Digital Communications with Emphasis on Data Modems

7M‐ary CODED MODULATION

7.1 INTRODUCTION

M‐ary coded waveforms operate in the power‐limited region of Shannon’s capacity curve shown in Figure 3.17 and obtain coding gain through bandwidth expansion. These codes do not provide forward error correction (FEC) as in convolutional and block codes, that achieve bandwidth expansion with the inclusion of parity‐check bits, instead, the M‐ary code waveforms achieve their performance using brute‐force bandwidth expansion with code symbols of length M = 2^k corresponding to k information bits.¹ For k ≤ 10 bits, the decoding complexity is manageable; however, for larger values, the decoding complexity becomes unwieldy. The complexity issue arises because a decision is made based on the code symbol having the highest correlation among all of the M possible hypotheses. For example, with k = 14 information bits per M‐ary code symbol the decoding must perform over 16,000 correlations before making an optimal decision.

The binary coded M‐ary sequences [1, 2] can be derived from: maximal length sequences (M‐sequences), pseudo‐random noise (PRN) generated sequences, and Hadamard or Walsh sequences. If the code symbols are sufficiently long, the correlation properties of the first two techniques will result in nearly orthogonal performance; however, the Hadamard [3] and Walsh [4, 5] sequences result in orthogonal codes with the following properties. Two continuous‐time unit‐energy waveforms s_i(t) and s_j(t) are orthogonal² if their normalized ...