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fft.jl
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fft.jl
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# This file is a part of Julia. License is MIT: http://julialang.org/license
# fft
a = rand(8) + im*rand(8)
@test norm(ifft(fft(a)) - a) < 1e-8
m4 = [16. 2 3 13;
5 11 10 8;
9 7 6 12;
4 14 15 1]
true_fft_m4 = [
34. 34. 34. 34.;
7. - 1.im -5. + 3.im -3. + 5.im 1. - 7.im;
16. -16. -16. 16.;
7. + 1.im -5. - 3.im -3. - 5.im 1. + 7.im ]
true_fftn_m4 = [
136. 0 0 0 ;
0. 20 8 + 8im 0 - 12im ;
0. 32 + 32im 0 32 - 32im ;
0. 0 + 12im 8 - 8im 20 ]
true_fftd2_m4 = [
34. 13 + 11im 4 13 - 11im ;
34. -5 - 3im -4 -5 + 3im ;
34. 3 + 5im -4 3 - 5im ;
34. -11 - 13im 4 -11 + 13im ]
b = rand(17,14)
b[3:6,9:12] = m4
sm4 = slice(b,3:6,9:12)
m3d = map(Float32,reshape(1:5*3*2, 5, 3, 2))
true_fftd3_m3d = Array(Float32, 5, 3, 2)
true_fftd3_m3d[:,:,1] = 17:2:45
true_fftd3_m3d[:,:,2] = -15
# use invoke to force usage of CTPlan versions even if FFTW is present
for A in (Array,SubArray)
for f in (:fft,:ifft,:plan_fft,:plan_ifft)
f_ = symbol(string(f, "_"))
@eval begin
$f_{T,N}(x::$A{T,N}) = invoke($f, Tuple{AbstractArray{T,N}}, x)
$f_{T,N,R}(x::$A{T,N},r::R) = invoke($f,Tuple{AbstractArray{T,N},R},x,r)
end
end
end
for (f,fi,pf,pfi) in ((fft,ifft,plan_fft,plan_ifft),
(fft_,ifft_,plan_fft_,plan_ifft_))
pm4 = pf(m4,1)
fft_m4 = f(m4,1)
fftd2_m4 = f(m4,2)
ifft_fft_m4 = fi(f(m4,1),1)
fftn_m4 = f(m4)
ifftn_fftn_m4 = fi(f(m4))
fft!_m4 = complex(m4); fft!(fft!_m4,1)
fft!d2_m4 = complex(m4); fft!(fft!d2_m4,2)
ifft!_fft_m4 = f(m4,1); ifft!(ifft!_fft_m4,1)
fft!n_m4 = complex(m4); fft!(fft!n_m4)
ifft!n_fftn_m4 = f(m4); ifft!(ifft!n_fftn_m4)
pfft_m4 = pf(m4,1)*m4
pfftd2_m4 = pf(m4,2)*m4
pifft_fft_m4 = pfi(fft_m4,1)*fft_m4
pfftn_m4 = pf(m4)*m4
pifftn_fftn_m4 = pfi(fftn_m4)*fftn_m4
pfft!_m4 = complex(m4); plan_fft!(pfft!_m4,1)*pfft!_m4
pfft!d2_m4 = complex(m4); plan_fft!(pfft!d2_m4,2)*pfft!d2_m4
pifft!_fft_m4 = f(m4,1); plan_ifft!(pifft!_fft_m4,1)*pifft!_fft_m4
pfft!n_m4 = complex(m4); plan_fft!(pfft!n_m4)*pfft!n_m4
pifft!n_fftn_m4 = f(m4); plan_ifft!(pifft!n_fftn_m4)*pifft!n_fftn_m4
sfftn_m4 = f(sm4)
psfftn_m4 = pf(sm4)*sm4
sfft!n_b = map(Complex128,b)
sfft!n_m4 = slice(sfft!n_b,3:6,9:12); fft!(sfft!n_m4)
psfft!n_b = map(Complex128,b)
psfft!n_m4 = slice(psfft!n_b,3:6,9:12); plan_fft!(psfft!n_m4)*psfft!n_m4
for i = 1:length(m4)
@test_approx_eq fft_m4[i] true_fft_m4[i]
@test_approx_eq fftd2_m4[i] true_fftd2_m4[i]
@test_approx_eq ifft_fft_m4[i] m4[i]
@test_approx_eq fftn_m4[i] true_fftn_m4[i]
@test_approx_eq ifftn_fftn_m4[i] m4[i]
@test_approx_eq fft!_m4[i] true_fft_m4[i]
@test_approx_eq fft!d2_m4[i] true_fftd2_m4[i]
@test_approx_eq ifft!_fft_m4[i] m4[i]
@test_approx_eq fft!n_m4[i] true_fftn_m4[i]
@test_approx_eq ifft!n_fftn_m4[i] m4[i]
@test_approx_eq pfft_m4[i] true_fft_m4[i]
@test_approx_eq pfftd2_m4[i] true_fftd2_m4[i]
@test_approx_eq pifft_fft_m4[i] m4[i]
@test_approx_eq pfftn_m4[i] true_fftn_m4[i]
@test_approx_eq pifftn_fftn_m4[i] m4[i]
@test_approx_eq pfft!_m4[i] true_fft_m4[i]
@test_approx_eq pfft!d2_m4[i] true_fftd2_m4[i]
@test_approx_eq pifft!_fft_m4[i] m4[i]
@test_approx_eq pfft!n_m4[i] true_fftn_m4[i]
@test_approx_eq pifft!n_fftn_m4[i] m4[i]
@test_approx_eq sfftn_m4[i] true_fftn_m4[i]
@test_approx_eq sfft!n_m4[i] true_fftn_m4[i]
@test_approx_eq psfftn_m4[i] true_fftn_m4[i]
@test_approx_eq psfft!n_m4[i] true_fftn_m4[i]
end
ifft!(sfft!n_m4)
