ch06 - 6 Frequency-based Methods for Time Series 6.1...

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Unformatted text preview: 6 Frequency-based 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 Semi-discrete Fourier Transform Definition 6.1: Suppose we have an infinite sequence { γ ( k ) : k ∈ Z } . The semi-discrete Fourier-Transform 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....
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This note was uploaded on 10/19/2009 for the course MATH 611 taught by Professor Jsdkasj during the Spring '09 term at Kansas.

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ch06 - 6 Frequency-based Methods for Time Series 6.1...

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