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Unformatted text preview: The Fourier Transform The Fourier Transform CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science The Fourier Transform Transforms General Idea of Transforms I Suppose that you have an orthonormal (orthogonal, unit length) basis set of vectors { e k } . I Any vector in the space spanned by this basis set can be represented as a weighted sum of those basis vectors: v = X k a k e k I To get the weights: a k = v e k I In other words: I The vector can be transformed into the weights a i . I Likewise, the transformation can be inverted by turning the weights back into the vector. The Fourier Transform Transforms Linear Algebra with Functions I The inner (dot) product of two vectors is the sum of the pointwise multiplication of each component: u v = X j u [ j ] v [ j ] I Cant we do the same thing with functions? f g = Z  f ( x ) g ( x ) dx I Functions satisfy all of the linear algebraic requirements of vectors. The Fourier Transform Transforms Transforms with Functions Just as we transformed vectors , we can also transform functions : Vectors { e k } Functions { e k ( t ) } Transform a k = v e k a k = f e k = X j v [ j ] e k [ j ] = Z  f ( t ) e k ( t ) dt Inverse v = X k a k e k f ( t ) = X k a k e k ( t ) The Fourier Transform The Fourier Transform Basis Set: Generalized Harmonics The set of generalized harmonics we discussed earlier form an orthonormal basis set for functions:...
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This note was uploaded on 03/02/2012 for the course C S 450 taught by Professor Morse,b during the Winter '08 term at BYU.
 Winter '08
 Morse,B

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