Wiley signals and systems e book TLFe BO 346

2
-3 -2 - 1

IT

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3

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4

k

iii igrirc 13.8: Series alk] ohtaiiircl from X ( z ) i i t E x a m p l ~13.7

(13.2%)
has two iiiimcwtor zeros (poles) outside the oiigiri of the cornplrx plarw: zL 2.5: + 1 :
0 gives z p l = 51 , uxp2 - 2 As the rcgiori of corrvrrgenct. lies outszdr a
circlc arountl the origin, we exp
riglit-mirtl series r [ k ]= Z ' { X (
possiblr to esprc'ss X(zj as the
f two terms (like (13.5j). my' can
term an (~xponc~iltirzl
oscillation. Diiect partial fraction expansion does nut lead t o
the goal a \ it prodiiccs ;t b ~ i i ^of
~ i the form

(13.30)
to cictrmiinr the exponeIit,ial teririi. Hcre it helps to expmd X ( :)/z

ir'ractions, instcad of X (2).

into partial

(13.31)

Determining t h~ partial fractioii cocfficienls. for exarnplr. by cqiiating coefficients.
I r ~ d str) Ro = 1, 6'1 = -2 a d B2 = 1. Multiplying b) t and trrm-b~--icrin
iriversion aided by Exainplr 13.2 yields the series
(13.32)