Docstring update: fft

1 files changed, 231 insertions, 234 deletions

diff --git a/numpy/fft/fftpack.py b/numpy/fft/fftpack.py
index fd973a123..d74307f5e 100644
--- a/numpy/fft/fftpack.py
+++ b/numpy/fft/fftpack.py
@@ -1,8 +1,7 @@
 """
-Discrete Fourier Transforms - FFT.py
+Discrete Fourier Transforms
 
-The underlying code for these functions is an f2c translated and modified
-version of the FFTPACK routines.
+Routines in this module:
 
 fft(a, n=None, axis=-1)
 ifft(a, n=None, axis=-1)
@@ -18,6 +17,18 @@ fft2(a, s=None, axes=(-2,-1))
 ifft2(a, s=None, axes=(-2, -1))
 rfft2(a, s=None, axes=(-2,-1))
 irfft2(a, s=None, axes=(-2, -1))
+
+i = inverse transform
+r = transform of purely real data
+h = Hermite transform
+n = n-dimensional transform
+2 = 2-dimensional transform
+(Note: 2D routines are just nD routines with different default
+behavior.)
+
+The underlying code for these functions is an f2c-translated and modified
+version of the FFTPACK routines.
+
 """
 __all__ = ['fft','ifft', 'rfft', 'irfft', 'hfft', 'ihfft', 'rfftn',
            'irfftn', 'rfft2', 'irfft2', 'fft2', 'ifft2', 'fftn', 'ifftn',
@@ -71,15 +82,16 @@ def _raw_fft(a, n=None, axis=-1, init_function=fftpack.cffti,
 
 def fft(a, n=None, axis=-1):
     """
-    Compute the one-dimensional discrete Fourier Transform
+    Compute the one-dimensional discrete Fourier Transform.
 
     This function computes the one-dimensional *n*-point discrete Fourier
-    Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm. [CT]
+    Transform (DFT) with the efficient Fast Fourier Transform (FFT)
+    algorithm [CT].
 
     Parameters
     ----------
     a : array_like
-        Input array, can be complex
+        Input array, can be complex.
     n : int, optional
         Length of the transformed axis of the output.
         If `n` is smaller than the length of the input, the input is cropped.
@@ -98,11 +110,11 @@ def fft(a, n=None, axis=-1):
     Raises
     ------
     IndexError
-        if `axes` is larger than the last axis of `a`
+        if `axes` is larger than the last axis of `a`.
 
     See Also
     --------
-    numpy.fft : for definition of the DFT and conventions used
+    numpy.fft : for definition of the DFT and conventions used.
     ifft : The inverse of `fft`.
     fft2 : The two-dimensional FFT.
     fftn : The *n*-dimensional FFT.
@@ -111,7 +123,6 @@ def fft(a, n=None, axis=-1):
 
     Notes
     -----
-
     FFT (Fast Fourier Transform) refers to a way the discrete Fourier
     Transform (DFT) can be calculated efficiently, by using symmetries in the
     calculated terms.  The symmetry is highest when `n` is a power of 2, and
@@ -128,10 +139,7 @@ def fft(a, n=None, axis=-1):
 
     Examples
     --------
-
-    >>> from numpy import arange, pi, exp
-    >>> from numpy.fft import fft
-    >>> fft(exp(2j*pi*arange(8)/8))
+    >>> np.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8))
     array([ -3.44505240e-16 +1.14383329e-17j,
              8.00000000e+00 -5.71092652e-15j,
              2.33482938e-16 +1.22460635e-16j,
@@ -141,11 +149,10 @@ def fft(a, n=None, axis=-1):
              1.14383329e-17 +1.22460635e-16j,
              -1.64863782e-15 +1.77635684e-15j])
 
-    >>> from numpy.fft import fft, fftfreq
     >>> import matplotlib.pyplot as plt
     >>> t = np.arange(256)
-    >>> sp = fft(np.sin(t))
-    >>> freq = fftfreq(t.shape[-1])
+    >>> sp = np.fft.fft(np.sin(t))
+    >>> freq = np.fft.fftfreq(t.shape[-1])
     >>> plt.plot(freq, sp.real, freq, sp.imag)
     >>> plt.show()
 
@@ -160,24 +167,24 @@ def fft(a, n=None, axis=-1):
 
 def ifft(a, n=None, axis=-1):
     """
-    Compute the one-dimensional inverse discrete Fourier Transform
+    Compute the one-dimensional inverse discrete Fourier Transform.
 
