# Question

A white Gaussian noise process, , is input to two filters with impulse responses, h1(t) and h2 (t) , as shown in the accompanying figure. The corresponding outputs are Y1 (t) and Y2 (t), respectively.

(a) Derive an expression for the cross- correlation function of the two outputs, RY1Y2 (τ).

(b) Derive an expression for the cross- spectral density of the two outputs, SY1Y2 (τ).

(c) Under what conditions (on the filters) are the two outputs independent when sampled at the same instants in time? That is, when are Y1 (to) and Y2 (to) and independent? Express your constraints in terms of the impulse responses of the filters and also in terms of their transfer functions.

(d) Under what conditions (on the filters) are the two outputs independent when sampled at different instants in time. That is, when are Y1 (t1) and Y2 (t2) and 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.

(a) Derive an expression for the cross- correlation function of the two outputs, RY1Y2 (τ).

(b) Derive an expression for the cross- spectral density of the two outputs, SY1Y2 (τ).

(c) Under what conditions (on the filters) are the two outputs independent when sampled at the same instants in time? That is, when are Y1 (to) and Y2 (to) and independent? Express your constraints in terms of the impulse responses of the filters and also in terms of their transfer functions.

(d) Under what conditions (on the filters) are the two outputs independent when sampled at different instants in time. That is, when are Y1 (t1) and Y2 (t2) and 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.

## Answer to relevant Questions

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