Discrete Fourier transforms

fft_c2c() computes the complex discrete Fourier transform of x or its inverse, fft_r2c() the transform of real input keeping only the non-redundant half of the last transformed axis, and fft_c2r() the inverse of fft_r2c(), returning real values. All three work over any subset of the dimensions of an array, in O(n log n) time for every length.

Usage

fft_c2c(
  x,
  inverse = FALSE,
  n = NULL,
  axes = NULL,
  norm = c("backward", "ortho", "forward", "none"),
  workers = NULL
)

fft_r2c(
  x,
  n = NULL,
  axes = NULL,
  norm = c("backward", "ortho", "forward", "none"),
  workers = NULL
)

fft_c2r(
  x,
  n = NULL,
  axes = NULL,
  norm = c("backward", "ortho", "forward", "none"),
  workers = NULL
)

Arguments

x: A double, integer, logical or complex vector or array. Integer and logical values are converted to double. fft_r2c() needs real input; fft_c2r() treats real input as complex.
inverse: TRUE for the inverse transform. 0 and 1 are accepted too, as in fftwtools.
n: Transform lengths, one per transformed axis in the order of axes (no recycling); with axes = NULL, one per dimension, in dimension order. Each axis of x is cropped or zero-padded to its length first. NULL keeps the lengths of x. For fft_c2r(), n gives the lengths of the real output; the halved axis of x must then hold n %/% 2 + 1 values and is cropped or padded to that. Its default is 2 * (m - 1) for an input extent m, which is always even: give n to recover an odd length.
axes: The dimensions to transform, 1-based, without repeats. NULL means all of them, in order; for fft_r2c() and fft_c2r() it means all of them in reverse order, so the first dimension is halved. With explicit axes, the real transforms halve the last axis listed, as SciPy does: axes = seq_along(dim(x)) halves the last dimension. Use axes = 1 for a column-wise transform of a matrix, as stats::mvfft() does.
norm: Where the scaling goes. "backward" (the default) leaves the forward transform unscaled and divides the inverse by N, the product of the transformed lengths; "forward" does the opposite; "ortho" divides both by sqrt(N); "none" scales neither direction, as FFTW, fftwtools and stats::fft() do.
workers: Threads to use. NULL uses the session default (1). Negative values count back from the number of available cores: -1 means all of them. Requests are capped at the available cores, and at 2 when _R_CHECK_LIMIT_CORES_ is set. Threads split the independent 1-D transforms of a multi-dimensional array, so they help only arrays with at least 32 768 elements and more than one line. Results agree to rounding whatever the number of threads; they are bit-identical on builds without fused multiply-add (CRAN's x86-64 builds).

Value

For fft_c2c() and fft_r2c() a complex vector or array, for fft_c2r() a double one. Arrays keep their dim, with the transformed extents, and the dimnames of axes that were not transformed; names are dropped.

Details

The conventions are those of scipy.fft (fft, fftn, rfftn, irfftn): n crops or zero-pads each transformed axis before the transform, and norm chooses where the scaling goes. They differ from SciPy in three ways: axes are 1-based; axes = NULL means every dimension rather than the last one; and with axes = NULL the real transforms halve the first dimension, not the last. R stores arrays column by column, so the first dimension is the contiguous one, and halving it is the fast choice (about 1.8 times faster than halving the last for a 2048 x 2048 matrix). It is also what fftwtools' 2-D and 3-D real transforms do.

Coming from fftwtools: fft_c2c(x) is fftw(x) and fftw_r2c(x), fft_r2c(x) is fftw_r2c(x, HermConj = 0), and an fftwtools inverse is the zufft inverse with norm = "none", because fftwtools, like stats::fft(), never scales.

Examples

x <- c(1, 2, 3, 4, 3, 2)
fft_c2c(x)
#> [1]  1.500000e+01+0i -4.000000e+00+0i -2.220446e-16+0i -1.000000e+00+0i
#> [5] -2.220446e-16+0i -4.000000e+00+0i
fft_r2c(x)
#> [1]  1.500000e+01+0i -4.000000e+00+0i -2.220446e-16+0i -1.000000e+00+0i
fft_c2r(fft_r2c(x), n = length(x))
#> [1] 1 2 3 4 3 2

# an inverse that matches fftwtools and stats::fft(inverse = TRUE)
fft_c2c(fft_c2c(x), inverse = TRUE, norm = "none") / length(x)
#> [1] 1+0i 2+0i 3+0i 4+0i 3+0i 2+0i

# columns of a matrix, and both axes
m <- matrix(rnorm(12), 4, 3)
fft_c2c(m, axes = 1)
#>                      [,1]                 [,2]                  [,3]
#> [1,] -3.587561+0.0000000i -0.2991786+0.000000i -0.4516224+0.0000000i
#> [2,]  1.037220-0.2608883i  2.4433704-1.395737i  0.3094998+0.9116875i
#> [3,] -4.087053+0.0000000i -2.1013512+0.000000i -1.1441756+0.0000000i
#> [4,]  1.037220+0.2608883i  2.4433704+1.395737i  0.3094998-0.9116875i
dim(fft_r2c(m))                 # halves the first dimension: 3 x 3
#> [1] 3 3
dim(fft_r2c(m, axes = 1:2))     # SciPy's choice, the last: 4 x 2
#> [1] 4 2
fft_c2r(fft_r2c(m), n = dim(m)) # back to m
#>              [,1]       [,2]       [,3]
#> [1,] -1.400043517  0.6215527 -0.2441996
#> [2,]  0.255317055  1.1484116 -0.2827054
#> [3,] -2.437263611 -1.8218177 -0.5536994
#> [4,] -0.005571287 -0.2473253  0.6289820

Usage

Arguments

Value

Details

See also

Examples