Upper bounds to the asymptotic performance of block quantizers

The problem of simultaneously estimating phase and decod-ing data symbols from baseband data is posed. The phase sequence isassumed to be a random sequence on the circle, and the symbols areassumed to be equally likely symbols transmitted over a perfectly equalizedchannel. A dynamic programming algorithm (Viterbi algorithm) is derivedfor decoding a maximum (Iposteriori (MAP) phase-symbol sequence on afinite dimensional phase-symbol trellis. A new and interesting principle ofoptima&y; ‘for simultaneously estimating phase and decoding phase-amplitude coded symbols leads to an efficient two-step decoding procedurefor decoding phase-symbol sequences. Simulation results for binary, &aty;phase shift keyed (PSK), and X-quadrature amplitude shift keyed (QASK)symbol sets transmitted over random walk and sinusoidal jitter channels arepresented and compared with results one may obtain with a decision-directedalgorithm or with the binary Viterbi algorithm introduced by Ungerboeck.When phase fluctuations are severe and when occasional large phasefluctuations exist, MAP phase-symbol sequence decoding on circles issuperior to Ungerboeck’s technique, which in turn is superior to decision-directed techniques. Manuscript received December 26, 1979; revised November 17, 1980.This work was supported in part by the Centre National de la RechercheScientifique; the Office of Naval Research, Statistics and ProbabilityBranch, Arlington, VA; and the Army Research Office, Research TrianglePark, NC.0. Mac&i; is with the CNRS Laboratoire des Signaux et Systemes,Plateau du Moulon, 91190 Gif-sur-Yvette, France.L. L. Scharf was with the CNRS Laboratoire des Signaux et Systemes,Gif-sur-Yvette, France and the Electrical Engineering Department, Col-orado State University, Fort Collins, CO. He is now with the ElectricalEngineering Department, University of Rhode Island, Kingston.