Discrete-time Random Signals

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1 Discrete-tie Rado Sigals Util ow, we have assued that the sigals are deteriistic, i.e., each value of a sequece is uiquely deteried. I ay situatios, the processes that geerate sigals are so cople as to ake precise descriptio of a sigal etreely difficult or udesirable. A rado or stochastic sigal is cosidered to be characterized by a set of probability desity fuctios.

2 Stochastic Processes Rado (or stochastic) process (or sigal) A rado process is a ideed faily of rado variables characterized by a set of probability distributio fuctio. A sequece [], <<. Each idividual saple [] is assued to be a outcoe of soe uderlyig rado variable X. The differece betwee a sigle rado variable ad a rado process is that for a rado variable the outcoe of a rado-saplig eperiet is apped ito a uber, whereas for a rado process the outcoe is apped ito a sequece.

3 Stochastic Processes (cotiue) Probability desity fuctio of []: Joit distributio of [] ad []: p ( ) p, (,, ), Eg., 1 [] = A cos(w+φ ), where A ad φ are rado variables for all < <, the 1 [] is a rado process.

4 Idepedece ad Statioary [] ad [] are idepedet iff p is a statioary process iff p for all k. (,,, ) = p(, ) p( ), (, k,, + k) = p(,,, ) + k + + k That is, the joit distributio of [] ad [] depeds oly o the tie differece.

5 Statioary (cotiue) Particularly, whe = for a statioary process: p (, + k) = p( ) + k, It iplies that [] is shift ivariat.

6 Stochastic Processes vs. Deteriistic Sigal I ay of the applicatios of discrete-tie sigal processig, rado processes serve as odels for sigals i the sese that a particular sigal ca be cosidered a saple sequece of a rado process. Although such a sigals are upredictable akig a deteriistic approach to sigal represetatio is iappropriate certai average properties of the eseble ca be deteried, give the probability law of the process.

7 Epectatio Mea (or average) { } = p( ), = ε d ε deotes the epectatio operator ε { g ( )} g( ) p(, ) = d For idepedet rado variables { y } { } ε{ y } ε = ε

8 Mea Square Value ad Variace Mea squared value ε{, = } ( ) p d Variace var { } = ε

9 Autocorrelatio ad Autocovariace Autocorrelatio φ {,} = = ε { } p (,,, ) d d Autocovariace γ = φ {,} = ε {,} {( )( ) } *

10 Statioary Process For a statioary process, the autocorrelatio is depedet o the tie differece. Thus, for statioary process, we ca write σ = = ε { } = ε {( ) } If we deote the tie differece by k, we have ( ) ( ) { } k = φ k = ε φ, + + k

11 Wide-sese Statioary I ay istaces, we ecouter rado processes that are ot statioary i the strict sese. If the followig equatios hold, we call the process wide-sese statioary (w. s. s.). σ = = ε { } = ε {( ) } ( ) ( ) { } k = φ k = ε φ, + + k

12 Tie Averages For ay sigle saple sequece [], defie their tie average to be [] = li [] L l + = L Siilarly, tie-average autocorrelatio is 1 1 [ ] [ ] [ ] + = li [ ] l 1 L L + 1 L + = L

13 Ergodic Process A statioary rado process for which tie averages equal eseble averages is called a ergodic process: [ ] = [ ] [ ] + = φ [ ]

14 Ergodic Process (cotiue) It is coo to assue that a give sequece is a saple sequece of a ergodic rado process, so that averages ca be coputed fro a sigle sequece. I practice, we caot copute with the liits, but istead the quatities. Siilar quatities are ofte coputed as estiates of the ea, variace, ad autocorrelatio. ˆ σ = = 1 L 1 L L 1 = 0 L 1 = 0 [] ( [] ˆ ) 1 L 1 [ ] [ ] + = [ + ] [ ] L L = 0

15 Properties of correlatio ad covariace sequeces φ γ φ γ y y [ ] { } = ε + { } [ ] = ε ( )( ) + [ ] { } = ε y + { } [ ] = ε ( )( y ) + y Property 1: γ γ y [ ] = φ [ ] [ ] [ ] = φ y y

16 Properties of correlatio ad covariace sequeces (cotiue) Property : φ γ y = E = Mea Squared Value [] 0 = σ = Variace [] 0 Property 3 φ γ [ ] [ ] [ ] = φ φ = φ [ ] [ ] [ ] [ ] = γ γ = γ [ ] y y y y

17 Properties of correlatio ad covariace sequeces (cotiue) Property 4: φ y [ ] φ [] 0 φ [] 0 yy γ y [ ] γ [] 0 γ [] 0 φ γ yy [ ] φ [ 0] [ ] γ [] 0

18 Properties of correlatio ad covariace sequeces (cotiue) Property 5: If y = 0 φ γ yy yy [ ] = φ [ ] [ ] = γ [ ]

19 Fourier Trasfor Represetatio of Rado Sigals Sice autocorrelatio ad autocovariace sequeces are all (aperiodic) oe-diesioal sequeces, there Fourier trasfor eist ad are bouded i w π. Let the Fourier trasfor of the autocorrelatio ad autocovariace sequeces be [ ] ( ) [ ] ( ) jw jw Φ e y Φ y e [ ] ( ) [ ] ( ) jw jw Γ e γ Γ e φ φ γ y y

