8.8.5 The autoregressive process  8.8 Examples  8.8.3 Pure random process

8.8.4 Effect of a filter

It is often useful to know what the effect of a filter will be on a variance spectrum, for example to predict how an instrument will respond when measuring the spectrum. Consider the filtered process Z(t) obtained from the time series X(t) and the filter or kernel W(y) by the transformation

$$Z(t) = \int_{-\infty}^\infty W(y)X(t-y)dy .$$ (8.40)

 

The filter W(y)=1/Y for $0\le y\le Y$ and 0 otherwise, for example, gives a running average of the values extending backward for an interval Y. While such a filter does not change the mean value of a stationary process, the autocovariance and spectral density will change.

The autocovariance function in terms of the filtered variables becomes

$${rl}V_{ZZ}(\tau) k = \langle Z(t)Z(t+\tau)\rangle$$

$$= \langle\int\int W(y^\prime)W(y)X(t-y^\prime)X(t+\tau-y)dy^\prime dy\rangle$$

$$= \int \int W(y^\prime)W(y)V_{XX}(\tau+y^\prime-y)dy^\prime dy$$

$$= \int\int\intW(y^\prime)W(y)\Gamma_{XX}(\nu)e^{i2\pi\nu(\tau+y^\prime -y)}d\nudy^\prime dy$$

$$= \int_{-\infty}^\infty\Gamma_{XX}(\nu)\left[\int_{-\infty}^\infty ......t]\left[\int_{-\infty}^\infty W(y)e^{-i2\pi\nuy}dy\right]e^{i2\pi\nu\tau} d\nu$$

$$= \int\Gamma_{XX}(\nu)\tilde W(\nu)\tilde W^*(\nu)e^{i2\pi\nu\tau}d\nu$$ (8.41)

 

The variance spectrum in terms of the filtered time series is then

 

$$\Gamma_{ZZ}(\nu) = \Gamma_{XX}(\nu)\vert\tilde W(\nu)\vert^2$$ (8.42)

 

or the original variance spectrum multiplied by the squared magnitude of the Fourier transform of the weighting functions.

In the case of a running average, calculation of the Fourier transform of the filter function gives

$$\vert W(\nu)\vert^2 = {{\sin^2(\pi\nu T)}\over{(\pi\nu T)^2}} \ .$$ (8.43)

 

The filter reduces frequencies higher than 1/T, but it does not have a sharp cutoff and it distorts the spectrum at frequencies below 1/T, so this is usually a poor choice for a filter.

   
 8.8.5 The autoregressive process  8.8 Examples  8.8.3 Pure random process