Continuous_Fourier_series

Continuous_Fourier_series

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Unformatted text preview: A dt = −t0 /2 At0 , T Note, however, that this is not enough to prove that they form an orthogonal basis for L2 (T ) since L2 (T ) is infinite-dimensional. Unlike in finite-dimensional vector spaces, it is not true that any infinite set of nonzero orthogonal vectors forms an orthogonal basis for L2 (T ). Proving the fact that signals φk do form an orthogonal basis for L2 (T ) is beyond the scope of this course. 3 s(t) A ttt ttt 0 − t2 −T 2 −T t0 2 T 2 T t Figure 1. Signal s(t) of Example 1. ak = = 1 T t0 /2 A exp − −t0 /2 j 2πkt T dt A T j 2πkt · exp − T −j 2πk T j πkt0 A1 · · exp πk 2j T A π kt0 = . sin πk T = t0 /2 −t0 /2 − exp − jπkt0 T (Note that this last formula is also valid for k = 0 if we define sin θ θ = 1.) θ =0 Another common way of decomposing CT periodic signals as linear combinations of sinusoidal signals is by using sines and cosines as basis functions, instead of complex exponentials. The following infinite collection of functions is also an orthogonal basis for L2 (T ), and is also called a Fourier basis: c0 (t) = 1, ck (t) = cos sk (t) = sin 2πkt , k = 1, 2 , . . . T 2πkt , k = 1, 2 , . . . T As we did previously, let us first prove that these functions are pairwise orthogonal, and find their energies over one period. We need to consider all pairwise inner products–which will now be integrals of products of trigonometric functions. We 4 will therefore need the following formulas: 1 (cos(α − β ) − cos(α + β )) 2 1 sin α cos β = (sin...
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This note was uploaded on 05/28/2012 for the course ECE 301 taught by Professor V."ragu"balakrishnan during the Fall '06 term at Purdue University-West Lafayette.

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