20 Fourier Trasfor Represetatio of Rado Sigals (cotiue) Cosider the iverse Fourier Trasfors: γ φ 1 [ ] π ( ) jw = Γ e π 1 dw [ ] ( ) jw jw = Φ e e dw π π π π e jw

21 Fourier Trasfor Represetatio of Rado Sigals (cotiue) Cosequetly, ε σ { []} [] 1 π ( ) jw = φ 0 = Φ e = γ P 1 π dw [] ( ) jw jw 0 = Γ e e dw Deote to be the power desity spectru (or power spectru) of the rado process. π π π π ( ) ( ) jw w = Φ e

22 Power Desity Spectru ε { []} 1 π = P ( w)dw π π The total area uder power desity i [ π,π] is the total eergy of the sigal. P (w) is always real-valued sice φ () is cojugate syetric For real-valued rado processes, P (w) = Φ (e jw ) is both real ad eve.

23 Mea ad Liear Syste Cosider a liear syste with frequecy respose h[]. If [] is a statioary rado sigal with ea, the the output y[] is also a statioary rado sigal with ea equalig to y [] ε{ y[] } = h[] k ε{ [ k] } = h[] k [ k] = k = k = Sice the iput is statioary, [ k] =, ad cosequetly, y = k = h [] ( ) j0 k = H e

24 Statioary ad Liear Syste If [] is a real ad statioary rado sigal, the autocorrelatio fuctio of the output process is φ, = ε = yy [ + ] = ε{ y[ ] y[ + ] } k= k= r= h h [][][ k h r k][ + r] [] k h[] r ε{ [ k][ + r] } r= Sice [] is statioary, ε{[ k][+ r] } depeds oly o the tie differece +k r.

25 Therefore, Statioary ad Liear Syste φ = yy = φ (cotiue) [, + ] k= yy h [ ] [] k h[] r φ [ + k r] r= The output power desity is also statioary. Geerally, for a LTI syste havig a wide-sese statioary iput, the output is also wide-sese statioary.

26 Power Desity Spectru ad Liear Syste By substitutig l = r k, where φ = yy [ ] = φ [ l][] h k h[][ k h l + k] l= c hh φ l= [ l] c () l A sequece of the for of c hh [l] is called a deteriistic autocorrelatio sequece. hh [] l = h[ k] h[ l + k] k = k=

27 Power Desity Spectru ad Liear Syste (cotiue) A sequece of the for of C hh [l] l = r k, Φ where C hh (e jw ) is the Fourier trasfor of c hh [l]. For real h, Thus yy ( ) ( ) ( ) jw jw jw e = C e Φ e hh c C hh hh hh [ l] = h[ l] h[ l] ( ) ( ) ( ) jw jw jw e = H e H e ( ) ( ) jw jw e H e C =

28 Power Desity Spectru ad Liear Syste (cotiue) We have the relatio of the iput ad the output power spectrus to be the followig: Φ yy ( ) ( ) ( ) jw jw jw e = H e Φ e ε { []} 1 π [] ( ) jw = φ 0 = Φ e π { []} 1 π [] ( ) ( ) jw jw y = φ 0 = H e Φ e ε yy π = total average power of π π the output dw = total average power of dw the iput

29 Power Desity Property Key property: The area over a bad of frequecies, w a < w <w b, is proportioal to the power i the sigal i that bad. To show this, cosider a ideal bad-pass filter. Let H(e jw ) be the frequecy of the ideal bad pass filter for the bad w a < w <w b. Note that H(e jw ) ad Φ (e jw ) are both eve fuctios. Hece, φ yy [ 0 ] = average power i output = 1 π w w b a H ( ) ( ) w ( ) ( b jw jw jw jw e Φ e dw + H e Φ e )dw 1 π w a

30 White Noise (or White Gaussia Noise) A white oise sigal is a sigal for which φ [ ] = σ δ [ ] Hece, its saples at differet istats of tie are ucorrelated. The power spectru of a white oise sigal is a costat Φ ( ) jw = σ e The cocept of white oise is very useful i quatizatio error aalysis.

31 White Noise (cotiue) The average power of a white-oise is therefore [] 1 π ( ) jw 1 π φ 0 = Φ e dw = σ dw = σ π π π π White oise is also useful i the represetatio of rado sigals whose power spectra are ot costat with frequecy. A rado sigal y[] with power spectru Φ yy (e jw ) ca be assued to be the output of a liear tie-ivariat syste with a white-oise iput. Φ yy ( ) ( ) jw jw e = H e σ

32 Cross-correlatio The cross-correlatio betwee iput ad output of a LTI syste: φ = ε y + = = y [ ] { [ ] [ ]} ε k = [] h[][ k + k] h [] k φ [ k] That is, the cross-correlatio betwee the iput output is the covolutio of the ipulse respose with the iput autocorrelatio sequece. k =

33 Cross-correlatio (cotiue) By further takig the Fourier trasfor o both sides of the above equatio, we have Φ y ( ) ( ) ( ) jw jw jw e = H e Φ e This result has a useful applicatio whe the iput is white oise with variace σ. y [ ] [ ] ( ) ( ) jw jw = σ h, Φ e σ H e φ = These equatios serve as the bases for estiatig the ipulse or frequecy respose of a LTI syste if it is possible to observe the output of the syste i respose to a white-oise iput. y

34 Reaied Materials Not Icluded Fro Chap. 4, the aterials will be taught i the class without usig slides