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Unformatted text preview: UCSB Fall 2007 Handout: Fourier Series Computation using Differentiation Fourier series expose the frequency content of periodic functions. For x ( t ) periodic with period T and fundamental frequency ω = 2 π T , we can write x ( t ) = ∞ summationdisplay k =∞ a k e jkω t (1) where the Fourier coefficients { a k } can be computed as a k = 1 T integraldisplay T x ( t ) e jkω t (2) where integraltext T denotes integration over any interval of length equal to the period T . When computing Fourier series coefficients, therefore, we can choose the interval of integration based on our convenience, as long as its length equals the period T . Terminology: For k ≥ 1, the complex exponentials e jkω t and e jkω t are the k th harmonics of the waveform x ( t ) (the fundamental frequency corresponds to k = 1, and is also called the first harmonic). The DC term corresponding to k = 0 is simply the average of the signal over a period. Notation: We use the following shorthand to denote the relation (1)(2) between a periodic waveform and its Fourier coefficients: x ( t ) ↔ { a k } . To completely specify this relationship, of course, we also have to specify the period T or the fundamental frequency ω . Given a periodic waveform x ( t ), we can directly compute the Fourier series coefficients using (2), and we have seen examples of this in class. However, we can often use shortcuts based on(2), and we have seen examples of this in class....
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This note was uploaded on 08/06/2008 for the course ECE 130A taught by Professor Madhow during the Fall '07 term at UCSB.
 Fall '07
 MADHOW
 Frequency

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