随机信号代写 ECE5606 Stochastic Signals and Systems:

ECE5606 Stochastic Signals and Systems:

随机信号代写 where x and y are stochastic processes and A, B, a and b are deterministic matrices and vectors, respectively.Is the process stationary

1. 
   

   Consider the moving average process of order one x(t) = e(t) + ce(t − 1) where e(t) t = . . . , −1, 0, 1, . . . } is a sequence of independent normal (0, 1) random variables. 随机信号代写

   
   

(a) Determine the covariance of x(t) and x(s).

(b) Is the process stationary, normal, Markovian, ergodic, singular?

(c) Does the process have the independent increments?

2. 
   

   Prove that

   
   

cov[Ax + a, By + b] = A{cov[x, y]}B T ,

where x and y are stochastic processes and A, B, a and b are deterministic matrices and vectors, respectively.

3. 
   

   Consider the singular stochastic process {x(t), 0 ≤ t < ∞} defifined by  随机信号代写

   
   

where the initial state is normal (0, 1). Is the process ergodic? Give a predictor for the process which predicts x(t + h) based on measurements of x(t).

4. 
   

   Consider the stochastic process

   
   

wheretheinitialstateisnormalwithzeromeanvalueandthecov ariance

Is the process ergodic? Give a predictor for the process which predicts x(t + h) based on observation of {x1(t), t0 ≤ s ≤ t}.

Hint: The solution of the state of the system is given by

x(t) = Φ(t, 0)x(0),

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