Signal Processing and Linear Systems-B.P.Lathi copy

T he ideal interpolation filter transfer function

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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 components from...
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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.

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