1. A linear filter of one random sequence (X.) into another sequence (Y.} is Y = Σ α.Χ. The Filter Theorem: Suppose X, i

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1. A linear filter of one random sequence (X.) into another sequence (Y.} is Y = Σ α.Χ. The Filter Theorem: Suppose X, is a stationary time series with spectral density fy(@). Let {a} be a sequence of real numbers such that E14/<. Then the process Y. = a.x... is a stationary time series with spectral density function fy(@) = |A(29f fx(@s) = a(m) fy(@), where A(z) is the filter generating function A(z)= ,-", Iasi. and a(m) = 4(e) is the transfer function of the linear filter. a) Use the above theorem to verify the previous results in Chapter 8 that the spectral density for the AR(1) process, X.-X-= : provided it satisfies the stationary condition, fr(@)= (1-26, cos(65)+442) JI 1 b) Using the theorem, find the spectral density function of the MA(2) process X, = 8, +0,8-1 +0,8,_. You may recall back to the notes to obtain the general formulae of MA processes for verification purpose.