Help 1 01 FOURIER ANALYSIS & FOURIER TRANSFORMS 02 03 Brian W James 04 University of Salford, Salford, M5 4WT, UK 05 06 Version 1.0 07 Copyright (c) 1995 08 09 In Fourier analysis any periodic function 10 can be represented by a sum of sine and 11 cosine functions. The effects of dispersion 12 and attenuation on the propagation of the 13 waveform can be investigated. The evolution 14 of a standing wave can be illustrated. 15 16 Non-periodic functions can be represented 17 by a integral sum of a spectrum of sine and 18 cosine functions of all frequencies called the 19 Fourier transform. One and two dimensional 20 discrete Fourier transforms are illustrated. 21 22 Press or click the mouse to continue. 23 F10 will select the menu. 24 25 Help 2 01 Fourier Analysis 02 03 Periodic waveforms are synthesized by a sum of 04 different amplitude sine and cosine terms. 05 06 F1 - The current HELP SCREEN 07 08 F2 - Run/Stop - toggles the automatic 09 addition of successive terms to graph 10 11 F3 - Step - alternately displays the next 12 term and adds the term to the graph 13 14 'Propagate' The changes in the Fourier 15 synthesized waveform can be investigated. 16 F2 and F3 now control the propagation. 17 18 'Evolve' The temporal variation of standing 19 waves can be investigated. 20 21 'Display setup' affects the screen resolution 22 and hence the lowest RMS Dif. to be found. 23 24 25 Help 3 01 One Dimensional Fourier Transform 02 03 Any non-periodic function can be represented by 04 an infinite sum of sine and cosine functions. 05 06 F1 - The current HELP SCREEN 07 08 F2 - The inverse Fourier transform or Fourier 09 transform of the display after filtering.10 11 F3 - Real and imaginary parts of the current 12 transform can be displayed. 13 14 F4 - The current transform is displayed as 15 an amplitude and phase plot by using 16 color to indicate phase. 17 18 F5 - Power spectrum of current transform 19 20 F10 - MENU - Used to select menu options:- 21 22 23 23 Press or click the mouse to continue. 24 25 Help 4 01 Two Dimensional Fourier Transforms 02 03 Any non-periodic function can be represented by 04 an infinite sum of sine and cosine functions. 05 06 F1 - The current HELP SCREEN 07 08 F4 - The Fourier transform or the inverse 09 Fourier transform of the current data. 10 11 F6 - Real part of current data plotted 12 13 F7 - Imaginary part of current data plotted 14 15 F8 - Surface plot of a) modulus or b) power. 16 Grey levels of c) real part or d) power. 17 18 F10 - MENU - Used to select menu options:- 19 Fourier - analysis of periodic functions 20 1-D DFT - 1-D Discrete Fourier Transforms 21 2-D DFT - 2-D Discrete Fourier Transforms 22 23 The SLIDERS can be used to select the view. 24 25 Help 5 01 Cauchy normal dispersion expression 02 03 The variation of refractive index, n, with 04 frequency, f, is conveniently described by the 05 expression introduced by Cauchy:- 06 07 n = A + B * (c/f)^2 08 09 where c is the velocity of light and A and B are 10 constants. In this simulation the Cauchy 11 dispersion expression is used to determine the 12 variation of velocity for the harmonics from the 13 Fourier analysis. 14 15 The attenuation of the propagated waves is also 16 usually frequencey dependent with higher 17 harmonics being attenuated more. 18 19 20 21 22 23 Press or click the mouse to continue. 24 25