Problem $16$: Random signals $($Page $104-$ IDENTIFICATION OF PHYSICAL SYSTEMS$-$ Applications to Condition Monitoring, Fault Diagnosis, Soft Sensor and Controller Design$)$

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We need to estimate from the input and the output data, the non$-$parametric model in terms of frequency response of a system H$($z$)$ given by:

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H$($z$)=$z$1/(11\mathrm{.}7387$z$1+0\mathrm{.}81$z$2)$

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For this purpose, the input is a chosen to be a zero$-$mean white noise process. The Fourier transforms of the input and the output are computed and the frequency response is estimated from a single realization of the random input and the output. The estimate of the non$-$parametric model expressed in terms of the frequency response: H$($f $)=$ Y $($f $)/$U $($f $)$

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$1)$ Perform the estimation of the system$$s parameters $3$ times $($take the same length for the input$).$

$2)$ For each estimation calculate the system$$s parameters.

$3)$ For each estimation evaluate the MSE of the frequency response

$4)$ Repeat the estimation by taking a input of length twice that of question $1),2)$ and $3)$

$5)$ Compare your results with the CRA command.

$6)$ Compare your results with ETFE resuts.