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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 Suppose that you have an orthonormal (orthogonal, unit length) basis set of vectors { e k } . Any vector in the space spanned by this basis set can be represented as a weighted sum of those basis vectors: v = k a k e k To get the weights: a k = v e k In other words: The vector can be transformed into the weights a i . Likewise, the transformation can be inverted by turning the weights back into the vector. The Fourier Transform Transforms Linear Algebra with Functions The inner (dot) product of two vectors is the sum of the point-wise multiplication of each component: u v = j u [ j ] v [ j ] Cant we do the same thing with functions? f g = - f ( x ) g ( x ) dx 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 = j v [ j ] e k [ j ] = - f ( t ) e k ( t ) dt Inverse v = k a k e k f ( t ) = 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: { e i 2 ut...

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