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Fourier Series

Module by: Don Johnson. E-mail the author

Summary: Signals can be composed by a superposition of an infinite number of sine and cosine functions. The coefficients of the superposition depend on the signal being represented and are equivalent to knowing the function itself.

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In Signal Decomposition , we have shown that we could express the square wave as a superposition of pulses. This superposition does not generalize well to other periodic signals: How can a superposition of pulses equal a smooth signal like a sinusoid? Because of the importance of sinusoids to linear systems, you might wonder whether they could be added together to represent a large number of periodic signals. You would be right and in good company as well. Euler and Gauss in particular worried about this problem, and Fourier got the credit even though tough mathematical issues were not settled until later. They worked on what is now known as the Fourier series.

Let st s t have period T T . We want to show that periodic signals, even those that have constant-valued segments like a square wave, can be expressed as sum of harmonically related sine waves.

st= a 0 + k =1 a k cos2πktT+ k =1 b k sin2πktT s t a 0 k 1 a k 2 π k t T k 1 b k 2 π k t T
(1)
The family of functions called basis functions cos2πktTsin2πktT 2 π k t T 2 π k t T form the foundation of the Fourier series. No matter what the periodic signal might be, these functions are always present and form the representation's building blocks. They do depend on the signal's period T T , and are indexed by k k . The frequency of each term is kT k T . For k=0 k 0 , the frequency is zero and the corresponding term a 0 a 0 is a constant. The basic frequency 1T 1 T is called the fundamental frequency because all other terms have frequencies that are integer multiples of it. These higher frequency terms are called harmonics : The term at frequency 1T 1 T is the fundamental and the first harmonic, 2T 2 T the second harmonic, etc. Thus, larger values of the series index correspond to higher harmonic-frequency sinusoids. The Fourier coefficients a k a k , b k b k depend on the signal's waveform. Because frequency is linked to index, the coefficients implicitly depend on frequency. Assuming we know the period, knowing the Fourier coefficients is equivalent to knowing the signal.

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