Introduction

In Chapters 2 and 3 we have focused on the receiver, assuming that the signal set was given to us. In this chapter we introduce the signal design.

The problem of choosing a convenient signal constellation is not as clean-cut as the receiver-design problem. The reason is that the receiver-design problem has a clear objective, to minimize the error probability, and one solution, namely the MAP rule. In contrast, when we choose a signal constellation we make trade-offs among conflicting objectives.

We have two main goals for this chapter: (i) to introduce the design parameters we care mostly about; and (ii) to sharpen our intuition about the role played by the dimensions of the signal space as we increase the number of bits to be transmitted. The continuous-time AWGN channel model is assumed.

Isometric transformations applied to the codebook

If the channel is AWGN and the receiver implements a MAP rule, the error probability is completely determined by the codebook C = {c0,…, cm−1}. The purpose of this section is to identify transformations to the codebook that do not affect the error probability. For the moment we assume that the codebook and the noise are real-valued. Generalization to complex-valued codebooks and complex-valued noise is straightforward but requires familiarity with the formalism of complex-valued random vectors (Appendix 7.9).

From the geometrical intuition gained in Chapter 2, it should be clear that the probability of error remains the same if a given codebook and the corresponding decoding regions are translated by the same n-tuple b ∈ ℝn.

A translation is a particular instance of an isometry. An isometry is a distancepreserving transformation. Formally, given an inner product space V, a : V → V is an isometry if and only if for any α ∈ V and β ∈ V, the distance between α and β equals that between a(α) and a(β). All isometries from ℝn to ℝn can be obtained from the composition of a reflection, a rotation, and a translation.

EXAMPLE 4.1 Figure 4.1 shows an original codebook C = {c0, c1, c2, c3} and three variations obtained by applying to C a reflection, a rotation, and a translation, respectively.