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# Discrete Fourier transforms (`scipy.fftpack`)

## Fast Fourier Transforms (FFTs)

`fft`(x[, n, axis, overwrite_x]) | Return discrete Fourier transform of real or complex sequence.
`ifft`(x[, n, axis, overwrite_x]) | Return discrete inverse Fourier transform of real or complex sequence.
`fft2`(x[, shape, axes, overwrite_x]) | 2-D discrete Fourier transform.
`ifft2`(x[, shape, axes, overwrite_x]) | 2-D discrete inverse Fourier transform of real or complex sequence.
`fftn`(x[, shape, axes, overwrite_x]) | Return multidimensional discrete Fourier transform.
`ifftn`(x[, shape, axes, overwrite_x]) | Return inverse multi-dimensional discrete Fourier transform of
`rfft`(x[, n, axis, overwrite_x]) | Discrete Fourier transform of a real sequence.
`irfft`(x[, n, axis, overwrite_x]) | Return inverse discrete Fourier transform of real sequence x.
`dct`(x[, type, n, axis, norm, overwrite_x]) | Return the Discrete Cosine Transform of arbitrary type sequence x.
`idct`(x[, type, n, axis, norm, overwrite_x]) | Return the Inverse Discrete Cosine Transform of an arbitrary type sequence.

## Differential and pseudo-differential operators

`diff`(x[, order, period, _cache]) | Return k-th derivative (or integral) of a periodic sequence x.
`tilbert`(x, h[, period, _cache]) | Return h-Tilbert transform of a periodic sequence x.
`itilbert`(x, h[, period, _cache]) | Return inverse h-Tilbert transform of a periodic sequence x.
`hilbert`(x[, _cache]) | Return Hilbert transform of a periodic sequence x.
`ihilbert`(x) | Return inverse Hilbert transform of a periodic sequence x.
`cs_diff`(x, a, b[, period, _cache]) | Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence.
`sc_diff`(x, a, b[, period, _cache]) | Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x.
`ss_diff`(x, a, b[, period, _cache]) | Return (a,b)-sinh/sinh pseudo-derivative of a periodic sequence x.
`cc_diff`(x, a, b[, period, _cache]) | Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence.
`shift`(x, a[, period, _cache]) | Shift periodic sequence x by a: y(u) = x(u+a).

## Helper functions

`fftshift`(x[, axes]) | Shift the zero-frequency component to the center of the spectrum.
`ifftshift`(x[, axes]) | The inverse of fftshift.
`fftfreq`(n[, d]) | Return the Discrete Fourier Transform sample frequencies.
`rfftfreq`(n[, d]) | DFT sample frequencies (for usage with rfft, irfft).

## Convolutions (`scipy.fftpack.convolve`)

`convolve` | convolve - Function signature:
`convolve_z` | convolve_z - Function signature:
`init_convolution_kernel` | init_convolution_kernel - Function signature:
`destroy_convolve_cache` | destroy_convolve_cache - Function signature:

## Other (`scipy.fftpack._fftpack`)

`drfft` | drfft - Function signature:
`zfft` | zfft - Function signature:
`zrfft` | zrfft - Function signature:
`zfftnd` | zfftnd - Function signature:
`destroy_drfft_cache` | destroy_drfft_cache - Function signature:
`destroy_zfft_cache` | destroy_zfft_cache - Function signature:
`destroy_zfftnd_cache` | destroy_zfftnd_cache - Function signature:

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