How to Utilize Gaussian Noise in Communication System Testing

A noise generator is a powerful tool for measuring the performance of communication systems. It allows the operator to add a controlled amount of thermal noise to a reference signal to determine the effect of the noise on system performance (e.g., bit error rate, BER). Thermal noise follows a Gaussian probability density distribution (PDF), making it easy to move from theoretical analysis to practical application. In most cases, the output of a noise generator is very close to actual (mathematical) Gaussian noise, making it suitable for performance analysis and test applications. The rest of this article explains how to use Gaussian noise in testing and how non-ideal Gaussian noise affects test results.

The ratio of signal energy to noise in a system is usually recorded as Eb _/ No ₍ or C/N, C/No _, SNR), which represents the ratio of signal strength to noise strength and is an important parameter for measuring communication channel performance. The method of calculating signal-to-noise ratio using additive white Gaussian noise is very mature and is widely used in various major communication standards (such as MIL-188-165a and ATSC A80).

White noise has the same intensity at all frequencies in the spectrum, making it an ideal choice for noise sources in system performance testing. The reason why the probability density of noise is Gaussian is that actual random signals follow a Gaussian distribution, or normal distribution. The noise in most communication channels (such as the noise introduced by the amplifier circuit) is thermal noise, which tends to be Gaussian. Moreover, the central limit theorem proves that if a sufficient number of random events occur simultaneously, no matter what distribution the individual events follow (uniform distribution, Gaussian distribution, or other), the limit value of their sum tends to infinity and is Gaussian.

The mathematical expression of Gaussian distribution is as follows:

The above formula gives a probability distribution function of a variable x with a mean of ? and a variance of Σ ^2. Mathematicians and statisticians generally call it a normal distribution, psychologists call it a Bell curve, and physicists and engineers call it a Gaussian distribution. This function mathematically describes the characteristics of the size of Gaussian noise fluctuating around its mean (Figure 1).

There are many ways to use noise to measure system performance. One method is to add noise to the channel under test and increase the intensity until the signal quality degrades until it cannot be detected. For example, "snow" can be added to a TV image as signal noise. The intensity of noise that causes the channel signal quality to degrade can be used to evaluate the capability and efficiency of the signal processing technology.

If a more quantitative analysis is needed, one method is to divide the system capacity into two parts: a signal with noise superimposed on it and a signal without noise. The signal without noise is easier to decompose (Figure 2), for example, a signal with voltage V0 _represents digital bit 0, and a signal with voltage V1 _represents digital bit 1. In actual electronic systems, there is always noise on the signal, and the signal amplitude will randomly fluctuate around V1 _or V0 _, and its probability density follows the Gaussian distribution given by formula 1.

If the two signals are far apart and do not overlap, there will be no problem in distinguishing them. However, the existence of Gaussian noise makes the signals always overlap to some extent (Figure 3). How to distinguish them at this time?

The solution is to set a threshold value (V ₁ --V ₀ )/2 between the two, which is less than V ₁ and greater than V _0. If the detected signal voltage is higher than the threshold, it is judged as 1, otherwise it is judged as 0. What will happen if the signal noise of bit 0 is large enough to exceed the threshold? Under the given judgment algorithm, 0 will be mistakenly judged as 1, and a bit error will occur.

A certain number of errors are unavoidable, so it is necessary to determine a measure of the severity of the problem for bit errors. It is possible to calculate the probability that when a 0 is transmitted, the signal level exceeds the threshold due to the presence of noise, or when a 1 is transmitted, the noise cancels the signal and the signal level drops below the threshold. According to Bayes' theorem, this probability can be expressed as:

The above formula shows that the total error probability is equal to the error probability of code 0 and code 1 multiplied by the sum of their occurrence probabilities. In a simple system, there are only two signals, 1 and 0, and the probability of 1 and 0 occurring is roughly the same (1 and 0 may each account for half), then Formula 2 can be rewritten as:

The error probability of code 0 is given by:

Where n represents the signal voltage with noise superimposed on it. The error probability of 1 code is:

Due to the symmetry of Gaussian distribution, the probability values ​​calculated according to the above two equations in Gaussian noise channel are equal and can be uniformly expressed as:

In the example above, the bit error rate of the system is equal to the probability that the noise intensity exceeds the threshold value. The statistical properties of the Gaussian distribution give the probability that the Gaussian variable x exceeds a given value a:

Where erfc is the complementary error function, erfc(x) = 1-erf(x), and erf is the error function.

The error function erf is widely used in various data analysis situations, including solving differential equations that describe the distribution of impurities in semiconductor materials. This equation has no analytical solution, but an approximate solution can be found by the Maclaurin series. Due to its importance, many textbooks list the value table of erf(x), and Microsoft Excel even includes erfc as part of its data analysis toolkit Toolpak.

Corresponding to the above example, the threshold value is (V ₁ – V ₀ )/2, and the statistical parameters of the noise voltage are zero mean and variance σ ² = V _n ² , where V _n ² is the RMS value of the noise voltage. Therefore, Equation 7 can be expressed as:

_{In order to obtain the expression of Eb} /N0 _, which represents the power ratio , we can transform Equation 8 into the square of voltage:

The above formula can be expressed in terms of power:

Where No _represents the noise power density. Since the power per bit Eb _is equal to the average of the two signal powers, the above formula can also be rewritten as:

So far we have derived a general formula for the bit error rate in a binary phase shift keying (BPSK) channel. The same derivation can be applied to quaternary phase shift keying (QPSK) and orthogonal QPSK (OQPSK) channels to produce the same results, with only slight variations of equation 11 for other modulation schemes. A detailed derivation of these modulation schemes is beyond the scope of this article, but it does serve to illustrate that the formula (and the “waterfall curve” derived from it) is derived from the intrinsic properties of the Gaussian PDF.