     This function computes the inverse of the one-dimensional *n*-point
     discrete Fourier transform computed by `fft`.  In other words,
-    `ifft(fft(a)) == a` to within numerical accuracy.
+    ``ifft(fft(a)) == a`` to within numerical accuracy.
     For a general description of the algorithm and definitions,
     see `numpy.fft`.
 
     The input should be ordered in the same way as is returned by `fft`,
-    i.e., `a[0]` should contain the zero frequency term,
-    `a[1:n/2+1]` should contain the positive-frequency terms, and
-    `a[n/2+1:]` should contain the negative-frequency terms, in order of
+    i.e., ``a[0]`` should contain the zero frequency term,
+    ``a[1:n/2+1]`` should contain the positive-frequency terms, and
+    ``a[n/2+1:]`` should contain the negative-frequency terms, in order of
     decreasingly negative frequency.  See `numpy.fft` for details.
 
     Parameters
     ----------
     a : array_like
-        Input array, can be complex
+        Input array, can be complex.
     n : int, optional
         Length of the transformed axis of the output.
         If `n` is smaller than the length of the input, the input is cropped.
@@ -197,18 +204,17 @@ def ifft(a, n=None, axis=-1):
     Raises
     ------
     IndexError
-        if `axes` is larger than the last axis of `a`
+        If `axes` is larger than the last axis of `a`.
 
     See Also
     --------
-    numpy.fft : An introduction, with definitions and general explanations
+    numpy.fft : An introduction, with definitions and general explanations.
     fft : The one-dimensional (forward) FFT, of which `ifft` is the inverse
-    ifft2 : The two-dimensional inverse FFT
-    ifftn : The n-dimensional inverse FFT
+    ifft2 : The two-dimensional inverse FFT.
+    ifftn : The n-dimensional inverse FFT.
 
     Notes
     -----
-
     If the input parameter `n` is larger than the size of the input, the input
     is padded by appending zeros at the end.  Even though this is the common
     approach, it might lead to surprising results.  If a different padding is
@@ -216,23 +222,20 @@ def ifft(a, n=None, axis=-1):
 
     Examples
     --------
-
-    >>> from numpy.fft import ifft
-    >>> ifft([0, 4, 0, 0])
+    >>> np.fft.ifft([0, 4, 0, 0])
     array([ 1.+0.j,  0.+1.j, -1.+0.j,  0.-1.j])
 
-    >>> from numpy import exp, pi, arange, zeros
+    Create and plot a band-limited signal with random phases:
+
     >>> import matplotlib.pyplot as plt
-    >>> t = arange(400)
-    >>> n = zeros((400,), dtype=complex)
-    >>> n[40:60] = exp(1j*np.random.uniform(0, 2*pi, (20,)))
+    >>> t = np.arange(400)
+    >>> n = np.zeros((400,), dtype=complex)
+    >>> n[40:60] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,)))
     >>> s = np.fft.ifft(n)
     >>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--')
     >>> plt.legend(('real', 'imaginary'))
     >>> plt.show()
 
-    Creates and plots a band-limited signal with random phases.
-
     """
 
     a = asarray(a).astype(complex)
@@ -256,10 +259,10 @@ def rfft(a, n=None, axis=-1):
     n : int, optional
         Number of points along transformation axis in the input to use.
         If `n` is smaller than the length of the input, the input is cropped.
-        If it is larger, the input is padded with zeros.  If `n` is not given,
+        If it is larger, the input is padded with zeros. If `n` is not given,
         the length of the input (along the axis specified by `axis`) is used.
     axis : int, optional
-        Axis over which to compute the FFT.  If not given, the last axis is
+        Axis over which to compute the FFT. If not given, the last axis is
         used.
 
     Returns
@@ -267,52 +270,50 @@ def rfft(a, n=None, axis=-1):
     out : complex ndarray
         The truncated or zero-padded input, transformed along the axis
         indicated by `axis`, or the last one if `axis` is not specified.
-        The length of the transformed axis is `n/2+1`.
+        The length of the transformed axis is ``n/2+1``.
 
     Raises
     ------
     IndexError
-        if `axis` is larger than the last axis of `a`
+        If `axis` is larger than the last axis of `a`.
 
