The colorful world of noise.(SIGNAL SOURCES)

Although much telecom testing involves additive white Gaussian noise (AWGN), white noise does not need to be Gaussian, nor is Gaussian noise necessarily white. White noise is defined by two characteristics: It has a zero mean value, and its autocorrelation is represented by a delta function. In other words, successive values are completely uncorrelated with previous values.

In the frequency domain, such a time-domain function has a constant power spectral density. This means that the spectrum of an ideal white noise source has constant power per cycle regardless of frequency.

Practical white noise sources are flat within some small deviation across a defined frequency band. For example, the Model WGN-1/200 White Noise Generator from dBm is specified as producing -87-dBm/Hz noise density with 0.5-dB flatness from 1 MHz to 200 MHz.

[FIGURE 1 OMITTED]

By definition, successive values of a truly random variable cannot be predetermined. Nevertheless, all of the values that occur within an arbitrarily large set of observations determine a distribution. The Gaussian or normal distribution is perhaps the most common and defined by the probability density function (PDF)

P(x) = [1/[[sigma][square root of (2[pi])]]] = [e.sup.[-(x-[mu])[.sup.2]]/[2[[sigma].sup.2]]] (1)

where [mu] = the mean

[[sigma].sup.2] = the variance

[sigma] = the standard deviation

When [sigma] = 1.0 and [mu] = 0.0, the definition simplifies to the standard form of the normal distribution

N(x) = [1/[square root of (2[pi])]] [e.sup.[-[x.sup.2]]/2] (2)

This equation describes the familiar bell-shaped curve shown in Figure 1. The probability density at the mean is 0.3989, at 1[sigma] larger or smaller = 0.2420, 2[sigma] = 0.0540, and 3[sigma] = 0.004432. Because of the square term in the exponent, the probability density falls off very quickly above 3[sigma] so that at 5[sigma] away from the mean N(x) = 0.000001487.

A large series of observed values can conform to a Gaussian distribution but occur in time in a deterministic, highly ordered manner. The signal would not have a flat spectrum and could not be used for noise testing. It would have a Gaussian distribution but would not be white--successive values would not be statistically independent.

Gaussian white noise has the benefit of a well-understood and compact mathematical description. Even if the actual distribution is not quite Gaussian, the Normal distribution often is assumed to apply because it simplifies further analysis.

The integral of the PDF is the cumulative probability density function (CDF), also shown in Figure 1. It indicates the probability that a value is to the left of any arbitrary point. For example, the probability that a sample within a Gaussian distribution has a value less than +1.0 is about 0.84. Obviously, the probability that a value lies between -5 and +5 is close to 1. …

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