Power of Random Processes 1/40

Transcription

1 Power of Random Processes 40

2 Power of a Random Process Recall : For deerminisic signals insananeous power is For a random signal, is a random variable for each ime. hus here is no single # o associae wih insananeous power. o ge he Epeced Insananeous Power i.e., on average we compue he saisical ensemble average of : 40

3 Power of a Random Process Ofen drop he Average erminology his is also called he mean square value of he process Avg. Power of : P X E{ } In general, can depend on ime. Bu we ll see i doesn for WSS & saionary processes 340

4 Relaionship of Power o ACF Recall: R X, +τ E { +τ } Clearly, seing τ 0 makes his equal o P X P X R X, If he Process is WSS or SS R X, R X 0 For WSS or SS P X R X 0 R X τ P X Power of process τ 440

5 Power vs. Variance Recall Variance: σ R E { [ ] } 0 for WSS P P X σ + Power & Variance are Equal for Zero mean Process AC Power DC Power Noe: If he WSS process is Zero Mean, hen: PX σ 540

6 640 Power Specral Densiy of a Random Process Recall: PSD for Deerminisic Signal : where lim j d e X X S

7 Power Specral Densiy of a Random Process For a random Process: each realizaion sample funcion of process has differen F and herefore a differen PSD. We again rely on averaging o give he Epeced PSD or Average PSD Bu Usually jus call i PSD. 740

8 Define PSD for WSS RP We define PSD of WSS process o be : S lim E X <Compare his o PSD for Deerminisic Signal> his definiion isn very useful for analysis so we seek an alernaive form he Wiener-Khinchine heorem provides his alernaive!!! 840

9 Weiner- Khinchine heorem Le be a WSS process w ACF R X τ and w PSD S X as defined in hen R X τ and S X form a F pair : S X F{ R X τ } or Equivalenly R X τ S X 940

10 040 Proof of WK heorem By definiion : d e X j For Real : d d e d e d e X X X j j j

11 40 Proof of WK heoremcon d hus : Move E {.} inside inegrals : lim d d e R S j lim d d e E S j

12 Proof of WK heoremcon d We were almos here BU we have one-oo-many inegrals. For convenience define: φ - R X - e-j- S lim φ d d 40

13 Proof of WK heoremcon d Change of Variables Change of Aes Insead of inegraing Row-by-Row as in we inegrae Diagonal-by-Diagonal. Le: τ τ + λ λ + λ τ 340

14 Proof of WK heoremcon d τ λ + 440

15 540 Proof of WK heoremcon d J λ τ λ τ Use Jacobian resul for -D change of variables From Calculus III

16 Proof of WK heoremcon d S lim φ τ dτdλ???? Q: As τ ranges over - τ how does λ range? J A: For each τ, λ mus be resriced o say inside original square see Figure on ne Char 640

17 Proof of WK heoremcon d τ τ 6 τ τ τ 4 λ + Noe: φ Doesn Change In he Direcion Of + Ais 740

18 Proof of WK heoremcon d τ τ + - τ λ τ τ λ + λ + - τ - τ When τ > 0 we need From Aes Figure: λ - τ - λ - + τ Works ou similarly for τ < 0 840

19 940 Proof of WK heoremcon d So τ λ τ φ τ τ d d S lim + - τ { } lim τ R d d I τ τ φ τ τ τ φ <End of Proof>

20 Some Properies of PSD & ACF he PSD is an even funcion of for a real process proof : since each sample funcion is real valued hen we know ha is even. is even: even even even So is S X w his is clear from 040

21 Some Prop. of PSD & ACF S X is real-valued and 0. proof : again from since w is realvalued & 0, so is S X w 3 R X τ is an even funcion of τ R τ E E R { { σ τ σ τ } τ } Also follows from IF{ Real & Even } Real & Even <Propery of he F> + 40

22 Graphical View of Prop #3 For WSS, ACF does no depend on absolue ime only relaive ime - τ +τ τ τ Doesn maer if you look forward by τ or look backward by τ 40

23 Compuing Power from PSD From i s name Power Specral Densiy we know wha o epec : P So le s Prove his!!!! S d π Well his par is no obvious!! 340

