592 lines
21 KiB
Julia
592 lines
21 KiB
Julia
# This file is a part of Julia. License is MIT: https://julialang.org/license
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module DFT
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# DFT plan where the inputs are an array of eltype T
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abstract type Plan{T} end
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import Base: show, summary, size, ndims, length, eltype,
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*, A_mul_B!, inv, \, A_ldiv_B!
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eltype(::Type{Plan{T}}) where {T} = T
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# size(p) should return the size of the input array for p
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size(p::Plan, d) = size(p)[d]
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ndims(p::Plan) = length(size(p))
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length(p::Plan) = prod(size(p))::Int
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##############################################################################
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export fft, ifft, bfft, fft!, ifft!, bfft!,
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plan_fft, plan_ifft, plan_bfft, plan_fft!, plan_ifft!, plan_bfft!,
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rfft, irfft, brfft, plan_rfft, plan_irfft, plan_brfft
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const FFTWFloat = Union{Float32,Float64}
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fftwfloat(x) = _fftwfloat(float(x))
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_fftwfloat(::Type{T}) where {T<:FFTWFloat} = T
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_fftwfloat(::Type{Float16}) = Float32
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_fftwfloat(::Type{Complex{T}}) where {T} = Complex{_fftwfloat(T)}
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_fftwfloat(::Type{T}) where {T} = error("type $T not supported")
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_fftwfloat(x::T) where {T} = _fftwfloat(T)(x)
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complexfloat(x::StridedArray{Complex{<:FFTWFloat}}) = x
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realfloat(x::StridedArray{<:FFTWFloat}) = x
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# return an Array, rather than similar(x), to avoid an extra copy for FFTW
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# (which only works on StridedArray types).
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complexfloat(x::AbstractArray{T}) where {T<:Complex} = copy1(typeof(fftwfloat(zero(T))), x)
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complexfloat(x::AbstractArray{T}) where {T<:Real} = copy1(typeof(complex(fftwfloat(zero(T)))), x)
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realfloat(x::AbstractArray{T}) where {T<:Real} = copy1(typeof(fftwfloat(zero(T))), x)
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# copy to a 1-based array, using circular permutation
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function copy1(::Type{T}, x) where T
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y = Array{T}(map(length, indices(x)))
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Base.circcopy!(y, x)
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end
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to1(x::AbstractArray) = _to1(indices(x), x)
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_to1(::Tuple{Base.OneTo,Vararg{Base.OneTo}}, x) = x
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_to1(::Tuple, x) = copy1(eltype(x), x)
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# implementations only need to provide plan_X(x, region)
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# for X in (:fft, :bfft, ...):
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for f in (:fft, :bfft, :ifft, :fft!, :bfft!, :ifft!, :rfft)
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pf = Symbol("plan_", f)
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@eval begin
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$f(x::AbstractArray) = (y = to1(x); $pf(y) * y)
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$f(x::AbstractArray, region) = (y = to1(x); $pf(y, region) * y)
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$pf(x::AbstractArray; kws...) = (y = to1(x); $pf(y, 1:ndims(y); kws...))
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end
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end
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"""
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plan_ifft(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)
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Same as [`plan_fft`](@ref), but produces a plan that performs inverse transforms
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[`ifft`](@ref).
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"""
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plan_ifft
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"""
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plan_ifft!(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)
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Same as [`plan_ifft`](@ref), but operates in-place on `A`.
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"""
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plan_ifft!
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"""
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plan_bfft!(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)
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Same as [`plan_bfft`](@ref), but operates in-place on `A`.
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"""
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plan_bfft!
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"""
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plan_bfft(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)
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Same as [`plan_fft`](@ref), but produces a plan that performs an unnormalized
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backwards transform [`bfft`](@ref).
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"""
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plan_bfft
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"""
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plan_fft(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)
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Pre-plan an optimized FFT along given dimensions (`dims`) of arrays matching the shape and
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type of `A`. (The first two arguments have the same meaning as for [`fft`](@ref).)
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Returns an object `P` which represents the linear operator computed by the FFT, and which
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contains all of the information needed to compute `fft(A, dims)` quickly.