plan_ifft!(psfft!n_m4)*psfft!n_m4
@test norm(sfft!n_m4 - m4) < 1e-8
@test norm(psfft!n_m4 - m4) < 1e-8
# The following capabilities are FFTW only.
# They are not available in MKL, and hence do not test them.
if Base.fftw_vendor() != :mkl
ifft3_fft3_m3d = fi(f(m3d))
fftd3_m3d = f(m3d,3)
ifftd3_fftd3_m3d = fi(fftd3_m3d,3)
fft!d3_m3d = complex(m3d); fft!(fft!d3_m3d,3)
ifft!d3_fftd3_m3d = copy(fft!d3_m3d); ifft!(ifft!d3_fftd3_m3d,3)
pfftd3_m3d = pf(m3d,3)*m3d
pifftd3_fftd3_m3d = pfi(fftd3_m3d,3)*fftd3_m3d
pfft!d3_m3d = complex(m3d); plan_fft!(pfft!d3_m3d,3)*pfft!d3_m3d
pifft!d3_fftd3_m3d = copy(fft!d3_m3d); plan_ifft!(pifft!d3_fftd3_m3d,3)*pifft!d3_fftd3_m3d
@test isa(fftd3_m3d, Array{Complex64,3})
@test isa(ifftd3_fftd3_m3d, Array{Complex64,3})
@test isa(fft!d3_m3d, Array{Complex64,3})
@test isa(ifft!d3_fftd3_m3d, Array{Complex64,3})
@test isa(pfftd3_m3d, Array{Complex64,3})
@test isa(pifftd3_fftd3_m3d, Array{Complex64,3})
@test isa(pfft!d3_m3d, Array{Complex64,3})
@test isa(pifft!d3_fftd3_m3d, Array{Complex64,3})
for i = 1:length(m3d)
@test_approx_eq fftd3_m3d[i] true_fftd3_m3d[i]
@test_approx_eq ifftd3_fftd3_m3d[i] m3d[i]
@test_approx_eq ifft3_fft3_m3d[i] m3d[i]
@test_approx_eq fft!d3_m3d[i] true_fftd3_m3d[i]
@test_approx_eq ifft!d3_fftd3_m3d[i] m3d[i]
@test_approx_eq pfftd3_m3d[i] true_fftd3_m3d[i]
@test_approx_eq pifftd3_fftd3_m3d[i] m3d[i]
@test_approx_eq pfft!d3_m3d[i] true_fftd3_m3d[i]
@test_approx_eq pifft!d3_fftd3_m3d[i] m3d[i]
end
end # if fftw_vendor() != :mkl ...
# rfft/rfftn
rfft_m4 = rfft(m4,1)
rfftd2_m4 = rfft(m4,2)
rfftn_m4 = rfft(m4)
prfft_m4 = plan_rfft(m4,1)*m4
prfftd2_m4 = plan_rfft(m4,2)*m4
prfftn_m4 = plan_rfft(m4)*m4
srfftn_m4 = rfft(sm4)
psrfftn_m4 = plan_rfft(sm4)*sm4
for i = 1:3, j = 1:4
@test_approx_eq rfft_m4[i,j] true_fft_m4[i,j]
@test_approx_eq rfftd2_m4[j,i] true_fftd2_m4[j,i]
@test_approx_eq rfftn_m4[i,j] true_fftn_m4[i,j]
@test_approx_eq prfft_m4[i,j] true_fft_m4[i,j]
@test_approx_eq prfftd2_m4[j,i] true_fftd2_m4[j,i]
@test_approx_eq prfftn_m4[i,j] true_fftn_m4[i,j]
@test_approx_eq srfftn_m4[i,j] true_fftn_m4[i,j]
@test_approx_eq psrfftn_m4[i,j] true_fftn_m4[i,j]
end
irfft_rfft_m4 = irfft(rfft_m4,size(m4,1),1)
irfft_rfftd2_m4 = irfft(rfftd2_m4,size(m4,2),2)
irfftn_rfftn_m4 = irfft(rfftn_m4,size(m4,1))
pirfft_rfft_m4 = plan_irfft(rfft_m4,size(m4,1),1)*rfft_m4
pirfft_rfftd2_m4 = plan_irfft(rfftd2_m4,size(m4,2),2)*rfftd2_m4
pirfftn_rfftn_m4 = plan_irfft(rfftn_m4,size(m4,1))*rfftn_m4
for i = 1:length(m4)
@test_approx_eq irfft_rfft_m4[i] m4[i]
@test_approx_eq irfft_rfftd2_m4[i] m4[i]
@test_approx_eq irfftn_rfftn_m4[i] m4[i]
@test_approx_eq pirfft_rfft_m4[i] m4[i]
@test_approx_eq pirfft_rfftd2_m4[i] m4[i]
@test_approx_eq pirfftn_rfftn_m4[i] m4[i]
end
if Base.fftw_vendor() != :mkl
rfftn_m3d = rfft(m3d)
rfftd3_m3d = rfft(m3d,3)
@test size(rfftd3_m3d) == size(fftd3_m3d)
irfft_rfftd3_m3d = irfft(rfftd3_m3d,size(m3d,3),3)
