9.4 Example Using the FFT Server Class

FILENAME: example3.cpp

Program:

/*

 * Example program using the Complex FFT server class.
 */

// Include the header file for complex FFT server class:
#include <rw/cfft.h>
// Header files for complex vectors:
#include <rw/math/mathvec.h>
#include <math.h>
#include <iostream.h>

main()
{
    // Set series length:
    unsigned npts = 12;

    // Allocate a server:
    DComplexFFTServer server;

/*
 * Create two series, one a cosine series, the
 * other a sine series. This is done by first
 * using a constructor for an RWMathVec<double> with
 * increasing elements, taking the cosine (or
 * sine) of this vector, then constructing a
 * complex vector from that.
 */

    // One cycle:
    RWMathVec<DComplex> a = 
       cos( RWMathVec<double>(npts,0,2.0*M_PI/npts) );

    // Two cycles:
    RWMathVec<DComplex> a2 = 
       sin( RWMathVec<double>(npts,0,4.0*M_PI/npts) );

/*
 * Calculate the superposition of the two. Note that we
 * are adding two complex vectors here with one
 * simple statement.
 */

   RWMathVec<DComplex> asum = a + a2;

   // Output the vectors:
   cout << "a:\n"    << a    << "\n";
   cout << "a2:\n"   << a2   << "\n";
   cout << "asum:\n" << asum << "\n";

/*
 * Print out the transforms, normalized by the 
 * number of points.
 */

   cout << "\nTransform of a:\n";
   cout << server.fourier(a)/DComplex(npts);

   cout << "\nTransform of a2:\n";
   cout << server.fourier(a2)/DComplex(npts);

   RWMathVec<DComplex> asumFFT = server.fourier(asum);

   cout << "\nTransform of asum:\n";
   cout << asumFFT/DComplex(npts);

/*
 * Check Parseval's Theorem. First, calculate the
 * variance of the series asum, using the global
 * function variance().
 */
 
   cout << "\nOriginal Variance: "
     << variance(asum) << "\n";

/*
 * Next, calculate the spectral variance of the Fourier
 * transformed series, using the global function
 * spectralVariance().
 */

   double var = spectralVariance(asumFFT/DComplex(npts));

   cout << "Spectral Variance:  " << var << "\n\n";

/*
 * Print out the normalized back transform of
 * asumFFT; this should equal the original asum.
 */

   cout << "\nBack Transform of asum:\n"
           <<server.ifourier(asumFFT)/DComplex(npts);
}

Sample Input:

None required.

Sample Output:

See file example3.out in the examples directory for the complete sample output. Note that the exact numbers depend on machine precision.

a:
[
( 1, 0) ( 0.866, 0) ( 0.5, 0) ( 0, 0) ...
]

a2:
[
( 0, 0) ( 0.866, 0) ( 0.866, 0) ( 0, 0) ...
]

asum:
[
( 1, 0) ( 1.732, 0) ( 1.366, 0) ( 0, 0) ...
]

Transform of a:
[
( 0, 0) ( 0.5, 0) ( 0, 0) ...
]

Transform of a2:
[
( 0, 0) ( 0, 0) ( 0, -0.5) ...

Transform of asum:
[
( 0, 0) ( 0.5, 0) ( 0, -0.5) ...
]

Original variance:  1
Spectral variance:  1

Back transform of asum:
[
( 1, 0) ( 1.732, 0) ( 1.366, 0) ( 0, 0) ...
]