Wiley signals and systems e book TLFe BO 305

290

11. Sampling and Periodic Signals

and Xi =
sa mplcd'!

8.
For which cases does aliasing occur?

Exercise 11.16

A sigiiad ~ ( t=)---hi
wq

.2

271.

What rases arc critically

(y )

is sampled at eqniciistant points in time i/T, v E

z

l o form a signal Z A ( ~ )of weighted delta impulses.

a) Give the origirial spectrurn

X ( j 0 j ) = F{z(L))(sketxh

b) Sketch tlie spcx:trimi X n ( j w ) a -

o Z A ( ~ )for

with Iabelled axes).

2n
the case T = -.
3w!?

Exercise 11.17
The sketched rectangular signal r(t) will be inriestigat ed

4n

1i a

a,) Whaf is the spectrum R ( j w ) = F ( r ( t ) }of the sigial for hoth ixra,gnitude
and phase? Sketch i R ( j w )1.
1)) The spec.trum R(.7w) is saiiiplcd at equidistant points spaced by uo.This
cieates the ilew sprctmm:
x

R(JW1) 6(w - Jm).

Ru(3.1.')= WO
u=-m

Sketch the magnitude of this new spectrum for lwl 5

$ with

a) w* = .x
47''
27r
4) WO = fl,

Y)

WO

=

F,

where the weighted delta iiiipulses arc rcpreseiitrtl by arrows of rorrwpondilig
length.
c) Slcetcli the hictioiis of time r , ( f ) , corrcspondirig to the eases t v ) , f i ) , y ) in
part, b).