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<html>
<head>
<title>
COLORED_NOISE - 1/F^alpha Power Law Noise Generation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
COLORED_NOISE <br> 1/F^alpha Power Law Noise Generation
</h1>
<hr>
<p>
<b>COLORED_NOISE</b>
is a C++ library which generates sequences that simulate
1/f^alpha power law noise. This includes white noise (alpha = 0),
pink noise (alpha = 1) and brown noise or Brownian motion (alpha = 2),
but also values of alpha between 0 and 2.
</p>
<p>
The original code listing by Kasdin referenced a number of functions from
the Numerical Recipes library (FOUR1, FREE_VECTOR, GASDEV, RAN1, REALFT,
VECTOR). Numerical Recipes is a proprietary library whose components
cannot be freely distributed. Moreover, the referenced functions
have some peculiarities of implementation (the FFT is not normalized;
the FFT data must be a power of 2 in order; the code uses single
precision real arithmetic).
</p>
<p>
In the code presented here, references to Numerical Recipes functions
have been replaced by references to, and the source code of,
nonproprietary code. In particular, the Fourier
transform is implemented by a "slow Fourier transform" method,
and by simple uniform and normal random number generators.
You are welcome to make an efficient code by replacing these
routines. The purpose of this posting is primarily to demonstrate
the method.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>COLORED_NOISE</b> is available in
<a href = "../../c_src/colored_noise/colored_noise.html">a C version</a> and
<a href = "../../cpp_src/colored_noise/colored_noise.html">a C++ version</a> and
<a href = "../../f77_src/colored_noise/colored_noise.html">a FORTRAN77 version</a> and
<a href = "../../f_src/colored_noise/colored_noise.html">a FORTRAN90 version</a> and
<a href = "../../m_src/colored_noise/colored_noise.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/black_scholes/black_scholes.html">
BLACK_SCHOLES</a>,
a C++ library which
implements some simple approaches to
the Black-Scholes option valuation theory,
by Desmond Higham.
</p>
<p>
<a href = "../../cpp_src/correlation/correlation.html">
CORRELATION</a>,
a C++ library which
contains examples of statistical correlation functions.
</p>
<p>
<a href = "../../cpp_src/ornstein_uhlenbeck/ornstein_uhlenbeck.html">
ORNSTEIN_UHLENBECK</a>,
a C++ library which
approximates solutions of the Ornstein-Uhlenbeck
stochastic differential equation (SDE) using the Euler method
and the Euler-Maruyama method.
</p>
<p>
<a href = "../../cpp_src/pink_noise/pink_noise.html">
PINK_NOISE</a>,
a C++ library which
computes a "pink noise" signal obeying a 1/f power law.
</p>
<p>
<a href = "../../cpp_src/sde/sde.html">
SDE</a>,
a C++ library which
illustrates the properties of stochastic differential equations (SDE's), and
common algorithms for their analysis,
by Desmond Higham;
</p>
<p>
<a href = "../../cpp_src/sftpack/sftpack.html">
SFTPACK</a>,
a C++ library which
implements the "slow" Fourier transform, intended as a teaching
tool and comparison with the fast Fourier transform.
</p>
<p>
<a href = "../../cpp_src/stochastic_rk/stochastic_rk.html">
STOCHASTIC_RK</a>,
a C++ library which
applies a Runge-Kutta scheme to a stochastic differential equation.
</p>
<p>
<a href = "../../cpp_src/uniform/uniform.html">
UNIFORM</a>,
a C++ library which
computes elements of a uniform pseudorandom sequence.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Martin Gardner,<br>
White and brown music, fractal curves and one-over-f fluctuations,<br>
Scientific American,<br>
Volume 238, Number 4, April 1978, pages 16-32.
</li>
<li>
Jeremy Kasdin,<br>
Discrete Simulation of Colored Noise and Stochastic Processes
and 1/f^a Power Law Noise Generation,<br>
Proceedings of the IEEE,<br>
Volume 83, Number 5, 1995, pages 802-827.
</li>
<li>
Edoardo Milotti,<br>
1/f noise: a pedagogical review,<br>
arXiv:physics/0204033.
