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Convolution of two probability distributions using FFT

  1. Nov 8, 2012 #1
    I've been trying to code an algorithm to compute the convolution of two probability distributions. using the FFT. This relies on the "convolution theorem":

    (p*q)[z] = FFT^{-1}(FFT(p) \cdot FFT(q))

    However, when I test it using the distributions

    p={0.1, 0.2, 0.3, 0.4}
    q={0.4, 0.3, 0.2, 0.1}

    the result that's output - {0.24, 0.22, 0.24, 0.3} - doesn't correspond to any definition or generalised definition of convolution that I can find.

    (I'm fairly sure that one of the 0.24s, for instance, results from p[0]q[2] + p[1]q[1] + p[2]q[0] + p[3]q[3] being computed and divided by n)

    Can someone well-versed in FFTs (and/or convolutions) shed any light on what might be going on? I enclose the source code, which uses C++ and the FFTW package, below:

    Code (Text):

    void convolution_of_two_probability_distributions_with_fft(long double* p, long double* q, long double* p_star_q, unsigned long int n)
    {
        fftwl_plan fpcp;
        fftwl_plan fpcq;
        fftwl_plan f_reverse;
        unsigned long int ulx;

        fftwl_complex* transformed_p = (fftwl_complex*)fftwl_malloc(sizeof(fftwl_complex) * n);
        fftwl_complex* transformed_q = (fftwl_complex*)fftwl_malloc(sizeof(fftwl_complex) * n);
        fftwl_complex* transformed_p_dot_transformed_q = (fftwl_complex*)fftwl_malloc(sizeof(fftwl_complex) * n);

        fpcp = fftwl_plan_dft_r2c_1d(n, p, transformed_p, FFTW_ESTIMATE);
        fpcq = fftwl_plan_dft_r2c_1d(n, q, transformed_q, FFTW_ESTIMATE);
        f_reverse = fftwl_plan_dft_c2r_1d(n, transformed_p_dot_transformed_q, p_star_q, FFTW_ESTIMATE);

        fftwl_execute(fpcp);
        fftwl_execute(fpcq);

        for (unsigned long int ulx=0; ulx < n; ulx++)
        {
            //transformed_p_dot_transformed_q[ulx] = transformed_p[ulx] * transformed_q[ulx];
            //There's no overloaded * for fftwl_complex.
            transformed_p_dot_transformed_q[ulx][0] = (transformed_p[ulx][0] * transformed_q[ulx][0]) - (transformed_p[ulx][1] * transformed_q[ulx][1]);
            transformed_p_dot_transformed_q[ulx][1] = (transformed_p[ulx][0] * transformed_q[ulx][1]) + (transformed_p[ulx][1] * transformed_q[ulx][0]);
        }

        fftwl_execute(f_reverse);

        for (ulx=0; ulx < n; ulx++)
        {
            p_star_q[ulx] /= n;
        }

        fftwl_destroy_plan(fpcp);
        fftwl_destroy_plan(fpcq);
        fftwl_destroy_plan(f_reverse);

        fftwl_free(transformed_p);
        fftwl_free(transformed_q);
        fftwl_free(transformed_p_dot_transformed_q);
    }
     
     
    Last edited: Nov 8, 2012
  2. jcsd
  3. Nov 9, 2012 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    Perhaps it's "circular convolution"

    [itex] (p*q)_n = \sum_{m=0}^3 { p_m q_{n-m} } [/itex]

    e.g.
    [itex] (p*q)_0 = \sum_{m = 0}^3 {p_m} q_{0-m} [/itex]
    [itex] = p_0 q_0 + p_1 q_{-1} + p_2 q_{-2} + p_3 q_{-3} [/itex]

    Where we regard [itex] q_{-1} = q_3 , q_{-2}= q_2 [/itex] etc

    [itex] = (0.1)(0.4) + (0.2)(0.1) + (0.3)(0.2) + (0.4)(0.3) = 0.24 [/itex]
     
  4. Nov 9, 2012 #3
    I think you're right - this is consistent with the existing results and with another pair of distributions I've tested it with. Thanks for that!
     
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