24 440 Compuing Power from PSD We Know: { } I π τ τ d e S S R j P X R X 0 Q.E.D.! 0 π π d S d e S P j X

25 Unis of PSD Funcion P S d π S Was Hz [ Was Hz ] 540

26 Using Symmery of S X w P 0 S d π Double o accoun for he - 0 par of inegral Inegrae only from 0 640

27 PSD for D Processes No much changes mosly, jus use DF insead of CF!! S X Ω DF { R X [m] } P S π Periodic in Ω wih period π π π Ω dω Need only look a -π Ω<π 740

28 Big Picure of PSD & ACF Narrow ACF Broad PSD Less Correlaed Sample-o-sample Process ehibis Rapid flucuaions i.e. High Frequencies have Large power conen Broad ACF Narrow PSD More Correlaed Sample-o-sample Process ehibis Slow flucuaions i.e. High Frequencies have Small power conen <<See Big Picure: Filered RP Chars in V-3 RP Eamples >> 840

29 Whie Noise he erm Whie Noise refers o a WSS process whose PSD is fla over all frequencies C- Whie Noise S X Convenion o use his form i.e. w division by N S X N Whie Noise Has Broades Possible PSD 940

30 C- Whie Noise NOE : C- whie noise has infinie Power : N d Can really eis in pracice bu sill a very useful Model for Analysis of Pracical Scenarios 3040

31 C- Whie Noise Q : wha is he ACF of C- whie Noise? A: ake he IF of he fla PSD : R τ I { N } R X τ N δ τ Dela funcion! Narrowes ACF Area N τ & are uncorrelaed for any Also. P X R X 0 N δ0 Infinie Power.. I Checks! 340

32 PSD is: S X Ω N Ω bu focus on Ω [-π,π] ACF is: R X [m] IDF {N } N δ [m] Dela sequence D- Whie Noise S X Ω N -π π R X [m] N Ω m Broades Possible PSD Narrowes ACF [k ] & [k ] are uncorrelaed for any k k 340

33 D- Whie Noise Noe: P P R π [0 ] π π N N dω was N was D- Whie Noise has Finie Power unlike C- Whie Noise 3340

34 Eamples of PSD Eample : BANDLIMIED WHIE NOISE his looks like whie noise wihin some bandwidh bu is PSD is zero ouside ha bandwidh hence he name. hus: S N rec 4πB -πb S X N πb And P B π N π πb d N π πb + πb NB 3440

35 Eample: BL Whie Noise he ACF using F pair for rec and sinc is: R X τ N B sinc πbτ Again we see P X N B, since R X 0 N B R X τ N B -3 B - B - B B B 3 B τ Noe: For τ > B & + τ are Approimaely Uncorrelaed 3540

36 Eample # of PSD Eample : SINUSOID WIH RANDOM PHASE A cos C + θ We eamined his RP before: R X τ A cos C τ So using F Pair for a Cosine gives he PSD: S X πa [δ + C + δ C ] Area πa S X Area πa - c c 3640

37 Eample # of PSD Noe : Can ge P X in wo ways:.. R π 0 S A d A <Sifing Propery> P X A 3740

38 Eample #3 of PSD Eample 3: FILERED D- RANDOM PROCESS < See Also: Class Noes on Filered RPs > [k] D- Filer y[k] [k] + [k +] Zero mean R X [m] σ δ[m] Inpu ACF Whie noise S X Ω σ² Ω Inpu PSD S X Ω σ² Ω 3840

39 Eample #3 of PSD For his case we showed earlier ha for his filer oupu he ACF is : R Y [m] σ² { δ[m] + δ[m-] + δ[m+] } So he Oupu PSD is: S Y Ω σ² [ + e -jω + e -jω ] σ² [cosω + ] Use he resul for DF of δ[m] and also ime-shif propery cos Ω By Euler 3940

40 Eample #3 of PSD S Y Ω σ² [cosω + ] Replicas S y Ω 4σ² Replicas -π -π π π Ω Remember: Limi View o [-π,π] General Idea Filer Shapes Inpu PSD: Here i suppresses High Frequency power 4040