ó î&]\c@`sÐdZddlmZmZmZddddddd d gZdd lmZmZdd lZd dl m Z ddl m Z dd l Z e je jƒe je jƒe je jƒe je jƒe je jƒe je jƒ[ d„Zd„Zd„Zd„Zd„Zd„Zd„Zieejejƒ6e jejejƒ6eejej ƒ6e j!ejej"ƒ6Z#ieejejƒ6e j$ejejƒ6Z%ieejej ƒ6e j&ejej"ƒ6eejejƒ6e j&ejejƒ6Z'd„Z(d„Z)d„Z*d de,d„Z-d de,d„Z.d de,d„Z/d de,d„Z0d„Z1d d e,d „Z2d!„Z3d d e,d"„Z4d d&e,d$„Z5d d'e,d%„Z6d S((s( Discrete Fourier Transforms - basic.py i(tdivisiontprint_functiontabsolute_importtffttiffttfftntifftntrffttirffttfft2tifft2(tswapaxestzerosNi(t_fftpack(t_init_nd_shape_and_axes_sortedcC`st|jj|ƒS(N(t issubclasstdtypettype(tarrt typeclass((s2lib/python2.7/site-packages/scipy/fftpack/basic.pytistypescC`sC||krtSt|tjƒ r6t|dƒr6tS|jdkS(s} Strict check for `arr` not sharing any data with `original`, under the assumption that arr = asarray(original) t __array__N(tFalset isinstancetnumpytndarraythasattrtbasetNone(Rtoriginal((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyt _datacopieds  "cC`s[t|ƒ}|dkrtSx/dD]'}x||dkrI||}q,Wq#W||d@ S(s· 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 iiii(ii(tinttTrue(tntc((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyt _is_safe_size3s   cO`sHt|ƒr"tj||||ŽStj||||ŽjtjƒSdS(N(R#R tcrffttzrffttastypeRt complex64(txR!tatkw((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyt _fake_crfftHs cO`sHt|ƒr"tj||||ŽStj||||ŽjtjƒSdS(N(R#R tcffttzfftR&RR'(R(R!R)R*((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyt _fake_cfftOs cO`sHt|ƒr"tj||||ŽStj||||ŽjtjƒSdS(N(R#R RtdrfftR&Rtfloat32(R(R!R)R*((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyt _fake_rfftVs cO`sZtjttt|ƒƒƒr4tj||||ŽStj||||Žjtj ƒSdS(N( RtalltlisttmapR#R tcfftndtzfftndR&R'(R(tshapeR)R*((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyt _fake_cfftnd]scC`sËt|dƒrf|jjtjdkrf|jtjkrPtj|dtjƒStj|d|jƒStj|ƒ}|jtjkrtj|dtjƒS|jjtjdkrÃtj|ƒS|SdS(s—Like 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.RtAllFloatN( RRtcharRt typecodesthalftasarrayR0tasfarray(R(tret((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyt _asfarray}s( cC`sÒt|jƒ}|||kretdƒgt|ƒ}td|ƒ||<|t|ƒ}|tfStdƒgt|ƒ}td||ƒ||<||| 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 stype %s is not supportedis1Invalid number of FFT data points (%d) specified.iÿÿÿÿiN(R@t _DTYPE_TO_FFTRtKeyErrorRJRRR't complex128RRR7RHRBR (R(R!RDRIttmpRLRM((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyRºs*K  $    %cC`spt|ƒ}yt|j}Wn$tk rCtd|jƒ‚nXt|tjƒpet|tjƒsqd}n|pƒt ||ƒ}|dkr¢|j |}n:||j |krÜt |||ƒ\}}|pÖ|}n|dkrûtd|ƒ‚n|dks |t |j ƒdkr6|||dd|ƒSt||dƒ}|||dd|ƒ}t||dƒS(sw 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 stype %s is not supportedis1Invalid number of FFT data points (%d) specified.iÿÿÿÿN(R@RPRRQRJRRR'RRRRR7RHRBR (R(R!RDRIRSRLRM((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyR#s*5  $    %cC`st|ƒ}tj|ƒs*tdƒ‚nyt|j}Wn$tk ratd|jƒ‚nX|ptt||ƒ}t |||d||ƒS(s£ Discrete Fourier transform of a real sequence. Parameters ---------- x : array_like, real-valued The data to transform. n : int, optional Defines the length of the Fourier transform. If `n` is not specified (the default) then ``n = x.shape[axis]``. If ``n < x.shape[axis]``, `x` is truncated, if ``n > x.shape[axis]``, `x` is zero-padded. axis : int, optional The axis along which the transform is applied. The default is the last axis. overwrite_x : bool, optional If set to true, the contents of `x` can be overwritten. Default is False. Returns ------- z : real ndarray The returned real array contains:: [y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2))] if n is even [y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2)),Im(y(n/2))] if n is odd where:: y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k*2*pi/n) j = 0..n-1 See Also -------- fft, irfft, numpy.fft.rfft Notes ----- Within numerical accuracy, ``y == rfft(irfft(y))``. Both single and double precision routines are implemented. Half precision inputs will be converted to single precision. Non floating-point inputs will be converted to double precision. Long-double precision inputs are not supported. To get an output with a complex datatype, consider using the related function `numpy.fft.rfft`. Examples -------- >>> from scipy.fftpack import fft, rfft >>> a = [9, -9, 1, 3] >>> fft(a) array([ 4. +0.j, 8.+12.j, 16. +0.j, 8.-12.j]) >>> rfft(a) array([ 4., 8., 12., 16.]) s"1st argument must be real sequencestype %s is not supportedi( R@Rt isrealobjt TypeErrort_DTYPE_TO_RFFTRRQRJRRO(R(R!RDRIRSRL((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyRvs:  cC`st|ƒ}tj|ƒs*tdƒ‚nyt|j}Wn$tk ratd|jƒ‚nX|ptt||ƒ}t |||d||ƒS(sM 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.]) s"1st argument must be real sequencestype %s is not supportediÿÿÿÿ( R@RRTRURVRRQRJRRO(R(R!RDRIRSRL((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyR¿sB  c C`s–|dk}t|||ƒ\}}|ryx6|D].}t||||ƒ\}}|p\|}q1W||||d|ƒSx9td|jdƒD]!} tj||| | ƒ}qWtt|j|j|jƒƒ} tj |jƒ} || | 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 i(t_raw_fftn_dispatch(R(R7R[RI((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyR6s:cC`sŸt|ƒ}yt|j}Wn$tk rCtd|jƒ‚nXt|tjƒpet|tjƒsqd}n|pƒt ||ƒ}t ||||||ƒS(Nstype %s is not supportedi( R@t_DTYPE_TO_FFTNRRQRJRRR'RRRR`(R(R7R[RIRKRSRL((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyRass  $ cC`st||||dƒS(s¶ 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 iÿÿÿÿ(Ra(R(R7R[RI((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyR‚siþÿÿÿcC`st||||ƒS(s 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(R7R[RI((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyR ¡s cC`st||||ƒS(sÿ 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(R7R[RI((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyR °s(iþÿÿÿiÿÿÿÿ(iþÿÿÿiÿÿÿÿ(7t__doc__t __future__RRRt__all__RR R tR tscipy.fftpack.helperRtatexittregistertdestroy_zfft_cachetdestroy_zfftnd_cachetdestroy_drfft_cachetdestroy_cfft_cachetdestroy_cfftnd_cachetdestroy_rfft_cacheRRR#R+R.R1R8RR0R%tfloat64R'R-RRRPR/RVR6RbR@RHRORRRRRRR`RRaRR R (((s2lib/python2.7/site-packages/scipy/fftpack/basic.pyts`             iSIP '=