Signals and Systems - Electrical Engineering

9.4 One-Sided Z-Transform 525

The Z-transform pairs of a cosine and a sine are, respectively,

cos(ω 0 n)u[n] ⇔

z(z−cos(ω 0 ))

(z−ejω^0 )(z−e−jω^0 )

ROC :|z|> 1 (9.14)

sin(ω 0 n)u[n] ⇔

zsin(ω 0 )

(z−ejω^0 )(z−e−jω^0 )

ROC :|z|> 1 (9.15)

The Z-transforms for these sinusoids have identical poles 1 e±jω^0 , but different zeros. The frequency of the

sinusoid increases from the lowest(ω 0 =0 rad)to the highest(ω 0 =πrad)) as the poles move along the unit

circle from 1 to− 1 in its lower and upper parts.

Consider then the signalx[n]=rncos(ω 0 n+θ)u[n], which is a combination of the above cases. As
before, the signal is expressed as a linear combination

x[n]=

[

ejθ(rejω^0 )n

2

+

e−jθ(re−jω^0 )n

2

]

u[n]

and it can be shown that its Z-transform is

X(z)=

z(zcos(θ)−rcos(ω 0 −θ))

(z−rejω^0 )(z−re−jω^0 )

(9.16)

The Z-transform of a sinusoid is a special case of the above (i.e., whenr=1). It also becomes clear
that as the value ofrdecreases toward zero, the exponential in the signal decays faster, and that
wheneverr>1, the exponential in the signal grows making the signal unbound.

The Z-transform pair

rncos(ω 0 n+θ)u[n] ⇔

z(zcos(θ)−rcos(ω 0 −θ))

(z−rejω^0 )(z−re−jω^0 )

(9.17)

shows how complex conjugate pairs of poles inside the unit circle represent the damping indicated by the

radiusrand the frequency given byω 0 in radians.

Double poles are related to the derivative ofX(z)or to the multiplication of the signal byn. If

X(z)=

∑∞

n= 0

x[n]z−n

its derivative with respect tozis

dX(z)

dz

=

∑∞

n= 0

x[n]

dz−n

dz

=−z−^1

∑∞

n= 0

nx[n]z−n