fourier_series

fourier_series - UCSB Fall 2007 Handout Fourier Series...

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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 0 = 2 π T , we can write x ( t ) = summationdisplay k = -∞ a k e jkω 0 t (1) where the Fourier coefficients { a k } can be computed as a k = 1 T integraldisplay T x ( t ) e - jkω 0 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 0 t and e - jkω 0 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 0 . 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
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