已知x(t),y(t)是相互独立的平稳过程,z(t)=x(t)y(t),求z(t)的自相关函数,用x(t)和y(t)的自相关函数表示.

Question

已知x(t),y(t)是相互独立的平稳过程,z(t)=x(t)y(t),求z(t)的自相关函数,用x(t)和y(t)的自相关函数表示.

Answer 1

Rz(t,t+τ)
=E(z(t)z(t+τ))
=E(x(t)y(t)x(t+τ)y(t+τ))
=E(x(t)x(t+τ)y(t)y(t+τ)) x(t)y(t)相互独立
=E(x(t)x(t+τ)E(y(t)y(t+τ))
=Rx(t,τ)Ry(t,τ)