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M ost o f t he s ignal e nergy is c ontained w ithin a c ertain b and o f B Hz, a nd t he
e nergy c ontributed b y t he c omponents b eyond B Hz is negligible. W,e c an t herefore
suppress t he s ignal s pectrum b eyond B H z w ith l ittle effect o n t he Signal s hape a nd
energy. T he b andwidth B is called t he e ssential b andwidth ~f t he sign~l. ! he
c riterion for s electing B d epends o n t he e rror t olerance in a p artIcular a pphcatlOn.
We may, for e xample, s elect B t o b e t hat b and w hich contai~s 95% o f t he s~g~al
e nergy.t T his f igure may b e h igher o r lower t han 95%, d ependmg o ? t he pre~lslon
n eeded. U sing s uch a c riterion, we c an d etermine t he e ssential bandWidth o f a signal.
T he e ssential b andwidth B for t he s ignal e atu(t), u sing 95% e nergy c riterion, was
d etermined i n E xample 4.16 t o b e 2.02a Hz.
For lowpass signals, the essential bandwidth may also be defin~ as a frequency at which the
value of the amplitude spectrum is a small fraction (about 1%) of lts peak value. In Example 4.16,
for instance, the peak value, which occurs at w = 0, is 1/a. ,p J(t) = J (t) * J(  t) (4.68b) F rom Eq. (4.68b) i t is clear t hat , pf(t) = j (t) * J (t) = , pf(t)
Therefore, for real J (t), a utocorrelation f unction ,p f (t) is a n even function of t. T he
Fourier t ransform o f Eq. (4.68b) yields
,pJ(t) <=> IF(w)12
(4.69)
Therefore, t he F ourier t ransform o f t he a utocorrelation f unction is its energy spectral d ensity IF(w)12. I t is clear t hat ,pf(t) p rovides t he s pectral i nformation of J (t)
directly.
T he d irect link of t he a utocorrelation function t o t he s pectral i nformation c an
be explained intuitively as follows. T he a utocorrelation function ,p f (t) is t he correlation of a signal w ith i tself delayed by t seconds. A signal J (t) c orrelates perfectly
with itself w ith zero delay. B ut as t he d elay increases, t he s imilarity decreases.
Thus, t he a utocorrelation f unction , pf(t) is a nonincreasing function of t . I f J (t)
is a slowly varying signal (low frequency signal), i t c hanges slowly w ith t. Consequently such a signal will show considerable similarity o r c orrelation w ith i tself even
for relatively large delay. T he a utocorrelation function ,p f ( t ) decays slowly w ith t
a nd h as a larger width. O n t he o ther h and, for a rapidly varying signal, t he signal
similarity will decrease rapidly w ith d elay t a nd ,p f (t) h as a s maller width. Thus,
t he s hape o f , pf(t) h as a direct link t o s pectral i nformation of J (t). 4.7 A pplication t o C ommunications: A mplitude Modulation M odulation causes a s pectral s hift in a signal a nd is used t o g ain certain advantages mentioned in Sec. 4.35. Broadly speaking, t here a re two classes of m odulation: a mplitude ( linear) m odulation a nd angle (nonlinear) modulation, which are
the s ubject of t he n ext two sections. I n t his s ection, we s hall discuss some practical
forms of a mplitude m odulation. 4.71 D ouble S ideband. S uppressed C arrier ( DSBSC) M...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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