     See Also
     --------
-    numpy.fft : for definition of the DFT and conventions used
-    irfft : The inverse of `rfft`
-    fft : The one-dimensional FFT of general (complex) input
+    numpy.fft : For definition of the DFT and conventions used.
+    irfft : The inverse of `rfft`.
+    fft : The one-dimensional FFT of general (complex) input.
     fftn : The *n*-dimensional FFT.
     rfftn : The *n*-dimensional FFT of real input.
 
     Notes
     -----
-
     When the DFT is computed for purely real input, the output is
     Hermite-symmetric, i.e. the negative frequency terms are just the complex
     conjugates of the corresponding positive-frequency terms, and the
     negative-frequency terms are therefore redundant.  This function does not
     compute the negative frequency terms, and the length of the transformed
-    axis of the output is therefore `n/2+1`.
+    axis of the output is therefore ``n/2+1``.
 
-    When `A = rfft(a)`, `A[0]` contains the zero-frequency term, which must be
-    purely real due to the Hermite symmetry.
+    When ``A = rfft(a)``, ``A[0]`` contains the zero-frequency term, which
+    must be purely real due to the Hermite symmetry.
 
-    If `n` is even, `A[-1]` contains the term for frequencies `n/2` and `-n/2`,
-    and must also be purely real.  If `n` is odd, `A[-1]` contains the term
-    for frequency `A[(n-1)/2]`, and is complex in the general case.
+    If `n` is even, ``A[-1]`` contains the term for frequencies ``n/2`` and
+    ``-n/2``, and must also be purely real.  If `n` is odd, ``A[-1]``
+    contains the term for frequency ``A[(n-1)/2]``, and is complex in the
+    general case.
 
     If the input `a` contains an imaginary part, it is silently discarded.
 
     Examples
     --------
-
-    >>> from numpy.fft import fft, rfft
-    >>> fft([0, 1, 0, 0])
+    >>> np.fft.fft([0, 1, 0, 0])
     array([ 1.+0.j,  0.-1.j, -1.+0.j,  0.+1.j])
-    >>> rfft([0, 1, 0, 0])
+    >>> np.fft.rfft([0, 1, 0, 0])
     array([ 1.+0.j,  0.-1.j, -1.+0.j])
 
     Notice how the final element of the `fft` output is the complex conjugate
-    of the second element, for real input.  For `rfft`, this symmetry is
-    exploited to compute only the nonnegative frequency terms.
+    of the second element, for real input. For `rfft`, this symmetry is
+    exploited to compute only the non-negative frequency terms.
 
     """
 
@@ -326,8 +327,8 @@ def irfft(a, n=None, axis=-1):
 
     This function computes the inverse of the one-dimensional *n*-point
     discrete Fourier Transform of real input computed by `rfft`.
-    In other words, `irfft(rfft(a), len(a)) == a` to within numerical accuracy.
-    (See Notes below for why `len(a)` is necessary here.)
+    In other words, ``irfft(rfft(a), len(a)) == a`` to within numerical
+    accuracy. (See Notes below for why ``len(a)`` is necessary here.)
 
     The input is expected to be in the form returned by `rfft`, i.e. the
     real zero-frequency term followed by the complex positive frequency terms
@@ -338,60 +339,56 @@ def irfft(a, n=None, axis=-1):
     Parameters
     ----------
     a : array_like
-        Input array
+        The input array.
     n : int, optional
         Length of the transformed axis of the output.
-        For `n` output points, `n/2+1` input points are necessary.  If the
+        For `n` output points, ``n/2+1`` input points are necessary.  If the
         input is longer than this, it is cropped.  If it is shorter than this,
         it is padded with zeros.  If `n` is not given, it is determined from
-        the length of the input (along the axis specified by `axis`)
-        as explained below.
+        the length of the input (along the axis specified by `axis`).
     axis : int, optional
         Axis over which to compute the inverse FFT.
 
     Returns
     -------
-    out : real ndarray
+    out : ndarray
         The truncated or zero-padded input, transformed along the axis
         indicated by `axis`, or the last one if `axis` is not specified.
         The length of the transformed axis is `n`, or, if `n` is not given,
-        `2*(m-1)` where `m` is the length of the transformed axis of the input.
-        To get an odd number of output points, `n` must be specified.
+        ``2*(m-1)`` where `m` is the length of the transformed axis of the
+        input. To get an odd number of output points, `n` must be specified.
 
     Raises
     ------
     IndexError
-        if `axis` is larger than the last axis of `a`
+        If `axis` is larger than the last axis of `a`.
 
