Definition and Examples 319 -Figure 13.5: Region of convergencc for the fiiriction Xz from Kxnmplp 13.Y region of tmivrrgencc is cirrular licrc too, and its iadius is equal to tlie mn
Trang 113.1 Definition and Examples 319
-Figure 13.5: Region of convergencc for the fiiriction X(z) from Kxnmplp 13.Y
region of tmivrrgencc is cirrular licrc too, and its iadius is equal to tlie mngiiitiide
of the pole
In ccrmptirison t o Examplc 13.2, wc irotice that the z-transform of thr riglit - sided cxponential sequence (13A) arid the left-sided cxponmtiaI sequence (13.7)
h v e the sarnt’ form and only thc regions of convergerice axe diffeierrt ( F i g i w s 13.3
M’ithout it a unique
IVc are familiar wiih this sit uation from the l ~ p l a c c transform In Exain-
p l ~ s 4.1 arid 4.2 we corisiclcred lcft-sided mid right-sidtvi continuous-time signals tlint arc’ likewise orily ( ~ s t i ~ ~ ~ i ~ i s ~ i t ~ t ~ by the region of corivergence (see Fignres 4.3 arid 4.4)
ne
Wi cari iriterprct the individual points of the :-plane in a similar way t o the s-
plane in Chapter 3.1.3 ti’ignrc 13.6 shon s the corresponding exponential secpen(*+’
Tht valucs z = eJb’ on t,lie w i t circle correspond to thc cqtonential serim eJc’kwitli corrstant amplitude: z = 1 lei& to a series with constant values because
eJok = l k =; 1, whilc z = - 1 is the highest repiesentable frequency, 1wc.ai.w
F ) x k = (- J ) k All other valiieq cm thc unit circle repr rit complex expotieni in1 oscillations ctf frcqiicncy 6 1 with ?r < i ic T c‘omplcx coiijngate values oi 3
a r t distingms2ied by the tlirectioii of rotation Values of z = I ’ c ~ ‘ ‘ witlirri ihc unit circle (/ < 1) b ~ l o n g to a decaying exltoncntial stqucnce arid vdiies out.;idc the unit circlc ( T 2 1) belong t o c? growiiig exponelitis1 seqiieiice In Figure 13.6 the exponential srries z h are each oiily slrom7n for k 2 0 This shooltf not lead t o the misintcrprelatiori that we are onlv dcalirig wit11 rnii1;uteral scquences since d s o for
k < 0, z k f 0