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FastFourierTransform.mo
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FastFourierTransform.mo
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within Modelica.Math;
package FastFourierTransform
"Library of functions for the Fast Fourier Transform (FFT)"
extends Modelica.Icons.Package;
import Modelica.Units.SI;
package Examples
"Examples demonstrating the usage of the Math.FastFourierTransform functions"
extends Modelica.Icons.ExamplesPackage;
model RealFFT1
"Example demonstrating the use of an FFT calculation during simulation (and storing both amplitudes and phases on file)"
import Modelica.Constants.pi;
extends Modelica.Icons.Example;
parameter SI.Frequency f_max = 4
"Maximum frequency of interest";
parameter SI.Frequency f_resolution = 0.2
"Frequency resolution";
parameter SI.Frequency f1 = 2 "Frequency of sine";
parameter SI.Frequency f2 = 3 "Frequency of cosine";
parameter String FFT_resultFileName = "RealFFT1_resultFFT.mat"
"File where FFT will be stored as [f,A,Phi], with f in {Hz] and A the amplitudes and Phi the phases in [rad]";
final parameter Integer nfi = max(1,min(integer(ceil(f_max/f_resolution))+1,nf))
"Number of frequency points of the interested frequency range (only up to f_max)";
final parameter SI.Frequency fi[nfi](each fixed=false)
"FFT frequencies of interested frequency points";
Real y(final start=0, final fixed=true)
"Signal from which FFT is computed";
final output Real Ai[nfi](each start=0, each fixed=true)
"FFT amplitudes of interested frequency points";
final output Real Phii[nfi](each start=0, each fixed=true)
"FFT phases of interested frequency points";
output Integer info(final start=0, final fixed=true)
"Information flag from FFT computation; = 0: FFT successfully computed";
protected
parameter Integer ns = realFFTsamplePoints(f_max, f_resolution, f_max_factor=5);
parameter SI.Frequency f_max_FFT = f_resolution*div(ns, 2)
"Maximum frequency used by FFT";
parameter Integer nf = div(ns,2) + 1 "Number of frequency points";
parameter SI.Time Ts = 1/(2*f_max_FFT) "Sample period";
parameter SI.Time T = (ns - 1)*Ts
"Simulation time for one FFT calculation";
Integer iTick(start=0, fixed=true);
Real y_buf[ns](start=vector([6.5; fill(0, ns - 1)]), each fixed=true);
initial equation
for i in 1:nfi loop
fi[i] = (i-1)*f_resolution;
end for;
algorithm
when sample(0,Ts) then
iTick :=pre(iTick) + 1;
y := 5 + 3*sin(2*pi*f1*time) + 1.5*cos(2*pi*f2*time);
if iTick >= 1 and iTick <= ns then
y_buf[iTick] := y;
end if;
if iTick == ns then
(info,Ai,Phii) := realFFT(y_buf, nfi);
Modelica.Math.FastFourierTransform.realFFTwriteToFile(time,FFT_resultFileName,f_max,Ai,Phii);
end if;
end when;
annotation (experiment(StopTime=6), preferredView="text",
Documentation(revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>",
info="<html>
<p>
In this example the signal y
</p>
<blockquote><pre>
y = 5 + 3*sin(2*pi*f1) + 1.5*cos(2*pi*f2)
</pre></blockquote>
<p>
is sampled and an FFT is computed from the sampled signal (default: f1 = 2 Hz, f2 = 3 Hz).
In the public part the FFT is stored up to f_max (internally in the protected part the FFT is stored up to 5*f_max).
With the default values for f_max (= 4 Hz) and f_resolution (= 0.2 Hz), the following results are achieved:
</p>
<blockquote><pre>
fi[0] = 0, Ai[0] = 5; // mean value of signal
fi[11] = 2, Ai[11] = 3; // frequency/amplitude of sine
fi[16] = 3, Ai[16] = 1.5; // frequency/amplitude of cosine
</pre></blockquote>
<p>
A plot of the resulting FFT is shown in the next image:
</p>
<blockquote>
<img src=\"modelica://Modelica/Resources/Images/Math/FastFourierTransform/FFT1.png\">
</blockquote>
<p>
Note, phases of small amplitudes (= smaller as 0.0001*maximalAmplitude) are explicitly set to zero, since the corresponding
\"phase\" is numerical noise (and would just be confusing).
