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Math Help - Linear filters autocorrelation and transfer functions

  1. #1
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    Linear filters autocorrelation and transfer functions

    Hi all,

    could you please help me out with this question:

    the AR(1) model is :
    X_n= aX_n-1 + Z_n;

    the linear filter is :

    Y_n= b X_n + c X_n-1 + b X_n-1;


    how can i determine the autocorrelation function of the filter Y_n? the transfer function of the filter as well and the PSD (power spectral density ) ?

    Thanks a million,
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Linear filters autocorrelation and transfer functions

    Quote Originally Posted by nerdygirl View Post
    Hi all,

    could you please help me out with this question:

    the AR(1) model is :
    X_n= aX_n-1 + Z_n;

    the linear filter is :

    Y_n= b X_n + c X_n-1 + b X_n-1;


    how can i determine the autocorrelation function of the filter Y_n? the transfer function of the filter as well and the PSD (power spectral density ) ?

    Thanks a million,
    In order to avoid confusion let's write the X and Y sequences as...

    x_{n}= a\ x_{n-1} + b_{n} (1)

    y_{n}= c\ x_{n}+ d\ x_{n-1} (2)

    The problem is well attached with the use of the Z-transform. Given a sequence x_{n},\ n=0,1,... its Z-transform is defined as...

    X(z)= \sum_{n=0}^{\infty} x_{n}\ z^{-n} (3)

    In term of Z-transform the (1) and (2) are...

    X(z)= \frac{B(z)}{1-a\ z^{-1}} (4)

    Y(z)= X(z)\ (c+d\ z^{-1}) (5)

    ... and combining (4) and (5) we obtain...

    Y(z)= \frac{c+d\ z^{-1}}{1-a\ z^{-1}}\ B(z) (6)

    Now y_{n} is found performing the inverse Z-transform of (6)...

    Kind regards

    \chi \sigma
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