Real data

Real input has a spectrum with Hermitian symmetry:  . Half of it is redundant, and the real transforms return only the other half. That halves the work and the memory.

using FFTL, Random
Random.seed!(5)

x = randn(1000, 300)
X = rfft(x)
size(X)

(501, 300)

The first transformed dimension is halved:     values for frequencies . The other dimensions keep their length. The complex array has the same type as a complex transform's output.

Going back

The inverse needs the length of the real array, because an odd and an even length can give the same half spectrum. irfft takes it explicitly:

irfft(X, 1000) ≈ x

true

y = randn(999)
Y = rfft(y)
size(Y), irfft(Y, 999) ≈ y

((500,), true)

brfft is the unnormalised inverse, the real counterpart of bfft. It differs from irfft by the product of the transformed lengths, the number of elements in the region; for one dimension, the length of the real array:

y1 = randn(999)
Y1 = rfft(y1)
brfft(Y1, 999) ≈ 999 .* irfft(Y1, 999)

true

Which dimension is halved

The half spectrum is taken along the first dimension of the region. To halve dimension 2 of a matrix, put it first in the region:

Z = randn(64, 128)
size(rfft(Z, (2, 1)))

(33, 128)

Plans

plan_rfft, plan_irfft and plan_brfft return plans with the same rules as the complex ones, including inv for the normalised inverse. Use them when the same real size is transformed many times; see Plans and reuse.