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To apply `P` to an array `A`, use `P * A`; in general, the syntax for applying plans is much
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like that of matrices. (A plan can only be applied to arrays of the same size as the `A`
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for which the plan was created.) You can also apply a plan with a preallocated output array `Â`
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by calling `A_mul_B!(Â, plan, A)`. (For `A_mul_B!`, however, the input array `A` must
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be a complex floating-point array like the output `Â`.) You can compute the inverse-transform plan by `inv(P)`
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and apply the inverse plan with `P \\ Â` (the inverse plan is cached and reused for
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subsequent calls to `inv` or `\\`), and apply the inverse plan to a pre-allocated output
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array `A` with `A_ldiv_B!(A, P, Â)`.
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The `flags` argument is a bitwise-or of FFTW planner flags, defaulting to `FFTW.ESTIMATE`.
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e.g. passing `FFTW.MEASURE` or `FFTW.PATIENT` will instead spend several seconds (or more)
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benchmarking different possible FFT algorithms and picking the fastest one; see the FFTW
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manual for more information on planner flags. The optional `timelimit` argument specifies a
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rough upper bound on the allowed planning time, in seconds. Passing `FFTW.MEASURE` or
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`FFTW.PATIENT` may cause the input array `A` to be overwritten with zeros during plan
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creation.
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[`plan_fft!`](@ref) is the same as [`plan_fft`](@ref) but creates a
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plan that operates in-place on its argument (which must be an array of complex
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floating-point numbers). [`plan_ifft`](@ref) and so on are similar but produce
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plans that perform the equivalent of the inverse transforms [`ifft`](@ref) and so on.
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"""
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plan_fft
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"""
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plan_fft!(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)
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Same as [`plan_fft`](@ref), but operates in-place on `A`.
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"""
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plan_fft!
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"""
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rfft(A [, dims])
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Multidimensional FFT of a real array `A`, exploiting the fact that the transform has
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conjugate symmetry in order to save roughly half the computational time and storage costs
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compared with [`fft`](@ref). If `A` has size `(n_1, ..., n_d)`, the result has size
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`(div(n_1,2)+1, ..., n_d)`.
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The optional `dims` argument specifies an iterable subset of one or more dimensions of `A`
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to transform, similar to [`fft`](@ref). Instead of (roughly) halving the first
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dimension of `A` in the result, the `dims[1]` dimension is (roughly) halved in the same way.
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"""
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rfft
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"""
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ifft!(A [, dims])
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Same as [`ifft`](@ref), but operates in-place on `A`.
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"""
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ifft!
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"""
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ifft(A [, dims])
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Multidimensional inverse FFT.
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A one-dimensional inverse FFT computes
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```math
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\\operatorname{IDFT}(A)[k] = \\frac{1}{\\operatorname{length}(A)}
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\\sum_{n=1}^{\\operatorname{length}(A)} \\exp\\left(+i\\frac{2\\pi (n-1)(k-1)}
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{\\operatorname{length}(A)} \\right) A[n].
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```
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A multidimensional inverse FFT simply performs this operation along each transformed dimension of `A`.
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"""
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ifft
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"""
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fft!(A [, dims])
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Same as [`fft`](@ref), but operates in-place on `A`, which must be an array of
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complex floating-point numbers.
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"""
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fft!
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"""
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bfft(A [, dims])
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Similar to [`ifft`](@ref), but computes an unnormalized inverse (backward)
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transform, which must be divided by the product of the sizes of the transformed dimensions
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in order to obtain the inverse. (This is slightly more efficient than [`ifft`](@ref)
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because it omits a scaling step, which in some applications can be combined with other
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computational steps elsewhere.)
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```math
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\\operatorname{BDFT}(A)[k] = \\operatorname{length}(A) \\operatorname{IDFT}(A)[k]
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```
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"""
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bfft
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"""
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bfft!(A [, dims])
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Same as [`bfft`](@ref), but operates in-place on `A`.
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"""
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bfft!
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# promote to a complex floating-point type (out-of-place only),
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# so implementations only need Complex{Float} methods
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for f in (:fft, :bfft, :ifft)
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pf = Symbol("plan_", f)
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@eval begin
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$f(x::AbstractArray{<:Real}, region=1:ndims(x)) = $f(complexfloat(x), region)
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$pf(x::AbstractArray{<:Real}, region; kws...) = $pf(complexfloat(x), region; kws...)