irfftn_rfftn_m3d = irfft(rfftn_m3d,size(m3d,1))
for i = 1:length(m3d)
@test_approx_eq rfftd3_m3d[i] true_fftd3_m3d[i]
@test_approx_eq irfft_rfftd3_m3d[i] m3d[i]
@test_approx_eq irfftn_rfftn_m3d[i] m3d[i]
end
fftn_m3d = fft(m3d)
@test size(fftn_m3d) == (5,3,2)
rfftn_m3d = rfft(m3d)
@test size(rfftn_m3d) == (3,3,2)
for i = 1:3, j = 1:3, k = 1:2
@test_approx_eq rfftn_m3d[i,j,k] fftn_m3d[i,j,k]
end
end # !mkl
end
# FFT self-test algorithm (for unscaled 1d forward FFTs):
# Funda Ergün, "Testing multivariate linear functions: Overcoming
# the generator bottleneck," Proc. 27th ACM Symposium on the Theory
# of Computing, pp. 407-416 (1995).
# Check linearity, impulse-response, and time-shift properties.
function fft_test{T<:Complex}(p::Base.DFT.Plan{T}, ntrials=4,
tol=1e5 * eps(real(T)))
ndims(p) == 1 || throw(ArgumentError("not a 1d FFT"))
n = length(p)
twopi_i = (-2 * convert(real(T), π)/n * (0:n-1)) * im
for trial = 1:ntrials
# linearity:
x = rand(T, n)
y = rand(T, n)
α = rand(T)
β = rand(T)
X = p * (α*x + β*y)
err = norm(α * (p*x) + β * (p*y) - X, Inf) / norm(X, Inf)
err <= tol || error("linearity error $err in $p")
# impulse-response:
z = zeros(T, n)
i = rand(0:n-1)
z[i+1] = 1
X = exp(twopi_i*i)
err = norm(p*z - X, Inf) / norm(X, Inf)
err <= tol || error("impulse-response error $err in $p")
# time-shift:
if n > 1
s = rand(1:n-1)
X = (p*x).*exp(twopi_i*s)
err = norm(p*circshift(x,s) - X, Inf) / norm(X, Inf)
err <= tol || error("time-shift error $err in $p")
end
end
end
for T in (Complex64, Complex128)
for n in [1:100; 121; 143; 1000; 1024; 1031; 2000; 2048]
x = zeros(T, n)
fft_test(plan_fft(x))
fft_test(plan_fft_(x))
end
end
# test inversion, scaling, and pre-allocated variants
for T in (Complex64, Complex128)
for x in (T[1:100;], reshape(T[1:200;], 20,10))
y = similar(x)
for planner in (plan_fft, plan_fft_, plan_ifft, plan_ifft_)
p = planner(x)
pi = inv(p)
p3 = 3*p
p3i = inv(p3)
@test eltype(p) == eltype(pi) == eltype(p3) == eltype(p3i) == T
@test vecnorm(x - p3i * (p * 3x)) < eps(real(T)) * 10000
@test vecnorm(3x - pi * (p3 * x)) < eps(real(T)) * 10000
A_mul_B!(y, p, x)
@test y == p * x
A_ldiv_B!(y, p, x)
@test y == p \ x
end
end
end
let
plan32 = plan_fft([1.0:2048.0;])
plan64 = plan_fft([1f0:2048f0;])
FFTW.flops(plan32)
FFTW.flops(plan64)
end
# issue #9772
for x in (randn(10),randn(10,12))
z = complex(x)
y = rfft(x)
@inferred rfft(x)
@inferred brfft(x,18)
@inferred brfft(y,10)
for f in (plan_bfft!, plan_fft!, plan_ifft!,
plan_bfft, plan_fft, plan_ifft,
fft, bfft, fft_, ifft)
p = @inferred f(z)
if isa(p, FFTW.Plan)
@inferred FFTW.plan_inv(p)
end
end
for f in (plan_bfft, plan_fft, plan_ifft,
plan_rfft, fft, bfft, fft_, ifft)
p = @inferred f(x)
if isa(p, FFTW.Plan)
@inferred FFTW.plan_inv(p)
end
end
# note: inference doesn't work for plan_fft_ since the
# algorithm steps are included in the CTPlan type
end