00001 #include "MusimatChapter9.h" 00002 MusimatChapter9Section(C091704c) { 00003 Print("*** Fractional Brownian Motion ***"); 00004 /***************************************************************************** 00005 00006 Fractional Brownian Motion 00007 00008 The preceding Brownian number generator produces a high degree 00009 of local similarity because subsequent points are constrained to remain relatively close to previous 00010 points. But because the random increment at each step is independent, Brownian motion typically only 00011 shows self-similarity in a region of its spectrum, so its fractal quality degenerates with scaling. 00012 00013 Fractional Brownian motion (fBm) is like Brownian motion, but the increments are no longer 00014 independent. Instead, just as low-frequency ocean waves extend their influence over many cycles 00015 of higher-frequency waves, in fBm, local rapidly fluctuating values are influenced by broader, 00016 slower-moving values extending proportionately over the entire spectrum. As fBm is magnified, 00017 it retains its statistically self-similar shape, and so it is fractal regardless of magnification. 00018 00019 Think of it this way. If we had an ideal tape recorder that accurately recorded all frequencies, 00020 and we gradually increased the speed of a tape recording of Brownian noise, the character of the 00021 noise would change (from a relatively low-frequency "whoosh" to a higher-frequency "whish"). 00022 But a recording of fBm noise will sound the same regardless of playback speed. All speeds sound 00023 the same because both the signal and the spectrum are self-similar at all levels of scale. A number 00024 of methods can be used to generate fBm noises. 00025 00026 *****************************************************************************/ 00027 } 00028 00030 /* $Revision: 1.2 $ $Date: 2006/09/05 06:32:03 $ $Author: dgl $ $Name: $ $Id: C091704c.cpp,v 1.2 2006/09/05 06:32:03 dgl Exp $ */ 00031 // The Musimat Tutorial © 2006 Gareth Loy 00032 // Derived from Chapter 9 and Appendix B of "Musimathics Vol. 1" © 2006 Gareth Loy 00033 // and published exclusively by The MIT Press. 00034 // This program is released WITHOUT ANY WARRANTY; without even the implied 00035 // warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 00036 // For information on usage and redistribution, and for a DISCLAIMER OF ALL 00037 // WARRANTIES, see the file, "LICENSE.txt," in this distribution. 00038 // "Musimathics" is available here: http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10916 00039 // Gareth Loy's Musimathics website: http://www.musimathics.com/ 00040 // The Musimat website: http://www.musimat.com/ 00041 // This program is released under the terms of the GNU General Public License 00042 // available here: http://www.gnu.org/licenses/gpl.txt 00043
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