p=fPd=A2+2∑(A+B) implying that the total power is distributed only among the dc term, the fundamental frequency 2/ and its harmonics koo, k> 1, with 2(Af+Bk) representing the power contained at the harmonic koo. For every periodic signal with finite power, since A,+0, Bx0, eventually the overharmonics become of decreasing The British physicist Schuster [Schuster, 1898] used this observation to suggest that the partial power Pk 2(Ak Bk) at frequency koos k=0-o0, be displayed as the spectrum. Schuster termed this method the periodogram, and information over a multiple of periods was used to compute the Fourier coefficients and/or to smooth the periodogram, since depending on the starting time, the periodogram may contain irregular and spurious peaks. a notable exception to periodogram was the linear regression analysis introduced by the British statistician Yule [Yule, 1927] to obtain a more accurate description of the periodicities in noisy data. Because he sampled periodic process x(k)=cos koo T containing a single harmonic component satisfies the recursive relation where a=2 cos @. T represents the harmonic component, its noisy version x(k)+ n(k)satisfies the recursion x(k)=ax(k-1)-x(k-2)+n(k) Yule interpreted this time series model as a recursive harmonic process driven by a ner generalized the above form to determine the periodicity in the sequence of sunspot numbers. Yule furt recursion to x(k)=ax(k-1)+bx(k-2)+n(k) where a and b are arbitrary, to describe a truly autoregressive process and since for the right choice of a, b the least-square solution to the above autoregressive equation is a damped sinusoid, this generalization forms the basis for the modern day parametric methods Modern Spectral analysis Norbert Wiener's classic work on Generalized Harmonic Analysis [ Wiener, 1930] gave random processes a firm statistical foundation, and with the notion of ensemble average several key concepts were then introduced. The formalization of modern day probability theory by Kolmogorov and his school also played an indispensable part in this development. Thus, if xt) represents a continuous-time stochastic (random)process, then for every fixed t, it behaves like a random variable with some probability density function f(x, t). The ensemble average or expected value of the process is given by (t)=E[x(t)]=xf (x,t)dx and the statistical correlation between two time instants t and s of the random process is described through its autocorrelation function Rx(t1,t2)=E[x(t1)x*(t2)]= x1x2f1x2(x,x2,1,t2 (t2,t1) c 2000 by CRC Press LLC© 2000 by CRC Press LLC implying that the total power is distributed only among the dc term, the fundamental frequency w0 = 2p/T0 and its harmonics kw0, k ³ 1, with 2(A2 k + B2 k ) representing the power contained at the harmonic kw0. For every periodic signal with finite power, since Ak Æ 0, Bk Æ 0, eventually the overharmonics become of decreasing importance. The British physicist Schuster [Schuster, 1898] used this observation to suggest that the partial power Pk = 2(A2 k + B2 k ) at frequency kw0, k = 0 Æ •, be displayed as the spectrum. Schuster termed this method the periodogram, and information over a multiple of periods was used to compute the Fourier coefficients and/or to smooth the periodogram, since depending on the starting time, the periodogram may contain irregular and spurious peaks. A notable exception to periodogram was the linear regression analysis introduced by the British statistician Yule [Yule, 1927] to obtain a more accurate description of the periodicities in noisy data. Because the sampled periodic process x(k) = cos kw0T containing a single harmonic component satisfies the recursive relation x(k) = ax(k – 1) – x(k – 2) where a = 2 cos w0T represents the harmonic component, its noisy version x(k) + n(k) satisfies the recursion x(k) = ax(k – 1) – x(k – 2) + n(k) Yule interpreted this time series model as a recursive harmonic process driven by a noise process and used this form to determine the periodicity in the sequence of sunspot numbers. Yule further generalized the above recursion to x(k) = ax(k – 1) + bx(k – 2) + n(k) where a and b are arbitrary, to describe a truly autoregressive process and since for the right choice of a, b the least-square solution to the above autoregressive equation is a damped sinusoid, this generalization forms the basis for the modern day parametric methods. Modern Spectral Analysis Norbert Wiener’s classic work on Generalized Harmonic Analysis [Wiener, 1930] gave random processes a firm statistical foundation, and with the notion of ensemble average several key concepts were then introduced. The formalization of modern day probability theory by Kolmogorov and his school also played an indispensable part in this development. Thus, if x(t) represents a continuous-time stochastic (random) process, then for every fixed t, it behaves like a random variable with some probability density function fx(x,t). The ensemble average or expected value of the process is given by and the statistical correlation between two time instants t1 and t2 of the random process is described through its autocorrelation function P T f t dt A Ak k B k T = = + + = • Ú Â 1 2 0 2 0 2 2 2 1 0 0 * ( )* ( ) mx x (t) = = E[x(t)] x f (x, t) dx -• • Ú R t t E x t x t x x f x x t t dx dx R t t xx x x xx ( , ) [ ( ) * ( )] * ( , , , ) * ( , ) – – 1 2 1 2 1 2 1 2 1 2 1 2 2 1 1 2 = = = • • -• • Ú Ú