Wiley signals and systems e book TLFe BO 315

300 12 TEic Spectriirn of Discrete Sigrittls Re ( x l k ]) Figure 12.5: Exartiples of complex t~porieuitialstxitas Example 12.2 The expoiicutial function in Figure 12.5 illustrates thr anibiguouu charactflu of (12.12): tlie uppcr curve has been calwlated with ari angular fr.eqiiency of ( I = ~ Incrcwing k by turns the v d i i r ~of the scries by ~r a d in the mathematicnlly positive direction (from the r ~ a part l t o f,he iriiagiuary part) For k = 10 Ihr samc direction as for h = is Icyeated l-wcniise 10sl - 27i; tlir absoliitc value is iednced to exp(-O.l x 10) = 1/e, i i o ~ e v ci~ecausc ~, ofthe danrpirig constant c = -0.1 111the lower cwve the angalar freqiient-yis $ = 1.97i The series rnakes almost one full t u n each time, f i is only racl ~ less hair a whole circle In comparison to the upper CUTVF", the Iowrr c i i ~ v edoes not seem to have a higher freqiioricy, but iiistead, R lower frequency, t ttiriirig in the oppositc dircctiori The exgonenliwl series with aiigiilar freqiicnry 12 = 1.9, iti identical to the exponeiittinl quency Q = 1.9, - 2~ = -0.171- e'\ already know this plieiioinerioii n movies, wlrcn the wheels 01 a wagon rtppcar t,o 1x3 turning biLclmmrds The ambiguousness of the fxquency o f discrete exponential hinctions is t h c reason for the oct'urrcnt'e of 'Aliasing' (compare Chapter 11.32) where samplirrg causes different freqiiencies t v oveilap 33ecaiisr of this rffect, for i-t spectral represcntatiori of di'sci ete signals, it is iiecessary to limit the r~recluericv-tlomairito the width ~