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434
Random Processes and Spectral Analysis Chap. 6
where the autocorrelation of the data is
R(k) = a n a n+k = a
I
(6–70c)
in which P i is the probability of getting the product (a n a nk ) i , of which there are I possible
values. F( f ) is the spectrum of the pulse shape of the digital symbol.
Note that the quantity in brackets in Eq. (6–70b) is similar to the discrete Fourier transform
(DFT) of the data autocorrelation function R(k), except that the frequency variable v is
continuous. Thus, the PSD of the baseband digital signal is influenced by both the “spectrum” of
the data and the spectrum of the pulse shape used for the line code. Furthermore, the spectrum
may also contain delta functions if the mean value of the data, aq n , is nonzero. To demonstrate
this result, we first assume that the data symbols are uncorrelated; that is,
R(k) = e a2 n , k = 0
a n a n+k , k Z 0 f = es2 a + m 2 a, k = 0
m 2 a, k Z 0 f
where, as defined in Appendix B, the mean and the variance for the data are m a = aq n and
s 2 a = (a n - m a ) 2 = a 2 n - m 2 a . Substituting the preceding equation for R(k) into Eq. (6–70b),
we get
x (f) = |F(f)|2
T s
cs a 2 + m a
2
From Eq. (2–115), the Poisson sum formula, we obtain
q
a e ;jkvT s
k = -q
where D = 1/T s is the baud rate. We see that the PSD then becomes
x (f) = |F(f)|2
T s
ca a 2 + m a 2 D a
q
Thus, for the case of uncorrelated data, the PSD of the digital signal is
x ( f) = s 2 a D|F(f)| 2 + (m a D) 2
Continuous spectrum
= D a
q
⎧
⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
⎧
⎪⎨⎪⎩
n =-q
n =-q
(6–70d)
For the general case where there is correlation between the data, let the data autocorrelation
function R(k) be expressed in terms of the normalized-data autocorrelation function r(k).
'
That is, let an
represent the corresponding data that have been normalized to have unity
variance and zero mean. Thus,
i = 1
(a n a n+k ) i P i
q
a
n =-q
q
a
k = -q
e jkvT s
d
d(f - nD)
d(f - nD) d
|F(nD)| 2 d (f - nD)
Discrete spectrum
and, consequently,
a n = s a a ' n + m a
R(k) = s a 2 r(k) + m a
2