(ph*3xSr/SQrSSKJr SSKJr SSjrSSjrSS jrSS jr SS jr SS jr SS jr SSjr g)z) Discrete Fourier Transforms - _basic.py )fftifftfftnifftnrfftirfftfft2ifft2) _pocketfft) _good_shapeNc4[R"XUSU5$)a8 Return discrete Fourier transform of real or complex sequence. The returned complex array contains ``y(0), y(1),..., y(n-1)``, where ``y(j) = (x * exp(-2*pi*sqrt(-1)*j*np.arange(n)/n)).sum()``. Parameters ---------- x : array_like Array to Fourier transform. n : int, optional Length of the Fourier transform. If ``n < x.shape[axis]``, `x` is truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The default results in ``n = x.shape[axis]``. axis : int, optional Axis along which the fft's are computed; the default is over the last axis (i.e., ``axis=-1``). overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. Returns ------- z : complex ndarray with the elements:: [y(0),y(1),..,y(n/2),y(1-n/2),...,y(-1)] if n is even [y(0),y(1),..,y((n-1)/2),y(-(n-1)/2),...,y(-1)] if n is odd where:: y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1 See Also -------- ifft : Inverse FFT rfft : FFT of a real sequence Notes ----- The packing of the result is "standard": If ``A = fft(a, n)``, then ``A[0]`` contains the zero-frequency term, ``A[1:n/2]`` contains the positive-frequency terms, and ``A[n/2:]`` contains the negative-frequency terms, in order of decreasingly negative frequency. So ,for an 8-point transform, the frequencies of the result are [0, 1, 2, 3, -4, -3, -2, -1]. To rearrange the fft output so that the zero-frequency component is centered, like [-4, -3, -2, -1, 0, 1, 2, 3], use `fftshift`. Both single and double precision routines are implemented. Half precision inputs will be converted to single precision. Non-floating-point inputs will be converted to double precision. Long-double precision inputs are not supported. This function is most efficient when `n` is a power of two, and least efficient when `n` is prime. Note that if ``x`` is real-valued, then ``A[j] == A[n-j].conjugate()``. If ``x`` is real-valued and ``n`` is even, then ``A[n/2]`` is real. If the data type of `x` is real, a "real FFT" algorithm is automatically used, which roughly halves the computation time. To increase efficiency a little further, use `rfft`, which does the same calculation, but only outputs half of the symmetrical spectrum. If the data is both real and symmetrical, the `dct` can again double the efficiency by generating half of the spectrum from half of the signal. Examples -------- >>> import numpy as np >>> from scipy.fftpack import fft, ifft >>> x = np.arange(5) >>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy. True N)r rxnaxis overwrite_xs G/var/www/html/venv/lib/python3.13/site-packages/scipy/fftpack/_basic.pyrr sX >>!dK 88c4[R"XUSU5$)a Return discrete inverse Fourier transform of real or complex sequence. The returned complex array contains ``y(0), y(1),..., y(n-1)``, where ``y(j) = (x * exp(2*pi*sqrt(-1)*j*np.arange(n)/n)).mean()``. Parameters ---------- x : array_like Transformed data to invert. n : int, optional Length of the inverse Fourier transform. If ``n < x.shape[axis]``, `x` is truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The default results in ``n = x.shape[axis]``. axis : int, optional Axis along which the ifft's are computed; the default is over the last axis (i.e., ``axis=-1``). overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. Returns ------- ifft : ndarray of floats The inverse discrete Fourier transform. See Also -------- fft : Forward FFT Notes ----- Both single and double precision routines are implemented. Half precision inputs will be converted to single precision. Non-floating-point inputs will be converted to double precision. Long-double precision inputs are not supported. This function is most efficient when `n` is a power of two, and least efficient when `n` is prime. If the data type of `x` is real, a "real IFFT" algorithm is automatically used, which roughly halves the computation time. Examples -------- >>> from scipy.fftpack import fft, ifft >>> import numpy as np >>> x = np.arange(5) >>> np.allclose(ifft(fft(x)), x, atol=1e-15) # within numerical accuracy. True N)r rrs rrr[sj ??1t[ 99rc4[R"XUSU5$)a Discrete Fourier transform of a real sequence. Parameters ---------- x : array_like, real-valued The data to transform. n : int, optional Defines the length of the Fourier transform. If `n` is not specified (the default) then ``n = x.shape[axis]``. If ``n < x.shape[axis]``, `x` is truncated, if ``n > x.shape[axis]``, `x` is zero-padded. axis : int, optional The axis along which the transform is applied. The default is the last axis. overwrite_x : bool, optional If set to true, the contents of `x` can be overwritten. Default is False. Returns ------- z : real ndarray The returned real array contains:: [y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2))] if n is even [y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2)),Im(y(n/2))] if n is odd where:: y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k*2*pi/n) j = 0..n-1 See Also -------- fft, irfft, scipy.fft.rfft Notes ----- Within numerical accuracy, ``y == rfft(irfft(y))``. Both single and double precision routines are implemented. Half precision inputs will be converted to single precision. Non-floating-point inputs will be converted to double precision. Long-double precision inputs are not supported. To get an output with a complex datatype, consider using the newer function `scipy.fft.rfft`. Examples -------- >>> from scipy.fftpack import fft, rfft >>> a = [9, -9, 1, 3] >>> fft(a) array([ 4. +0.j, 8.