Furthermore, note that the FFT phases are with respect to a cos(..) signal.
</p>
</html>"));
end RealFFT1;
model RealFFT2
"Example demonstrating the use of an FFT calculation during simulation (and storing only amplitudes on file)"
import Modelica.Constants.pi;
extends Modelica.Icons.Example;
parameter SI.Frequency f_max = 4
"Maximum frequency of interest";
parameter SI.Frequency f_resolution = 0.2
"Frequency resolution";
parameter SI.Frequency f1 = 2 "Frequency of sine";
parameter SI.Frequency f2 = 3 "Frequency of cosine";
parameter String FFT_resultFileName = "RealFFT2_resultFFT.mat"
"File where FFT will be stored as [f,A,Phi], with f in {Hz] and A the amplitues and Phi the phases in [rad]";
final parameter Integer nfi = max(1,min(integer(ceil(f_max/f_resolution))+1,nf))
"Number of frequency points of the interested frequency range (only up to f_max)";
final parameter SI.Frequency fi[nfi](each fixed=false)
"FFT frequencies of interested frequency points";
Real y(final start=0, final fixed=true)
"Signal from which FFT is computed";
final output Real Ai[nfi](each start=0, each fixed=true)
"FFT amplitudes of interested frequency points";
output Integer info(final start=0, final fixed=true)
"Information flag from FFT computation; = 0: FFT successfully computed";
protected
parameter Integer ns = realFFTsamplePoints(f_max, f_resolution, f_max_factor=5);
parameter SI.Frequency f_max_FFT = f_resolution*div(ns, 2)
"Maximum frequency used by FFT";
parameter Integer nf = div(ns,2) + 1 "Number of frequency points";
parameter SI.Time Ts = 1/(2*f_max_FFT) "Sample period";
parameter SI.Time T = (ns - 1)*Ts
"Simulation time for one FFT calculation";
Integer iTick(start=0, fixed=true);
Real y_buf[ns](start=vector([6.5; fill(0, ns - 1)]), each fixed=true);
initial equation
for i in 1:nfi loop
fi[i] = (i-1)*f_resolution;
end for;
algorithm
when sample(0,Ts) then
iTick :=pre(iTick) + 1;
y := 5 + 3*sin(2*pi*f1*time) + 1.5*cos(2*pi*f2*time);
if iTick >= 1 and iTick <= ns then
y_buf[iTick] := y;
end if;
if iTick == ns then
(info,Ai) := realFFT(y_buf, nfi);
Modelica.Math.FastFourierTransform.realFFTwriteToFile(time,FFT_resultFileName,f_max,Ai);
end if;
end when;
annotation (experiment(StopTime=6), preferredView="text",
Documentation(revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>",
info="<html>
<p>
This is the same example as <a href=\"modelica://Modelica.Math.FastFourierTransform.Examples.RealFFT1\">Examples.RealFFT1</a>
with the only difference that just the amplitudes of the FFT are stored on file (but not the phases).
