Continuous Time Consider a continuous time function x(t) defined over an interval 0≤≤tT. We apply the method of OLS to find the finite sum of the form xt a a kTtb kTtKkkkK() cos sin=+ +⎛⎝⎜⎞⎠⎟=∑0122ππ that best approximates x(t). This sum has a constant term a0 and sines and cosines at radian frequencies which are integer multiples (harmonics) of ωπ02= T . This ω is the frequency of a sine wave that has exactly one cycle in the interval between 0 and T. 0 We wish to minimize the square error []SE x t x t dtKkT=−∫() ()20. We optimize in the usual way, substituting the first equation into the second and setting derivatives equal to zero. Spectral Theory  1Consider the following a3 equation. ∂∂ωω ωωSEax t tdt a tdt a t k tdtKkTKTT300 0 0 0010023 3 3=− − −⎡⎣⎢⎤⎦⎥∫∑∫∫()cos$cos$cos cos −+⎡⎣⎢⎤⎦⎥=∫∑∫∑23 3 00000000101$cos cos$cos cosa t k tdt b t k tdtTKTKωω ωω Conveniently, most of the integrals vanish. For example, when k and m are different integers, cos coskt mtdtTωω0000=∫ because the integrand is a product of two orthogonal functions. When all the algebra is finished, the minimizing coefficients are shown to be: $()aTxtdtT001=∫, $()cosaTxt k dtkT=∫200ω , and $()sinbTxt k tdtkT=∫200ω k = 1, 2, … Spectral Theory  2An important property of approximation by sums of harmonic sinusoids is that the SEK decreases as K increases. In fact, limKKSE→∞=0. Normally we combine the sine and cosine terms into a single sinusoid through the identity: cit aitbitiiiicos( ) cos sinωϕωω000+=+ with ⎟⎟⎠⎞⎜⎜⎝⎛−=+=−iiiiiiabandbac122tanϕ. The ci are referred to as the amplitudes, and when the graph of these against k is referred to as the Amplitude Spectrum. The ϕi are called phase angles, and the graph of these against k is called the Phase Spectrum. Together they form a kind of “catalog” of the sinusoids in the Fourier Series of a given function. This allows the Fourier Series to be written in the notation xt a c k tkik( ) cos( )=+ +=∞∑001ωϕ. Spectral Theory  3Consider the following example: Graph of f(x)=3 sin t-3030.00 1.57 3.14 4.71 6.28Radiansf(x) Its spectra look like this: ϕk−π2k k 11 3 ck Spectral Theory  4Complex Fourier Series xt zekik t()=−∞∞∑ω0 in which the complex coefficients zk are given by zTxte dtzekiktTkizk==−∠∫100()ω The equivalence with the previous form is given by azczkzzzkkkkkk00201122===−−=∠=−, , , , , ,...*ϕ This makes zk the amplitude and ∠=zkkϕ the phase. Spectral Theory  5Discrete Time The discrete time Fourier Transform pair is: XxexXednninnin()( ) , ..., , , , ,...ωπωωωωππ===−−−∞∞−∑∫121012 It can be shown that X X andXX() ( )() ( ).ωωωω=−∠=−∠− Spectral Theory 