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$f(x::AbstractArray{<:Complex{<:Union{Integer,Rational}}}, region=1:ndims(x)) = $f(complexfloat(x), region)
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$pf(x::AbstractArray{<:Complex{<:Union{Integer,Rational}}}, region; kws...) = $pf(complexfloat(x), region; kws...)
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end
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end
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rfft(x::AbstractArray{<:Union{Integer,Rational}}, region=1:ndims(x)) = rfft(realfloat(x), region)
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plan_rfft(x::AbstractArray, region; kws...) = plan_rfft(realfloat(x), region; kws...)
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# only require implementation to provide *(::Plan{T}, ::Array{T})
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*(p::Plan{T}, x::AbstractArray) where {T} = p * copy1(T, x)
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# Implementations should also implement A_mul_B!(Y, plan, X) so as to support
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# pre-allocated output arrays. We don't define * in terms of A_mul_B!
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# generically here, however, because of subtleties for in-place and rfft plans.
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##############################################################################
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# To support inv, \, and A_ldiv_B!(y, p, x), we require Plan subtypes
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# to have a pinv::Plan field, which caches the inverse plan, and which
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# should be initially undefined. They should also implement
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# plan_inv(p) to construct the inverse of a plan p.
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# hack from @simonster (in #6193) to compute the return type of plan_inv
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# without actually calling it or even constructing the empty arrays.
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_pinv_type(p::Plan) = typeof([plan_inv(x) for x in typeof(p)[]])
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pinv_type(p::Plan) = eltype(_pinv_type(p))
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inv(p::Plan) =
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isdefined(p, :pinv) ? p.pinv::pinv_type(p) : (p.pinv = plan_inv(p))
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\(p::Plan, x::AbstractArray) = inv(p) * x
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A_ldiv_B!(y::AbstractArray, p::Plan, x::AbstractArray) = A_mul_B!(y, inv(p), x)
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##############################################################################
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# implementations only need to provide the unnormalized backwards FFT,
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# similar to FFTW, and we do the scaling generically to get the ifft:
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mutable struct ScaledPlan{T,P,N} <: Plan{T}
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p::P
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scale::N # not T, to avoid unnecessary promotion to Complex
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pinv::Plan
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ScaledPlan{T,P,N}(p, scale) where {T,P,N} = new(p, scale)
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end
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ScaledPlan{T}(p::P, scale::N) where {T,P,N} = ScaledPlan{T,P,N}(p, scale)
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ScaledPlan(p::Plan{T}, scale::Number) where {T} = ScaledPlan{T}(p, scale)
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ScaledPlan(p::ScaledPlan, α::Number) = ScaledPlan(p.p, p.scale * α)
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size(p::ScaledPlan) = size(p.p)
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show(io::IO, p::ScaledPlan) = print(io, p.scale, " * ", p.p)
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summary(p::ScaledPlan) = string(p.scale, " * ", summary(p.p))
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*(p::ScaledPlan, x::AbstractArray) = scale!(p.p * x, p.scale)
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*(α::Number, p::Plan) = ScaledPlan(p, α)
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*(p::Plan, α::Number) = ScaledPlan(p, α)
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*(I::UniformScaling, p::ScaledPlan) = ScaledPlan(p, I.λ)
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*(p::ScaledPlan, I::UniformScaling) = ScaledPlan(p, I.λ)
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*(I::UniformScaling, p::Plan) = ScaledPlan(p, I.λ)
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*(p::Plan, I::UniformScaling) = ScaledPlan(p, I.λ)
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# Normalization for ifft, given unscaled bfft, is 1/prod(dimensions)
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normalization(T, sz, region) = one(T) / Int(prod([sz...][[region...]]))
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normalization(X, region) = normalization(real(eltype(X)), size(X), region)
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plan_ifft(x::AbstractArray, region; kws...) =
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ScaledPlan(plan_bfft(x, region; kws...), normalization(x, region))
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plan_ifft!(x::AbstractArray, region; kws...) =
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ScaledPlan(plan_bfft!(x, region; kws...), normalization(x, region))
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plan_inv(p::ScaledPlan) = ScaledPlan(plan_inv(p.p), inv(p.scale))
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A_mul_B!(y::AbstractArray, p::ScaledPlan, x::AbstractArray) =
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scale!(p.scale, A_mul_B!(y, p.p, x))
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##############################################################################
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# Real-input DFTs are annoying because the output has a different size
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# than the input if we want to gain the full factor-of-two(ish) savings
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# For backward real-data transforms, we must specify the original length
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# of the first dimension, since there is no reliable way to detect this
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# from the data (we can't detect whether the dimension was originally even
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# or odd).