</p>
</html>"));
end RealFFT2;
annotation (Documentation(revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>"));
end Examples;
function realFFTinfo
"Print information about real FFT for given f_max and f_resolution"
extends Modelica.Icons.Function;
import Modelica.Utilities.Streams.print;
input SI.Frequency f_max "Maximum frequency";
input SI.Frequency f_resolution "Frequency resolution";
input Integer f_max_factor(min=1)=5
"Maximum FFT frequency >= f_max*f_max_factor (sample frequency = 2*Maximum FFT Frequency)";
protected
Integer ns = realFFTsamplePoints(f_max, f_resolution, f_max_factor);
Integer nf = div(ns,2)+1;
SI.Frequency f_max_used = f_resolution*div(ns, 2);
SI.Frequency fs = f_max_used*2;
SI.Time Ts= 1/(2*f_max_used) "Sample period";
SI.Time T=(ns - 1)*Ts "Simulation time for FFT calculation";
Integer e2, e3, e5;
Boolean success;
algorithm
(success,e2,e3,e5) :=Internal.prime235Factorization(ns);
print("\n... Real FFT properties");
print(" Desired:");
print(" f_max = " + String(f_max) + " Hz");
print(" f_resolution = " + String(f_resolution) + " Hz");
print(" f_max_factor = " + String(f_max_factor));
print(" Calculated:");
print(" Number of sample points = " + String(ns) + " (= 2^"+String(e2)+"*3^"+String(e3)+"*5^"+String(e5)+")");
print(" Sampling frequency = " + String(fs) + " Hz (= " + String(f_resolution) + "*" + String(ns) + ")");
print(" Sampling period = " + String(Ts) + " s (= " + "1/" + String(fs) + ")");
print(" Maximum FFT frequency = " + String(f_max_used) + " Hz (= " + String(f_resolution) + "*" + String(ns) + "/2; "
+ "f={0," + String(f_resolution) + ","
+ String(2*f_resolution) + ",...,"
+ String(f_max_used) + "} Hz)");
print(" Number of frequency points = " + String(nf) + " (= " + String(ns) + "/2+1)");
print(" Simulation time = " + String(T) + " s");
annotation (Documentation(revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>",
info="<html>
<h4>Syntax</h4>
<blockquote><pre>
<strong>realFFTinfo</strong>(f_max, f_resolution, f_max_factor=5);
</pre></blockquote>
<h4>Description</h4>
<p>
From the maximum interested frequency f_max (in [Hz]) and the frequency resolution f_resolution (in [Hz]) the
function computes the key FFT data as used by the FFT blocks and prints them to the output window.
</p>
<h4>Example</h4>
<blockquote>
realFFTinfo(f_max=170, f_resolution=0.3)
</blockquote>
<p>
results in the following output:
</p>
<blockquote><pre>
... Real FFT properties
Desired:
f_max = 170 Hz
f_resolution = 0.3 Hz
f_max_factor = 5
Calculated:
Number of sample points = 5760 (= 2^7*3^2*5^1)
Sampling frequency = 1728 Hz (= 0.3*5760)
Sampling period = 0.000578704 s (= 1/1728)
Maximum FFT frequency = 864 Hz (= 0.3*5760/2; f={0,0.3,0.6,...,864} Hz)
Number of frequency points = 2881 (= 5760/2+1)
Simulation time = 3.33275 s
</pre></blockquote>
<h4>See also</h4>
<p>
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTsamplePoints\">realFFTsamplePoints</a>,
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFT\">realFFT</a>,
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTwriteToFile\">realFFTwriteToFile</a>
</p>
</html>"));
end realFFTinfo;
function realFFTsamplePoints "Return number of sample points for a real FFT"
extends Modelica.Icons.Function;
input SI.Frequency f_max "Maximum frequency of interest";
input SI.Frequency f_resolution "Frequency resolution";
input Integer f_max_factor(min=1)=5
"Maximum FFT frequency >= f_max*f_max_factor (sample frequency = 2*Maximum FFT Frequency)";
output Integer ns
"Number of sample points that can be expressed as ns = 2^i*3^j*5^k and ns is even";
protected
Integer ns1;
algorithm
// Check input arguments
assert(f_resolution > 0, "f_resolution > 0 required");
assert(f_max > f_resolution, "f_max > f_resolution required");
// Compute best ns according to f_max*f_max_factor and f_resolution = roundAgainstInfinity(2*f_max*f_max_factor/f_resolution)
ns1 :=2*integer(ceil(f_max*f_max_factor/f_resolution));
// If necessary, enlarge ns1 so that it is even and can be expressed as 2^i*3^j*5^k
