Wiley signals and systems e book TLFe BO 420

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17.2. Expected Values

To characterise randoni signals deterininistically, we use statzstzcal averages
a,lso called just averages. They c m be classificd into characteristics of random
processes which hold for a. corriplet,c: ensemble of random signals (expected value),
and tzme-averages, which are found by averagirig one sample function along the
tinie-axis. In the next section we will deal wit,h expected values and time-avera.ges.

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17.2.1 Expected Value and Ensemble Mean
Several different sarnplc functions of a process arc reprcseiited in Figure 17.1. M'e
can imagine that they are noise signals. that are measnred at the same time on
various amplifiers of the same modcl. As expected value (also ensemblr mean) we
define the nieaxi value that is trt)tained at thc same time from all sample functions
of the same process:
(17.1)
A s v,e can obtairr differeiit means at, different times, the expected value is in gent.ra1
time-dgycaderit :
(17.2)

can bp observed that the meail or the fimctions
1 2 than at, time t l .
Sincc the averaging in Figure 17.1 runs in the direction of the tladiecl linrs. the
expected valiie i.i an avrrage a(~05.sthe pro

. In contrast, tlie tirne-average i s
taken in the direction of the timr-axis, and is an ave~ageulanq the process.
The definition of the expected value in (17 1) should be understood ac, a tormal tiescription aud not as a tiictliotl for its cal
ion. It says that it shonld
be deterrninrd from all sample functions of a pr
which is in practiccl an impossible task. The expected valiie identifies the whole process. not just srlectcd
sample fiinctions. If we wish to dttiially cvalute an expected value, there are three
wailable methods.

In Figure

17.1 it

.rl(t),2 2 ( t ) . . . . , .c,(t) takcs anothcr value at time

* from precisc knowledge of the process the cxpected valiie can be calculaled
without averaging sample functions. We need tools from mathematic*alprobability, however, that, we did not want as pre-requisites.
e

h e r a g i n g a, finite niimber of sample ftnwtions can give an approximation
of tlie expected value from (17.1). This is equivalent to the liniit in (17.1)
being only part i dly carried out. Thc approxirnation becomes mor e acciu ate
as more sample lunctiorls are included.