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Showing content from https://github.com/JuliaArrays/FFTViews.jl below:

JuliaArrays/FFTViews.jl: Julia package for fast fourier transforms and periodic views

A package for simplifying operations that involve Fourier transforms. An FFTView of an array uses periodic boundary conditions for indexing, and shifts all indices of the array downward by 1.

Let's create a random signal:

julia> using FFTViews

julia> a = rand(8)
8-element Array{Float64,1}:
 0.720657
 0.42337
 0.207867
 0.959567
 0.371366
 0.907781
 0.852526
 0.689934

Now let's take its Fourier transform, and wrap the result as an FFTView:

julia> afft = fft(a)
8-element Array{Complex{Float64},1}:
   5.13307+0.0im
 -0.183898+0.796529im
   0.03163+0.31835im
   0.88248-0.492787im
 -0.828236+0.0im
   0.88248+0.492787im
   0.03163-0.31835im
 -0.183898-0.796529im

julia> v = FFTView(afft)
FFTViews.FFTView{Complex{Float64},1,Array{Complex{Float64},1}} with indices FFTViews.URange(0,7):
   5.13307+0.0im
 -0.183898+0.796529im
   0.03163+0.31835im
   0.88248-0.492787im
 -0.828236+0.0im
   0.88248+0.492787im
   0.03163-0.31835im
 -0.183898-0.796529im

Now we can easily look at the zero-frequency bin:

julia> v[0]
5.133068739504999 + 0.0im

julia> sum(a)
5.133068739504998

or negative as well as positive frequencies:

julia> v[-4:3]
8-element Array{Complex{Float64},1}:
 -0.828236+0.0im
   0.88248+0.492787im
   0.03163-0.31835im
 -0.183898-0.796529im
   5.13307+0.0im
 -0.183898+0.796529im
   0.03163+0.31835im
   0.88248-0.492787im

Perhaps even more interestingly, one can also simplify the process of convolution. Let's create a "delta-function" signal:

julia> b = zeros(8); b[3] = 1; b  # the signal
8-element Array{Float64,1}:
 0.0
 0.0
 1.0
 0.0
 0.0
 0.0
 0.0
 0.0

and then create the kernel using an FFTView:

julia> kernel = FFTView(zeros(8))
FFTViews.FFTView{Float64,1,Array{Float64,1}} with indices FFTViews.URange(0,7):
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0

julia> kernel[-1:1] = rand(3)
3-element Array{Float64,1}:
 0.16202
 0.446872
 0.649135

julia> kernel
FFTViews.FFTView{Float64,1,Array{Float64,1}} with indices FFTViews.URange(0,7):
 0.446872
 0.649135
 0.0
 0.0
 0.0
 0.0
 0.0
 0.16202

Now compute the convolution via the FFT:

julia> real(ifft(fft(b).*fft(kernel)))
8-element Array{Float64,1}:
  0.0
  0.16202
  0.446872
  0.649135
  0.0
 -5.55112e-17
  0.0
 -6.93889e-17

or alternatively

julia> irfft(rfft(b).*rfft(kernel),8)
8-element Array{Float64,1}:
  0.0
  0.16202
  0.446872
  0.649135
  0.0
 -2.77556e-17
  0.0
 -5.55112e-17

This simplifies the process of remembering how to pack your kernel.

In Julia, almost all other view types are composable: you can make a ReshapedArray of a SubArray of a StaticArray of a .... In contrast, FFTViews are not safe when placed inside other containers. The reason is that the *fft methods are specialized for FFTViews, and strip off the outer container; this does not happen if you wrap an FFTView inside of some other array type. If you do wrap FFTViews, you might see strange off-by-1 bugs due to the FFTView translating the indices.

Another way of saying the same thing is the following: for a general vector x, its FFT is defined as

Here x[n] is defined with periodic boundary conditions, so that if the indices of x are not naturally from 1 to N, this formula still holds.

However, if y = FFTView(x), then in terms of y we have

which is shifted by 1. Since FFTViews use a different definition of the FFT compared to all other array types, they need to be used with caution. It's recommended that the FFTView wrapper be applied only for the process of setting up or analyzing the result of the transform; for all other operations, pass the parent array (obtainable from parent(y) or just by reference to x itself).

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