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Sec. 6–2 Power Spectral Density 435
where
Substituting this expression for R(k) into Eq. (6–70b) and using R(-k) = R(k), we get
x (f) = |F(f)|2 q
2
cs
T a a r(k)e -jkvT q
s 2
+ m a a e -jkvT s
d
s
Thus, for the general case of correlated data, the PSD of the digital signal is
where
x ( f) = s 2 a D|F(f)| 2 p (f) + (m a D) 2
Continuous spectrum
r (f)
r(k) = [a ' na ' n + k]
(6–70e)
(6–70f)
is a spectral weight function obtained from the Fourier transform of the normalized autocorrelation
impulse train
q
a
k = -q
k = -q
q
a
k = -q
r(k)e -j2pkfT s
r(k)d(t - kT s )
This demonstrates that the PSD of the digital signal consists of a continuous spectrum that
depends on the pulse-shape spectrum F( f ) and the data correlation. Furthermore, if m a 0
and F(nD) 0, the PSD will also contain spectral lines (delta functions) spaced at harmonics
of the baud rate D.
Examples of the application of these results are given in Sec. 3–5, where the PSD for
unipolar RZ, bipolar, and Manchester line codes are evaluated. (See Fig. 3–16 for a plot of the
PSDs for these codes.) Examples of bandpass digital signaling, such as OOK, BPSK, QPSK,
MPSK, and QAM, are given in Sections 5–9 and 5–10.
q
a
⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
⎧ ⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
n = - q
k = -q
|F(nD)| 2 d (f - nD)
Discrete spectrum
Example 6–5 PSD FOR A BIPOLAR NRZ LINE CODE USING
THE DATA AUTOCORRELATION
Using Eq. (6–70b) and referring to Fig. (3–15e), evaluate and plot the PSD for a bipolar NRZ line
code with A = 1.414 and R = 1. Also, evaluate and plot the PSD for a bipolar RZ line code with
A = 2 and R = 1. (The A’s are selected so that the NRZ and RZ line codes have the same average
energy per bit, E b .) See Example6_05.m for the solution. Compare these results with Fig. 3–16d.
White-Noise Processes
DEFINITION. A random process x(t) is said to be a white-noise process if the PSD is
constant over all frequencies; that is,
x (f) = N 0
2
(6–71)