On the Exact BER of Bit-Wise Demodulators for One-Dimensional Constellations

TL;DR

This paper derives exact closed-form expressions for the bit-error rate of optimal bit-wise demodulators in 4-PAM and 8-PAM systems, revealing that for practical BER levels, optimal and symbol-wise demodulators perform identically.

Contribution

It provides new closed-form BER formulas for 4-PAM with any labeling and for 8-PAM with specific labelings, enhancing understanding of demodulator performance.

Findings

No BER difference between optimal and symbol-wise demodulators below 0.1.

Closed-form BER expressions enable precise performance evaluation.

Analysis covers various labelings, including Gray and binary codes.

Abstract

The optimal bit-wise demodulator for M-ary pulse amplitude modulation (PAM) over the additive white Gaussian noise channel is analyzed in terms of uncoded bit-error rate (BER). New closed-form BER expressions for 4-PAM with any labeling are developed. Moreover, closed-form BER expressions for 11 out of 23 possible bit patterns for 8-PAM are presented, which enable us to obtain the BER for 8-PAM with some of the most popular labelings, including the binary reflected Gray code and the natural binary code. Numerical results show that, regardless of the labeling, there is no difference between the optimal demodulator and the symbol-wise demodulator for any BER of practical interest (below 0.1).