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

498 RANDOM PROCESSES AND SPECTRAL ANALYSIS
Fi g ure 9.25
Phasor
representation of
a sinusoid and a
narrowband
Gaussian noise.
}
n
ns
Sinusoidal Signal in Noise
Another case of interest is a sinusoid plus a narrowband Gaussian noise. If A cos (w e t + ({J)
is a sinusoid mixed with n(t), a Gaussian bandpass noise centered at We, then the sum y(t) is
given by
y(t) = A cos (w e t + <p) + n(t)
Using Eq. (9.66) to represent the bandpass noise, we have
y(t) = [A + ne(t)] cos (w e t + <p) + ils(t) sin (w e t + ({J)
= E(t) cos [w e t + 0(t) + ({J]
(9.82a)
(9.82b)
where E(t) is the envelope [E(t) > 0] and 0(t) is the angle shown in Fig. 9.25,
E(t) = J[A + nc(t)] 2 + n;(t)
Ils(t)
0(t) = - tan - 1 ---
A + Ile(t)
(9.83a)
(9.83b)
Both nc (t) and ns(t) are Gaussian, with variance a 2 . For white Gaussian noise, a 2 = 2NB
[Eq. (9.8Ob)]. Arguing in a manner analogous to that used in deriving Eq. (8.57), and observing
that
n 2 + n 2 = E 2 - A 2 - 2An
C S C
= E 2 - 2A (A + Ile) + A 2
we have
= E 2 - 2AE cos 0(t) + A 2 (9.84)
where a 2 is the variance of nc (or ns) and is equal to 2NB for white noise. From Eq. (9.84)
we have
PE (E) = 1_: PEe (E, 0) d0
= E e- (E 2 +A 2 ) /
2a [-1- f 2 n e<AE / a 2 ) cos 0 do]
2
a 2:rr -n
(9.85)
The bracketed term on the right-hand side of Eq. (9.85) defines I o (AE/a 2 ), where I o is the
modified zero-order Bessel function of the first kind. Thus,
(9.86a)