Wiley signals and systems e book TLFe BO 315

300

12. TEic Spectriirn of Discrete Sigrittls

4

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 = 0 . 2 ~ . Incrcwing k by 1 turns the v d i i r ~of the scries by 0 . 2 ~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 = 0 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 = 1.97i. The series rnakes almost
one full t u n each time, f i is only 0 . 1 racl
~ less 1 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 2 ~ .