This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 6 Frequencybased Methods for Time Series 6.1 Stationary Processes in the Frequency Domain If a time series has a periodic component, this can be modelled as X t = R cos( ωt + φ ) + Z t . (6.1) More than one periodic components can be modelled as X t = k summationdisplay j =1 R j cos( ω j t + φ j ) + Z t . (6.2) This can be rewritten as X t = k summationdisplay j =1 a j cos( ω j t ) + b j sin( ω j t ) + Z t . (6.3) For discrete time processes measured at unit time intervals, t ∈ Z , the highest observable frequency is π . This frequency is called the Nyquist rate . For a process measured at intervals of Δ t , the Nyquist rate is π/ Δ t . 6.2 Semidiscrete Fourier Transform Definition 6.1: Suppose we have an infinite sequence { γ ( k ) : k ∈ Z } . The semidiscrete FourierTransform of { γ ( k ) } is f ( ω ) = 1 π ∞ summationdisplay k =∞ γ ( k ) e iωk , π < ω < π (6.6) and its inverse is γ ( k ) = 1 2 integraldisplay π π f ( ω ) e iωk dω, k ∈ Z (6.7)(6....
View
Full
Document
This note was uploaded on 10/19/2009 for the course MATH 611 taught by Professor Jsdkasj during the Spring '09 term at Kansas.
 Spring '09
 jsdkasj

Click to edit the document details