Random Process Examples 1/23

Transcription

1 ando Process Eaples /3

2 E. #: D-T White Noise Let ] be a sequence of V s where each V ] in the sequence is uncorrelated with all the others: E{ ] ] } 0 for This DEFINES a DT White Noise Also called Uncorrelated Process Physically, uncorrelated eans that nowing ] provides no insight into what value ] for ) will be liely to tae roll a die; the value you get provides no insight into what you epect to get on any future roll) /3

3 E. #: D-T White Noise ecall: Gaussian process are very coon, so Also, let ] be Gaussian with Zero ean and Variance σ² Define P s PDF; also called Noral X] ~ N 0,σ²) Mean Variance PDF : p ; ) e πσ σ Note: Does not depend on at least the st order PDF is tie invariant 3/3

4 E. #: D-T White Noise TASK : We have a odel. Find the ean, ACF, and chec if WSS also find variance of process) MEAN of Process : E { ] } 0 CONSTANT By definition! ACF: X, )E { ]. ] } By our definition of white noise,.this is 0 if 4/3

5 E. #: D-T White Noise Now for : Thus,, ) { } σ, ) E ] σ 0, σ, δ ] By definition of variance for zero-ean case lie τ for cont-tie ACF) ACF for DT White P ) σ δ ] 5/3

6 E. #: D-T White Noise ACF displays lac of correlation between any pair of any tie instants: σ² X ] σ²δ] Now since we have constant ean and ACF depends only on - WSS 6/3

7 E. #: D-T White Noise Variance σ 0] 0] 0 σ For this case: Variance of the Process Variance of the V 7/3

8 E. #: Filtered D-T P Start with White P ] in previous eaple ecall : Zero Mean Process X ] σ² δ] WSS ] D-T Filter hn] ] y] ] -] Two Tap FI filter Taps ] 8/3

9 E. #: Filtered D-T P TASK: Is y] WSS? need to find ean & ACF MEAN: Using filter output epressions gives E { } { } y ] E E ] { } { } ] E ] E {y]} 0 ] 9/3

10 0/3 E. #: Filtered D-T P ACF : { } { } { } { } { } { } )) ) ) ) ) ] ] ] ] ] ] ] ] ]) ] ]) ] ] ] ), y E E E E E y y E Plug in Eq. for output

11 /3 E. #: Filtered D-T P Y ) σ²δ] δ-] δ] ) ] ] ] ] ) ) ) ) ) ), y δ σ δ σ δ σ δ σ where ACF for -Tap Filtered White P y] is WSS

12 E. #: Filtered D-T P σ² σ² y ] Note : Filter introduces correlation between adjacent saples - but still no correlation for saples or ore saples apart for this filter) /3

13 Big Picture: Filtered P Filters can be used to change the correlation structure of a P: ACF ] of Input Input P One Saple Function) White Noise ] ACF y ] of Output D-T Filter hn] ] y] ] -] Output P One Saple Function) /3

14 Big Picture: Filtered P cont) ACF y ] of Output Output P One Saple Function) Filter: ] ACF y ] of Output Output P One Saple Function) Filter: ones,0) ACF y ] of Output Output P One Saple Function) Filter: ones,5) /3

15 Filtered Ps: Insight Our study of the ACFs of filtered rando processes and the degree of soothness of the saple functions shows the following general result: Narrow ACF apid Fluctuations Broad ACF Slow Fluctuations 5/3

16 E. #3: Ep. Sig. White Define a P as: Noise y ] a w ] Deterinistic Function a deter. # ando Variable piced once randoly and then fied) White Noise get a new rando value at each uncorrelated fro saple-to-saple) 6/3

17 E. #3: Ep. Sig. WN Further Definition of this P: w] is white noise Each w] is a Gaussian V with w ] ~ N0, σ w ) X is a Gaussian V with X ~ N0, σ X ) V X is independent of each V w] E{ w]} E{} E{w]} 0 The # a is a deterinistic nuber 7/3

18 E. #3: Ep. Sig. WN TASK: Is this WSS? 0 0 MEAN: E{ y] } a E{X} E{ w] } 0 Constant) ACF: y,) E{ y] y] } E {a X w]) a X w])} a E{X } a E{X w]} a E{X w]} σ X E {w] w]} σ w δ] 0 Due to Indep. 8/3

19 E. #3: Ep. Sig. WN y, ) σ a σ w δ ] In General: Depends on not just Not WSS!!! But If a ± then a KM a K a M a M Then it is WSS 9/3

20 E. #3: Ep. Sig. WN If a σ w σ σ X ] If a - -3 σ w σ - σ X ] 3 - σ 0/3

21 E. #4: Weird Func of WN Let z] be white noise with zero ean & variance of σ z Define n] Z intn/)] Then 0] z0] ] z0] ] z] 3] z] intd 3 D D.d d d 3 d 4 ) D 3 D D Always appear in pairs /3

22 E. #4: Weird Func of WN TASK: Is this P WSS? MEAN: E { n] } E { z] } 0 X n, n) E{ n] n] } n/ if n is even n-)/ if n is odd If then n] & n] are two different z.] values uncorrelated X n,n) 0 for /3

23 E. #4: Weird Func of WN If 0 then X n, n ) E { z ] } σ z If and n is even: X n, n) and n is odd: X n, n) 0 σ z n dependence causes it not to be WSS n 3/3