Why is Eb/N0 the natural figure of merit in digital communications?

In digital communications, the measure of the signal to noise ratio of a signal is \(E_b/N_0\). Why is that? To figure it out, let’s start from the classical SNR, which is the ratio of the average signal power \(S\) to the average noise power \(N\), given by

\[\frac{S}{N}.\]

For analog communications, the SNR is clearly a good way of describing the quality of the received signal. The numerator represents the useful power of the signal, and the denominator represents the power that degrades the signal. For digital communications, however, a few adjustments must be made.

First, the energy per bit is a much more natural measure for the “strength” of the signal than the power. In itself, signal power is not very enlightening in digital communications, because the useful information is now divided into bits instead of being continuous. This means that the same average power can represent wildly different signal qualities for different bitrates, making it quite uninformative. Hence, it is better to work with the energy per bit, which can be computed by integrating the power over the length of a bit \(T\), as \(E_b=ST\). In practice, the parameter that is used is typically the bit rate \(R=1/T\), which results in \(E_b=S/R\).

Second, the noise spectral density \(N_0=N/W\), where \(W\) is the bandwidth of the signal, is better to describe the noise. Its main advantage is that the actual bandwidth is taken out of the equation, since \(N_0\) is simply the noise power in a 1 Hz bandwidth. This allows comparing different modulation schemes without having to worry about their spectral properties.

Together, this results in

\[\frac{E_b}{N_0}=\frac{S/R}{N/W},\]

which can be rewritten most clearly as

\[\frac{E_b}{N_0}=\frac{S}{N}\frac{W}{R}.\]

This last expression emphasizes that \(E_b/N_0\) is simply an SNR that is normalized for bandwidth and bit rate, to make it suitable for the very specific application that is digital communications.

Submitted by Tom Roelandts on 29 June 2016

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