ns :=if mod(ns1, 2) == 0 then ns1 else ns1 + 1;
while true loop
ns1 :=ns;
while mod(ns1,2) == 0 loop ns1 :=div(ns1, 2);end while;
while mod(ns1,3) == 0 loop ns1 :=div(ns1, 3);end while;
while mod(ns1,5) == 0 loop ns1 :=div(ns1, 5);end while;
if ns1 <= 1 then
break;
end if;
ns :=ns + 2;
end while;
annotation (Documentation(revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>",
info="<html>
<h4>Syntax</h4>
<blockquote><pre>
ns = <strong>realFFTsamplePoints</strong>(f_max, f_resolution, f_max_factor=5);
</pre></blockquote>
<h4>Description</h4>
<p>
From the maximum interested frequency f_max (in [Hz]) and the frequency resolution f_resolution (in [Hz]) the
function computes the number of sample points ns that is as small as possible and fulfills the following criteria:
</p>
<ul>
<li> Maximum FFT frequency ≥ f_max_factor*f_max (= the largest frequency value of the frequency vector).</li>
<li> Frequency axis resolution is f_resolution.</li>
<li> The number of sample points is expressed as 2^a*3^b*5^c
(and a,b,c are appropriate Integers).</li>
<li> The number of sample points is even.</li>
</ul>
<p>
Note, in the original publication about the efficient computation of FFT (Cooley and Tukey, 1965),
the number of sample points must be 2^a. However, all newer FFT algorithms do not have
this strong restriction and especially not the open source software
<a href=\"http://sourceforge.net/projects/kissfft/\">KissFFT</a> from Mark Borgerding
used in this function
</p>
<h4>References</h4>
<dl>
<dt>Mark Borgerding (2010):</dt>
<dd> <strong>KissFFT, version 1.3.0</strong>.
<a href=\"http://sourceforge.net/projects/kissfft/\">http://sourceforge.net/projects/kissfft/</a>.
<br>
</dd>
<dt>James W. Cooley, John W. Tukey (1965):</dt>
<dd> <strong>An algorithm for the machine calculation of complex Fourier series</strong>.
Math. Comput. 19: 297-301. doi:10.2307/2003354.
<br>
</dd>
<dt>Martin R. Kuhn, Martin Otter, Tim Giese (2015):</dt>
<dd> <strong>Model Based Specifications in Aircraft Systems Design</strong>.
Modelica 2015 Conference, Versailles, France,
pp. 491-500, Sept.23-25, 2015.
Download from:
<a href=\"http://www.ep.liu.se/ecp/118/053/ecp15118491.pdf\">http://www.ep.liu.se/ecp/118/053/ecp15118491.pdf</a>
</dd>
</dl>
<h4>Example</h4>
<blockquote>
ns = realFFTinfo(f_max=170, f_resolution=0.3)
</blockquote>
<p>
results in the following output:
</p>
<blockquote><pre>
ns = 5760
</pre></blockquote>
<h4>See also</h4>
<p>
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTinfo\">realFFTinfo</a>,
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFT\">realFFT</a>,
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTwriteToFile\">realFFTwriteToFile</a>
</p>
</html>"));
end realFFTsamplePoints;
function realFFT "Return amplitude and phase vectors for a real FFT"
extends Modelica.Icons.Function;
input Real u[:]
"Signal for which FFT shall be computed (size(nu,1) MUST be EVEN and should be an integer multiple of 2,3,5, that is size(nu,1) = 2^a*3^b*5^c, with a,b,c Integer >= 0)";
input Integer nfi
"Number of frequency points that shall be returned in amplitudes and phases (typically: nfi = max(1,min(integer(ceil(f_max/f_resolution))+1,nf))); the maximal possible value is nfi=div(size(u,1),2)+1)";
output Integer info
"Information flag (0: FFT computed, 1: nu is not even, 3: another error)";
output Real amplitudes[nfi] "Amplitudes of FFT";
output Real phases[nfi] "Phases of FFT in [deg]";
protected
Integer nu = size(u,1);
Integer nf = div(size(u,1),2)+1;
Real u_DC;
Real u2[size(u,1)];
Real A[div(size(u,1),2)+1];
Real Phi[div(size(u,1),2)+1];
Real Aeps;
algorithm
assert(nfi > 0 and nfi <= div(size(u,1),2)+1, "Argument nfi is out of range");
u_DC :=sum(u)/nu;
u2 :=u - fill(u_DC, nu);
(info, A, Phi) :=Internal.rawRealFFT(u2);
amplitudes :=A[1:nfi];
phases :=Modelica.Units.Conversions.to_deg(Phi[1:nfi]);
Aeps :=0.0001*max(amplitudes);
amplitudes[1] :=u_DC;
phases[1] := 0.0;
// Set phases[i] explicitly to zero, if the correspondion amplitude is < Aeps (= 0.0001*Amax; = numerical noise).
