Both strategies on imaginary and real parts are like each other. Equation (18) can be simplified as

(19)

and

(20)

(21)

If and t has a Gaussian distribution

(22)

Now we propose a computational method to generate a proper function of, set where we select as large as possible, considering small positive value for, we have

for

while

if

end

end

end

Where is a large number which, Figure 1 shows nonlinear function synthesizing to generate uniform PDF in [–0.5, 0.5] interval. It is clear that generated function guarantees the Equations (20) and (21).

4. Some Examples of Generating Asymmetric and PDF with Desired PSD

4.1. One Dimensional Real Signal

Consider the representation of a WSS Rayleigh random process with the variance of, mean given by

, and a given continuous PSD. To represent and generate this process we need determining a nonlinear transform function to convert Gaussian distribution of to Rayleigh PDF, the PDFs of the amplitude and frequency can be calculated after the first approximation of desired PSD. Figure 2 shows a nonlinear transform function to convert Gaussian with shaped spectrum to Rayleigh one. Figure 3 depicts the Rayleigh PDF, the blue curve in Figure 4 shows the first approximated power spectral density and the red dashed line shows desired spectrum. We define the following error function

(23)

This error function can be decreased by the use of sinusoidal terms

(24)

where

and

(25)

Therefore, the frequency PDF can be calculated as the following relation

(26)

also

(27)

The average of 10,000 realizations of a periodogram along with the true PSD (indicated by the black curve) is shown in Figure 4. It is interesting to note that the generated PDF has less accuracy than what is calculated before adding sinusoidal term and it is depicted in Figure 5. First approximation play an important role in having a more accurate results.

4.2. Two Dimensional Complex Signal

Consider the WSS complex signal with ar-

Figure 1. Proper nonlinear transform function, and.

Figure 2. Nonlinear transform function converting non zero mean Gaussian PDF to Rayleigh random process.

Figure 3. Generated Rayleigh random process after applying g(t) on Gaussian PDF.

Figure 4. Power spectral density synthesizing.

Figure 5. First order probability density function of Z[n], red line depicts Rayleigh PDF.

bitrary power spectral density, where and are independent from each other and represent the real and complex parts of respectively. Both and have desired PDF and every one generates a part of total power spectrum which depends on its variance. Therefore, where is the PDF of a complex signal and, represent the PDF of real and imaginary parts of the signal. Figure 6 shows a joint PDF of and. If we consider

(28)

and

(29)

where

(30)

After applying the algorithm that was mentioned in Section 3 we can achieve to proper nonlinear transform function. Figure 7 shows generated PDF and Figure 8 depicts imaginary and real parts of this discrete signal. The power spectrum can be converged to its correct response by considering proper PDF for both and values as follows

(31)

(32)

Therefore

(33)

(34)

Generated and desired power spectral density also has been shown in Figure 9.

5. Conclusion

A new method has been presented which can generate a complex WSS random process with a desired PSD and a given first-order PDF. One of the advantages of this method is that we have no limitation on PDF to be infi-

Figure 6. Joint PDF of x and y.

Figure 7. Generated joint PDF of Re{Z} and Im{Z}.

Figure 8. Discrete imaginary and real parts of generated signal.

Figure 9. Synthesized power spectral density.

nitely divisible or symmetric. Also with simple computational method we can calculate proper transform function to generate desired PDF from Gaussian PDF with shaped spectrum. Furthermore, this computational method guarantees generated spectrum to be under the wanted spectrum for the first approximation. The error function is reduced to zero by the means of sinusoidal component with proper frequency and amplitude PDF, but with small deviation from exact desired PDF. In other words, there is some tradeoff between exact PDF and PSD. The ergodicity in the mean and variance has been proven under certain conditions. We have avoided bottleneck calculation of Hankel transform function that [8] faces directly to it. Because of negligible deviation from exact PDF it can be used in any practical system.

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Appendix

A) Mean value

and

while

similarly

therefore

B) Autocorrelation

the above relation can be simplified as

similarly

on the other hand

and

therefore

C) PDF of A and B Assume are constant values, with this assumption

if we consider

then

and

and

Therefore, in domain can be modeled as

by considering Bessel function characteristics we have

Therefore

suddenly

similarly