Techniques in Signal Processing CSC 508 Generating Random Numbers with a Computer

2.3 Generating Random Numbers Before we begin this section, it is important to realize that a digital (VonNeumann architecture) computer cannot generate sequences of random numbers. Only after you accept and understand this will we proceed to generate sequences of random numbers with a computer. At this point some explanation is in order: A conventional digital computer is perfectly deterministic and, therefore, is incapable of exhibiting any sort of random behavior (unless it is broken). Computation methods exist t generate sequences of numbers that appear to be random and that pass certain standard tests for randomness. These are called pseudo-random sequences. You must appreciate this difference to avoid embarassing and sometimes costly errors and simulation and analysis. Any simulation or other computer-based analysis which relies on the randomness of a pseudo-random sequence, is susceptable to a special kind of error. Improperly used, a pseudo-random sequence can produce structure or artifacts in your data that are a product of the pseudo-random number generator itself rather than a real feature of the system being modeled. The most important thing to remember is that every pseudo-random number sequence generated on a computer will eventually begin to repeat itself. We must never draw from such a sequence past this point.

Method of Congruence One of the most popular methods of generating pseudo-random sequences is called the method of congruence. A sequence of n pseudo-random number X n ={x 0,x 1,..., x n ) may be generated by the recurrence relation, where we establish the following relationship between the values a,b and T: (1) T is the maximum possible value in Xn, (a function of our computer). (2) b is relatively prime to T (i.e. they have no common factors) (3) a = 1 mod p where p is a prime factor of T. (4) a = 1 mod 4 if 4 is a factor of T. For example, if we let T=2q then b must be odd and we must ensure that a=1 mod 4. If we let T=10q then b cannot be divisible by 2 or 5 (the prime factors of T) and we must choose a value of a so that a = 1 mod 20. To answer the question building in your mind. No. We cannot rely on the built-in random number generators provided with most lanuage compilers. These ready-to-go random number generators are notoriously flawed and inadequate for serious computer research.

A simple Ada function using the method of congruence is shown below. This function generates a uniformly distributed pseudo-random sequence. An inital seed value for XVAL (say 141) is needed to start the sequence. Each time RAN(MAXVAL) is called an integer between 0 and MAXVAL is returned. If our computer can represent integers up to in value then we must make sure that A*XVAL+B does not exceed this value. As a test we note that for XVAL 0 =141, T=2 10, A=2 4 +1=17 and B=1 The maximum value of XVAL is T-1 or 1023 and that A*XVAL+B never exceeds which is <

In general the value of MAXVAL must not be set larger than T/2, otherwise some values in the range (0..MAXVAL) cannot be generated by RAN. This is due to the truncation in the conversion from floating point back to integer type in the RAN function. Now that we have established the maximum size for MAXVAL, we should consider how many values can be drawn from RAN before we are in danger of obtaining a repeated sequence. A good rule of thumb is to never exceed the square root of the maximum number of values which can be represented by the computer system we are using. In our case we are limited to /2 =181. Those of you with programming experience may be interested in testing the random number generator built into your favorite programming language for the length of repeating sequences. (Or maybe not.) We can use a uniform random number generator to generate random numbers over any range of values. For example, we can use RAN to produce a source of random floating- point numbers U [ ] in the range [-1.0,+1.0] with the following scaling transormation, where MAXVAL is the largest integer allowed as an upper limit for the function RAN. In general, a uniform random distribution over any range can be obtained through a process of scaling and shifting the values produced by RAN.

Consider the graph on the left in which a random deviate x in the range [0,MaxVal] is used in the transform- ation function R(x) to compute a new uniform deviate in the range [-1.0,+1.0]. This is called the inverse method of generating random deviates and can be used to generate other types of random distributions such as the normal distribution from a uniform distribution. To use the inverse method to produce normally distributed random numbers we need to find a function N(x) to replace R(x). This function must map uniformly distributed values U(x) into normally distributed values N(x).

N(x) is the functional inverse of the integral of the Gaussian PDF. This integral is called the cumlative distribution function (CDF) and has no closed-form solution. We can approximate N(x) using a method after Abramowitz and Stegun, as shown in this chart. This approximation produces values of N(x) which are good to within +/ _ 4.5x10 -4 when modeling a normal distribution with zero mean and unit sigma. You can then convert these normal deviates to any mean and standard deviation by multiplying each times the desired  and adding the desired . N'(x) =  N(x) + 

The Direct Method of Generating Normal Deviates Another method for generating normal deviates is called the direct method. Given two draws from a uniform random distribution U 1 and U 2 with range [0.0,1.0], we generate two independent normal random deviates N 1 and N 2 by, Remember that the implied mean of the normal distribution is 0.0 and the standard deviation is 1.0 for the normal distribution modeled using the direct method. What values of U 1 and U 2 produce the largest positive and largest negative values for N? What values of U 1 and U 2 result in N=0.0? This is the preferred method when the cost of computing the logarthims and trigonometric functions is not prohibitive. With the nearly universal availability of CPUs with built-in math coprocessors, the direct method of computing normal random deviates is becoming the standard.