tutorial Statistical reliability analysis on Rayleigh probability distributions

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1 tutorial Statistical reliability analysis on Rayleigh probability distributions These techniques for determining a six-sigma capability index for a Rayleigh distribution use sample mean and deviation. By Russell J. Hoppenstein Statistical analysis is a useful tool for obtaining the reliability information of a device or process based on a limited number of samples. Sample sets are analyzed referenced to a known distribution function. Once a distribution function is assumed, the sample mean and variance are used to determine reliability information. Most processes follow the general trend of a Gaussian distribution. Gaussian statistics are most often used to characterize performance. The Gaussian random variable is generated by the infinite sum of a uniform random variable, per the Central Limit Theorem. The Gaussian probability distribution function (pdf) is Figure 1. Gaussian pdf. shown in Figure 1. It is bounded by plus and minus infinity and is symmetrical about its mean. The distribution is essentially defined by two parameters: its mean and standard deviation, sigma. Not all events can be characterized by a Gaussian distribution. Other types of distributions surface that behave in unique ways for which applying a Gaussian type of analysis does not fit the data set. For those types of distributions, the typical reliability equations must be altered to accurately relate the true characteristics of the process. Rayleigh pdf One common pdf is called the Rayleigh distribution, which characterizes processes that are determined by two independent, orthogonal, Gaussian random variables. The Rayleigh random variable is created by the following: R X + Y The variables X and Y are independent, zero mean Gaussian variables. The variable R is a Rayleigh random variable. The Rayleigh variable is essentially the distance from the origin to a point defined by two orthogonal parameters. The Rayleigh pdf is a twodimensional probability function, which can be difficult to analyze; however, the function can be simplified if the coordinate axes are transformed into cylindrical coordinates. As such, the pdf of the random variable R is symmetrical with respect to angle theta and can be expressed as: r r fr( r) exp a a The Rayleigh pdf is illustrated in Figure. Within this form, the function is more easily analyzed. The variable a is a constant used to determine the proper shape of the distribution. The distribution varies from zero to infinity. The mean and standard deviation of the distribution are not independent; they are both related to the variable a by: µ R a The variable a is derived from the standard deviation of the independent Gaussian functions that define it, which is: a σ X σ Y π π σ R a The Rayleigh random variable is used to define a process that varies as a Gaussian distribution within independent directions of X and Y. In the physical world this can be thought of as the position of an arrow on a target shot by an archer. The archer will aim for the bull s eye, but inevitable random errors in horizontal and vertical motion will cause the arrow to randomly hit the target. The position of the arrow with respect to the bull s eye is a form of a Rayleigh distribution. The Rayleigh distribution also characterizes more pertinent measurements within the communications field. Examples include measurements for input return loss, modulation sideband rejection, carrier suppression, and RF fading. These measurements consist of two independent and orthogonal parameters, expressed as either the amplitude and phase, or the real and imaginary parts. The magnitude of the vector from the starting point to the given point is defined as the Rayleigh random variable. continued on page October 000

2 Reliability The general purpose of applying statistical analysis is to determine reliability information. Because the Gaussian distribution is most common, the equations and terminology for reliability are generally based on those distributions; however, those equations do not apply to different types of random variables. First, the Gaussian reliability information and terminology are defined. The reliability information is expressed as a probability of finding a measurement value within certain limits. For example, the probability of finding a value within +/- one Gaussian sigma of the mean is 68.3% or stated mathematically as: P[ µ σ ] X P[ µ + σ] 68. 3% Furthermore, the error probability is calculated from the remaining percentage: 31.7% in the example above. Recall that the Gaussian distribution is symmetrical about its mean and each side or tail moves asymptotically toward zero; therefore, the error region within each tail contributes equally to the error probability. The terminology of achieving sixsigma reliability has become fairly renowned in the industry, but what does it mean? The six-sigma standard is initially confusing because the defined error probability is actually associated with a 4.5σ process. The error rate is determined by the probability of finding a point within one of the tails that is greater than 4.5σ away from the mean. The six-sigma nomenclature is introduced because the mean of the process is allowed to shift by 1.5σ from nominal as shown in Figure 3. Figure. Rayleigh pdf as a Function of Radius, r. Regardless of any mean shift, the defect rate derived from a 4.5σ process is kept constant. Table 1 shows probabilities for given deviations around the mean and the appropriate defect rate for a Gaussian distribution. Notice that no finite window encompasses every 76 October 000

