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Lab9 - L303-9.R3 Drexel University Electrical and Computer...

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L303-9.R3 8-1 Drexel University Electrical and Computer Engineering Dept. Electrical Engineering Laboratory III, ECEL 303 E. L. Gerber FOURIER ANALYSIS and FILTER DESIGN Object The object of this experiment is to become familiar with some properties of the Fourier series. You will calculate the Fourier series of a periodic signal and investigate the effect that the number of terms in the series has on the accuracy of the series. Then you will design a filter and observe the effect that the filter has on the frequency content of a periodic signal passing through. Both computer simulation and real circuits will be examined. Introduction Most periodic functions can be decomposed into an infinite series of sinusoidal or expo- nential terms. The terms consist of frequencies that are integer multiples of the funda- mental frequency of the periodic function. In practice we only consider a finite number of terms to approximate the periodic function. Maple can be used to calculate the Fourier coefficients of a periodic function and display the sum of a finite number. When a filter’s transfer function is frequency dependent, the output signal is affected by the filter’s behavior. When a Fourier decomposition is performed on the input signal, the effect of the transfer function H(j ω ), on each harmonic may be calculated term by term at each frequency. The sum of the harmonics at the output yields the resulting output function of the system. Last day to hand in reports: Monday, June 8, 2009 not Tuesday.
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L303-9.R3 8-2 Theory I - The Fourier Series: * A periodic wave of period T may be expressed in the form of a Fourier series. The Fourier series contains a constant term and a number of periodic terms. The constant term is equal to the average or DC value of the given wave (which may be zero). The periodic terms may be expressed in sinusoidal form. The Fourier series of a periodic function f(t) in sinusoidal form is given as, f ( t ) = a 0 + a n cos( n ! 0 t ) n " + b n sin( n ! 0 t ) n " (1) The lowest frequency of the periodic function is the fundamental frequency, f 0 = 1/T. Other terms have frequencies that are integer multiples of the fundamental frequency, nf 0 , called harmonics . The coefficients of Eq. 1, a n and b n are evaluated from, a n = 2 T f ( t )cos( n ! o 0 T " t ) dt and b n = 2 T f ( t )sin( n ! o 0 T " t ) dt (2) Maple can calculate these coefficients. And once they are found Maple can sum a finite number of the terms, as in Eq. 1. A plot of the summation will show the Fourier series approximation of the function. The more terms taken the closer the series approaches the function. The expression below is the sum of the sinusoidal terms of a unit square wave. The expression is written for the first ten non-zero terms of the unit square wave, ± 1 for ω 0 = 1 r/s.
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