Thread: 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,

Originally Posted by nerdygirl

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...

(1)

(2)

The problem is well attached with the use of the Z-transform. Given a sequence its Z-transform is defined as...

(3)

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

(4)

(5)

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

(6)

Now is found performing the inverse Z-transform of (6)...

Kind regards