tity P/N is called the signal-to-noise ratio, or SNR. The funda- mental result in Eq. (14.13), which establishes that the capacity of a "noisy channel" is proportional to log(1 + SNR) is probably the most well known and encompassing Gaussian channel and Shannon-Hartley theorem n(P+N) Figure 14.1 (a) Location of input message X, defined by vector x, and likely location of output message y, defined by vector y, on the surface of a hypersphere of dimension n; (b) as (a), but for n = 2. conclusion of all Shannon's information theory. Given the SNR characteristics of a com- munication channel with noise and signal power limitation, it provides an upper bound for the code rate at which information can be transmitted with arbitrarily small errors. Before expanding on the implications of this result for communication systems, it is worth providing a plausibility argument for the existence of the corresponding codes with low error probability. The argument immediately following can be found with different variants in most IT textbooks. Consider an input message (or codeword) consisting of a sequence of n discrete waveform samples with random amplitudes Xi of zero mean and mean power (x2) = P. This message defines a vector x = (x1, x2, Xn) in an n-dimensional space, which is characterized by a length = + x2 = nP. Thus, we can view the input message as lying on any point x = (x1, X2, Xn), which is most probably located close to the surface of a hypersphere centered about the origin and having a radius of VnP. Likewise, the received or output message corresponds to an n-vector y=(y1,y2, = yn) with yi x x i t zi, where (Zi) = 0 and (z?) = N. Thus, the output vector 1 has the length y|=Vn(P+N) = with deviation n N. The output message is likely to fall on any point y = (y1, yn) inside a small hypersphere centered about X = (x1, , X2, Xn) and having a radius of Vn N, as illustrated in Fig. 14.1. For any output message y to represent nonconfusable input messages X (i.e., most likely to correspond to a unique input message x), the