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Unformatted text preview: ro beyond the first cycle of F(w) (see
Fig. 5.5b). According to t he Paley-Wiener criterion, i t is impossible t o realize even
this filter. T he o nly a dvantage in this case is t hat t he required filter can be closely
approximated w ith a smaller time delay. This fact indicates t hat i t is impossible
in practice t o recover a bandlimited signal f (t) exactly from its samples, even if
the sampling r ate is higher t han t he Nyquist rate. However, as t he sampling rate
increases, t he recovered signal approaches t he desired signal more closely.
F (ro) IFI > F s/2 Hz; 2. T he r eappearance of this tail inverted o r folded onto t he s pectrum. Note t hat
t he s pectra cross a t frequency F s/2 = 1 /2T Hz. This frequency is called t he
f olding f requency. T he s pectrum, therefore, folds onto itself a t t he folding
frequency. For instance, a component of frequency
+ F x shows up as o r
"impersonates" a c omponent of lower frequency
Fx in t he r econstructed
signal. Thus, t he c omponents of frequencies above F s/2 r eappear as components of frequencies below F s /2. T his tail inversion, known as s pectral
f olding or a liasing, is shown shaded in Fig. 5.6. In this process of aliasing,
not only are we losing all t he c omponents of frequencies above F s/2 Hz, b ut
these very components reappear (aliased) as lower frequency components. This
reappearance destroys t he i ntegrity of t he lower frequency components also, as
depicted in Fig. 5.6. '1'1- - Aliasing problem is analogous t o t hat of a n a rmy with a platoon t hat has secretly
defected to t he enemy side. T he p latoon is, however, ostensibly loyal t o t he army.
The army is in double jeopardy. First, t he a rmy has lost this platoon as a fighting
force. I n a ddition, during actual fighting, t he a rmy will have t o c ontend with t he
s abotage by t he defectors, and will have t o find another loyal platoon t o n eutralize
the defectors. Thus, t he a rmy has lost two platoons in nonproductive activity.
A Solution: T he Antialiasing Filter ro,
Lost tail gets
folded back L ost tail 1- Fig. 5 .6 Aliasing effect.
T he Treachery o f Aliasing There is a nother f undamental practical difficulty in reconstructing a signal from
its samples. T he sampling theorem was proved on t he a ssumption t hat t he signal
f (t) is b andlimited. A ll p ractical s ignals a re t imelimited; t hat is, they are of I f you were t he c ommander of t he b etrayed army, t he s olution t o t he problem
would be obvious. As soon as t he c ommander gets wind of t he defection, he would
incapacitate, by whatever means, t he defecting platoon before the fighting begins.
T his way he loses only one (the defecting) platoon. This is a p artial solution t o t he
double jeopardy of betrayal, a solution t hat p artly rectifies t he problem a nd reduces
t he losses t o half.
We follow exactly t he s ame procedure. T he p otential defectors are all t he frequency components beyond
= 2~ Hz. We should eliminate (suppress) these
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