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Sec. 6–7 Bandpass Processes 463
f R (R)
0.6
f R (R)=
(R/Í 2 )eR 2 /2Í 2 , R ≥ 0
0 , R < 0
0.4
where Í=1
0.2
(a) PDF for the Envelope
R
f¨ (¨)
1
2∏
2∏
¨
(b) PDF for the Phase
Figure 6–14 PDF for the envelope and phase of a Gaussian process.
This is called a uniform PDF. Sketches of these PDFs are shown in Fig. 6–14.
The random variables R = R(t 1 ) and u = u(t 1 ) are independent, since f R (R, u) = f R (R) f(u).
However, R(t) and u(t) are not independent random processes because the random variables
R = R(t 1 ) and u = u(t 1 + t) are not independent for all values of t. To verify this statement, a fourdimensional
transformation problem consisting of transforming the random variables (x(t 1 ), x(t 2 ),
y(t 1 ), y(t 2 )) into (R(t 1 ), R(t 2 ), u (t 1 ), u (t 2 )) needs to be worked out, where t 2 - t 1 = t (Davenport
and Root, 1958).
The evaluation of the autocorrelation function for the envelope R(t) generally requires
that the two-dimensional density function of R(t) be known, since R R (t) = R(t)R(t + t) .
However, to obtain this joint density function for R(t), the four-dimensional density function
of (R(t 1 ), R(t 2 ), u(t 1 ), u(t 2 )) must first be obtained by a four-dimensional transformation, as
discussed in the preceding paragraph. A similar problem is worked to evaluate the autocorrelation
function for the phase u(t). These difficulties in evaluating the autocorrelation function
for R(t) and u(t) arise because they are nonlinear functions of v(t). The PSDs for R(t) and u(t)
are obtained by taking the Fourier transform of the autocorrelation function.