Wiley signals and systems e book TLFe BO 418

e

All of the cont,inuous and discrete signals that, we have considered so far in the
timeclomain arid freqitcnry-doincri W P L signals
~
that could be described by xiiatliematical luncIions. 1% could calculate the valoes of the signal, add and subtract
signals, clday signals. and torrn derivatives ancl integrals. We found integration
vcry uscful, foi convolution, for Fourier arid Laplace tiansfornis, ~ n t wc~
l also u s t d
complex iiitegrrttion for the inverse Laplace and inverse z-transfornris. For discrete
signals summation is iised in d of irittegration. Shis was all possiblc bcc*ause
we l m l assiimcd that every s
I had one definite value at avenj’ poiiit in time,
md that every signal coiilil he described by a mathematical formula, however
complicated that formula might be.
M M Lsignals
~
that occur in practise do not conform to this assumption. It
~voulclbe theoretically possible to desrribe the sprcch signal from Figure 1.1 wit 2.1
the properties of the human vocal tract by superiiiiposirig various waves, but this
would not lead to a tecliiiically realistic solution. It is cornpletely impossible to
assign functions to iioise signals, or Sigriiils made np of
c ascillations. A i i ~ w
coiiccpt nnist be found to represent such irregular pro
Just unclerstwding
that, a. signal waveform can have aii uiiprcdictablc val
is t h c b r efore rtid[)IT1,
does not actually help much. To deal with system inputs and outputs in the way we
are nsed to, random signals must he described by non-random. or ‘deterministic’
quantities. This can be prrtorrn.d by the so-called expected values, which arc
iritrodncc~din the next section. Then we will cleal with stationary and ergo&;

raiidoni processes, for w2iich a significantly simpler calculation i s possi blt with
cxpertetl valuts, An important clabs of exp
tl values arc correlation functions.
which will also be disc-ussad. All of tlicsc forms for describing random signals will
be introduced for continuous signals, a i d the chapter concludes by cxteudiiig t h t
concepts to dibcrete rantloni signals.

.1.1

at Are Ran

The signals that we have been working with 60 far are called ddernimzstzr sz.gnala,
which means that a signal has a ksiowri unambiguous value at every poiiit ill
time. A signal ctui also be deterministic when it cannot, he described by siinple
mathematiral functions but i i i s t e d for example, by an iiifiiiite Fourier series,