B &]\U @sdZddlmZmZmZddddddd d gZdd lmZmZdd lZd dl m Z ddl m Z dd l Z e e je e je e je e je e je e j[ ddZddZddZddZddZddZddZeejeeeje jeej eeej!e j"iZ#eejeeeje j$iZ%eej eeej!e j&eejeeeje j&iZ'ddZ(d d!Z)d"d#Z*d3d&dZ+d4d'dZ,d5d(dZ-d6d)dZ.d*d+Z/d7d,dZ0d-d.Z1d8d/dZ2d9d1d Z3d:d2d Z4d S);z( Discrete Fourier Transforms - basic.py )divisionprint_functionabsolute_importfftifftfftnifftnrfftirfftfft2ifft2)swapaxeszerosN)_fftpack)_init_nd_shape_and_axes_sortedcCst|jj|S)N) issubclassdtypetype)arrZ typeclassr2lib/python3.7/site-packages/scipy/fftpack/basic.pyistypesrcCs0||kr dSt|tjs&t|dr&dS|jdkS)z} Strict check for `arr` not sharing any data with `original`, under the assumption that arr = asarray(original) FZ __array__N) isinstancenumpyZndarrayhasattrbase)rZoriginalrrr _datacopieds rcCsJt|}|dkrdSx&dD]}x||dkr6||}q WqW||d@ S)z Is the size of FFT such that FFTPACK can handle it in single precision with sufficient accuracy? Composite numbers of 2, 3, and 5 are accepted, as FFTPACK has those rT)r)int)ncrrr _is_safe_size3s r#cOs<t|rtj||f||Stj||f||tjSdS)N)r#rZcrfftzrfftastyper complex64)xr!akwrrr _fake_crfftHsr*cOs<t|rtj||f||Stj||f||tjSdS)N)r#rZcfftzfftr%rr&)r'r!r(r)rrr _fake_cfftOsr,cOs<t|rtj||f||Stj||f||tjSdS)N)r#rr drfftr%rfloat32)r'r!r(r)rrr _fake_rfftVsr/cOsHtttt|r(tj||f||Stj||f||tj SdS)N) ralllistmapr#rZcfftndzfftndr%r&)r'shaper(r)rrr _fake_cfftnd]sr5cCst|drH|jjtjdkrH|jtjkr8tj|tjdStj||jdSt|}|jtjkrntj|tjdS|jjtjdkrt|S|SdS)zLike numpy asfarray, except that it does not modify x dtype if x is already an array with a float dtype, and do not cast complex types to real.rZAllFloat)rN) rrcharrZ typecodesZhalfZasarrayr.Zasfarray)r'Zretrrr _asfarray}s    r7cCst|j}|||krJtdgt|}td|||<|t|}|dfStdgt|}td||||<|||<t||jj}||t|<|dfSdS)z6 Internal auxiliary function for _raw_fft, _raw_fftnd.NrFT)r1r4slicelentuplerrr6)r'r!axissindexzrrr _fix_shapes    r?cCs|dkr|j|}n&||j|kr:t|||\}}|p8|}|dkrNtd||dksh|t|jdkrz|||||d}n(t||d}|||||d}t||d}|S)z8 Internal auxiliary function for fft, ifft, rfft, irfft.Nrz1Invalid number of FFT data points (%d) specified.) overwrite_x)r4r? ValueErrorr9r )r'r!r; directionrA work_function copy_maderrrr_raw_ffts   rGr@FcCst|}yt|j}Wn"tk r8td|jYnXt|tjsVt|tjsVd}|pbt ||}|dkrx|j |}n&||j |krt |||\}}|p|}|dkrtd||dks|t |j dkr|||dd|St ||d}|||dd|}t ||dS)a 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 -------- >>> from scipy.fftpack import fft, ifft >>> x = np.arange(5) >>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy. True ztype %s is not supportedrNz1Invalid number of FFT data points (%d) specified.r@r)r7 _DTYPE_TO_FFTrKeyErrorrBrrr& complex128rr4r?r9r )r'r!r;rAtmprDrErrrrs*K  cCst|}yt|j}Wn"tk r8td|jYnXt|tjsVt|tjsVd}|pbt ||}|dkrx|j |}n&||j |krt |||\}}|p|}|dkrtd||dks|t |j dkr|||dd|St ||d}|||dd|}t ||dS)aw 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 ztype %s is not supportedrNz1Invalid number of FFT data points (%d) specified.r@)r7rHrrIrBrrr&rJrr4r?r9r )r'r!r;rArKrDrErrrr#s*5  cCslt|}t|stdyt|j}Wn"tk rJtd|jYnX|pXt||}t |||d||S)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, numpy.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 related function `numpy.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.]) z"1st argument must be real sequenceztype %s is not supportedr) r7r isrealobj TypeError_DTYPE_TO_RFFTrrIrBrrG)r'r!r;rArKrDrrrr vs: cCslt|}t|stdyt|j}Wn"tk rJtd|jYnX|pXt||}t |||d||S)aM 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, numpy.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 related function `numpy.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.]) z"1st argument must be real sequenceztype %s is not supportedr@) r7rrLrMrNrrIrBrrG)r'r!r;rArKrDrrrr sB c Cs.|dk}t|||\}}|rVx(|D] }t||||\}}|p@|}q"W|||||dSx.td|jdD]} t||| | }qhWtt|j|j|j} t|j} || | <x4tt | D]$} t||| | | \}}|p|}qW||| ||d} x0tt |ddD]} t| | || } q W| S)z.Internal auxiliary function for fftnd, ifftnd.N)rArrr@) rr?rangesizerr r1ndimZonesr9) r'r<axesrCrArDZnoaxesZaxrEiZwaxesr4rFrrr _raw_fftnds&    rTcCst||||dS)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-dimensional) 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 ``scipy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the i-th 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-dimensional numpy array The (n-dimensional) 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 -------- >>> from scipy.fftpack import fftn, ifftn >>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16)) >>> np.allclose(y, fftn(ifftn(y))) True r)_raw_fftn_dispatch)r'r4rRrArrrr6s:cCsvt|}yt|j}Wn"tk r8td|jYnXt|tjsVt|tjsVd}|pbt ||}t ||||||S)Nztype %s is not supportedr) r7_DTYPE_TO_FFTNrrIrBrrr&rJrrT)r'r4rRrArCrKrDrrrrUssrUcCst||||dS)a Return inverse multi-dimensional 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 r@)rU)r'r4rRrArrrrsr@cCst||||S)z 2-D discrete Fourier transform. Return the two-dimensional discrete Fourier transform of the 2-D argument `x`. See Also -------- fftn : for detailed information. )r)r'r4rRrArrrr s cCst||||S)z 2-D discrete inverse Fourier transform of real or complex sequence. Return inverse two-dimensional discrete Fourier transform of arbitrary type sequence x. See `ifft` for more information. See also -------- fft2, ifft )r)r'r4rRrArrrr s)Nr@F)Nr@F)Nr@F)Nr@F)NNF)NNF)NrWF)NrWF)5__doc__Z __future__rrr__all__rr rrZscipy.fftpack.helperratexitregisterZdestroy_zfft_cacheZdestroy_zfftnd_cacheZdestroy_drfft_cacheZdestroy_cfft_cacheZdestroy_cfftnd_cacheZdestroy_rfft_cacherrr#r*r,r/r5rr.Zfloat64r$r&rJr+rHr-rNr3rVr7r?rGrrr r rTrrUrr r rrrrsZ               i S I P' =