Diss / 10

The upper bound of the jth channel can be presented in the following form:

corresponding to the middle between the levels gj and gj−1.

The principal flowchart of a pdf measurer is shown in Figure 14.12. The functioning principles are the following. The input realization x(t) of a stochastic process comes in at the ADC input. After sampling in time and amplitude, the input signal comes in at the input of input data register and, after that, at the ALU input, in which a determination of the pdf is carried out according to algorithm. The final result, i.e., the ALU output, changes information in RAM for output data, where the values of probability density function estimates for all ν channels are stored. The value of the pdf is reproduced by a monitor. Initial data required to compute the pdf estimations are stored by ROM in the form of total number of samples N, interval between the channels (or levels) x, and interval [c, d] of possible values of investigated stochastic process. The channel number where we can find the reading x[i] = xi can be defined as

where {…} means the integer number. The algorithm to compute the pdf estimate is presented in Figure 14.13. There is a need to note that all blocks of the structural block diagram can be realized by microprocessor systems.

The previously mentioned methods to measure the pdf are based on uniform division of the range of possible stochastic process values. However, this procedure, as a rule, is not optimal, since peculiarities of measured characteristic variations are not taken into consideration, which leads finally to information excess and high errors under definition of characteristics of argument values. Additionally, there is an inverse problem—to define the argument values by the given pdf values.

Consider the method of argument definition by the given characteristic values applied to the probability distribution function estimate in the case of sampled stochastic process. Division of probability distribution function domain on discrete values Fj must be done in accordance with the applied problem. The most favorable action is a uniform division from zero to unit using the

Information about p*j

j = j +1

FIGURE 14.13  Algorithm of pdf computation.

equal intervals δF. The method to define an argument of probability distribution function is the following. In the case of arbitrary argument, according to the rule given by (14.20), the probability distribution function estimate F*(x) is computed and compared with the given value Fj. As a result of comparison, we obtain the signal

based on which it is possible to measure the level x. The value x j at which the signal given by (14.156) is equal to zero is the estimate of probability distribution function argument. In practice, a nonzero measure is chosen as a criterion that the estimate F*(x) is very close to Fj. The simplest way to define this measure is to compare the module of the signal given by (14.165) with the previously given value ε. In this case, the level satisfying the condition

can be considered as the estimate x j . The probability distribution function can be given either in the form of deterministic numbers Fj within the limits of the interval [0,1] or using the values of corresponding transformations of reference stochastic process samples with the known probability distribution function [4].

Consider the measurement method when the probability distribution function is given as a set of discrete numbers Fj. In this case, the current estimate x j of argument for jth value of probability distribution function can be defined using the recurrent relationship (12.342) that can be transformed and applied to the considered method of measurement in the following form:

In doing so, the current estimate of the probability distribution function Fj [N] at the nth step is defined as

N

Fj [N] = 1 ∑ηx[i], (14.159)

N i=1

where we use the transformation by analogy with (14.19)

Transformation (14.160) means that the current sample of investigated stochastic process at the ith iteration step is compared with the argument estimate obtained at the previous (i − 1) th iteration step. Initial argument value x j [0] = a is chosen in arbitrary way from the range of possible values [c, d]. The factor γ[N] defining the next iteration step must satisfy the condition given by (12.343). As a rule, it is inversely proportional to the number of iterations, i.e., γ[N] = κN−1, where κ > 0 characterizes the value of the first iteration step. The probability distribution function estimate given by (14.159), by analogous way with (12.347), can be presented in the following­ form:

The iteration number N is defined based on the selected rule. The most widely used rule can be presented in the following form:

where α > 0 is the given number. When the condition (14.162) is satisfied, the iteration process is stopped. The structure of algorithm to compute the argument estimates for the given values of probability function distribution is shown in Figure 14.14.

Computing a ratio of the end differences of probability distribution function and its arguments, we can define the pdf estimate of the investigated stochastic process. As applied to the probability distribution function estimate by three of its values x j −1, x j , and x j +1, the pdf estimate for the argument x*j can be presented in the following form:

j = j +1

FIGURE 14.14  Algorithm of the argument estimate computation.

