Answer to Question #280305 in Statistics and Probability for Dek

Question

Define a filter in time series analysis. Find the effect of a linear filter on the time series

X_t=cosθt+isinθt

Answer 1

Filtering in time series mean removal of spectral power at certain chosen frequencies while retaining others.

A high-pass filter retains higher frequencies while removing low frequencies; a low-pass filter does the opposite. A band-pass filter removes all frequencies outside a prespecified band.

for X_t =cos$\theta t+ sin\theta t$

filtering involves convolution between two series

X[_t]=cos$\theta [t] + sin\theta [t]$

the convolution of the series is the sum of the product between series forward and the series going backwards

X[_t] = {cos$\theta$ + sin$\theta$ }_t $\forall t$

introduce the product in both left hand side and right hand side .

$\\\prod X$ [_t]=$\prod$ {cos$\theta$ +sin$\theta$ }_t $\forall t$