# ch07 - 7 Spectral Analysis 7.1 A Simple Sinusoidal Model...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 7 Spectral Analysis 7.1 A Simple Sinusoidal Model Suppose we suspect that a given time series, with observations made at unit time intervals, contains a sinusoidal component of frequency ω plus a random error term. We will consider the following model: X t = μ + α cos( ωt ) + β sin( ωt ) + Z t , t = 1 , . . ., N (7.1) Here Z i is a purely random process, and ( μ, α, β ) are to be estimated from data x 1 , . . ., x N . To make the algebra easy, we only consider the case of N even. We represent (7.1) in matrix notation by E ( X ) = Aθ where x T = ( x 1 , . . ., x n ) , θ T = ( μ, α, β ) and A N × 3 = 1 cos( ω ) sin( ω ) 1 cos(2 ω ) sin(2 ω ) . . . . . . . . . 1 cos( Nω ) sin( Nω ) . The least squares estimator is given by ˆ θ = ( A T A )- 1 A T x where A T A = N ∑ N k =1 cos( kω ) ∑ N k =1 sin( kω ) ∑ N k =1 cos( kω ) ∑ N k =1 cos 2 ( kω ) ∑ N k =1 cos( kω ) sin( kω ) ∑ N k =1 sin( kω ) ∑ N k =1 cos( kω ) sin( kω ) ∑ N k =1 sin 2 ( kω ) . They become very simple if ω is restricted to one of the values ω p = 2 πp/N, p = 1 , . . ., N/ 2....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

ch07 - 7 Spectral Analysis 7.1 A Simple Sinusoidal Model...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online