Wiley signals and systems e book TLFe BO 420

Expected Values 405 To characterise randoni signals deterininistically, we use statzstzcal averages a,lso called just averages.. They c m be classificd into characteristics of random pr

17.2 Expected Values 405

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 ạ 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-averạges

17.2.1

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:

Expected Value and Ensemble Mean

(17.1)

A s v,e can obtairr differeiit means at, different times, the expected value is in gent.ra1 time-dgycaderit :

(17.2)

In Figure 17.1 it can bp observed that the meail or the fimctions

.rl(t), 2 2 ( t ) , c,(t) takcs anothcr value at time 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 ịi an avrrage ẵ05.s the pro In contrast, tlie tirne-average i s taken in the direction of the timr-axis, and is an a v e ~ a g e ulanq the process

The definition of the expected value in (17 1) should be understood ac, a tor- mal 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 im- possible 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

* from precisc knowledge of the process the cxpected valiie can be calculaled without averaging sample functions We need tools from mathematic*al prob- ability, 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 d l y carried out Thc approxirnation becomes mor e acciu ate

as more sample lunctiorls are included