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 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 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 ~ 011