for i in 2:nfi loop
if amplitudes[i] < Aeps then
phases[i] :=0.0;
end if;
end for;
annotation (Documentation(revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>",
info="<html>
<h4>Syntax</h4>
<blockquote><pre>
(info, amplitudes, phases) = <strong>realFFT</strong>(u);
</pre></blockquote>
<h4>Description</h4>
<p>
The input argument of this function is a Real vector u. size(u,1) <strong>must</strong> be even. An efficient computation
is performed, if size(u,1) = 2^a*3^b*5^c (a,b,c Integer ≥ 0).
An appropriate length of vector u can be computed with function
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTsamplePoints\">realFFTsamplePoints</a>.
Function <strong>realFFT</strong> computes a real FFT (Fast Fourier Transform) of u and returns the result
in form of the outputs amplitudes and phases. Argument info provides additional information:
</p>
<blockquote><pre>
info = 0: Successful FFT computation.
info = 1: size(u,1) is not even.
info = 3: Another error.
</pre></blockquote>
<p>
Note, in the original publication about the efficient computation of FFT (Cooley and Tukey, 1965),
the number of sample points must be 2^a. However, all newer FFT algorithms do not have
this strong restriction and especially not the open source software
<a href=\"http://sourceforge.net/projects/kissfft/\">KissFFT</a> from Mark Borgerding
used in this function.
</p>
<p>
The function returns the FFT such that amplitudes[1] is the mean value of u (= sum(u)/size(u,1)), and
amplitudes[i] is the amplitude of a sine-function at the i-th frequency.
</p>
<h4>References</h4>
<dl>
<dt>Mark Borgerding (2010):</dt>
<dd> <strong>KissFFT, version 1.3.0</strong>.
<a href=\"http://sourceforge.net/projects/kissfft/\">http://sourceforge.net/projects/kissfft/</a>.
<br>
</dd>
<dt>James W. Cooley, John W. Tukey (1965):</dt>
<dd> <strong>An algorithm for the machine calculation of complex Fourier series</strong>.
Math. Comput. 19: 297-301. doi:10.2307/2003354.
<br>
</dd>
<dt>Martin R. Kuhn, Martin Otter, Tim Giese (2015):</dt>
<dd> <strong>Model Based Specifications in Aircraft Systems Design</strong>.
Modelica 2015 Conference, Versailles, France,
pp. 491-500, Sept.23-25, 2015.
Download from:
<a href=\"http://www.ep.liu.se/ecp/118/053/ecp15118491.pdf\">http://www.ep.liu.se/ecp/118/053/ecp15118491.pdf</a>
</dd>
</dl>
<h4>Example</h4>
<blockquote>
(info, A) = realFFT({0,0.1,0.2,0.4,0.5, 0.6})
</blockquote>
<p>
See also <a href=\"modelica://Modelica.Math.FastFourierTransform.Examples.RealFFT1\">Examples.RealFFT1</a>
which is a complete example where an FFT is computed during simulation and stored on file.
