Abstract :
Turin [1] has studied the problem of optimal sequential detection over a channel with feedback when the signals used are subject to peak and average power constraints. With special additional constraints on the signals he was able to obtain a series solution for the first passage probability density and hence find optimal signals when the ratio of peak power to average power was infinity or one. Horstein,[2] concurrently with this study, was able to find optimal signals for arbitrary ratios of peak to average power when the signals were subject to the same special constraints. This paper presents a simpler and more direct approach for obtaining optimal signals which minimize the expected time to decision for arbitrary peak power constraints and various average power constraints. Explicit closed form solutions are derived for the expected energies and times to decision without having to first obtain the probability distribution of the first passage time. No special constraints are required on the signals. For communication with white noise in the feedback link, optimal stationary signals are found for peak and average power constraints. It is shown that communication rates up to but strictly less than channel capacity are possible with noisy feedback. However, it is also noted that for the schemes considered here, the stationary signals and white noise in the feedback link imply infinite feedback power.