3 Figure 3. Shifted mean Gaussian distribution. point on the distribution (i.e. probability is never 100%). The probability of a value falling outside 4.5σ per each side is or parts per million (ppm). This probability is small, but not zero; from here forth, six-sigma quality will be equated with an error probability fewer than 3.4 failures in a million tries. We can use the probability information of the Gaussian process as a function of its standard deviation to determine an equivalent expression for the Rayleigh distribution as a function of its standard deviation. The Rayleigh cumulative distribution function (cdf) is equated to the desired Gaussian probability: r 1 exp P + [ X µ nσ a ] G 1 Pe[ nσ G] where n represents any real positive number. Rearranging the equation and substituting the Rayleigh standard deviation in place of variable a yields the following: This equation provides a real scaling factor to equate a nσ Gaussian probability to an equal kσ Rayleigh probabilir σ R - ln Pe[ n G σ ] π October 000

4 Figure 4. Shifted mean Rayleigh distribution. ty. Table provides the scaling factors for commonly used probabilities. To achieve the equivalent error probability associated with a 4.5 Gaussian sigma for a Rayleigh distribution, the sample standard deviation must be scaled by Equivalently, the expression 4.5σ G 7.64σ R could be used. Capability index Reliability information is often expressed as a capability index. The capability index relates how well a sample data set meets a six-sigma probability for success. For a Gaussian distribution, the peak capability index, C pk, is given by: G min[ µ LSL, USL µ ] 3σ the mean could shift up to 1.5σ and still achieve the defect rate associated with a 4.5σ process. The scaled value of the Rayleigh standard deviation can be substituted into the above equation to generate a revised equation: R 15. min( USL µ R) 766. R µ R Shifted mean Rayleigh distribution Recall that the Rayleigh random variable is created from two independent, zero mean Gaussian variables. What happens if the contributing variables are not zero mean functions? In this case, the Rayleigh distribution becomes more complicated. With zero mean contributing variables, the Rayleigh distribution is symmetrical in variable theta within cylindrical coordinates. If one or both of the contributing variables has a shifted mean and then the Rayleigh distribution loses its symmetry, the distribution function becomes skewed when expressed as a function of r. Proper analysis would require the use of two-dimensional probability density functions, which negate the simple scaling factors used above; however, applying certain assumptions, the capability index can be approximated. For a skewed-mean Rayleigh distribution, the function is still bound by zero and infinity. The skewing effect essentially spreads out the distribution. This causes the mean and the standard deviation to increase. That, in turn, would push out the point for which a six-sigma probability is reached, but that point is unknown. Using the data from the sample set, some initial approximations can be made. Using the sample standard deviation, the equivalent non-skewed mean where LSL and USL represent lowerand upper-spec limits respectively. The C pk calculation explicitly uses the minimum distance in the numerator to provide the worst-case scenario. For convenience, the C pk equation can be scaled such that the denominator is equal to 4.5σ: Figure 5: Input return loss histogram and shifted mean distribution. G 15. min[ µ LSL, USL µ ] 45. σ It is evident that a 4.5σ probability of success occurs if the distance from the mean to the limit is equal to or greater than the distance of mean to the 4.5σ point; therefore, a C pk of value 1.5 or greater represents a process that meets a 4.5σ probability of success. Note, for a Gaussian distribution, a C pk value of or greater signifies that for Rayleigh distributions. Recall that for Rayleigh distributions, the sampled data are only valid for real positive numbers; so the mean should always be below the upper specification limit. Obtaining a C pk value of 1.5 using the above equation with a Rayleigh distributed sample set represents the same probability of success as that of a Gaussian distributed sample set meeting a six sigma defect rate. can be computed. This is the standard Rayleigh distribution with an equivalent variance as that of the sample data. Then, the non-skewed distribution is shifted such that its mean is equal to that of the sample mean. The result is a distribution that is more conservative than the actual data. The distance from the 4.5σ point to the mean can be approximated by the distance of the 4.5σ point of a standard Rayleigh distribution of given standard 80 October 000

