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Baseband Pulse and Digital Signaling Chap. 3
where F(f) is the Fourier transform of the pulse shape f(t) and R(k) is the autocorrelation of the
data. This autocorrelation is given by
(3–36b)
where a n and a n + k are the (voltage) levels of the data pulses at the nth and (n + k)th symbol
positions, respectively, and P i is the probability of having the ith a n a n + k product. Note that
Eq. (3–36a) shows that the spectrum of the digital signal depends on two things: (1) the pulse
shape used and (2) statistical properties of the data.
Using Eq. (3–36), which is the stochastic approach, we can evaluate the PSD for the
various line codes shown in Fig. 3–15.
Unipolar NRZ Signaling. For unipolar signaling, the possible levels for the a’s are +A
and 0 V. Assume that these values are equally likely to occur and that the data are independent.
Now, evaluate R(k) as defined by Eq. (3–36b). For k = 0, the possible products of a n a n are
A * A= A 2 and 0 * 0 = 0, and consequently, I = 2. For random data, the probability of having
A 2 1
1
is and the probability of having 0 is , so that
2
For k Z 0, there are I = 4 possibilities for the product values: A A, A 0, and
0 * A, 0 *
1
0. They all occur with a probability of Thus, for k Z 0,
Hence,
R(k) = a
4
R(0) = a
2
i=1
i=1
I
R(k) = a (a n a n+k ) i P i
2
i=1
(a n a n ) i P i = A 2 # 1 2 + 0 # 1 2 = 1 2 A2
4 .
(a n a n+k )P i = A 2 # 1 4 + 0 # 1 4 + 0 # 1 4 + 0 # 1 4 = 1 4 A2
1
2 A2 , k = 0
R unipolar (k) = d
t
1
4 A2 , k Z 0
(3–37a)
For rectangular NRZ pulse shapes, the Fourier transform pair is
f(t) =ßa t sin pf(T b )
b 4 F(f) = T (3–37b)
T b
.
b pfT b
Using Eq. (3–36a) with T s = T b , we find that the PSD for the unipolar NRZ line code is