Fourer Transforms傅里叶变换.pptVIP

Fourier Transforms Fourier series To go from f(? ) to f(t) substitute To deal with the first basis vector being of length 2? instead of ?, rewrite as Fourier series The coefficients become Fourier series Alternate forms where Complex exponential notation Euler’s formula Euler’s formula Taylor series expansions Even function ( f(x) = f(-x) ) Odd function ( f(x) = -f(-x) ) Complex exponential form Consider the expression So Since an and bn are real, we can let and get Complex exponential form Thus So you could also write Fourier transform We now have Let’s not use just discrete frequencies, n?0 , we’ll allow them to vary continuously too We’ll get there by setting t0=-T/2 and taking limits as T and n approach ? Fourier transform Fourier transform So we have (unitary form, angular frequency) Alternatives (Laplace form, angular frequency) Fourier transform Ordinary frequency Fourier transform Some sufficient conditions for application Dirichlet conditions f(t) has finite maxima and minima within any finite interval f(t) has finite number of discontinuities within any finite interval Square integrable functions (L2 space) Tempered distributions, like Dirac delta Fourier transform Complex form – orthonormal basis functions for space of tempered distributions Next time: Affine Transformations Topic: How do we represent the rotations, translations, etc. needed to build a complex scene from simpler objects? Read: ? Watt, Section 1.1.Optional: ? Foley, et al, Chapter 5.1-5.5. ? David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, 2nd Ed., McGraw-Hill, New York, 1990, Chapter 2. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Phasor notation: