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For an LTI system with stochastic input prove that autocorrelation of output is given by convolution of cross-correlation(between input-output) and LTI system impulse response.

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For an LTI system with stochastic input prove that autocorrelation of output is given by convolution of cross-correlation(between input-output) and LTI system impulse response.

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written 7.5 years ago by |

Let {X(t)} and {Y(t)} denote the input and output respectively.

If h(t) is the system weighing function, then

Y(t)= $∫^\infty_{-\infty} h(β). X(t-β) dβ Autocorrelation of output is given as $R_{YY}$($t_1$,$t_2$) = E{Y($t_1$.Y*($t_2$)}

=E$∫^\infty_{-\infty} h(β). X({t_1}-β) dβ {Y^*}({t_2})$

=$∫^\infty_{-\infty} E{X({t_1}-β){Y^*}({t_2})} h(β).dβ$

= $∫^\infty_{-\infty} {R_{XY}}({t_1}-β,{t_2})h(β).dβ$

=$R_{XY} ({t_1},{t_2})*h({t_1})$

Thus the autocorrelation of output is given by convolution of cross-correlation( between input-output ) and LTI system impulse response.

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