Under the use of reference stochastic process with the known pdf, the structure of algorithm to measure the arguments of probability distribution function is changed. Denote discrete values of realizations of investigated and reference stochastic processes as xi and yi, respectively. As applied to the reference stochastic process, we can theoretically define the argument value y by the given value of probability distribution function F(y) = Fj. If we apply the following transformation

with respect to samples yi of reference stochastic process, then the unbiased probability distribution function estimate F*(y) of random sample yi is defined by (14.20). Comparing the probability distribution function estimate of reference stochastic process with the probability distribution function estimate F*(x) of the investigated stochastic process, we obtain the signal

N

ε(x) = F (y) − F (x) = N1 ∑zi , (14.165)

i=1

FIGURE 14.16  Algorithm of computation of the probability distribution function argument estimations.

14.7  SUMMARY AND DISCUSSION

The pdf is defined experimentally like the average value of the pdf within the limits of the interval [x ± 0.5dx]. Because of this, from the viewpoint of approximation of the measured pdf to its true value at the level x, it is desirable to decrease the interval x. However, a decrease in the x at the fixed observation time interval leads to a decrease in time of the stochastic process within the limits of the interval x and, consequently, to an increase in the variance of the pdf estimate owing to decreasing the statistical data sample based on which the decision about the pdf magnitude is made. Therefore, there is an optimal value of the interval x, under which the dispersion of the pdf estimate takes the minimal magnitude at the fixed observation time.

There are several ways to measure the total time of a stochastic process observation below the given level or within its limits. The first way is the direct measurement of total time when a magnitude of the realization of continuous ergodic process is below the fixed level x. In this case, the realization of continuous ergodic process is transformed into the sequence of rectangular pulses (see the Figure 14.1b) with the unit amplitude and duration equal to the time when a magnitude of the realization of continuous ergodic process is below the fixed level. The second method to define and measure the probability distribution function is based on computation of the number of quantized by amplitude and duration of the sampled pulses. In this case, at first, the continuous realization of stochastic process is transformed by corresponding pulse modulator into sampled

sequence and comes in at the input of the threshold device with the threshold level. Then, a ratio of the number of pulses that do not exceed the threshold to the total number of pulses corresponding to the observation time interval is equal by value to the probability distribution function estimate.

The root-mean-square deviation σ{F*(x)} of measurements of the probability distribution function of Gaussian stochastic process as a function of the normalized level for various values of p is presented in Figure 14.3. As we can see from Figure 14.3, the maximal values of σ{F*(x)} are obtained at the zero level. Formula (14.63) demonstrates that the variance of the probability distribution function estimate F*(x) of stochastic process by correlated samples increases in comparison with the variance of the probability distribution function estimate F*(x) of stochastic process by uncorrelated samples of the same sample size N.

The dependences Var{f (x)}N p(x) versus the normalized interval between the levels z and various values of the normalized level z under investigation of the Gaussian (the solid line) and Rayleigh (the dashed line) stochastic processes in the case of uncorrelated samples are shown in Figure 14.8. As we can see from Figure 14.8, as opposed to the bias of pdf estimate, the variance of the pdf estimate is decreased with an increase in the interval between levels. However, under experimental measurement of the pdf the interval between the levels x must be defined based on the condition of minimization of dispersion of the pdf estimate. The normalized resulting errors of

measurement of the probability density function of Gaussian stochastic process D{p (x)} p(x) as a function of the normalized interval between levels z at z = 0 and N = 104 and N = 105 are shown in Figure 14.9. As we can see from Figure 14.9, for the given number of samples and at z = 0, the minimal magnitude of the resulting error is decreased with a decrease the values z under an increase in the number of samples.

REFERENCES

1.Haykin, S. and M. Moher. 2007. Introduction to Analog and Digital Communications. 2nd edn. New York: John Wiley & Sons, Inc.

2.Gradshteyn, I.S. and I.M. Ryzhik. 2007. Tables of Integrals, Series and Products. 7th edn. London, U.K.: Academic Press.

3.Sheddon, I.N. 1951. Fourier Transform. New York: McGraw Hill, Inc.

4.Domaratzkiy, A.N., Ivanov, L.N., and Yu. Yurlov. 1975. Multipurpose Statistical Analysis of Stochastic Signals. Novosibirsk, Russia: Nauka.