期末测试5

《随机信号分析与处理》期末自我测评 试题（五） 考试形式：    闭卷      考试时间：  150      分钟  满分：    100    分 考试对象：              学员队别： 　　大队 　       学号：            学员姓名：                     各专业必修　　       得分 一、填空题（共15小题，每小题1分，共15分）   (1) A random process may be viewed as a _______________________, with time t as a parameter. (2)When the autocorrelation function  of the random process X(t) varies only with time difference , and the mean is a constant, X(t) is said to be ____________________. (3)A white noise is applied to a linear time-invariant system; the output process is __________________. (4) The autocorrelation function defines how much a signal is _________ a time-shift version of itself. White noise is known to correlate only with an exact replica of itself. (5) If X(t) has a periodic component, then __________________ will have a periodic component with the same period. (6) A random process X(t) is ergodic if all of its statistics can be determined from _______________of the process. (7) According to Wiener-Khinchin Theorem, the autocorrelation function of a wss process and its _________________ constitute a Fourier transform pair. (8) For a Gaussian process X(t), If it is stationary in the wide-sense, it is also __________. (9) For a detection problem, the observation is ___________ which is generated according to some probability law. (10) X(t) is called ____________ process if its PSD is concentrated a small frequency band and the frequency band is much less than center frequency. (11)The decision problem in the case of two hypotheses essentially consists in partitioning _______________ Z into two regions Z0 and Z1. (12) The estimate to minimize the mean square error is called _________________. (13) Decision tests based on several optimum criteria fall into the general class of________________. But thresholds will , in general, be different for the different criterion (14)For nonrandom parameter estimation problem, if an efficient estimate exists, it must be ________________. (15) For a linear mean square error estimate, __________ is orthogonal to the observation. 得分 二、选择题（共5小题，每小题3分，共15分） For questions from (1) to (5), choose a correct answer among four alternatives.    (1) Consider a detection problem. Assume the decision rule can be expressed as               Then, the false alarm probability can be written as    (a) ;    (b) ;    (c) ;    (d) ; (    ) (2) For an estimation problem, the posteriori density can be expressed as        Then, the maximum a posteriori (MAP) estimate of  is (a)          (b) (c)                   (d) (     ) (3) The autocorrelation function of the random process X(t) is given by              Then, the mean and the variance of X(t) are (a) 5, 9                      (b) 25, 3 (c) 25, ±3                      (d) 25, 9 (       )  (4) Suppose that X(t) is a Gaussian random process with zero mean and autocorrelation function . Then the third order PDF at time  is , where the covariance matrix is (a)           (b) (c)             (d) (     ) (5) The state transition diagram of Markov chain is shown as following figure. Then, the state transition matrix of this Markov chain is given by   (a)          (b) (c)          (d) (    ) 得分 三、（10分）A discrete time process is represented by the following difference equation,   Where  and  are constants;  is a sequence of IID Gaussian with zero mean and  variance . Find the autocorrelation function and power spectral density (PSD) of .   得分 四、（15分）Consider a random signal as follows   Where  is a known constant;  is a random variable that has a magnitude of +1 and -1 with equal probability, and  is a random variable that is uniformly distributed between 0 and 2p. Assume that random variables A and  are independent. (1) Find the mean, variance and autocorrelation function of X(t). Is X(t) a wide-sense stationary? Why? (2) Find the power spectral density (PSD) of X(t).     五、（15分）A white noise process  with autocorrelation function  is applied to a filter with impulse response INCLUDEPICTURE "http://s4.gfkd.mtn/zdkc/sjxhcl/wwwroot/course/modules1/mod311000/index_5_clip_image008_0004.gif" \* MERGEFORMATINET . (1) Determine the power spectral density (PSD) of the output process. (2) Determine the autocorrelation function  of the output process. (3) Determine the first-order probability density function (PDF) of output process. (Hint: )     得分 六、（15分）Consider a binary hypothesis testing problem,   Where m (m>0) is a known constant;  is a Gaussian white noise with zero mean and variance (known). We assume the prior probabilities and are known. The costs  are also known. (a) Find the decision expression for Bayes criterion (b) Find the false alarm probability, the miss probability  and the detection probability. (c) If  , and , find the total  probability of error .         得分 七、（15分）Consider following estimation problem. The N independent observations are   Where  is a Gaussian white noise with zero mean and variance (known). (1) Find the ML estimate of A; (2) Is  unbiased estimate?  Is  an efficient estimate?  Why? (3)What is the variance of .   <查看解答> 一、填空题（共15小题，每小题1分，共15分） (1) random variable  (collection of random variable) (2) wide-sense stationary (3) Gaussian process (4) similar to (correlated to ) (5) autocorrelation function (6) a sample function (7) power spectral density (8) strictly stationary (9) a random variable (10) a Narrow-band (11) the observation space (12) the minimum mean square error estimate (MMSE) ) (13) likelihood ratio test (14) maximum likelihood estimate (15) the error of the estimate 二、选择题（共5小题，每小题3分，共15分） (1) (a) (2) (c) (3) (a) (4) (b) (5) (d) 三、（10分） Solution Method 1：                                   (1 point)          (1 point)           (2 points)           (2 points)                                         (2 points)                      (2 points)                          Method 2                                         (1 point)                                              (1 point)                                    (1 point)                  ( 1 point)                         (2 points)                                        (2 points)                                 (2 points) 四、（15分） Solution (1)                                (2 points )                                           (1 point)                                    (1 point)                                     (1 point)                     (1 point)                   (1 point)                                   (1 point)                                (2 points) Since the mean is a constant; the autocorrelation function is only dependent on the time difference; X(t) is a wide-sense stationary                        (1 points) (2)                                      (1 point)                                        (3 points) 五、（15分） Solution (1)                                  (1 point)                                                     (1 point)                                           (2 points)                                             (1 point) (2)                                                     (1 point)                                        (4 points) (3) Since input process is a white noise and system is a linear time-invariant system, the output process is a Gaussian process.                                 (1 point)    The mean of the output y(t) is zero and the variance of the output y(t) is .                                               (2 points) Therefore, the PDF of y(t) is                   (2 points) 六、（15分） Solution (1)                               (1 point)                                  (1 point)                                             (2 points)                              (3points) (2)                              (1 point)                             (1 point)                     (1 point)               (1 point)                                                  (1 point) (3) ,                                           (1 point)                                                                                                                    (1 point)             (1 point) 七、（15分） Solution (1)                                        (2 points)                          (1 point) Let , we obtain                           (2 points) (2) Since, so  is an unbiased estimate.  (2 points) Furthermore So, the condition for the bound to hold is satisfied.                 (2 point) Therefore   is an efficient estimate.                         (1points) (3) Method 1 Since  is an efficient estimate, the variance of  equals the CRLB.   Or                          (1 point) But                                        (2 points) Therefore                                        (2 points) Method 2                           (2 points)                               (3 points)