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Chapter 2: Statistics, Probability and Noise

Digital Noise Generation

Random noise is an important topic in both electronics and DSP. For example, it limits how small of a signal an instrument can measure, the distance a radio system can communicate, and how much radiation is required to produce an xray image. A common need in DSP is to generate signals that resemble various types of random noise. This is required to test the performance of algorithms that must work in the presence of noise.

The heart of digital noise generation is the random number generator. Most programming languages have this as a standard function. The BASIC statement: X = RND, loads the variable, X, with a new random number each time the command is encountered. Each random number has a value between zero and one, with an equal probability of being anywhere between these two extremes. Figure 2-10a shows a signal formed by taking 128 samples from this type of random number generator. The mean of the underlying process that generated this signal is 0.5, the standard deviation is 1/√12 = 0.29, and the distribution is uniform between zero and one.

Algorithms need to be tested using the same kind of data they will encounter in actual operation. This creates the need to generate digital noise with a Gaussian pdf. There are two methods for generating such signals using a random number generator. Figure 2-10 illustrates the first method. Figure (b) shows a signal obtained by adding two random numbers to form each sample, i.e., X = RND+RND. Since each of the random numbers can run from zero to one, the sum can run from zero to two. The mean is now one, and the standard deviation is 1/√6 (remember, when independent random signals are added, the variances also add). As shown, the pdf has changed from a uniform distribution to a triangular distribution. That is, the signal spends more of its time around a value of one, with less time spent near zero or two.

Figure (c) takes this idea a step further by adding twelve random numbers to produce each sample. The mean is now six, and the standard deviation is one. What is most important, the pdf has virtually become a Gaussian. This procedure can be used to create a normally distributed noise signal with an arbitrary mean and standard deviation. For each sample in the signal: (1) add twelve random numbers, (2) subtract six to make the mean equal to zero, (3) multiply by the standard deviation desired, and (4) add the desired mean.

The mathematical basis for this algorithm is contained in the Central Limit Theorem, one of the most important concepts in probability. In its simplest form, the Central Limit Theorem states that a sum of random numbers becomes normally distributed as more and more of the random numbers are added together. The Central Limit Theorem does not require the individual random numbers be from any particular distribution, or even that the random numbers be from the same distribution. The Central Limit Theorem provides the reason why normally distributed signals are seen so widely in nature. Whenever many different random forces are interacting, the resulting pdf becomes a Gaussian.

In the second method for generating normally distributed random numbers, the random number generator is invoked twice, to obtain R1 and R2. A normally distributed random number, X, can then be found:

Just as before, this approach can generate normally distributed random signals with an arbitrary mean and standard deviation. Take each number generated by this equation, multiply it by the desired standard deviation, and add the desired mean.

Random number generators operate by starting with a seed, a number between zero and one. When the random number generator is invoked, the seed is passed through a fixed algorithm, resulting in a new number between zero and one. This new number is reported as the random number, and is then internally stored to be used as the seed the next time the random number generator is called. The algorithm that transforms the seed into the new random number is often of the form:

In this manner, a continuous sequence of random numbers can be generated, all starting from the same seed. This allows a program to be run multiple times using exactly the same random number sequences. If you want the random number sequence to change, most languages have a provision for reseeding the random number generator, allowing you to choose the number first used as the seed. A common technique is to use the time (as indicated by the system's clock) as the seed, thus providing a new sequence each time the program is run.

From a pure mathematical view, the numbers generated in this way cannot be absolutely random since each number is fully determined by the previous number. The term pseudo-random is often used to describe this situation. However, this is not something you should be concerned with. The sequences generated by random number generators are statistically random to an exceedingly high degree. It is very unlikely that you will encounter a situation where they are not adequate.

Next Section: Precision and Accuracy