ffts(A [, dims])

Result is semantically equivalent to fftshift(fft(A, dims), dims) However, the shift is done with ShiftedArrays and therefore doesn't allocate memory.

See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

ffts!(A [, dims])

Result is semantically equivalent to fftshift(fft!(A, dims), dims). A is in-place modified. However, the shift is done with ShiftedArrays and therefore doesn't allocate memory.

See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

iffts(A [, dims])

Result is semantically equivalent to ifft(ifftshift(A, dims), dims). A is in-place modified. However, the shift is done with ShiftedArrays and therefore doesn't allocate memory.

See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

rffts(A [, dims])

Calculates a rfft(A, dims) and then shift the frequencies to the center. dims[1] is not shifted, because there is no negative and positive frequency. The shift is done with ShiftedArrays and therefore doesn't allocate memory.

See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

irffts(A, d, [, dims])

Calculates a irfft(A, d, dims) and then shift the frequencies back to the corner. dims[1] is not shifted, because there is no negative and positive frequency. The shift is done with ShiftedArrays and therefore doesn't allocate memory.

See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

fftshift_view(A [, dims])

Result is semantically equivalent to fftshift(A, dims) but returns a view instead.

ifftshift_view(A [, dims])

Result is semantically equivalent to fftshift(A, dims) but returns a view instead.

rfftshift_view(A, dims)

Shifts the frequencies to the center expect for dims[1] because there os no negative and positive frequency.

irfftshift_view(A, dims)

Shifts the frequencies back to the corner except for dims[1] because there os no negative and positive frequency.

ft(A [, dims])

Digital Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(fft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with both coordinate systems in real and Fourier-space being centered at position CtrFT == size÷2+1

The following identities are true:

julia> sz = (5,5)
(5, 5)

julia> ft(ones(sz)) ≈ prod(sz) .* δ(sz)
true

julia> ft(δ(sz)) ≈ ones(sz)
true

See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

ift(A [, dims])

Digital inverse Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(ifft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with both coordinate systems in real and Fourier-space being centered at position CtrFT == size÷2+1

The following identities are true:

julia> sz = (5,6,7)
(5, 6, 7)

julia> ift(ones(sz)) ≈ δ(sz)
true

julia> ift(δ(sz)) ≈ ones(sz) ./ prod(sz)
true

See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

rft(A [, dims])

Digital real-valued Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(rfft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with the coordinate systems in real space centered at CtrFT == size÷2+1 and in (half) Fourier-space being centered at CtrRFT == setindex(size÷2 +1,1,1).

The following identities are true:

julia> sz = (6,6)
(6, 6)

julia> rft(δ(sz)) ≈ ones(rft_size(sz))
true

See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

irft(A, d, [, dims])

Digital real-valued inverse Fourier-transformation centered in both spaces. The result is semantically equivalent to fftshift(irfft(ifftshift(A, dims), dims), dims) This is a digital Fourier transformation with the coordinate systems in real space centered at CtrFT == size÷2+1 and in (half) Fourier-space being centered at CtrRFT == setindex(size÷2 +1,1,1). Note that the size d of the first transform direction [1] is a required argument.

The following identities are true:

julia> sz = (6,6)
(6, 6)

julia> irft(ones(rft_size(sz)),sz[1]) ≈ δ(sz)
true

See also: ft, ift, rft, irft, ffts, iffts, ffts!, rffts, irffts,

fftpos(L, N)

Construct a range from -L/2 to L/2. However, we ensure that everything is centered around the center in a way that a FFT interpretes it correctly. For odd sequences it is indeed in the real center. For even sequences the center is at N/2 + 1.

Examples

julia> collect(fftpos(1, 4))
4-element Array{Float64,1}:
 -0.5
 -0.25
  0.0
  0.25
julia> collect(fftpos(1, 5))
5-element Array{Float64,1}:
 -0.5
 -0.25
  0.0
  0.25
  0.5

fft_center(x)

Returns the center of a size in Fourier sense and Julia 1-based indices.

rft_size(sz::NTuple{Int})

Returns the size of an rft or rfft performed on the data x, without performing the rfft. sz: corresponding real space size to obtain the rft size for

rft_size(arr)

Returns the size of an rft or rfft performed on the data x, without performing the rfft.

arr: array to optain the corresponding rft size for

rfft_size(size, dims)

Returns the size rfft would return if applied to a real array. size is the input size to rfft and dims the dimensions the rfft transforms over. Actually we only would need first(dims).

julia> using FFTW

julia> rfft((ones((4,3,2))), (2,3)) |> size
(4, 2, 2)

julia> FourierTools.rfft_size((4,3,2), (2, 3))
(4, 2, 2)

ft_center_diff(s [, dims])

Calculates how much each dimension must be shifted that the center frequency is at the Fourier center. This if for a normal fft

rft_center_diff(s [, dims])

Calculates how much each dimension must be shifted that the center frequency is at the Fourier center. This is for rfft. The dims[1] must be therefore not shifted!

center_pos(x)

Calculate the position of the center frequency. Size of the array is x

Examples

julia> FourierTools.center_pos(3)
2
julia> FourierTools.center_pos(4)
3