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for f in (:brfft, :irfft)
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pf = Symbol("plan_", f)
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@eval begin
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$f(x::AbstractArray, d::Integer) = $pf(x, d) * x
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$f(x::AbstractArray, d::Integer, region) = $pf(x, d, region) * x
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$pf(x::AbstractArray, d::Integer;kws...) = $pf(x, d, 1:ndims(x);kws...)
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end
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end
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for f in (:brfft, :irfft)
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@eval begin
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$f(x::AbstractArray{<:Real}, d::Integer, region=1:ndims(x)) = $f(complexfloat(x), d, region)
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$f(x::AbstractArray{<:Complex{<:Union{Integer,Rational}}}, d::Integer, region=1:ndims(x)) = $f(complexfloat(x), d, region)
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end
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end
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"""
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irfft(A, d [, dims])
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Inverse of [`rfft`](@ref): for a complex array `A`, gives the corresponding real
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array whose FFT yields `A` in the first half. As for [`rfft`](@ref), `dims` is an
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optional subset of dimensions to transform, defaulting to `1:ndims(A)`.
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`d` is the length of the transformed real array along the `dims[1]` dimension, which must
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satisfy `div(d,2)+1 == size(A,dims[1])`. (This parameter cannot be inferred from `size(A)`
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since both `2*size(A,dims[1])-2` as well as `2*size(A,dims[1])-1` are valid sizes for the
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transformed real array.)
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"""
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irfft
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"""
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brfft(A, d [, dims])
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Similar to [`irfft`](@ref) but computes an unnormalized inverse transform (similar
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to [`bfft`](@ref)), which must be divided by the product of the sizes of the
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transformed dimensions (of the real output array) in order to obtain the inverse transform.
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"""
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brfft
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function rfft_output_size(x::AbstractArray, region)
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d1 = first(region)
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osize = [size(x)...]
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osize[d1] = osize[d1]>>1 + 1
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return osize
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end
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function brfft_output_size(x::AbstractArray, d::Integer, region)
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d1 = first(region)
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osize = [size(x)...]
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@assert osize[d1] == d>>1 + 1
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osize[d1] = d
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return osize
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end
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plan_irfft(x::AbstractArray{Complex{T}}, d::Integer, region; kws...) where {T} =
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ScaledPlan(plan_brfft(x, d, region; kws...),
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normalization(T, brfft_output_size(x, d, region), region))
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"""
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plan_irfft(A, d [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)
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Pre-plan an optimized inverse real-input FFT, similar to [`plan_rfft`](@ref)
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except for [`irfft`](@ref) and [`brfft`](@ref), respectively. The first
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three arguments have the same meaning as for [`irfft`](@ref).
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"""
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plan_irfft
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##############################################################################
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export fftshift, ifftshift
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fftshift(x) = circshift(x, div.([size(x)...],2))
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"""
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fftshift(x)
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Swap the first and second halves of each dimension of `x`.
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"""
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fftshift(x)
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function fftshift(x,dim)
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s = zeros(Int,ndims(x))
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for i in dim
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s[i] = div(size(x,i),2)
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end
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circshift(x, s)
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end
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"""
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fftshift(x,dim)
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Swap the first and second halves of the given dimension or iterable of dimensions of array `x`.
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"""
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fftshift(x,dim)
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ifftshift(x) = circshift(x, div.([size(x)...],-2))
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"""
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ifftshift(x, [dim])
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Undoes the effect of `fftshift`.
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"""
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ifftshift
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function ifftshift(x,dim)
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s = zeros(Int,ndims(x))
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for i in dim
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s[i] = -div(size(x,i),2)
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end
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circshift(x, s)
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end
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##############################################################################
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# FFTW module (may move to an external package at some point):
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"""
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fft(A [, dims])
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Performs a multidimensional FFT of the array `A`. The optional `dims` argument specifies an
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iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along.
