Wiley signals and systems e book TLFe BO 311

12. The Spectrum of Discrete Signals

296

inr-ervads T ant1 is translated 8,s a numerical value :r[ k ]:
.E[q

= %(X.T),

kG

z.

(12.1)

Here it i s essential t o distinguish the discrete signal z[kj from the sampled signal

introduced in Chapter 11. Thc dependent variable f is tfefincd for every point in IR,
although ~ ~ (is1zero
) tor almost all of these time-points (except for C = AT, X; E Z).
In contrast, s[k]is only defined where the index k is rtri integer, so intcgration of
~ [ k i ]s not possible, and the Fomier transform trom Chapter 9 cannot be usecl.
Tn the following sectioiis wc will deal with discrete signals in depth without being
lirriitecl to sqnences of sainplecl values, in arl w e will consider general tirnriseries r[k:. We will return to rising sampling in Section 12 4. The values x [ k ]

Figure 12.1: R.epresent,atioriof a discrete-tirnc signal :c[k]

are t*hemselvescontinuous and in geneial. they could also be conipltx r [ k ] E C.
Striclly spcaking, this 1s not the case if the series is going to bc proce
a computer. The finite word length means that numbers cari only be
within thc limits of tlie nurriber representation being i i s ~ d ,and the values . ~ [ k ]

signals
tlien become themselves discrete. 'Vo distinguish tlieiri fIoin the di
with coiit inuoiis values, we refei to thcsc signals as d291td szgnals.
a n d i.; called quantrsutzon.
Rounding to discrc vtzlues i s a non-linear proc
Processes involving digital signals therefore cannot>be described by LTI-systems.
A cornputer with sufficiently great word length can work with digital signals that
are so finely quanl i d , how(\ver, that they call approximate discrete signals with
contiriiious values. Mie will therefore restrict ourselves to discrete signals from 1 1 0 ~
011.