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D.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS THE FOURIER DECOMPOSITION OF A TIME SERIES In spite of the notion that a regular trigonometrical function is an inappropriate means for modelling an economic cycle other than a seasonal fluctuation, there are good reasons for explaining a data sequence in terms of such functions. The Fourier decomposition of a series is a matter of explaining the series entirely as a composition of sinusoidal functions. Thus it is possible to represent the generic element of the sample as (1) y t = n j =0 α j cos( ω j t ) + β j sin( ω j t ) . Assuming that T = 2 n is even, this sum comprises T functions whose frequen- cies (2) ω j = 2 πj T , j = 0 , . . . , n = T 2 are at equally spaced points in the interval [0 , π ]. As we might infer from our analysis of a seasonal fluctuation, there are as many nonzeros elements in the sum under (1) as there are data points, for the reason that two of the functions within the sum—namely sin( ω 0 t ) = sin(0) and sin( ω n t ) = sin( πt )—are identically zero. It follows that the mapping from the sample values to the coefficients constitutes a one-to-one invertible transformation. The same conclusion arises in the slightly more complicated case where T is odd. The angular velocity ω j = 2 πj/T relates to a pair of trigonometrical com- ponents which accomplish j cycles in the T periods spanned by the data. The highest velocity ω n = π corresponds to the so-called Nyquist frequency. If a component with a frequency in excess of π were included in the sum in (1), then its effect would be indistinguishable from that of a component with a frequency in the range [0 , π ] To demonstrate this, consider the case of a pure cosine wave of unit am- plitude and zero phase whose frequency ω lies in the interval π < ω < 2 π . Let ω = 2 π ω . Then (3) cos( ωt ) = cos (2 π ω ) t = cos(2 π )cos( ω t ) + sin(2 π )sin( ω t ) = cos( ω t ); which indicates that ω and ω are observationally indistinguishable. Here, ω [0 , π ] is described as the alias of ω > π . 1
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FOURIER DECOMPOSITION For an illustration of the problem of aliasing, let us imagine that a person observes the sea level at 6am. and 6pm. each day. He should notice a very gradual recession and advance of the water level; the frequency of the cycle being f = 1 / 28 which amounts to one tide in 14 days. In fact, the true frequency is f = 1 1 / 28 which gives 27 tides in 14 days. Observing the sea level every six hours should enable him to infer the correct frequency.
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