Variable-rate data transmission schemes in which constellation points are selected according to a nonuniform probability distribution are studied. When the criterion is one of minimizing the average transmitted energy for a given average bit rate, the best possible distribution with which to select constellation points is a Maxwell-Bdtzmann distribution. In principle, when constellation points are selected according to a Maxwell-Boltzmann distribution, the ultimate shaping gain (7re/6 or 1.53 dB) can be achieved in any dimension. Nonuniform signaling schemes can be designed by mapping simple variable-length prefix codes onto the constellation. Using the Huffman procedure, prefix codes can be designed that approach the optimal performance. These schemes provide a fixed-rate primary channel and a variable-rate secondary channel, and are easily incorporated into standard lattice-type coded modulation schemes.
Absfrucf-This paper presents a unified study of partial-response signaling (PRS) systems and extends previous work on the comparison of PRS schemes. A PRS system model is introduced which enables the investigation of PRS schemes from the viewpoint of spectral properties such as bandwidth, nulls, and continuity of derivatives. Several desirable properties of PRS systems and their relation to system functions are indicated and a number of useful schemes, some of them not previously analyzed, are presented. These systems are then compared using as figures of merit speed tolerance, minimum eye width, and signal-to-noise ratio (SNR) degradation over ideal binary transmission. A new definition of speed tolerance, which takes into account multilevel outputs and the effect of sampling time, is introduced and used in the calculation of speedtolerance figures. It is shown that eye width, a performance measure that has not been used previously in comparing PRS systems, can be calculated analytically in many cases. Exact values as well as bounds on the SNR degradation for the systems under consideration are presented. The effect of precoding on system performance is also analyzed. P
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