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Unformatted text preview: ------(-bl---t---- f(t-::-:f-I _ ________ Fig. 3.5 Physical explanation of the correlation function.
against the channel noise and o ther imperfections. However, as mentioned earlier,
because of some o ther reasons, different schemes such as a n o rthogonal scheme,
where e n = 0, a re also used even when t hey provide a smaller margin (from 0 t o 1
a nd vice versa) in distinguishing t he pulses.
Some a spects of pulse dispersion were discussed in Sees. 2.7-5 and 2.7-6. I n
c hapter 4, we s hall discuss pulse distortion during transmission. Calculation of error
probability in t he presence of noise a nd o ther i mperfections are beyond t he scope
of this work. For a detailed t reatment of this subject, t he r eader may refer t o t he
a uthor's b ook on communication systems. 1 3 .2-2 Correlation Functions Consider a n a pplication of correlation t o signal detection in a radar, where
a signal pulse is t ransmitted in order t o d etect a suspected target. I f a t arget is
p resent, t he pulse will be reflected by it. I n c ontrast, if t he t arget is n ot p resent,
there will b e no reflected pulse, j ust noise. B y d etecting t he presence or absence of
the reflected pulse we confirm t he presence or absence of t he t arget. By measuring
the time delay between t he t ransmitted a nd received (reflected) pulse we d etermine
t he d istance of t he t arget. Let t he t ransmitted a nd t he reflected pulses b e d enoted
by g(t) a nd I (t), respectively, as shown in Fig. 3.5. I f we were to use Eq. (3.25)
directly to measure t he c orrelation coefficient en, we would o btain en = 1 ~ y EjE g ;00 f(t)g(t) dt = 0 (3.29) - 00 Thus, t he c orrelation is zero because t he pulses are disjoint (nonoverlapping in
time). T he i ntegral (3.29) will yield zero value even when t he pulses are identical
b ut w ith relative time shift. To avoid this difficulty, we c ompare t he received pulse
f (t) w ith a delayed pulse g(t) for various values of delay. I f for some value of delay
p arameter t here is a s trong correlation, we n ot only detect t he presence of t he pulse
b ut we also d etect t he relative time shift of f (t) w ith respect t o g(t). For this
reason, instead of using t he i ntegral on the right-hand, we use t he modified integral
1/!jg(t), t he c rosscorrelation function of two real signals f (t) a nd g(t) defined b yt
t For c omplex signals we define
'l/!fg(t) == 1: f *(r)g(r - t) d r 182 3 Signal Representation by Orthogonal Sets 1/Jfg(t) == [ : f (7)g(7 - t)d7 3.3 Signal representation by Orthogonal Signal Set 183 (3.30) Here 7 is a d ummy variable, a nd t he pulse 9 (7 - t) is t he pulse 9 (7) delayed by
t seconds with respect to t he f (7) pulse. Therefore, 1/;fg(t) is a n indication of
similarity (correlation) of the f pulse with 9 pulse delayed by t seconds. Thus,
1/J fg(t) m easures t he similarity of pulses even if they are disjoint. In t he case of
signals in Fig. 3.5, 1/J fg(t) will show significant correlation around t = T . T his
observation allows us not only t o d etect t he presence of t he t arget, b ut also t o
calculate its distance.
Convolution a nd Correlation We now e xamine t he close connectio...
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