</p>
<h4>See also</h4>
<p>
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTinfo\">realFFTinfo</a>,
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTsamplePoints\">realFFTsamplePoints</a>,
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTwriteToFile\">realFFTwriteToFile</a>
</p>
</html>"));
end realFFT;
function realFFTwriteToFile "Write real FFT computation to file"
extends Modelica.Icons.Function;
import Modelica.Utilities.Streams.print;
import Modelica.Units.Conversions.to_deg;
import Modelica.Utilities.Streams.writeRealMatrix;
input Real t_computed "Time instant at which the FFT was computed";
input String fileName
"File where FFT shall be stored (if it exists, it is deleted and then re-created)";
input SI.Frequency f_max "Maximum frequency";
input Real amplitudes[:] "Amplitudes of FFT";
input Real phases[:] = fill(0.0,0)
"Phases of FFT (either provide no argument, or a vector with the same length as amplitudes)";
input String format = "4"
"MATLAB MAT-file version: \"4\" -> v4, \"6\" -> v6, \"7\" -> v7"
annotation(choices(choice="4" "MATLAB v4 MAT-file",
choice="6" "MATLAB v6 MAT-file",
choice="7" "MATLAB v7 MAT-file"));
output Boolean success "true if successful";
protected
Integer nA = size(amplitudes,1);
Real fA[3*size(amplitudes,1),if size(phases,1)==0 then 2 else 3];
Real f;
algorithm
assert(size(phases,1) == 0 or size(phases,1) == size(amplitudes,1), "Input vector phases has the wrong size");
// Remove file, if it exists
Modelica.Utilities.Files.removeFile(fileName);
// Construct output matrix
if size(phases,1) == size(amplitudes,1) then
for i in 1:nA loop
f := f_max*(i-1)/(nA-1);
fA[1+3*(i-1):1+3*(i-1)+2,:] :=[f,0,0; f,amplitudes[i],phases[i]; f,0,0];
end for;
else
for i in 1:nA loop
f := f_max*(i-1)/(nA-1);
fA[1+3*(i-1):1+3*(i-1)+2,:] :=[f,0; f,amplitudes[i]; f,0];
end for;
end if;
// Write matrix on file and print message
success := writeRealMatrix(fileName, "FFT", fA, format=format);
if success then
print("... FFT result computed at time = " + String(t_computed) + " s stored on file: " + Modelica.Utilities.Files.fullPathName(fileName));
end if;
annotation (Documentation(info="<html>
<h4>Syntax</h4>
<blockquote><pre>
success = <strong>realFFTwriteToFile</strong>(t_computed, fileName, f_max, amplitudes, phases, format);
</pre></blockquote>
<h4>Description</h4>
<p>
This functions stores the result of an FFT computation on file, so that it can
be easily plotted. <strong>amplitudes</strong> and <strong>phases</strong> are the vectors that hold the
amplitudes and phases values of an FFT computation. If the size of the phases vector is zero,
no phases will be stored on file. Otherwise, phases must have the same dimension as the amplitudes
vector. The frequency vector f is constructed within the function from the dimension of the
amplitudes vector and the information that amplitudes[end] is at frequency <strong>f_max</strong>.
The <strong>format</strong> argument defines the file format (for details see
<a href=\"modelica://Modelica.Utilities.Streams.writeRealMatrix\">writeRealMatrix</a>).
Argument <strong>t_computed</strong> is the actual time instant when the FFT was computed.
It is used in the print message after the result was stored on file.
</p>
<p>
The matrix on file has the following structure:
</p>
<ul>
<li> First column: Equidistant frequency vector f in Hz from 0 Hz ... f_max Hz.</li>
<li> Second column: Amplitudes[:]</li>
<li> Optional third column: Phases[:]</li>
</ul>
<h4>Example</h4>
<p>
See detailed example model:
<a href=\"modelica://Modelica.Math.FastFourierTransform.Examples.RealFFT1\">Examples.RealFFT1</a>.