</li>
<li>
Sophocles Orfanidis,<br>
Introduction to Signal Processing,<br>
Prentice-Hall, 1995,<br>
ISBN: 0-13-209172-0,<br>
LC: TK5102.5.O246.
</li>
<li>
William Press,<br>
Flicker Noises in Astronomy and Elsewhere,<br>
Comments on Astrophysics,<br>
Volume 7, Number 4, 1978, pages 103-119.
</li>
<li>
Miroslav Stoyanov, Max Gunzburger, John Burkardt,<br>
Pink Noise, 1/f^alpha Noise, and Their Effect on Solutions
of Differential Equations,<br>
International Journal for Uncertainty Quantification,<br>
Volume 1, Number 3, pages 257-278, 2011.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "colored_noise.cpp">colored_noise.cpp</a>, the source code.
</li>
<li>
<a href = "colored_noise.hpp">colored_noise.hpp</a>, the include file.
</li>
<li>
<a href = "colored_noise.sh">colored_noise.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "colored_noise_prb.cpp">colored_noise_prb.cpp</a>
a sample calling program.
</li>
<li>
<a href = "colored_noise_prb.sh">colored_noise_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "colored_noise_prb_output.txt">colored_noise_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>ALPHA</b> is a sequence of 9 sets of data, for ALPHA = 0.00 to ALPHA
= 2.00, each time computing a sequence of 128 values, and using the
same sequence of random numbers. This makes the effect of ALPHA
very clear.
<ul>
<li>
<a href = "make_plots.m">make_plots.m</a>
a MATLAB program which reads the "txt" files and makes plots.
</li>
<li>
<a href = "alpha_0.00.txt">alpha_0.00.txt</a>
values for ALPHA = 0.00.
</li>
<li>
<a href = "alpha_0.00.png">alpha_0.00.png</a>,
a plot.
</li>
<li>
<a href = "alpha_0.25.txt">alpha_0.25.txt</a>
values for ALPHA = 0.25.
</li>
<li>
<a href = "alpha_0.25.png">alpha_0.25.png</a>,
a plot.
</li>
<li>
<a href = "alpha_0.50.txt">alpha_0.50.txt</a>
values for ALPHA = 0.50.
</li>
<li>
<a href = "alpha_0.50.png">alpha_0.50.png</a>,
a plot.
</li>
<li>
<a href = "alpha_0.75.txt">alpha_0.75.txt</a>
values for ALPHA = 0.75.
</li>
<li>
<a href = "alpha_0.75.png">alpha_0.75.png</a>,
a plot.
</li>
<li>
<a href = "alpha_1.00.txt">alpha_1.00.txt</a>
values for ALPHA = 1.00.
</li>
<li>
<a href = "alpha_1.00.png">alpha_1.00.png</a>,
a plot.
</li>
<li>
<a href = "alpha_1.25.txt">alpha_1.25.txt</a>
values for ALPHA = 1.25.
</li>
<li>
<a href = "alpha_1.25.png">alpha_1.25.png</a>,
a plot.
</li>
<li>
<a href = "alpha_1.50.txt">alpha_1.50.txt</a>
values for ALPHA = 1.50.
</li>
<li>
<a href = "alpha_1.50.png">alpha_1.50.png</a>,
a plot.
</li>
<li>
<a href = "alpha_1.75.txt">alpha_1.75.txt</a>
values for ALPHA = 1.75.
</li>
<li>
<a href = "alpha_1.75.png">alpha_1.75.png</a>,
a plot.
</li>
<li>
<a href = "alpha_2.00.txt">alpha_2.00.txt</a>
values for ALPHA = 2.00.
</li>
<li>
<a href = "alpha_2.00.png">alpha_2.00.png</a>,
a plot.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>F_ALPHA</b> generates a 1/F^ALPHA noise sequence.
</li>
<li>
<b>R8_NORMAL_01</b> returns a unit pseudonormal R8.
</li>
<li>
<b>R8_TO_STRING</b> converts an R8 to a C++ string.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_PRINT_PART</b> prints "part" of an R8VEC.
</li>
<li>
<b>R8VEC_SFTB</b> computes a "slow" backward Fourier transform of real data.
</li>
<li>
<b>R8VEC_SFTF</b> computes a "slow" forward Fourier transform of real data.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 09 June 2010.
</i>
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