Error Correcting Codes & Real Channels 7.1.1 Motivation for Gaussian Channel

The noisy channel coding theorem that we have proved shows that there exist`very good' error correcting codes for any noisy channel. In this chapter we address two questions. First, many practical channels have real, rather than discrete, inputs and outputs. How should digital signals be mapped into analogue waveforms, and vice-versa? Secondly, how are practical error correcting codes made, and what is achieved in practice, relative to the possibilities proved by Shannon? 7.1 The Gaussian Channel The most popular continuous channel model is the Gaussian channel. The Gaussian Channel has a real input x and a real output y. The conditional distribution of y given x is a Gaussian distribution: P(yjx) = 1 p 2 2 exp h ?(y ? x) 2 =2 2 i : (7.1) I will not introduce extra notation to distinguish probability densities from probabilities.] Note that this channel has a continuous input and output but is discrete in time. As shown below, certain continuous{time channels are in fact equivalent to the discrete time Gaussian channel. Notation. If a Gaussian variable y has mean x and variance 2 then we may write: y Normal(x; 2); or P(y) = Normal(y; x; 2): (7.2) As with discrete channels, we will discuss what rate of error-free information communication can be achieved over this channel. Consider a real (electrical, say) channel with inputs and outputs that are continuous in time. We put in x(t), which is some sort of band-limited signal, and out comes y(t) = x(t) + n(t). Our transmission has a power cost. The average power of a transmission of length T may be constrained thus: Z T 0 dt x(t)] 2 =T P: