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Sec. 6–2 Power Spectral Density 433
General Formula for the PSD of Digital Signals
We now derive a general formula for the PSD of digital signals. The formulas for the PSD in
Example 6–4 are valid only for polar signaling with a n = ; 1 and no correlation between the
bits. A more general result can be obtained in terms of the autocorrelation of the data, a n , by
starting with Eq. (6–56). As illustrated in Figs. 3–12 and 3–14, the data may be binary or multilevel.
The duration (width) of the symbol pulse f(t) is T s . For binary data, T s T b , where T b
is the duration of 1 bit. We define the autocorrelation of the data by
R(k) = a n a n+k (6–68)
Next, we make a change in the index in Eq. (6–58), letting m n k. Then, by using Eq. (6–68)
and T (2N 1) T s , Eq. (6–58) becomes
Replacing the outer sum over the index n by 2N 1, we obtain the following expression.
[This procedure is not strictly correct, since the inner sum is also a function of n. The correct
procedure would be to exchange the order of summation, in a manner similar to that used in
Eq. (6–47) through (6–50), where the order of integration was exchanged. The result would
be the same as that as obtained below when the limit is evaluated as N → q.]
or
x (f) = ƒ F(f) ƒ
2
x (f) = ƒ F(f) ƒ 2
But because R(k) is an autocorrelation function, R(-k) = R(k), and Eq. (6–69) becomes
In summary, the general expression for the PSD of a digital signal is
An equivalent expression is
T s
= ƒ F(f) ƒ 2
T s
x (f) = ƒ F(f) ƒ 2
q
cR(0) + a R(-k)e -jkvT q
s
+ a R(k)e jkvT s
d
T s
x (f) = ƒ F(f) ƒ 2
q
cR(0) + a R(k)(e jkvT s
+ e -jkvT s
) d
T s
(2N + 1)
lim c
N: q (2N + 1) a
= ƒ F(f) ƒ 2
T s
cR(0) + ā
1
x (f) = ƒ F(f) ƒ 2
q
cR(0) + 2 a R(k) cos(2pkfT s ) d
T s
k = q
a
k =-q
R(k)e jkvT s
k =-q
k = 1
k = 1
k = 1
q
x (f) = |F(f)|2 c
T a
s
R(k)e jkvT q
s
+ a R(k)e jkvT s
d
k = -q
n = N
lim c 1
N: q (2N + 1)T a s
k = N- n
k =-N - n
k = N-n
a
n = -N k = -N-n
R(k)e jkvT s
d
R(k)e jkvT s
d
k = 1
k = 1
R(k)e jkvT s
d
(6–69)
(6–70a)
(6–70b)