lecture 15-static-color-04.pdf - Trigonometric Polynomials Applications of the FFT Trigonometric Polynomials Applications of the FFT Outline Numerical

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Trigonometric Polynomials Applications of the FFT Numerical Analysis and Computing Lecture Notes #15 — Approximation Theory — The Fast Fourier Transform, with Applications Peter Blomgren, [email protected] Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182-7720 Fall 2014 Peter Blomgren, [email protected] The Fast Fourier Transform, w/Applications — (1/45) Trigonometric Polynomials Applications of the FFT Outline 1 Trigonometric Polynomials Trigonometric Interpolation: Introduction Historical Perspective the FFT 2 Applications of the FFT Recap, Notes, & Historical Perspective 1080p Video, ..., Image Processing Peter Blomgren, [email protected] The Fast Fourier Transform, w/Applications — (2/45) Trigonometric Polynomials Applications of the FFT Trigonometric Interpolation: Introduction Historical Perspective the FFT Trigonometric Polynomials: Least Squares Interpolation. Last Time: We used trigonometric polynomials, i.e. linear combinations of the functions: Φ 0 ( x ) = 1 2 Φ k ( x ) = cos( kx ) , k = 1 , . . . , n Φ n + k ( x ) = sin( kx ) , k = 1 , . . . , n 1 to find least squares approximations (where n < m ) to equally spaced data (2 m points) in the interval [ π, π ], at the node points x j = π + ( j π/ m ) , j = 0 , 1 , . . . , (2 m 1) . This Time: We will find the interpolatory ( n = m ) trigonometric polynomials... and we will figure out how to do it fast! Peter Blomgren, [email protected] The Fast Fourier Transform, w/Applications — (3/45) Trigonometric Polynomials Applications of the FFT Trigonometric Interpolation: Introduction Historical Perspective the FFT Why use Interpolatory Trigonometric Polynomials? Interpolation of large amounts of equally spaced data by trigonometric polynomials produces very good results (close to optimal, c.f. Chebyshev interpolation). Some Applications Digital Filters (Lowpass, Bandpass, Highpass) Signal processing/analysis Antenna design and analysis Quantum mechanics Optics Spectral methods numerical solutions of equations. Image processing/analysis Peter Blomgren, [email protected] The Fast Fourier Transform, w/Applications — (4/45)
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Trigonometric Polynomials Applications of the FFT Trigonometric Interpolation: Introduction Historical Perspective the FFT Interpolatory Trigonometric Polynomials Let x i be 2 m equally spaced node points in [ π, π ], and f i = f ( x i ) the function values at these nodes. We can find a trigonometric polynomial of degree m : P ( x ) ∈ T m which interpolates the data: S m ( x ) = a 0 2 + a m 2 cos( mx ) + m - 1 k =1 [ a k cos( kx ) + b k sin( kx )] , where a k = 1 m 2 m - 1 j =0 f j cos( kx j ) b k = 1 m 2 m - 1 j =0 f j sin( kx j ) . The only difference in this formula compared with the one corresponding to the least squares approximation, S n ( x ) , n < m is the division by two of the a m coefficient.
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