lect8 - ORTHOGONAL FUNCTIONS AND FOURIER SERIES Two...

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ORTHOGONAL FUNCTIONS AND FOURIER SERIES Two functions, ) ( t f and ) ( t g , are said to be orthogonal if = 0 ) ( ) ( dt t g t f (8.1) where the integral is over the domain of the functions. A typical domain might be the interval ] , 0 [ T , in which case the orthogonality condition becomes 0 ) ( ) ( 0 = T dt t g t f . (8.2) An infinite set of functions, ), ( , ), ( ), ( ), ( 3 2 1 t t t t n j j j j is called an orthogonal set if they are all mutually orthogonal, i.e. = j i all for dt t t j i ; 0 ) ( ) ( j j . (8.3) The set is called orthonormal if they are also individually normalized to unit energy, = i all for dt t i ; 1 ) ( 2 j . (8.4) Combining (8.3) and (8.4), the condition for orthonormality can be written - = ) ( ) ( ) ( j i dt t t j i d j j . (8.5)
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An often-used procedure in signal processing is the expansion of a given function as a linear combination of a given set of orthonormal functions. For example, we may try to express a funtion, ) ( t f , in terms of the ) ( t i j as follows: = = 1 ) ( ) ( n n n t f t f j . (8.6) For a given set of orthonormal functions, it may be that only a certain class of functions can be expanded in this way, while other functions can not be. Example 8.1 The set = = , 3 , 2 , 1 ; 2 cos 2 ) ( n t T n T t n p j is an orthonormal set on the interval ] 2 / , 2 / [ T T - , as the student can easily show. Notice that each of these functions has zero average value on the interval, and also that each of them is an even function of t . It easily follows that any linear combination of these functions, as in (8.6) must also be an even function with zero average value. Therefore, it would be impossible to express an odd function, say, as such a linear combination of these cosines. As it turns out (we state here without proof), any even function with zero average value on the interval [0, T] can be expressed correctly as a linear combination of this set of cosines, in the manner of (8.6). Thus, we say that this set of cosines is a complete orthonormal set (COS) for the class of even functions with zero
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average value. The set is not, however, complete for the class of all functions on that interval. If a function, ) ( t f , can be exressed as a linear combination of the form (8.6) then it is easy to find the coefficients. Multiplying both sides of (8.6) by ) ( t k j and integrating yields k n n n k n n k f k n f dt t t f dt t t f = - = = = = 1 1 ) ( ) ( ) ( ) ( ) ( d j j j . Thus, = dt t t f f n n ) ( ) ( j . (8.7) Example 8.2 – Standard Fourier Series For < < - n let T t for e T t t T n j n = 0 ; 1 ) ( 2 p j . The student can easily show that this set is orthonormal. It turns out ( but not proved here) that this set is, in fact, a COS for almost all functions of interest to us, real or complex on the interval T t 0 . Thus, given a function ) ( t f on that interval we can express it as T t for e f T t f n t T n j n = -∞ = 0 ; 1 ) ( 2 p
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where - = T t T n j n dt e t f T f 0 2 ) ( 1 p .
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lect8 - ORTHOGONAL FUNCTIONS AND FOURIER SERIES Two...

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