随机信号处理考试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

(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.

(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 .

(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 .

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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)