SPECTRAL - D.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS...

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Unformatted text preview: D.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS STATISTICAL FOURIER ANALYSIS The Fourier Representation of a Sequence According to the basic result of Fourier analysis, it is always possible to approximate an arbitrary analytic function defined over a finite interval of the real line, to any desired degree of accuracy, by a weighted sum of sine and cosine functions of harmonically increasing frequencies. The accuracy of approximation increases with the number of functions within the sum. Similar results apply in the case of sequences, which may be regarded as functions mapping from the set of integers onto the real line. For a sample of T observations y , . . . , y T − 1 , it is possible to devise an expression in the form (1) y t = n X j =0 n α j cos( ω j t ) + β j sin( ω j t ) o , wherein ω j = 2 πj/T is a multiple of the fundamental frequency ω 1 = 2 π/T . Thus, the elements of a finite sequence can be expressed exactly in terms of sines and cosines. This expression is called the Fourier decomposition of y t and the set of coefficients { α j , β j ; j = 0 , 1 , . . . , n } are called the Fourier coefficients. When T is even, we have n = T/ 2; and it follows that (2) sin( ω t ) = sin(0) = 0 , cos( ω t ) = cos(0) = 1 , sin( ω n t ) = sin( πt ) = 0 , cos( ω n t ) = cos( πt ) = ( − 1) t . Therefore, equation (1) becomes (3) y t = α + n − 1 X j =1 n α j cos( ω j t ) + β j sin( ω j t ) o + α n ( − 1) t . When T is odd, we have n = ( T − 1) / 2; and then equation (1) becomes (4) y t = α + n X j =1 n α j cos( ω j t ) + β j sin( ω j t ) o . In both cases, there are T nonzero coefficients amongst the set { α j , β j ; j = 0 , 1 , . . . , n } ; and the mapping from the sample values to the co- efficients constitutes a one-to-one invertible transformation. In equation (3), the frequencies of the trigonometric functions range from ω 1 = 2 π/T to ω n = π ; whereas, in equation (4), they range from ω 1 = 2 π/T to ω n = π ( T − 1) /T . The frequency π is the so-called Nyquist frequency....
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This note was uploaded on 03/02/2012 for the course EC 7087 taught by Professor D.s.g.pollock during the Fall '11 term at Queen Mary, University of London.

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SPECTRAL - D.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS...

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