Signal Detection and Estimation

Transcription

1 394 based on the criteria o the unbiased and minimum variance estimator but rather on minimizing the squared dierence between the given data and the assumed signal data. We concluded the chapter with a brie section on recursive leastsquare estimation. PROBLEMS 6.1 Let Y 1 Y Y be the observed random variables such that Y a bx Z 1 The constants 1 are nown while the constants a and b are x not nown. The random variables 1 are statisticall independent each with zero mean and variance estimate o (a b). Z nown. Obtain the ML 6. Let Y be a Gaussian random variable with mean zero and variance. (a) Obtain the ML estimates o and. (b) Are the estimates eicient? 6.3 Let Y 1 and Y be two statisticall independent Gaussian random variables such that E[ Y1 ] m E[ Y ] 3m and var[ Y 1] var[ Y ] 1; m is unnown. (a) Obtain the ML estimates o m. (b) I the estimator o m is o the orm a1y1 b1y determine a 1 and a so that the estimator is unbiased. 6.4 The observation sample o the envelope o a received signal is given b the ollowing exponential distribution 1 Y ( ) exp 1 is an unnown parameter and the observations are statisticall independent. (a) Obtain the ML estimate o. (b) Is the estimator unbiased? (c) Determine the lower bound on the estimator. (d) Is the estimator consistent?

2 Parameter Estimation Let the observation Y satis the binomial law such that the densit unction o Y is (a) Find an unbiased estimate or p. (b) Is the estimate consistent? Y n n ( ) p p (1 ) n 6.6 Obtain the ML estimates o the mean m and variance observations Y 1 Y Y such that 1 m Y ( ) exp 1 or the independent 6.7 Let x be an unnown deterministic parameter that can have an value in the interval [ 11]. Suppose we tae two observations o x with independent samples o zero-mean Gaussian noise and with variance each o the observations. (a) Obtain the ML estimate o x. (b) Is xˆ ml unbiased? superimposed on 6.8 LetY 1 Y Y be independent observed random variables each having a Poisson distribution given b Y ( ) e! 0 1. The parameter is unnown. (a) Obtain the ML estimate o. (b) Veri that the estimator is unbiased and determine the lower bound. 6.9 Let Y 1 Y Y be independent and identicall distributed observations. The observations are uniorml distributed between and where is an unnown parameter to be estimated. (a) Obtain the MLE o. (b) How is the estimator unbiased?

3 Let Y 1 Y Y be independent variables with P( Y 1) p and P( Y 0) 1 p where p 0 p 1 is unnown. (a) Obtain the ML estimate. (b) Determine the lower bound on the variance o the estimator assuming that the estimator is unbiased Find xˆ ms the minimum mean-square error and xˆ map the maximum a posteriori estimators o rom the observations Y N and N are random variables with densit unctions 1 ( x) [( x 1) ( x 1)] and N 1 x exp 6.1 The conditional densit unction o the observed random variable Y given a random parameter is given b Y xe ( x) 0 x 0 and 0 x 0 The a priori probabilit densit unction o is r ( x) ( r) 0 x r1 e x x 0 x 0 where is a parameter r is a positive integer and (r ) is the gamma unction. (a) Obtain the a priori mean and variance o. (b) For Y given 1. Obtain the minimum mean-square error estimate o.. What is the variance o this estimate? (c) Suppose we tae independent observations o Y 1 such that Y ( xe x) 0 x 0 and 0 x 0

4 Parameter Estimation Determine the minimum mean-square error estimate o.. What is the variance o this estimate? (d) Veri i the MAP estimate equals the MMSE estimate Consider the problem where the observation is given b Y ln N where is the parameter to be estimated. is uniorml distributed over the interval [ 01] and N has an exponential distribution given b N e 0 n n 0 Obtain (a) The mean-square estimate xˆ ms. (b) The MAP estimate xˆ map. (c) The MAVE estimate xˆ mave The observation Y is given b Y N where and N are two random variables. N is normal with mean one and variance and is uniorml distributed over the interval [0 ]. Determine the MAP estimate o the parameter Show that the mean-square estimation ˆ ms E[ ] commutes over a linear transormation Suppose that the joint densit unction o the observation Y and the parameter is Gaussian. The means m and m are assumed to be zero. can then be expressed as a linear orm o the data. Determine an expression or the conditional densit Y ( ) Consider the problem o estimating a parameter rom one observation Y. Then Y N where and the noise N are statisticall independent with and 0 N n 0 0 n Determine ˆ blue the best linear unbiased estimate o.

5 398 Reerences [1] Van Trees H. L. Detection Estimation and Modulation Theor Part I New Yor: John Wile and Sons 1968 p. 95. [] Vaseghi S. V. Advanced Digital Signal Processing and Noise Reduction New Yor: John Wile and Sons 000. [3] Sorenson H. W. Parameter Estimation: Principles and Problems New Yor: Marcel Deer Selected Bibliograph Dudewicz E. J. Introduction to Statistics and Probabilit New Yor: Holt Rinehart and Winston Gevers M. and L Vandendorpe Processus Statistiques Estimation et Prédiction Université Catholique de Louvain Hain S. Adaptive Filter Theor Englewood Clis NJ: Prentice Hall Helstrom C. W. Elements o Englewood Clis NJ: Prentice Hall a S. M. Fundamentals o Statistical Signal Processing: Estimation Theor Englewood Clis NJ: Prentice Hall Lewis T. O. and P. L. Odell Estimation in Linear Models Englewood Clis NJ: Prentice Hall Mohant N. Signal Processing: Signals Filtering and Detection New Yor: Van Nostrand Reinhold Sage A. P. and J. L. Melsa Estimation Theor with Applications to Communications and Control New Yor: McGraw-Hill Shanmugan. S. and A. M. Breipohl Random Signals: Detection Estimation and Data Analsis New Yor: John Wile and Sons Srinath M. D. and P.. Rajasearan An Introduction to Statistical Signal Processing with Applications New Yor: John Wile and Sons Star H. and J. W. Woods Probabilit Random Processes and Estimation Theor or Engineers Englewood Clis NJ: Prentice Hall 1986 Urowitz H. Signal Theor and Random Processes Dedham MA: Artech House Whalen A. D. Detection o Signals in Noise New Yor: Academic Press 1971.