# Legacy discrete Fourier transforms (scipy.fftpack)¶

Warning

This submodule is now considered legacy, new code should use scipy.fft.

## 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 multidimensional discrete Fourier transform. 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. dctn(x[, type, shape, axes, norm, overwrite_x]) Return multidimensional Discrete Cosine Transform along the specified axes. idctn(x[, type, shape, axes, norm, overwrite_x]) Return multidimensional Discrete Cosine Transform along the specified axes. dst(x[, type, n, axis, norm, overwrite_x]) Return the Discrete Sine Transform of arbitrary type sequence x. idst(x[, type, n, axis, norm, overwrite_x]) Return the Inverse Discrete Sine Transform of an arbitrary type sequence. dstn(x[, type, shape, axes, norm, overwrite_x]) Return multidimensional Discrete Sine Transform along the specified axes. idstn(x[, type, shape, axes, norm, overwrite_x]) Return multidimensional Discrete Sine Transform along the specified axes.

## Differential and pseudo-differential operators¶

 diff(x[, order, period, _cache]) Return kth 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. 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). next_fast_len(target) Find the next fast size of input data to fft, for zero-padding, etc.

Note that fftshift, ifftshift and fftfreq are numpy functions exposed by fftpack; importing them from numpy should be preferred.

## Convolutions (scipy.fftpack.convolve)¶

 convolve(x,omega,[swap_real_imag,overwrite_x]) Wrapper for convolve. convolve_z(x,omega_real,omega_imag,[overwrite_x]) Wrapper for convolve_z. Wrapper for init_convolution_kernel.