Wiley signals and systems e book TLFe BO 419

17. Drscribing Ilaridoni Signals

3c14

Sigiials that ltevr unknown behavioin arc called rwt-deter m 1 r 1 1
.siochastic supals or runrlorrh szqnals. Examples of .r;mdorn signals in tlectroiiict. are
~ ~ ~ t ~ ~ rsignals
f r r ~ like
i i (antenna,
~ ~ ~ noise, ax
er distort ion or tf.lrrmaI resistawe
noise, hut uuefril signals c a i ~also he stoch
1x1 c.oinrnunications. it is pointless
already krrowii to the rcwiver. Jn fact, for a receiver, the less
that can bc predicted, the greatrr t h i ~~ ~ f ~ r i con4,eiit
~ ~ ~ t ~ino i i
I.

TOdescribe rantiorri signals we cnn first of all try to Start wirh tlir signals theinstIlvcs. When the signal form it,sdf caiilrnot be iiiathematically dcscribed, it raii still
be mexsiir~dand a graph can be nbtaiiied, for examplc, in Figurcs 17.1 mid 17.2.
It is mt known whether Fourier, 1,aplare or convolut ioii irifegrals of t h t random
signalh exist, or whether the iiicthods nsrd to cdciilalc~the sp
funrtiovi art> weii tlefined Even whc
t existetice of an integral is certain, the r +
sulting .;pcctruni or outpiit signal is
again it randoni v:iriablc whose behaviour
we caiinot rriakr m y genmal statements about.
IVe can. for exampie, iiiterprrt the signal in Figiirc 17.1 as the noise of an
~ ~ n r ~ larid
i ~ ec*alctrlatc
r

the response of a post-connect ed sy In. Thcs kntm&dge
of this output sigrial, however. c'anitot he transtcrrrd to other sitnations, as another
amplifirr of the same kind would produce another noise signal, .rz ( t ) . The first
amplifier would aiso never repeat the noise 4gnaI x ~ ( t )so
, we cnn do very little
with an output sigiiai calculated from i t .
Thc solution to this problc~iiis found not? by considering individual random
signals. hut instead by ttnalysiiig the procew that, procliacrs the signals. In oiir
exampie it2nieaiis that, we should derive general st#alemeutsabottt the noire bc;havioiir. Of coursc, it is inipossiblc to know the miise iignals in individual amplifirm in advance. liistcad, the typical noise ponver can be given, for example. This
h w two Ggnifisaiit admritages:
the given noise power is a determinis;tic property which can br cdnilat.ed irt
a norrnal way,
it, is tiic same for all amplifiers of a particrtlar iiiotlel.

We need to introdiicc some nrw terms to extexid our discussion to general random
signals. A process that prodrtces random sign& will be (died a randoin pmctss.
The emttixety of all random signals that it, can prodixccb is caIIed an rnsernble of
vandoni signds. Iiidividiial rantioixr signals (e.g., x l ( t ) ,x z ( t ) , x7(t)in Figure 17.1)
a r called
~
sample fur ons or retzlrsatzons of a random p~ocess.We will concentrate
OKI 1 he rmdorn process that prodncrs t k signals as only it giws iiifoi~~~iatiO~i
0x1
all its sample fimctions.