Step by Step solution about How to resolve the algorithm Fast Fourier transform step by step in the Julia programming language

This code defines a Julia function, fft, which computes the Fast Fourier transform (FFT) of a complex or real-valued array. To do so, it uses a recursive algorithm to compute the FFT in place. First, we ensure that the input array is of type Array{Complex{Float64},1}, i.e., an array of complex numbers with 64-bit floating-point precision and one dimension. If the input array is not of the correct type, we convert it using the convert function.

Then, we initialize an array y of the correct size to store the complex fourier coefficients of the array. We also initialize the array w of weights used in the FFT algorithm.

Next, we iterate over the elements of the array a, and for each element, we compute the corresponding complex fourier coefficient and store it in the array y.

Finally, we return the array y containing the complex fourier coefficients of the array a.

using FFTW # or using DSP

fft([1,1,1,1,0,0,0,0])


8-element Array{Complex{Float64},1}:
      4.0+0.0im
  1.0-2.41421im
      0.0+0.0im
 1.0-0.414214im
      0.0+0.0im
 1.0+0.414214im
      0.0+0.0im
  1.0+2.41421im


function fft(a)
    y1 = Any[]; y2 = Any[]
    n = length(a)
    if n ==1 return a end
    wn(n) = exp(-2*π*im/n)
    y_even = fft(a[1:2:end])
    y_odd = fft(a[2:2:end])
    w = 1
    for k in 1:Int(n/2)
        push!(y1, y_even[k] + w*y_odd[k])
        push!(y2, y_even[k] - w*y_odd[k])
        w = w*wn(n)
    end
    return vcat(y1,y2)
end