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Unformatted text preview: Fourier Series Much of this course concerns the problem of representing a function as a sum of its different frequency components. Expressing a musical tone as a sum of a fundamental tone and various harmonics is such a representation. So is a spectral decomposition of light waves. The ability to isolate the signal of a single radio or television station from the dozens that are being simultaneously received depends on being able to amplify certain frequencies and suppress others. We start our look at the theory of Fourier series with two questions: Question #1 Which functions f ( t ) have a representation as a sum of constants times cos( kt )s and sin( kt )s? Since cos( kt ) and sin( kt ) can be written in terms of complex exponentials (1) using cos( kt ) = 1 2 bracketleftbig e ikt + e ikt bracketrightbig sin( kt ) = 1 2 i bracketleftbig e ikt e ikt bracketrightbig and, conversely, e ikt can be written in terms of cos( kt )s and sin( kt )s using e ikt = cos( kt ) i sin( kt ) it is equivalent to ask which functions f ( t ) have a representation f ( t ) = summationdisplay k = c k e ikt (1) for some constants c k . Because it will save us considerable writing we shall start with this form of the question and return to sines and cosines later. First, observe that, for every integer k , e ikt = cos( kt )+ i sin( kt ) is periodic with period 2 . So the right hand side is necessarily periodic of period 2 . Unless f ( t ) is periodic with period 2 , it cannot possibly have a representation of the form (1). We shall shortly state a result that says that, on the other hand, every sufficiently continuous (details later) function of period 2 has a representation (1). In this course, we shall never fully justify this claim. On the other hand, it is fairly easy justify an analogous claim for Discrete Fourier Series, which is the version of Fourier series for functions f ( t ) that are only defined for t = n , with n running over the integers and a fixed spacing. This is done in the notes DiscreteTime Fourier Series and Fourier Transforms. Before giving the detailed answer to this question, we consider Question #2 Suppose that we know that some specific function f ( t ) has a representation of the form (1). What are the values of the coefficients c k ? With a little trickery, we shall be able to answer this question completely and easily. We wish to solve (1) for the c k s. At first this task looks somewhat daunting because (1) is really a system of infinitely many equations (one equation for each value of t ) in infinitely many unknowns (the c k s). The trick will allow us to reduce this system to a single equation in any one unknown. Suppose, for example, that we wish to solve for c 17 . The index 17 has been chosen at random. Then we use the orthogonality relation that, when k negationslash = 17, integraldisplay e ikt e i 17 t dt = integraldisplay...
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- Spring '10
- Fourier Series