随机信号和系统代写 ECE5606 Stochastic Signals and Systems

ECE5606 Stochastic Signals and Systems: HW 1

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.  随机信号和系统代写

(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?

1. Prove that

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

1. Consider the singular stochastic process {x(t), 0 ≤ t < ∞} defifined by 随机信号和系统代写

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).

1. Consider the stochastic process  随机信号和系统代写

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)随机信号和系统代写

where