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4.7 SUMMARY AND OMISSIONS
In this chapter we have covered a wide range of problems. The central theme that related them was an additive Gaussian noise component. Using this theme as a starting point, we examined different types of problems and studied their solutions and the implications of these solutions. It turned out that the formal solution was the easiest part of the problem and that investigating the implications consumed most of our efforts. It is worthwhile to summarize some of the more general results.
The simplest detection problem was binary detection in the presence of white Gaussian noise. The optimum receiver could be realized as a matched filter or a correlation receiver. The performance depended only on the normalized distance between the two signal points in the decision
space. This distance was characterized by the signal energies, their correlation coefficient, and the spectral height of the additive noise. For equal energy signals, a correlation coefficient of — 1 was optimum. In all cases the signal shape was unimportant. The performance was insensitive to the detailed assumptions of the model.
The solution for the M signal problem followed easily. The receiver structure consisted of at most M — 1 matched filters or correlators. Except for a few special cases, performance calculations for arbitrary cost assignments and a priori probabilities were unwieldy. Therefore we devoted our attention to minimum probability of error decisions. For arbitrary signal sets the calculation of the probability of error was still tedious. For orthogonal and nonorthogonal equally-correlated signals simple expressions could be found and evaluated numerically. Simple bounds on the error probability were derived that were useful for certain ranges of parameter values. The question of the optimum signal set was discussed briefly in the text and in more detail in the problems. We found that for large M, orthogonal signals were essentially optimum.
The simple detection problem was then generalized by allowing a nonwhite additive Gaussian noise component. This generalization also included known linear channels. The formal extension by means of the whitening approach or a suitable set of observable coordinates was easy. As we examined the result, some issues developed that we had not encountered before. By including a nonzero white noise component we guaranteed that the matched filter would have a square-integrable impulse response and that perfect (or singular) detection would be impossible. The resulting test was stable, but its sensitivity depended on the white noise level. In the presence of a white noise component the performance could always be improved by extending the observation interval. In radar this was easy because of the relatively long time between successive pulses. Next we studied the effect of removing the white noise component. We saw that unless we put additional “smoothness” restrictions on the signal shape our mathematical model could lead us to singular and/or unstable tests.
The next degree of generalization was to allow for uncertainties in the signal even in the absence of noise. For the case in which these uncertainties could be parameterized by random variables with known densities, the desired procedure was clear. We considered in detail the random phase case and the random amplitude and phase case. In the random phase problem, we introduced the idea of a simple estimation system that measured the phase angle and used the measurement in the detector. This gave us a method of transition from the known signal case to situations, such as the radar problem, in which the phase is uniformly distributed. For binary signals we found that the optimum signal set depended on the
376 4.7 Summary and Omissions
quality of the phase measurement. As we expected, the optimum correlation coefficient ranged from p = — 1 for perfect measurement to p = 0 for the uniform density.
The random amplitude and phase case enabled us to model a number of communication links that exhibited Rayleigh and Rician fading. Here we examined no-measurement receivers and perfect measurement receivers. We found that perfect measurement offered a 6-db improvement. However, even with perfect measurement, the channel fading caused the error probability to decrease linearly with Er/N0 instead of exponentially as in a nonfading channel.
We next considered the problem of multiple channel systems. The vector Karhunen-Loeve expansion enabled us to derive the likelihood ratio test easily. Except for a simple example, we postponed our discussion of vector systems to later chapters.
The basic ideas in the estimation problem were similar, and the entire formulation up through the likelihood function was identical. For linear estimation, the resulting receiver structures were identical to those obtained in the simple binary problem. The mean-square estimation error in white noise depended only on E/N0.