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Rational Spectra

Of  special interest in engineering applications is the class of one-dimensional random processes that can be realized as the stationary output of a linear filter to white noise excitation. These processes have a spectral density of the form


where and are polynomial operators of order n and d respectively. The interest in this class of processes stems from the fact that a necessary and sufficient condition for a process to be realizable as a finite dimensional Markovian process is that its spectral density function be of the form expressed by equation (2.35) (Kree and Soize, 1986). Loosely speaking, the Markovian  property of a process implies that the effect of the infinite past on the present is negligible. That is, the process has a finite memory. For stationary process, equation (2.33) becomes,

When the domain covers the whole real line, the above equation is equivalent to the Wiener-Hopf integral equation , the solution of which may be found explicitly (Paley and Wiener, 1934; Noble, 1958). The case where is finite, however, is more relevant to the context of the monograph, and the solution of the associated integral equation is next detailed. Taking into consideration the one-dimensional form of equation (2.4), and substituting for from equation (2.35), equation (2.36) becomes,

Differentiating equation (2.37) twice with respect to is equivalent to multiplying the integrand by . Thus, applying the differential operator to this equation yields

where denotes the dirac delta function. Equation (2.38) can be viewed as a reformulation of equation (2.36) as a homogeneous differential equation. This equation may be solved in terms of the parameter and of 2d arbitrary constants which are calculated by backsubstituting the resulting solution into equation (2.36). Note, parenthetically, that explicit expressions for the transcendental characteristic equation associated with the differential equation (2.38), for a number of kernel functions, are given in Youla (1957). In the remainder of this section, the preceding treatment of equation (2.36) is applied to the important kernel representing the first order Markovian process. This kernel has been used extensively to model processes in a variety of fields (Yaglom, 1962). Further, it is noted that higher order Markovian kernels may be expressed as linear combinations of first order ones. This important kernel  is given by the equation

where b is a parameter with the same units as x and is often termed the correlation length, since it reflects the rate at which the correlation decays between two points of the process. It is assumed that the process is defined over the one dimensional interval . Clearly, can be made rapidly attenuating versus by selecting a suitable value of the parameter b. In case the domain of the problem is the one-dimensional segment , the eigenfunctions and eigenvalues of the covariance function given by equation (2.39) are the solutions to the following integral equation (Van Trees, 1968)


where . Equation (2.40) can be written as


Differentiating equation (2.41) with respect to and rearranging gives


Differentiating once more with respect to , the following equation is obtained

Introducing the new variable

equation (2.43) becomes


To find the boundary conditions associated with the differential equation (2.45), equations (2.41) and (2.42) are evaluated at x = -a and x = +a. After rearrangement, the boundary conditions become

Thus, the integral equation given by equation (2.45) is transformed into the ordinary differential equation (2.45) with appended boundary conditions given by equations (2.46) and (2.47). It can be shown that is the only range of for which equation (2.45) is solvable, the solution being given by the equation,


Further, applying the boundary conditions specified by equations (2.46) and (2.47), gives

Nontrivial solutions exist only if the determinant of the homogeneous system in equation (2.49) is equal to zero. Setting this determinant equal to zero gives the following transcendental equations

Denoting the solution of the second of these equations by , the resulting eigenfunctions are

and

for even n and odd n respectively. The corresponding eigenvalues are

and


where and are defined by equation (2.50). Thus, a process with covariance function given by equation (2.39) can be expanded as

 
Figure 1.1: Eigenfunctions , , n = 1 , 2 , 3 , 4; Exponential Covariance, Correlation Length=1.

 
Figure 1.2: Trends of the Eigenvalues of the Exponential Kernel for Various Values of the Correlation Length b; assumes values for only.


 
Figure 1.3: Exact Covariance Surface versus and ; , ; Correlation Length = 1.


 
Figure 1.4: 4-Term Approximation of Covariance Surface versus and ; , ; Correlation Length = 1.

 
Figure 1.5: 4-Term Relative Error Surface of Covariance Approximation versus and ; , ; Maximum Error = 0.1126; Correlation Length = 1.

 
Figure 1.6: 10-Term Approximation Covariance Surface versus and ; , ; Correlation Length = 1.

 
Figure 1.7: 10-Term Relative Error Surface of Covariance Approximation versus and ; , ; Maximum Error = 0.0425; Correlation Length = 1.


Figure (2.1) shows the first 4 eigenfunctions as defined by equations (2.51) and (2.52), for a value of a equal to and a value of b equal to 1. Figure (2.2) shows the eigenvalues as given by equation (2.54) for various values of b. Note that the smaller the value of b, the more contribution should be expected from terms associated with smaller eigenvalues. Figures (2.3)-(2.7) show the exact kernel, its four term approximation, its ten term approximation and the associated errors, respectively.



next up previous
Next: Nonrational Spectra Up: Solution of the Previous: Preliminary Remarks



roger g ghanem
Fri Oct 27 18:43:58 EDT 1995