082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

818 Chapter 24 The Fourier transform
The received and demodulated signal is represented by f[n] and the transmitted
signal prior to modulation is represented byg[n]. Consider the case whereg[n] is a
square pulse comprising 10 samples. Let the time between each sample beT. Both
signalsareshownplottedinFigure24.34.Notethatthisisanidealizedcaseinwhich
thereis no noise on the received signal.
1.0
0.8
0.6
g[n]
0.4
f [n]
0.2
20 40 60 80 100 120 140
Figure24.34
Plotoftransmitted and received digital signals foraradar system.
n
The returned signal is delayed due to the time taken for it to travel to a distant
target and back. Examining Figure 24.34 we note that the returned signal is delayed
by 65 samples when compared with the transmitted signal. This equates to a time
delayof65T.Weknowthespeedthatthetransmittedpulsetravelsisc,thespeedof
light.Using
distance =time × velocity
gives total distance travelled = 65T ×c. But the total distance travelled is twice the
distance between the transmitter and the target,thatis2d. Henced = 65Tc
2 .
We now confirm the number of samples delay by calculating the circular crosscorrelation
of f[n] andg[n]. We wish to calculatec[n] = f ○∗ gand plot this on a
graphforanalysis.Thiscanbedoneeitherdirectlybycalculatingthecircularcrosscorrelation
or by using the circular correlation theorem. Owing to the large number
ofcalculationsinvolvedtheyarenotpresentedhere.Normallysuchaprocesswould
be carriedout usingacomputer.
TheresultisasequenceofsampleswhichareplottedinFigure24.35.Thelargest
value of c[n] occurs at 65, which corresponds to the delay in samples between the
two signals. It agrees with our initial observation where the difference was quite
straightforward tosee by inspecting the two signals.