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# recitation3 notes - Filtering 1 14.384 Time Series Analysis...

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Filtering 1 14.384 Time Series Analysis, Fall 2007 Recitation by Paul Schrimpf Supplementary to lectures given by Anna Mikusheva September 21, 2007 Recitation 3 ± ± ± Filtering In lecture 4, we introduced ﬁltering. Here we’ll spend a bit more time deriving some common ﬁlters and showing how to use them. Recall that an ideal band-pass ﬁlter has B ( e ) = 1 [ ω l , ω h ] 0 otherwise and can be written as B ( e ) = ± j e i±j −∈ where 1 ± = e i±j j 2 ω | ± [ l , h ] 1 = e i±j + e i±j 2 ω l , h h , l 1 = e i h j e i l j + e i l j e i h j 2 ωij sin( j h ) sin( j l ) j ± = 0 = j h l j = 0 Baxter-King Baxter and King (1999) proposed approximating the ideal ﬁlter with one of order J by solving 1 ± B ( e ) | 2 min | B ( e B () 2 ω s.t. B (1) = β

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Baxter-King 2 ± ± ± ± ± ± ± ± where the constraint may or may not be present. We might want to impose B (1) = 0 so that the ﬁltered series is stationary, or if we’re constructing a low-pass ﬁlter, we might want
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recitation3 notes - Filtering 1 14.384 Time Series Analysis...

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