Noise generators can serve as useful test tools in evaluating communications systems performance. They allow an operator to add a controlled amount of thermal noise to a reference signals to determine the effect of noise on system performance, such as bit-error rate (BER). Thermal noise conforms to a Gaussian probability density function (PDF) allowing a smooth transition from theoretical analysis to a test bench. For the most part, the output of a noise generator is approximately "close enough" to true (mathematically true) Gaussian noise to be useful for analysis and testing. What follows is an explanation of how to work with Gaussian noise in test applications and how to gauge the impact on the test when nonperfect Gaussian noise is used.

The signal energy in a system relative to its noise, usually given as E_b/N_o (or in its other guises C/N, C/N_o, SNR) is an expression of a signal's strength relative to the noise surrounding it, and an important figure of merit in testing communications channels. The use of additive white Gaussian noise in generating this ratio is well established, with techniques being called out in major standards governing communications practices (e.g., MIL-188-165a and ATSC A80).

White noise is suitable for testing because it represents equal energy at all frequencies in a spectrum. It is Gaussian because randomness in nature exhibits a Gaussian or normal distribution. Most of the noise in communications channels (such as noise introduced by an amplifier) has a thermal characteristic, and so tends toward a Gaussian distribution. Furthermore, the Central Limit Theorem shows that if enough random distributions are taken together, regardless of their individual distributions (flat, Gaussian, or otherwise), the limit as the sum tends to infinity is that of a Gaussian distribution.

Mathematically, a Gaussian distribution is expressed as:

This gives the distribution of a variate x, with mean µ and variance Σ². Mathematicians and statisticians tend to call this the normal distribution; psychologists refer to it as the bell curve; and physicists and engineers talk about the Gaussian distribution. Gaussian noise then is a fluctuation with this mathematical description about a mean (Fig. 1).

Noise can be used for testing system performance in several ways. One way is to add noise to a channel and increase the noise level until an unacceptable degradation in the signal quality is reached. For example, a television picture shows increased "snow" as noise is added to the signal. The amount of noise that causes the degradation is a measure of the channels signal strength or effectiveness of its signal processing.

What if a more quantitative measure is desired? One method involves the capability of the system to resolve two signals with and without noise. Without noise, the signals are distinct and easy to resolve (Fig. 2), for example as the digital bit 0 for a signal with voltage V₀ and digital bit1 for a signal with voltage V₁. In reality, some noise will be added to these signals as part of an electronic system, random fluctuations about the mean V₁ or V₀ that follow a Gaussian distribution as given by Eq. 1.

The ability to differentiate the two signals will not a problem if they are sufficiently far apart not to overlap. Yet, with a Gaussian distribution there will always be some measure of an overlap (Fig. 3) . So how is it possible to tell one signal from another?

This can be done by setting a threshold value at a midpoint between the two mean signal levels. This threshold takes the voltage value of (V₁ — V₀)/2 below V₁ (or above V₀). Voltages detected above the threshold produce a digital value of 1 while voltages below it are said to produce 0. What happens to a 0 when the noise is sufficient to raise the signal level over the threshold? Given the decision-making algorithm, the 0 will be mistaken for a 1, resulting in a bit error.

Some amount of error is inevitable, so some measure of the bit errors is necessary to determine the severity of the problem. It is possible to look for the probability that, while a 0 is being transmitted the noise increases the signal level above the threshold, and that while a 1 is being transmitted the noise decreases the level below the threshold. Bayes Theorem then leads to: