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R18 - ENEE 241 02 READING ASSIGNMENT 18 Thu 04/16 Lecture...

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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 18 Thu 04/16 Lecture Topics: periodic signals in continuous time; sums of harmonically related sinusoids; introduction to Fourier series Key Points: • Most periodic signals in continuous time can be expressed as sums of Fourier sinusoids. • For a fundamental period T and (correspondingly) fundamental frequency Ω = 2 π/T , the k th complex Fourier sinusoid (where k ∈ Z ) is given by v ( k ) ( t ) = e jk Ω t The frequency k Ω is referred to as the k th harmonic (of Ω ). • The complex Fourier series of period T with coefficients { S k } is defined by s ( t ) = ∞ X k =-∞ S k e jk Ω t , provided the infinite sum converges. It is real-valued if and only if ( ∀ k ∈ Z ) S- k = ( S k ) * Theory and Examples: 1 . We have seen that the DFT synthesis equation s [ n ] = 1 L L- 1 X k =0 S [ k ] e jk (2 π/L ) n , n = 0 : L- 1 can be extended outside the time index range n = 0 : L- 1 to produce an infinite periodic sequence having period L . This also means that every discrete-time sequence that is periodic with period L can be expressed as a sum of L Fourier sinusoids with coefficients given by the (scaled) DFT of the segment of the sequence corresponding to time indices n = 0 : L- 1. Not surprisingly, the L Fourier frequencies ω k = k (2 π/L ) (where k = 0 : L- 1) are the only distinct values of ω in [0 , 2 π ) such that x [ n ] = e jωn is periodic with period L . 2 . Most periodic signals encountered in continuous time have a similar structure. In other words, they can be written as a sum of continuous-time Fourier sinusoids, whose coefficients are obtained by an operation that resembles the DFT....
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R18 - ENEE 241 02 READING ASSIGNMENT 18 Thu 04/16 Lecture...

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