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FourierSeries_06

FourierSeries_06 - Periodic Signals and Systems From the...

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Periodic Signals and Systems From the Difference and Differential Equation Solutions of LTI systems the general form of the system response to an exponential input has been determined. If the systems are stable or satisfy the dominance condition , then when initials conditions are applied at the output will contain only the steady-state response at −∞ any finite time. That is, ( ) () For Continuous-Time Signals For Discrete-Time Signals st st nn eH s e zH z z Eigenfunctions and Eigenvalues If the system response to a signal ( x ( t ) or x ( n )) is a replica of the input multiplied by a constant, the signal is said to be an eigenfunction of the system and the constant its eigenvalue. That is x ( t ) or x ( n ) are eigenfunctions if ( ) ( ) () () Continuous-Time Discrete-Time xt Axt xn Axn Most signals are not eigenfunctions Ex: System response to an impulse ( ) ( ) th t nh n δ In general h ( t ) and h ( n ) are not impulses and so the impulse function is not an eigenfunction. However, for LTI systems the exponential signals are eigenfunctions and and s tn ez are the eigenvalues respectively. Eigenfunctions will be shown to play ( ) ( ) and Hs Hz a very important role in the analysis of LTI systems.

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2 Fourier Series Expansion --- Continuous-Time Consider a set of complex exponential signals which are special cases of the exponential e st , i.e., ( ) 0 for k 0, 1, 2, jk t k te ω φ == ± ± " where is a real constant. Then the above functions form an infinite set of 0 eigenfunctions. Next consider the set of all periodic signals, x ( t ), where ( ) ( ) for all xt T xt t += It can be shown that almost all periodic signals can be expanded in a sum of weighted exponentials of the type given above, i.e., () () 0 j kt kk k X t Xe ∞∞ =−∞ =−∞ ∑∑ where and X k are constants. Here T is the fundamental period of x(t) and is 0 2 T π = 0 called its fundamental frequency (radians/sec). The preceding sum is called the Exponential Fourier Series . The constants are called the Fourier Coefficients. k X Determination of the Fourier Coefficients Starting with the Fourier Series () 0 j k k =−∞ = multiply both sides of this equation by , where m is an integer constant. Then 0 j mt e 0 000 0 0 jk m t jm t jm t jk t jk t jm t k k e xt e e ωωω −− =−∞ =−∞ =−∞ ===
3 Now observe that the left side of this equation is periodic with period T , () 0 0 22 2 2 jm t T jm t jm t jm t T jm TT T jm t e xt T e e xt T ex t ππ π ω −+ += + = Here T is the fundamental period of the periodic signal. Now integrate both sides of the previous equation over any period from t 0 to t 0 +T , then 00 0 0 0 tT tt j kmt j jm t kk t t d t X e d t X e d t ωω ++ + ∞∞ −− =−∞ =−∞ == ∑∑ ∫∫ Now denote 0 0 0 jk m t km t gt e d t + = Then () () 0 0 0 jm t m k t t d t X g t + =−∞ = and 0 0 0 0 0 0 0 0 1 1 1 1 jk m t km t t t T jk m T e d t e ee + +  =−  But . Therefore, when the exponential in the bracket is 1 and the whole 0 2 T = expression is equal to 0. When the above expression is indeterminate. However, = examining the integral directly when

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FourierSeries_06 - Periodic Signals and Systems From the...

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