Chapter 10 Advanced Topics in Random Processes

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1 ery Stark ad Joh W. Woods, Probability, Statistics, ad Radom Variables for Egieers, 4th ed., Pearso Educatio Ic.,. ISBN Chapter Advaced opics i Radom Processes Sectios. Mea-Square (m.s.) Calculus 635 Stochastic Cotiuity ad Derivatives [-] 635 Further Results o m.s. Covergece [-] 645. Mea-Square Stochastic Itegrals 65.3 Mea-Square Stochastic Differetial Equatios Ergodicity [-3] Karhue Lo`eve Expasio [-5] Represetatio of Badlimited ad Periodic Processes 67 Badlimited Processes 67 Badpass Radom Processes 674 WSS Periodic Processes 677 Fourier Series for WSS Processes 68 Summary 68 Appedix Itegral Equatios 68 Existece heorem 683 Problems 686 Refereces 699 B.J. Bazui, Fall 6 of ECE 38

2 .4 Ergodicity From Cooper ad McGillem Ergodicity deals with the problem of determiig the statistics of a esemble based o measuremets from a sample fuctio of the esemble. For ergodic processes, all the statistics ca be determied from a sigle fuctio of the process. his may also be stated based o the time averages. For a ergodic process, the time averages (expected values) equal the esemble averages (expected values). hat is to say, x f x dx lim t dt Note that ergodicity caot exist uless the process is statioary! A Process for Determiig Statioarity ad Ergodicity a) Fid the mea ad the d momet based o the probability b) Fid the time sample mea ad time sample d momet based o time averagig. c) If the meas or d momets are fuctios of time o-statioary d) If the time average mea ad momets are ot equal to the probabilistic mea ad momets or if it is ot statioary, the it is o ergodic. Example Computatios for meas ad d momet x f x dx ad x f x dx x ˆ lim x t dt ad x lim x t dt B.J. Bazui, Fall 6 of ECE 38

3 Back to our textbook Based o the theoretical developmet skipped over i sectio. of the stochastic itegral, we ca defie the time average of a radom process as ˆ lim t dt If this is equivalet to the probabilistic mea, the the R.V. is ergodic i the mea. his provides a way of coectig statistical measuremets with a probabilistic equivalet. I establishig whether or ot this property exists, a mea-square test ca be applied. he mea-square calculus defied i the earlier sectios of the chapter is ecessary to perform this operatio. herefore, to get deeper ito the subject, the material is available! Examples of o ergodic Radom variable processes ad sequeces First t A For A, a radom variable, the time average will be the measured value ad ca ot represet possible values for a esemble based o the pdf. Ay time a amplitude or magitude takes o a radom value for all time the other possible esemble values ca ot be assessed based o a time average. Secod t A cos f t B si f t c c For A ad B radom variables, we have the problem of amplitudes or magitudes existig for all time. Eve if E A EB ad E A B, the process would be ergodic i the mea but ot i power! herefore, defiitios of ergodic i the mea ad ergodic i mea-square ca ad are defied. If we pass these tests, we ca have a WSS R.V. that is also ergodic i mea ad power. B.J. Bazui, Fall 6 3 of ECE 38

4 Defiitio.4- A WSS radom process is ergodic i the mea if the time average coverges to the esemble mea lim t dt as From statistics, we ca defie so we would expect for WSS ad ˆ E lim t dt E M ˆ lim E t E M dt ˆ lim dt lim dt E M E lim Computatio of the variace of the statistical mea has bee previously doe! If we are to determie that this is correct i a mea squared sese, we must determie that lim ˆ lim E M his is where the mea-square calculus derivatio comes i! As may be expected, this is related to the covariace based o K t t dt dt this leads to heorem.4- B.J. Bazui, Fall 6 4 of ECE 38

5 heorem.4- fuctio satisfies A WSS radom process (t) is ergodic i the mea iff the covariace lim K d For ergodicity of higher momets, we have ergodic i mea square ad ergodic i correlatio. Defiitio.4- A WSS radom process is ergodic i the mea square if the time average coverges to the d momet lim t dt R E t as Defiitio.4- A WSS radom process is ergodic i correlatio if the time average coverges to the autocorrelatio as lim * t t dt R as o establish the ecessary ad sufficiet coditios, heorem.4- is provided. he derivatio is based o the covariace of the autocorrealtio, which is a forth power process requirig fourth momet statioarity. B.J. Bazui, Fall 6 5 of ECE 38

6 Example.4 Radom amplitude cosie Where A is N(,) ad theta is U(-pi,pi) From before we ca expect t A cos f t ergodic i the mea ot ergodic i power ot ergodic i correlatio Explicitly t EA cos f t E E t EA Ecos f t ad lim t dt as lim A cos f t A lim cos f t dt dt A lim si f t f A lim si f si f f f f A lim cos si4 f For appropriate, such as 4 ad the summig all periods as ifiity f A f cos lim si4 f B.J. Bazui, Fall 6 6 of ECE 38

7 he text provides a aalysis of the heorem.4- to validate this computatio. o asses ergodic i power lim t dt R E t as lim A cos f t dt A lim cos f t dt From this equatio, the R.V A remais ad we really do ot have to cotiue it caot be ergodic i power. Ad if it is ot ergodic i power, there is o way to be ergodic i correlatio. B.J. Bazui, Fall 6 7 of ECE 38

8 .5 Karhue Lo`eve Expasio his sectio provides a alterate derivatio for the matched filter. It uses a orthoormal expasio whose coefficiets are ucorrelated radom variables. heorem.5 A complete orthoormal basis set is defied for a covariace as K he we ca decompose the radom process usig with t, t t dt t, t t t, for t t t * dt he resultig coefficiets are statistically orthogoal such that With a orthoormal set defied as E * m m m * t t dt m Now based o the theory of itegrals kow as Mercer s theorem, K t t t *, t B.J. Bazui, Fall 6 8 of ECE 38

9 Example.5 4 Matched Filter We defie two hypotheses; there is a sigal preset i oise or just oise. t t m t W W t Usig the theorem preseted, the cotiuous waveforms ca be replaced by simpler scalars as m W W where the m ad W are K-L coefficiets. Now the W represet a idepedet radom sequece. If we take the first basis elemet to be based o m(t), scaled so it is ormalized where c t c m t m t dt he oly will have a relatioship to m(t) such that c W W o compute, usig K-L it is just c t m t dt he zeroth lag of the correlatio of m(t) either with m(t) + oise or oise. he correlatio ca be equivaletly performed usig a filter operatios a matched filter! B.J. Bazui, Fall 6 9 of ECE 38

10 .6 Represetatio of Badlimited ad Periodic Processes I ECE 455 you will be taught about the samplig theorem ad covertig betwee a cotiuous ad discreet time domai. his sectio is providig a similar traslatio betwee the cotiuous autocorrelatio ad power spectral desity to a bad-limited discrete time sample domai. I geeral R R R t t For a badlimited autocorrealtio fuctio with a sample spacig of w f f his is defied as the Nyquist samplig rate for a badlimited sigal. Recostructio of the cotiuous time autocorrealtio is performed as R R t R t R si w w si f f t t he relatioship of the Power Spectral Desity fuctios become t t S w S w, a a w B.J. Bazui, Fall 6 of ECE 38

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