Wiley signals and systems e book TLFe BO 310

In Chapters 1to 10 we gcjt to know wmc powertul tools for dealirig with contiiiuous
signals arid systems We carried out the convei sion of red-world coritiiiiioirs signals
t o sanipled signals in Chapter 1 I, which is necessary foi digital processing. The
sampled signals were tieated as conitiniioiis-tiinevariables so thal the rools we had
learnt , like the Foiirier transforin, could A l l br iised.
In a coiiipiater, we can only work with a sequence of niuriberi, that are defined
over a discrete range by a numrrical value (the index). Examples of such an index
might be the S ~ ~ I T ~nuniber
P~C
fox a digital aiidio signal, or the pixel address within
a digital image. In addition there are discrete sigrials t h t haw not arisen from
sampling a continuous signal. Example:, of these are represented in Figurc~1.3
and 1.4. W e need iiew tools to descrihe such secpien , arid w e will concentrate or1
them in Chsptcrs 12 - 14. This chapter dcals with dzscrrte signulr and the discrete
form of the Foiirier transtorm, the J, trun.$fm-m. This traiisforni is also rcferrcd
to as dhwrete-tme Fourqrr traiLcsfOrm (DTFT). The two following (liaptrrs deal
with dzscretr systems and we will leain the discrete counterpart to the Lsplace
transform , the z-lrunsform.
In Swtions 12.1 arid 12 2 wtl will coiisider diicrete signals together with sonic
exaniples. The discrete-time Fowler transform will also be introduced, which we
will iisc to exaniirie discrete signals in the frcqueiicy-domain. We will sec that i t
has sirriilai properties to the Foiarier transform for coiitinuous sign&. At the end
of this chapter we will investigate the relationship between coiitiiiiioiis signals and
their disrrete cquivaleiit as a series of samples.

42.11
A discrete-time signal is represented by a sequence of iiuriihcrs that is callrd a time
series. There is no sniooth transition between the nurnl-jers. Figure 12.1 shows
the conventions we will use to reprrsent such signals; in order to distinguish them
from continuous-time signal$, we put the independent vaf iable in square brackt.
In inany technical appiications, a cliscrete-time signal arises from the sarnplzrry

of a continuoiis-tiiii~ signal 5 ( i ) , wherc a sample is taken kern P ( t ) at regular