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Most efficient if the size of `A` along the transformed dimensions is a product of small
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primes; see `nextprod()`. See also `plan_fft()` for even greater efficiency.
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A one-dimensional FFT computes the one-dimensional discrete Fourier transform (DFT) as
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defined by
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```math
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\\operatorname{DFT}(A)[k] =
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\\sum_{n=1}^{\\operatorname{length}(A)}
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\\exp\\left(-i\\frac{2\\pi
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(n-1)(k-1)}{\\operatorname{length}(A)} \\right) A[n].
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```
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A multidimensional FFT simply performs this operation along each transformed dimension of `A`.
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!!! note
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* Julia starts FFTW up with 1 thread by default. Higher performance is usually possible by
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increasing number of threads. Use `FFTW.set_num_threads(Sys.CPU_CORES)` to use as many
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threads as cores on your system.
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* This performs a multidimensional FFT by default. FFT libraries in other languages such as
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Python and Octave perform a one-dimensional FFT along the first non-singleton dimension
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of the array. This is worth noting while performing comparisons. For more details,
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refer to the [Noteworthy Differences from other Languages](@ref)
|
||
section of the manual.
|
||
"""
|
||
fft
|
||
|
||
"""
|
||
plan_rfft(A [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)
|
||
|
||
Pre-plan an optimized real-input FFT, similar to [`plan_fft`](@ref) except for
|
||
[`rfft`](@ref) instead of [`fft`](@ref). The first two arguments, and the
|
||
size of the transformed result, are the same as for [`rfft`](@ref).
|
||
"""
|
||
plan_rfft
|
||
|
||
"""
|
||
plan_brfft(A, d [, dims]; flags=FFTW.ESTIMATE; timelimit=Inf)
|
||
|
||
Pre-plan an optimized real-input unnormalized transform, similar to
|
||
[`plan_rfft`](@ref) except for [`brfft`](@ref) instead of
|
||
[`rfft`](@ref). The first two arguments and the size of the transformed result, are
|
||
the same as for [`brfft`](@ref).
|
||
"""
|
||
plan_brfft
|
||
|
||
module FFTW
|
||
import ..DFT: fft, bfft, ifft, rfft, brfft, irfft, plan_fft, plan_bfft, plan_ifft,
|
||
plan_rfft, plan_brfft, plan_irfft, fft!, bfft!, ifft!, plan_fft!, plan_bfft!, plan_ifft!,
|
||
Plan, rfft_output_size, brfft_output_size, plan_inv, normalization, ScaledPlan
|
||
|
||
export r2r, r2r!, plan_r2r, plan_r2r!
|
||
|
||
"""
|
||
plan_dct!(A [, dims [, flags [, timelimit]]])
|
||
|
||
Same as [`plan_dct`](@ref), but operates in-place on `A`.
|
||
"""
|
||
function plan_dct! end
|
||
|
||
"""
|
||
plan_idct(A [, dims [, flags [, timelimit]]])
|
||
|
||
Pre-plan an optimized inverse discrete cosine transform (DCT), similar to
|
||
[`plan_fft`](@ref) except producing a function that computes
|
||
[`idct`](@ref). The first two arguments have the same meaning as for
|
||
[`idct`](@ref).
|
||
"""
|
||
function plan_idct end
|
||
|
||
"""
|
||
plan_dct(A [, dims [, flags [, timelimit]]])
|
||
|
||
Pre-plan an optimized discrete cosine transform (DCT), similar to
|
||
[`plan_fft`](@ref) except producing a function that computes
|
||
[`dct`](@ref). The first two arguments have the same meaning as for
|
||
[`dct`](@ref).
|
||
"""
|
||
function plan_dct end
|
||
|
||
"""
|
||
plan_idct!(A [, dims [, flags [, timelimit]]])
|
||
|
||
Same as [`plan_idct`](@ref), but operates in-place on `A`.