5 ±σ Probability P e (per side) % 15.9% % 0.135% % % % % Table 1. Gaussian probabilities. deviation to that of its associated mean. That distance is then compared to the distance from the upper limit to the actual sample mean. The capability index is then found to be: 15. ( USL µ S 766. σ S µ 15. ( USL µ σ S ( USL µ σ S n P e [nσ G ] k % % 5.549% ppm ppm Table. Rayleigh scaling factors. where the subscript S is used to represent the parameters from a sample set. Recall that, for a standard Rayleigh distribution, the standard deviation and mean are not independent functions, so the denominator of the above equation can be simplified to a scaled quantity of sigma. The above equation will always provide a pessimistic estimate in which the error decreases as the means of the contributing variables approach zero. Experimental results The above techniques can be illustrated with an example and verified with experimental data. Simulated data were used to generate a skewedmean Rayleigh distribution. The distribution data were generated by two independent Gaussian variables with means that deviated from zero. Figure 4 illustrates three curves related to the data. The first is a histogram of the 5,000 data points. The second is a continuous Rayleigh distribution curve that has an equivalent standard deviation as the data set. The third curve represented the previous distribution that is shifted so that the mean matches the data set. Observe that the tail of the shifted mean distribution is higher than that of the data set; so error probabilities associated with the data set will be lower than that of the shifted mean distribution. Next, input return loss data from a 1900 MHz linear power amplifier was analyzed using the standard Gaussian equations and the modified Rayleigh equations. The measurements are originally recorded in decibels so the initial Gaussian equations are applied to those parameters. To apply the Rayleigh analysis, the data must first be converted to the linear scale. Table 3 shows the statistical data and the calculated capability index for the measurement. Note, the resultant C pk, using Gaussian techniques on the raw data, is less than unity. As such, the process would be incorrectly deemed less than six-sigma quality. On the other hand, using the more relevant Rayleigh analysis yields a C pk greater than 1.5, which characterizes a process below the six-sigma defect rate. To further enhance the notion that the data follows a Rayleigh distribution, the histogram of more than,000 data points and the equivalent shifted mean distribution are shown in Figure 5. Note that this sample set has a small mean shift which is imperceptible on the graph. Nevertheless, the distribution follows the general trend of a Rayleigh distribution. Conclusion Statistical analysis can be beneficial if the techniques and equations are used correctly. The modified capability Mean Std. Dev. Limit C pk Gaussian analysis db 3.7 db 8.0 db 0.80 Rayleigh analysis Table 3. Gaussian and Rayleigh C pk comparisons. index for Rayleigh distributed variables offers an additional tool for analyzing non-gaussian processes. Because the nature of the contributing variables cannot be known, the C pk for the shifted mean distribution should be used. It is guaranteed to provide reliability performance at a minimum, achieving six-sigma quality. Most of the parameters to which the Rayleigh analysis would apply are expressed in decibels. As expected, the transformation of the pdfs from the linear scale to the decibel scale is non-linear. Generating equivalent expressions for mean, standard deviation, and C pk for a Rayleigh distribution in log scale is not possible; therefore, the sample data must always be transformed to linear scale for the analysis to be valid. References [1] Williams, Richard H., Electrical Engineering Probability. St. Paul, MN: West Publishing Company, [] Papoulis, Athanasios, Probability, Random Variables, and Stochastic Processes. New York : McGraw Hill, About the author Russell J. Hoppenstein is the lead electrical engineer for Motorola in Ft. Worth, Texas. Hoppenstein acquired his undergraduate degree in electrical engineering at the University of Texas at Austin in 199 and a MSEE at the University of Texas at Arlington in He has worked in the development and design of cellular base stations at Motorola in Ft. Worth for eight years. Through the development and release of cellular products he has had extensive exposure to reliability and quality management. Currently, he is working in the Linear Power Amplifier group on 3G CDMA amplifiers. He can be reached at ; qrh00@ .mot.com. 8 October 000