+12.j, 16. +0.j, 8.-12.j]) >>> rfft(a) array([ 4., 8., 12., 16.]) N)r rfft_fftpackrs rrrst  " "1t[ AArc4[R"XUSU5$)a Return inverse discrete Fourier transform of real sequence x. The contents of `x` are interpreted as the output of the `rfft` function. Parameters ---------- x : array_like Transformed data to invert. n : int, optional Length of the inverse Fourier transform. If n < x.shape[axis], x is truncated. If n > x.shape[axis], x is zero-padded. The default results in n = x.shape[axis]. axis : int, optional Axis along which the ifft's are computed; the default is over the last axis (i.e., axis=-1). overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. Returns ------- irfft : ndarray of floats The inverse discrete Fourier transform. See Also -------- rfft, ifft, scipy.fft.irfft Notes ----- The returned real array contains:: [y(0),y(1),...,y(n-1)] where for n is even:: y(j) = 1/n (sum[k=1..n/2-1] (x[2*k-1]+sqrt(-1)*x[2*k]) * exp(sqrt(-1)*j*k* 2*pi/n) + c.c. + x[0] + (-1)**(j) x[n-1]) and for n is odd:: y(j) = 1/n (sum[k=1..(n-1)/2] (x[2*k-1]+sqrt(-1)*x[2*k]) * exp(sqrt(-1)*j*k* 2*pi/n) + c.c. + x[0]) c.c. denotes complex conjugate of preceding expression. For details on input parameters, see `rfft`. To process (conjugate-symmetric) frequency-domain data with a complex datatype, consider using the newer function `scipy.fft.irfft`. Examples -------- >>> from scipy.fftpack import rfft, irfft >>> a = [1.0, 2.0, 3.0, 4.0, 5.0] >>> irfft(a) array([ 2.6 , -3.16405192, 1.24398433, -1.14955713, 1.46962473]) >>> irfft(rfft(a)) array([1., 2., 3., 4., 5.]) N)r irfft_fftpackrs rrrsD  # #A$k BBrcL[XU5n[R"XUSU5$)a Return multidimensional discrete Fourier transform. The returned array contains:: y[j_1,..,j_d] = sum[k_1=0..n_1-1, ..., k_d=0..n_d-1] x[k_1,..,k_d] * prod[i=1..d] exp(-sqrt(-1)*2*pi/n_i * j_i * k_i) where d = len(x.shape) and n = x.shape. Parameters ---------- x : array_like The (N-D) array to transform. shape : int or array_like of ints or None, optional The shape of the result. If both `shape` and `axes` (see below) are None, `shape` is ``x.shape``; if `shape` is None but `axes` is not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the ith dimension is truncated to length ``shape[i]``. If any element of `shape` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints or None, optional The axes of `x` (`y` if `shape` is not None) along which the transform is applied. The default is over all axes. overwrite_x : bool, optional If True, the contents of `x` can be destroyed. Default is False. Returns ------- y : complex-valued N-D NumPy array The (N-D) DFT of the input array. See Also -------- ifftn Notes ----- If ``x`` is real-valued, then ``y[..., j_i, ...] == y[..., n_i-j_i, ...].conjugate()``. Both single and double precision routines are implemented. Half precision inputs will be converted to single precision. Non-floating-point inputs will be converted to double precision. Long-double precision inputs are not supported. Examples -------- >>> import numpy as np >>> from scipy.fftpack import fftn, ifftn >>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16)) >>> np.allclose(y, fftn(ifftn(y))) True N)r r rrshapeaxesrs rrrs&v $ 'E ??1T4 ==rcL[XU5n[R"XUSU5$)am Return inverse multidimensional discrete Fourier transform. The sequence can be of an arbitrary type. The returned array contains:: y[j_1,..,j_d] = 1/p * sum[k_1=0..n_1-1, ..., k_d=0..n_d-1] x[k_1,..,k_d] * prod[i=1..d] exp(sqrt(-1)*2*pi/n_i * j_i * k_i) where ``d = len(x.shape)``, ``n = x.shape``, and ``p = prod[i=1..d] n_i``. For description of parameters see `fftn`. See Also -------- fftn : for detailed information. Examples -------- >>> from scipy.fftpack import fftn, ifftn >>> import numpy as np >>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16)) >>> np.allclose(y, ifftn(fftn(y))) True N)r r rrs rrrTs'8 $ 'E   AdD+ >>rc[XX#5$)a 2-D discrete Fourier transform. Return the 2-D discrete Fourier transform of the 2-D argument `x`. See Also -------- fftn : for detailed information. Examples -------- >>> import numpy as np >>> from scipy.fftpack import fft2, ifft2 >>> y = np.mgrid[:5, :5][0] >>> y array([[0, 0, 0, 0, 0], [1, 1, 1, 1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3], [4, 4, 4, 4, 4]]) >>> np.allclose(y, ifft2(fft2(y))) True )rrs rrrts2  ))rc[XX#5$)a 2-D discrete inverse Fourier transform of real or complex sequence. Return inverse 2-D discrete Fourier transform of arbitrary type sequence x. See `ifft` for more information. See Also -------- fft2, ifft Examples -------- >>> import numpy as np >>> from scipy.fftpack import fft2, ifft2 >>> y = np.mgrid[:5, :5][0] >>> y array([[0, 0, 0, 0, 0], [1, 1, 1, 1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3], [4, 4, 4, 4, 4]]) >>> np.allclose(y, fft2(ifft2(y))) True )rrs rr r s8  **r)NF)NNF)N)r"F)__doc____all__ scipy.fftr _helperr rrrrrrrr rrr)sK ! L9^5:p:BzBCJ<>~?@*8+r