Optimal communication systems with noiseless feedback for a binary source and white Gaussian channel

Abstract

The sequential and nonsequential feedback systems for a binary source, first proposed by Turin, are considered. It is shown that in both cases the optimal signals belong to the family <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\pm A (y)[1 + \exp (\pm y)]^{-1}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(y)</tex> is arbitrary. In the sequential scheme the bound on transmission rate is achieved for any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{e}</tex> by all members of this family, and in the nonsequential scheme for small <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{e}</tex> only. The optimal signals within this family, i.e., signals for which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha =</tex> peak power/average power is minimal, result if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(y) = \pm \sqrt{\alpha P_{av}} (1 -b e^{-|y|})</tex> . In the sequential case, the optimal scheme is compared with two suboptimal schemes, and the advantages of the first are demonstrated. Finally, the problem of sequential detection with time-varying thresholds is presented, and its equivalence to a problem with fixed thresholds is proved.