How Can You Implement the Numerical Fourier Transform in MATLAB or FORTRAN?

[zetafunction]

hi, i would need some info on how can implement in MATLAB or FORTRAN (g90) the Numerical evaluation of the integral

$$\int_{-\infty}^{\infty}dxf(x)exp(iux)$$

and the evaluation of the ivnerse Fourier transform

$$\int_{-\infty}^{\infty}du\frac{F(u))}{\int_{-\infty}^{\infty}g(x)exp(iux)dx}$$

the idea is to implement some numerical tool to evaluate Fourier transform integral, in order to apply them to solve Wiener-Hopf equation.

 

[junglebeast]

It's not possible to implement the complete Fourier transform numerically because it has infinite domain and range. You need to use the Discrete Fourier Transform instead.

 

[quicknote]

Sure, I can provide some information on how to implement the numerical Fourier transform in MATLAB or FORTRAN. The Fourier transform is a mathematical operation that decomposes a function into its frequency components, and it is commonly used in signal processing and image analysis.

To implement the numerical Fourier transform in MATLAB, you can use the "fft" function. This function takes the discrete Fourier transform of a vector or matrix and returns a vector or matrix of the same size. For example, if your function is defined as "f(x)" and you want to evaluate the Fourier transform at a specific frequency "u", you can use the following code:

F = fft(f); % compute the Fourier transform of f
u = 2*pi/linspace(-N/2,N/2,N); % define the frequency vector
F_u = F .* exp(-1i*u*x(1)); % multiply the Fourier transform by the exponential term
f_u = ifft(F_u); % compute the inverse Fourier transform to get f(u)

The "fft" function in MATLAB uses the fast Fourier transform (FFT) algorithm, which is a computationally efficient method for calculating the Fourier transform. It requires the number of data points "N" to be a power of 2, so you may need to pad your data with zeros if it is not already a power of 2.

In FORTRAN, you can use the "dfft" function in the FFTPACK library to perform the Fourier transform. This function also requires the data to be a power of 2, so you may need to pad your data with zeros before using the function. The code would look something like this:

call dfftf(N, f, wsave) % compute the Fourier transform of f
u = 2*pi/linspace(-N/2,N/2,N) % define the frequency vector
do i = 1, N
F_u(i) = f(i) * exp(-1i*u*x(1)) % multiply the Fourier transform by the exponential term
end do
call dfftb(N, F_u, wsave) % compute the inverse Fourier transform to get f(u)

The "dfftf" and "dfftb" functions in FORTRAN use the FFTPACK library to perform the FFT algorithm. You can find more information on how to use these functions in the FFTPACK documentation.

To solve the Wiener-Hopf equation, you can use the numerical Fourier transform to evaluate the