Linear filters autocorrelation and transfer functions

• Nov 28th 2011, 03:25 AM
nerdygirl
Linear filters autocorrelation and transfer functions
Hi all,

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,
• Nov 28th 2011, 09:18 AM
chisigma
Re: Linear filters autocorrelation and transfer functions
Quote:

Originally Posted by nerdygirl
Hi all,

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...

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

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

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

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

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

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

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

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

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$