</p>
<h4>See also</h4>
<p>
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTinfo\">realFFTinfo</a>,
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFTsamplePoints\">realFFTsamplePoints</a>,
<a href=\"modelica://Modelica.Math.FastFourierTransform.realFFT\">realFFT</a>
</p>
</html>"));
end realFFTwriteToFile;
package Internal
"Internal library that should not be used directly by a user"
extends Modelica.Icons.InternalPackage;
pure function rawRealFFT "Compute raw Fast Fourier Transform for real signal vector"
extends Modelica.Icons.Function;
input Real u[:]
"Signal for which FFT shall be computed (size(nu,1) MUST be EVEN and should be an integer multiple of 2,3,5, that is size(nu,1) = 2^a*3^b*5^c, with a,b,c Integer >= 0)";
output Integer info
"Information flag (0: FFT computed, 1: nu is not even, 2: nwork is wrong, 3: another error)";
output Real amplitudes[div(size(u,1),2)+1] "Amplitudes of FFT";
output Real phases[ div(size(u,1),2)+1] "Phases of FFT";
protected
Real work[3*size(u,1) + 2*(div(size(u,1),2)+1)];
external "C" info = ModelicaFFT_kiss_fftr(u, size(u,1), work, size(work,1), amplitudes, phases)
annotation(Include="#include \"ModelicaFFT.h\"", Library="ModelicaExternalC");
annotation (Documentation(revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>",
info="<html>
<h4>Syntax</h4>
<blockquote><p>
(info, amplitudes, phases) = <strong>rawRealFFT</strong>(u);
</p></blockquote>
<h4>Description</h4>
<p>
Raw interface to a function of the Kiss_FFT package to compute the FFT of a real, sampled signal.
The input argument of this function is a Real vector u. size(u,1) <strong>must</strong> be even. An efficient computation
is performed, if size(u,1) = 2^a*3^b*5^c (a,b,c Integer ≥ 0).
The function computes a real FFT (Fast Fourier Transform) of u and returns the result
in form of the outputs amplitudes and phases. Argument info provides additional information:
</p>
<blockquote><pre>
info = 0: Successful FFT computation.
info = 1: size(u,1) is not even.
info = 2: size(work,1) is not correct (= a protected utility array).
info = 3: Another error.
</pre></blockquote>
<p>
Note, in the original publication about the efficient computation of FFT (Cooley and Tukey, 1965),
the number of sample points must be 2^a. However, all newer FFT algorithms do not have
this strong restriction and especially not the open source software
<a href=\"http://sourceforge.net/projects/kissfft/\">KissFFT</a> from Mark Borgerding
used in this function.
</p>
<h4>References</h4>
<dl>
<dt>Mark Borgerding (2010):</dt>
<dd> <strong>KissFFT, version 1.3.0</strong>.
<a href=\"http://sourceforge.net/projects/kissfft/\">http://sourceforge.net/projects/kissfft/</a>.
<br>
</dd>
<dt>James W. Cooley, John W. Tukey (1965):</dt>
<dd> <strong>An algorithm for the machine calculation of complex Fourier series</strong>.
Math. Comput. 19: 297-301. doi:10.2307/2003354.
<br>
</dd>
<dt>Martin R. Kuhn, Martin Otter, Tim Giese (2015):</dt>
<dd> <strong>Model Based Specifications in Aircraft Systems Design</strong>.
Modelica 2015 Conference, Versailles, France,
pp. 491-500, Sept.23-25, 2015.
