Wiley signals and systems e book TLFe BO 317

302

12. The Sgec*trumof Discrete Signals

iscrete-Time Fonrier
This section containb the Fourier transforms of the simple sequences discussed
in Section 12.2. Thcse were the uriit impulse, unit stcp tunction and bilateral
exponential series. As a concluding example we will deterrilirre the spectrum of a
rectangle ftmction.

+F*Transform of a

.3.1

iserete Unit Impulse

'li, calculate the 7%
traiisform of a unit iiiipirlse x [ k ] = 6 [ k ]we start with the
defining equation of the ;C, transform, and insert the unit impulse. The result caii

he given immediately, using the selcetive property of the unit impiilse:

X(&)

= C G [ k j e - ' ~ ~ k= I

.

(12.17)

k

We thus obtain the transform pair
(12 18)
I

Ark - K ] leads to a linear change

T h e transform of a shifted unit impiilse .L[/C)
of phase as for a contiiiuous-time signal:

q/C- fFje-3SZk

= e- J $ Z h ,

(12.19)

k

and so the transform pair is

We can

PP

3.3.2

that, tbe transform pAr (12.18) is

B

special rase for

K = 0.

.F* Transform of an Undamped Complex Exponential Series

To find the F*transform of an uiidaniped bilateral exponcntial scries
start wit,h the interesting relationship

e3"Ok,

we

(12.21)
fc

U

(,hat we had derived from equation (11.7) in thc last clmpter. Now we are using it
to determine the spcctruni of ,r[k]= eJnrlk: