Fourie Analysis George Mason University傅里叶分析乔治梅森大学.pptVIP

Fourier Analysis PSCI 702 November 2, 2005 Even and Odd Functions Even Functions Odd Functions Even and Odd Functions Kronecker’s Rule Periodic Functions Trigonometric System Trigonometric System of period 2a Fourier Series The basic Idea behind Fourier series is to express a periodic function in terms of trigonometric system using the orthogonality relations. Fouriser Coefficient Even Odd Extensions Let the function f be defined on (0,a). The even extension fe and odd extension fo of f are the following functions Square Wave Example Complex version of the Fourier expansion Fourier Transform Let f is a piecewise smooth function defined over R. Since f may not be periodic, we define fL as the peridic extension of f over (-L,L). fL can be expressed as: Fourier Transform Fourier Transform Fourier Transform: Properties #1 Fourier Transform: Properties #2 Convolution illustrated Duality Parseval’s Equality Suppose that the f is piecewise continous function then: Famous Fourier Transforms Famous Fourier Transforms Famous Fourier Transforms FFT of DHM FFT of DHM FFT of DHM Measuring multiple frequencies Measuring multiple frequencies Fast Fourier Transform A direct calculation of N-point DFT requires (N-1)2 multiplications and N(N-1) addition. The FFT is begun by noting that W0.5N=-1 and splitting the DFT in to two sums Fast Fourier Transform Now we halve {fk} into two subsequences, according to whether k is even or odd FFT FFT FFT Some useful links /fourier/ Fourier series java applet Famous Fourier Transforms Sine wave Delta function Sinc function Square wave Exponential Lorentzian Gaussian Gaussian Example problem Find the Fourier transform of Example problem: Answer. Find the Fourier transform of Using the Fourier transforms of Π and Λ and the linearity and scaling properties, F(u) = 4sinc(4u) - 2sinc2(2u) + .5sinc2(u) f(x) = Π(x /4) – Λ(x /2) + .5Λ(x) Example problem: Alternative Answer. Find the Fourier transform of Using the Fourier transforms