On the design of signals for sequential and nonsequential detection systems with feedback

The design of signals for binary communication systems employing feedback has previously been considered by Turin. A delayless, infinite-bandwidth forward channel disturbed by additive, white, Gaussian noise is assumed. At each instant of time, the log likelihood ratio of the two possible signals is fed back to the transmitter via a noiseless and delayless feedback channel. The forward-channel signals are said to be optimally designed when the feedback information is so utilized that the average (for sequential detection) or fixed (for nonsequential detection) transmission time is minimized, subject to a specified probability of error. Average and peak power constraints are also placed on the signals. Turin has solved the signal design problem for extreme values (i.e., very large or equal to one) of the peak-to-average power constraint ratio. These results are extended in this paper to arbitrary values of the power constraint ratio, for both sequential and nonsequential detection.