# 随机信号作业代写 ECE5606 Stochastic Signals and Systems

## ECE5606 Stochastic Signals and Systems

1. ### Let {u(t), t ∈ T} and {y(t), t ∈ T} be stochastic processes related through the equation  随机信号作业代写

y(t) + a(t 1)y(t 1) = u(t)

show that

Ruy(s, t) + a(t 1)Ruy(s, t 1) = Ru(s, t)

Ryu(s, t) + a(s 1)Ryu(s 1, t) = Ru(s, t)

1. ### A stationary discrete time stochastic process has the spectral density

Perform the spectral factorization and determine the pulse transfer function of a stable system such that the output has the spectral density S when the input

is white noise.

1. ### Consider a normal stationary process y(t) which is generated by  随机信号作业代写

y(t) = x1(t) + x2(t)

where

x1(t + 1) = ax1(t) + v1(t) (1)

x2(t + 1) = bx2(t) + v2(t) (2)

where {v1(t)} and {v2(t)} are sequences of independent normal (0, σ1) respec tively (0, σ2) stochastic variables. Show that a stochastic process with the same

spectral density can be represented by

where q is the shift operator (qy(t) = y(t + 1)), the same as z in the notes, and {e(t)} is a sequence of independent normal (0, 1) stochastic variables. Deter

mine the parameters λ and c随机信号作业代写

1. ### Consider the moving average process

y(t) = e(t)+4e(t 1)

where {e(t), t = ..., 1, 0, 1, ...} is a sequence of independent normal (0, 1) ran dom variables. Show that a process with the same spectral density can be generated by

y(t) = λ[ (t) + c(t 1)]

where { (t), t = ..., 1, 0, 1, ...} is a sequence of independent normal (0, 1) stochastic variables and |c| < 1. Furthermore, determine c and λ.