|
||
"""
|
||
function plan_idct! end
|
||
|
||
"""
|
||
dct(A [, dims])
|
||
|
||
Performs a multidimensional type-II discrete cosine transform (DCT) of the array `A`, using
|
||
the unitary normalization of the DCT. The optional `dims` argument specifies an iterable
|
||
subset of dimensions (e.g. an integer, range, tuple, or array) to transform along. Most
|
||
efficient if the size of `A` along the transformed dimensions is a product of small primes;
|
||
see [`nextprod`](@ref). See also [`plan_dct`](@ref) for even greater
|
||
efficiency.
|
||
"""
|
||
function dct end
|
||
|
||
"""
|
||
idct(A [, dims])
|
||
|
||
Computes the multidimensional inverse discrete cosine transform (DCT) of the array `A`
|
||
(technically, a type-III DCT with the unitary normalization). The optional `dims` argument
|
||
specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to
|
||
transform along. Most efficient if the size of `A` along the transformed dimensions is a
|
||
product of small primes; see [`nextprod`](@ref). See also
|
||
[`plan_idct`](@ref) for even greater efficiency.
|
||
"""
|
||
function idct end
|
||
|
||
"""
|
||
dct!(A [, dims])
|
||
|
||
Same as [`dct!`](@ref), except that it operates in-place on `A`, which must be an
|
||
array of real or complex floating-point values.
|
||
"""
|
||
function dct! end
|
||
|
||
"""
|
||
idct!(A [, dims])
|
||
|
||
Same as [`idct!`](@ref), but operates in-place on `A`.
|
||
"""
|
||
function idct! end
|
||
|
||
"""
|
||
r2r(A, kind [, dims])
|
||
|
||
Performs a multidimensional real-input/real-output (r2r) transform
|
||
of type `kind` of the array `A`, as defined in the FFTW manual.
|
||
`kind` specifies either a discrete cosine transform of various types
|
||
(`FFTW.REDFT00`, `FFTW.REDFT01`, `FFTW.REDFT10`, or
|
||
`FFTW.REDFT11`), a discrete sine transform of various types
|
||
(`FFTW.RODFT00`, `FFTW.RODFT01`, `FFTW.RODFT10`, or
|
||
`FFTW.RODFT11`), a real-input DFT with halfcomplex-format output
|
||
(`FFTW.R2HC` and its inverse `FFTW.HC2R`), or a discrete
|
||
Hartley transform (`FFTW.DHT`). The `kind` argument may be
|
||
an array or tuple in order to specify different transform types
|
||
along the different dimensions of `A`; `kind[end]` is used
|
||
for any unspecified dimensions. See the FFTW manual for precise
|
||
definitions of these transform types, at http://www.fftw.org/doc.
|
||
|
||
The optional `dims` argument specifies an iterable subset of
|
||
dimensions (e.g. an integer, range, tuple, or array) to transform
|
||
along. `kind[i]` is then the transform type for `dims[i]`,
|
||
with `kind[end]` being used for `i > length(kind)`.
|
||
|
||
See also [`plan_r2r`](@ref) to pre-plan optimized r2r transforms.
|
||
"""
|
||
function r2r end
|
||
|
||
"""
|
||
r2r!(A, kind [, dims])
|
||
|
||
Same as [`r2r`](@ref), but operates in-place on `A`, which must be
|
||
an array of real or complex floating-point numbers.
|
||
"""
|
||
function r2r! end
|
||
|
||
"""
|
||
plan_r2r!(A, kind [, dims [, flags [, timelimit]]])
|
||
|
||
Similar to [`plan_fft`](@ref), but corresponds to [`r2r!`](@ref).
|
||
"""
|
||
function plan_r2r! end
|
||
|
||
"""
|
||
plan_r2r(A, kind [, dims [, flags [, timelimit]]])
|
||
|
||
Pre-plan an optimized r2r transform, similar to [`plan_fft`](@ref)
|
||
except that the transforms (and the first three arguments)
|
||
correspond to [`r2r`](@ref) and [`r2r!`](@ref), respectively.
|
||
"""
|
||
function plan_r2r end
|
||
|
||
(Base.USE_GPL_LIBS || Base.fftw_vendor() == :mkl) && include(joinpath("fft", "FFTW.jl"))
|
||
end
|
||
|
||
importall .FFTW
|
||
export FFTW, dct, idct, dct!, idct!, plan_dct, plan_idct, plan_dct!, plan_idct!
|
||
|
||
##############################################################################
|
||
|
||
end
|