Derive an expression for the cross-correlation function of

Question

A white Gaussian noise process  is input to two filters with impulse responses,  as shown in the accompanying figure. The corresponding outputs  respectively.

(a) Derive an expression for the cross-correlation function of the two outputs, 

(b) Derive an expression for the cross-spectral density of the two outputs 

 (c) Under what conditions (on the filters) are the two outputs independent when sampled at the same instants in time? That is, when are  Independent for arbitrary t₁ and t₂ Express your constraints in terms of the impulse responses of the filters and also in terms of their transfer functions.