Consider an LTI system subjected to a wide-sense stationary input {x(n)}, which is a white noise sequence. The cross-correlation ϕ_{x}[m] between the input x(n), and output y(n) is:

Where \({{\rm{\Phi }}_{xx}}\left[ m \right] = \sigma _x^2\delta \left[ m \right]\) and h[⋅] is the impulse response.

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ESE Electronics 2015 Paper 1: Official Paper

Option 1 : \(\sigma _x^2h\left[ m \right]\)

CT 3: Building Materials

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__Derivation__:

The cross-correlation function between the input and output processes is given by:

R_{xy} (t_{1}, t_{2}) = E{X(t_{1}) Y*(t_{2})}

\( = E\left\{ {X\left( {{t_1}} \right)\mathop \smallint \limits_{ - \infty }^\infty {X^*}\left( {{t_2} - \alpha } \right){h^*}\left( \alpha \right)d\alpha } \right\}\)

\( = \mathop \smallint \limits_{ - \infty }^\infty {R_{xx}}\left( {{t_1},{t_2} - \alpha } \right){h^*}\left( \alpha \right)d\alpha \)

= R_{xx} (t_{1}, t_{2}) * (h* (t^{2}))

∴ The output cross-correlation between the input and the output is the convolution of the autocorrelation function of the input with the conjugate of the impulse response of the system.

__Analysis__:

Given the autocorrelation of the input wide-sense stationary signal as:

\({\phi _{xx}}\left[ m \right] = \sigma _x^2\delta \left( m \right)\)

The cross-correlation will be:

ϕ_{x}[m] = ϕ_{xx} [m] * h[⋅]

\( = \sigma _x^2\delta \left( m \right) \otimes h\left[ \cdot \right]\)

Since x(n) * δ(n) = x(n)

\({\phi _x}\left( m \right) = \sigma _x^2h\left( m \right)\)