Wiley signals and systems e book TLFe BO 301

286

11. Sampling and Pcriodic Signals

Sanipling iii t,he frequency-domain is again idealized in (11.57) as mult,i~~licati(j~i
wit,h a delta impulse train. This corresponds to convoliitioii (11.58), that causes
periodic continuation of $(t) in the time-domain. If the signal G(t) is limited
in time and its duration is less than T , sampling in the frequency-domain does
riot lead to overlapping in the time-domain. 'rhen y ( t ) ca.n be recovered from
y ( t ) by multiplication with a time window of leng%liT . In the frequency-domain,
t,his correspoiids bo a11 interpolation (convolution) with the spectrum si (uT/2),
that transforms the line spectrum into the smooth spectrum belonging to S i t ) . If
overlapping occurs in the time-domain, an error-free reconstruction is impossible.
In this context we use the term tzm~-domnza&using.
Saarrpling in the frequency-domain is always carried out if values of a. spectrum
are t o be stored in t,he memory of a. cornpiiter, where continuous functions cannot
exist, oiily a series of numbers. The necessary restvict,iorito time-limited functions
is, for example, achieved by splitting the signel into sections with finite duration.
The relationship bctween a series of finite length and their spectra is represented
by the discrete E'ourier tmrisfo~m,(DFT).

xercises
xereise 11.1
Write the sketched signal with a sum of delta impulses as well a? with the _UL
symbol.

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5

10

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Exercise 11.2
Determine the Foiirier transform X ( j w ) of ~ ( k )= UL(at) and sketch ~ ( tand
)
X (jw)for a = a == 1 and U = 3. Note: use tmnsform pair (11.12)

i,

Exercise 11.3
Coiisider a delta impulse train shifted by to: