Random Signals and Noise Winter Semester 2017 Problem Set 12 Wiener Filter Continuation

Transcription

1 Radom Sigals ad Noise Witer Semester 7 Problem Set Wieer Filter Cotiuatio Problem (Sprig, Exam A) Give is the sigal W t, which is a Gaussia white oise with expectatio zero ad power spectral desity fuctio S WW ht u t u t T, give i the figure: ht N W t is passed through the filter T t Let us deote by X t the output of the filter, amely: X t W t* ht a Fid the expectatio of the process X t b Fid the autocorrelatio fuctio of X t, c Draw RXX XX R We would like to predict the future of the process from observig its mometary value a For X t X t from, fid the optimal MMSE estimator of b What is the achieved mea squared error? 3 Now, we would like to predict the future of the process from observig its mometary value ad a additioal value from the past a For, D, fid the optimal MMSE estimator of pair of samples X t, X t D X t from the b Are there values of D for which the estimators from sectio ad 3 are idetical? If so, what are they? Explai your aswer

2 A bright egieer foud the optimal estimator of X t from all the samples from the past together: X t t t 4 For what values of is this estimator: a idetical to the estimator from sectio? Explai! b idetical to the estimator from sectio 3? Explai! There is o eed to fid the estimator Problem : Two JWSS radom processes, autocorrelatio ad cross-correlatio fuctios N, R N, R N, N spectrums S S, S, respectively N, N N, N N t N t are give, each with expectatio zero, R ad Fid the optimal liear estimator that uses all the sample sigal N t : Write dow the optimal Nˆ t ht t N t H dt Now, it is give that N t V t g t N t V t g t, where process with expectatio zero ad spectrum: S V a The frequecy resposes of the filters are give by: G, G Explai why ideed N t Nt calculate S S, S N, N N, N V t is a WSS, are JWSS with expectatio zero ad 3 Calculate the estimator you foud i sectio ad its mea squared error 4 For what values of the parameters, is the estimatio error zero? Prove mathematically ad explai ituitively 5 For what values of the parameters, is the estimatio error maximal? What is the estimator i this case ad what is the estimatio error?

3 Problem 3: Give is a discrete-time WSS Gaussia radom process, X, with expectatio X ad autocorrelatio fuctio RX k Let us assume the sigal Y X Z is give, where ad idepedet of Z is Gaussia white oise with expectatio zero, PSD S e X What is the optimal MMSE estimator of X from Y? What is the optimal MMSE estimator of X from Y? Z Z Problem 4: Give is a discrete-time WSS radom process SX e, show i the figure, ad it is give that oise with PSD N, idepedet of X SX e X with expectatio zero ad PSD Y X Z, where Z is white A Calculate RX k Calculate the optimal liear estimator of X from Y, ad the mea squared error, as a fuctio of the delay Fid the value of for which the error is miimal ad explai the behavior of the error as a fuctio of 3 Calculate the mea squared error of the optimal liear estimator of X from Y, Y 5 4 Fid the optimal filter for estimatig X from Y squared error ad calculate its mea 5 Compare the errors of the estimators from sectios,3,4

4 Problem 5: Two idepedet WSS Gaussia radom processes X[ ], Z[ ], are give The two processes have zero expectatio ad the followig Power Spectral Desities:, Z, SX e S e We are iterested i calculatig X [ ] out of Y[ ] X[ ] Z[ ] Are X [ ] ad Y [] JWSS? Calculate the optimal MMSE estimator of the process X [ ] from the process Y [] 3 What is the mea squared error (MSE) of the estimator from sectio? We defie the processes ' [ ] Y X [ ] Z [ ] '' ad Y 4 Are X [ ] ad Y '' [ ] JWSS? ' [ ] Y [ ] 5 What is the optimal estimator (ot ecessarily LTI) of the process X [ ] from the process 6 Are X [ ] ad Y '' [ ]? Y ' [ ] JWSS? 7 What is the optimal estimator (ot ecessarily LTI) of the process X [ ] from the process Y ' [ ]? Problem 6: Give are three idepedet radom processes Xt, () N () t ad N () t The three processes are statioary, with expectatio zero ad spectrums S X, S ad S, respectively Also a RV P is give, idepedet of the three processes Xt, () N () t ad N () t, with the followig distributio: w p p P p w p p where p is costat

5 N t Y t P X t N t X t P Y t P X t N t N t Show that the three processes Xt, () Y() t ad Y () t are JWSS i pairs (meaig that each pair of them is JWSS) ˆX t, which is the optimal liear MMSE estimator of the process Calculate Xt () from the process Y() t 3 Calculate ˆX t, which is the optimal liear MMSE estimator of the process Xt () from the process Y () t We ow defie a ew estimator ( t) X ( t) Xˆ ( t) Xˆ ( t) Xˆ ( t) Xˆ t ad the estimatio error 4 Show that the cross-correlatio betwee () t ad ay oe of the two processes Y() t, Y () t is zero 5 Deduce that X ˆ () t is the optimal liear estimator of the process Xtfrom () both processes t Y t Y t t,

6 Wieer Process Problem 7: (based o a questio from a exam) t X is Wieer radom process with parameter Let us defie costat times t,t, ad assume t t We will ow defie the radom variable (for a real costat c): Y X t c X t Fid a costat c such that Y will be idepedet of the sample X t Let us ow cosider two more samples of the process: X t X 3, t that: t t t t3 Based o the costat c foud i sectio : a Is Y idepedet of X t 3? b Is Y idepedet of X t?, such Problem 8: W t is Wieer radom process with parameter We defie its sig process t sigw t Y Calculate Yt expectatio ad autocorrelatio fuctios Express your aswer usig Q-fuctio Q e Prove that R t, t t, t Y 3 Prove that t, t lim R Y t x dx