[PDF] 6 Spectral Analysis -- Smoothed Periodogram Method

Any obvious trend should also be removed prior to spectral estimation. Trend produces aspectral peak at zero frequency, and this peak can dominate the spectrum such that otherimportant features are obscured. After detrending, the next steps are computation of the Fouriertransform, computation of the raw periodogram, and smoothing of the periodogram.Discrete Fourier transform. Say x , x , , x0 1 n is an arbitrary time series of length n. The1time series can be expressed as the sum of sinusoids at the Fourier frequencies of the series:x A ( 0 ) 2 ( ) c o s 2 ( ) s in 2t A f f t B f f t j j j j 0 jn / 2 A ( f ) c o s 2 f t , t 0 ,1, , n 1n / 2 n / 2where the summation is over Fourier frequenciesjf , j 1, 2 , ( n 1) / 2 ,jnand the last term in braces is included only if n is even (Bloomfield 2000, p. 38) Note that thetotal number of coefficients is n whether n is even or odd. The coefficients in (1) are given byn 12A ( f ) x c o s 2fttnt 0n 12B ( f ) x s in 2 ft.tnt 0Equations (2) are sine and cosine transforms that transform the time series x into two series oftcoefficients of sinusoids. The relationships in (2) can be more succinctly expressed in complexnotation by making use of the Euler relationixe co s x i sin x(3)and its inverse1ix ix ix ixco s x e e , sin x e e (4)2In general, observed data are strictly real-valued, but they may be regarded as complex numberswith zero imaginary parts. Suppose x , x , , x0 1 n is such a real-valued time series expressed as1complex numbers. The discrete Fourier transform (DFT) of x is given in complex notation byt(1)(2)n 11 2 ift (5)td ( f ) x ent 0Periodogram. The relationships (2) transform the time series into a series of coefficients atits Fourier frequencies. The discrete Fourier transform is the complex expression of thesecoefficientsd ( f )A ( f ) B ( f )i2 2 (6)where A and B are identical to the quantities defined in (2).The original data can be recovered from the DFT using the inverse transform2 if j tx d ( f ) e tjwhich is the complex equivalent of equation (1). (7)jNotes_6, GEOS 585A, Spring 2013 4