B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

References 499
Figure 9.26
Rican PDF.
This is known as the Rice density, or Ricean density. For a large sinusoidal signal (A » a),
it can be shown that 4
and
Io ( AE ) 2 - :::::: -- e
AE/a 2
a 2 2rrAE
p ( E ) :::::: ✓ E e-<E-A) 2 /za 2
E
2rrAa 2
(9.86b)
Because A » a, E :::::: A, andpE(E) in Eq. (9.86b) is very nearly a Gaussian density with mean
A and variance a,
(9.86c)
Figure 9.26 shows the PDF of the normalized RV E/a. Note that for Aja = 0, we obtain the
Rayleigh density.
From the joint PDF P E e (E, 0), we can also obtain P e
(0), the PDF of the phase e, by
integrating the joint PDF with respect to E,
P e (0) = fo 00 PEe (E, 0) dE
Although the integration is straightforward, there are a number of involved steps, and for this
reason it will not be repeated here. The final result is
P e (0) = 2
e-A2 ; 2a 2 { 1 + ✓ 2 n cos 0 2 cos 2 0/2a2 [ 1 - Q ( A c ; s 0 ) ] }
(9.86d)
REFERENCES
1. B. P. Lathi, An Introduction to Random Signals and Communication Theory, International Textbook
Co., Scranton, PA, 1968.
2. J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering, Wiley, New York,
1965.