Download from:
<a href=\"http://www.ep.liu.se/ecp/118/053/ecp15118491.pdf\">http://www.ep.liu.se/ecp/118/053/ecp15118491.pdf</a>
</dd>
</dl>
<h4>Example</h4>
<blockquote>
(info, A, phases) = realFFT({0,0.1,0.2,0.4,0.5, 0.6})
</blockquote>
</html>"));
end rawRealFFT;
function prime235Factorization "Factorization of an integer in prime numbers 2,3,5"
extends Modelica.Icons.Function;
input Integer n;
output Boolean success "= true, if factorization in 2,3,5 is possible";
output Integer e2 "n = 2^e2*3^e3*5^e5";
output Integer e3 "n = 2^e2*3^e3*5^e5";
output Integer e5 "n = 2^e2*3^e3*5^e5";
protected
Integer ns1 = n;
algorithm
e2:=0;
e3:=0;
e5:=0;
while mod(ns1,2) == 0 loop
ns1 :=div(ns1, 2);
e2 :=e2 + 1;
end while;
while mod(ns1,3) == 0 loop
ns1 :=div(ns1, 3);
e3 := e3+1;
end while;
while mod(ns1,5) == 0 loop
ns1 :=div(ns1, 5);
e5 :=e5 + 1;
end while;
success :=ns1 <= 1;
annotation (Documentation(revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>",
info="<html>
<h4>Syntax</h4>
<blockquote><p>
(success, e2, e3, e5) = <strong>prime235Factorization</strong>(n);
</p></blockquote>
<h4>Description</h4>
<p>
Compute the factorization of input Integer n in prime numbers 2, 3, and 5. If this is possible, success = true and
e2 is the number of prime numbers2, e3 the number of prime numbers 3 and e5 the number of prime numbers 5.
If this is not possible, success = false, and e2, e3, e5 are dummy values.
</p>
<h4>Example</h4>
<blockquote><pre>
(success, e2, e3, e5) = prime235Factorization(60) // success=true, e2=2, e3=1, e5=1 (= 2^2*3^1*5^1)
(success, e2, e3, e5) = prime235Factorization(7) // success=false
</pre></blockquote>
</html>"));
end prime235Factorization;
annotation (Documentation(revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>"));
end Internal;
annotation (Documentation(info="<html>
<p>
This package provides functions to compute the Fast Fourier Transform (FFT).</p>
<p>
For an example see <a href=\"modelica://Modelica.Math.FastFourierTransform.Examples.RealFFT1\">Examples.RealFFT1</a>
where the following signal is computed during simulation
</p>
<blockquote><pre>
y = 5 + 3*sin(2*pi*2) + 1.5*cos(2*pi*3)
</pre></blockquote>
<p>
the continuous-time signal y is sampled and the FFT is computed with a call to realFFT(f_max=4, f_resolution=0.2),
resulting in:
</p>
<blockquote>
<img src=\"modelica://Modelica/Resources/Images/Math/FastFourierTransform/FFT1.png\">
</blockquote>
<h4>References</h4>
<dl>
<dt>Mark Borgerding (2010):</dt>
<dd> <strong>KissFFT, version 1.3.0</strong>.
<a href=\"http://sourceforge.net/projects/kissfft/\">http://sourceforge.net/projects/kissfft/</a>.
<br>
</dd>
<dt>James W. Cooley, John W. Tukey (1965):</dt>
<dd> <strong>An algorithm for the machine calculation of complex Fourier series</strong>.
Math. Comput. 19: 297-301. doi:10.2307/2003354.
<br>
</dd>
<dt>Martin R. Kuhn, Martin Otter, Tim Giese (2015):</dt>
<dd> <strong>Model Based Specifications in Aircraft Systems Design</strong>.
Modelica 2015 Conference, Versailles, France,
pp. 491-500, Sept.23-25, 2015.
Download from:
<a href=\"http://www.ep.liu.se/ecp/118/053/ecp15118491.pdf\">http://www.ep.liu.se/ecp/118/053/ecp15118491.pdf</a>
</dd>
</dl>
</html>",
revisions="<html>
<table border=\"1\" cellspacing=\"0\" cellpadding=\"2\">
<tr><th>Date</th> <th align=\"left\">Description</th></tr>
<tr><td> Nov. 29, 2015 </td>
<td>
Initial version implemented by
Martin R. Kuhn and Martin Otter
(<a href=\"http://www.dlr.de/rmc/sr/en\">DLR Institute of System Dynamics and Control</a>.</td></tr>
</table>
</html>"), Icon(graphics={
Line(points={{-60,20},{-60,-80}}, color={95,95,95}),
Line(points={{-20,60},{-20,-80}}, color={95,95,95}),
Line(points={{20,40},{20,-80}}, color={95,95,95}),
Line(points={{60,-20},{60,-80}}, color={95,95,95})}));
end FastFourierTransform;