered. An optimization algorithm based on Chernoff bounds for the error probability is developed. For the standard Poisson regime with constant pulse heights this algorithm derives correctly the logarithmic local functions predicted rigorously for maximum likelihood detection. For random pulse heights the functions resemble smoothly truncated logarithmic functions of the transmitted signal intensities, and the logarithmic singularity for zero values of the signal is either removed or greatly weakened. Calculations for several classes of pulse height distributions show that the degradation in performance caused by pulse height fluctuations can be predicted with fair accuracy from an "excess noise factor" defined in terms of the ratio of pulse height variance to the square of its mean.

The performance of a maximum likelihood receiver whose input is the photoelectric current created by a modulated light beam is studied for a number of discrete modulation schemes. The performance is compared with that of a classical microwave receiver, i.e., one designed to optimize detection in a Gaussian channel. Such a receiver has the advantages of availability, simplicity, ease of implementation, and greater tolerance to slow photodetector response and is an obvious candidate for suboptimum detection. Procedures for calculating receiver performance in the Poisson regime by the tilted probability density technique are described and applied specifically to modulation schemes employing one or more subcarriers that are biphase or quadriphase modulated. Results indicate that the classical receiver falls noticeably short of the optimum only at or near 100 percent modulation, where it may require at worst up to 30 percent more light power in order to equal the maximum likelihood receiver. When modulation is incomplete, or when background radiation is significant, the difference in performance is generally much smaller.