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ft.slides.printing.6 - The Fourier Transform The Fourier...

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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 ] ± Can’t we do the same thing with functions? f · g = ² -∞ 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 ] = ² -∞ 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 π ut
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ft.slides.printing.6 - The Fourier Transform The Fourier...

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