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AM, FM, and Digital Modulated Systems Chap. 5
This equation is plotted in Fig. 5–40a, where it is apparent that the autocorrelation
function for the PN waveform is periodic with triangular pulses of width 2T c repeated every NT c
seconds and that a correlation level of -1N occurs between these triangular pulses.
Furthermore, since the autocorrelation function is periodic, the associated PSD is a line
spectrum. That is, the autocorrelation is expressed as the Fourier series
q
R c (t) = a r n e j2pnf 0t
(5–126)
n=-q
where f 0 = 1(NT c ) and {r n } is the set of Fourier series coefficients. Thus, using Eq. (2–109)
yields
q
c (f) = [R c (t)] = a r n d(f - nf 0 )
(5–127)
n=-q
where the Fourier series coefficients are evaluated and found to be
R c ()
1.0
1
N
T c
T c
NT c
(a) Autocorrelation Function
0
p c (f )
NT c
t
Weight =
( N + 1
2
( sin(∏n/N)
N 2 ∏n/N
Weight = 1/N 2
(
(
3 2 1 0 1
2
3
Tc Tc Tc
Tc Tc Tc
f 0 =1/(NT c )
f
(b) Power Spectral Density (PSD)
Figure 5–40
Autocorrelation and PSD for an m-sequence PN waveform.