     See Also
     --------
-    numpy.fft : for definition of the DFT and conventions used
+    numpy.fft : For definition of the DFT and conventions used.
     rfft : The one-dimensional FFT of real input, of which `irfft` is inverse.
-    fft : The one-dimensional FFT
+    fft : The one-dimensional FFT.
     irfft2 : The inverse of the two-dimensional FFT of real input.
     irfftn : The inverse of the *n*-dimensional FFT of real input.
 
     Notes
     -----
-
     Returns the real valued `n`-point inverse discrete Fourier transform
-    of `a`, where `a` contains the nonnegative frequency terms of a
+    of `a`, where `a` contains the non-negative frequency terms of a
     Hermite-symmetric sequence. `n` is the length of the result, not the
     input.
 
     If you specify an `n` such that `a` must be zero-padded or truncated, the
     extra/removed values will be added/removed at high frequencies. One can
     thus resample a series to `m` points via Fourier interpolation by:
-    `a_resamp = irfft(rfft(a), m)`.
+    ``a_resamp = irfft(rfft(a), m)``.
 
 
     Examples
     --------
-
-    >>> from numpy.fft import ifft, irfft
-    >>> ifft([1, -1j, -1, 1j])
+    >>> np.fft.ifft([1, -1j, -1, 1j])
     array([ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j])
-    >>> irfft([1, -1j, -1])
+    >>> np.fft.irfft([1, -1j, -1])
     array([ 0.,  1.,  0.,  0.])
 
     Notice how the last term in the input to the ordinary `ifft` is the
@@ -410,31 +407,46 @@ def irfft(a, n=None, axis=-1):
 
 def hfft(a, n=None, axis=-1):
     """
-    Compute the fft of a signal which spectrum has Hermitian symmetry.
+    Compute the FFT of a signal whose spectrum has Hermitian symmetry.
 
     Parameters
     ----------
-    a : array
-        input array
-    n : int
-        length of the hfft
-    axis : int
-        axis over which to compute the hfft
+    a : array_like
+        The input array.
+    n : int, optional
+        The length of the FFT.
+    axis : int, optional
+        The axis over which to compute the FFT, assuming Hermitian symmetry
+        of the spectrum. Default is the last axis.
+
+    Returns
+    -------
+    out : ndarray
+        The transformed input.
 
     See also
     --------
-    rfft
-    ihfft
+    rfft : Compute the one-dimensional FFT for real input.
+    ihfft : The inverse of `hfft`.
 
     Notes
     -----
-    These are a pair analogous to rfft/irfft, but for the
+    `hfft`/`ihfft` are a pair analogous to `rfft`/`irfft`, but for the
     opposite case: here the signal is real in the frequency domain and has
-    Hermite symmetry in the time domain. So here it's hermite_fft for which
-    you must supply the length of the result if it is to be odd.
+    Hermite symmetry in the time domain. So here it's `hfft` for which
+    you must supply the length of the result if it is to be odd:
+    ``ihfft(hfft(a), len(a)) == a``, within numerical accuracy.
 
-    ihfft(hfft(a), len(a)) == a
-    within numerical accuracy.
+    Examples
+    --------
+    >>> signal = np.array([[1, 1.j], [-1.j, 2]])
+    >>> np.conj(signal.T) - signal   # check Hermitian symmetry
+    array([[ 0.-0.j,  0.+0.j],
+           [ 0.+0.j,  0.-0.j]])
+    >>> freq_spectrum = np.fft.hfft(signal)
+    >>> freq_spectrum
+    array([[ 1.,  1.],
+           [ 2., -2.]])
 
     """
 
@@ -510,7 +522,7 @@ def _raw_fftnd(a, s=None, axes=None, function=fft):
 
 def fftn(a, s=None, axes=None):
     """
-    Compute the N-dimensional discrete Fourier Transform
+    Compute the N-dimensional discrete Fourier Transform.
 
     This function computes the *N*-dimensional discrete Fourier Transform over
     any number of axes in an *M*-dimensional array by means of the Fast Fourier
@@ -519,7 +531,7 @@ def fftn(a, s=None, axes=None):
     Parameters
     ----------
     a : array_like
-        Input array, can be complex
+        Input array, can be complex.
     s : sequence of ints, optional
         Shape (length of each transformed axis) of the output
         (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.).
@@ -529,7 +541,7 @@ def fftn(a, s=None, axes=None):
         if `s` is not given, the shape of the input (along the axes specified
         by `axes`) is used.
     axes : sequence of ints, optional
-        Axes over which to compute the FFT.  If not given, the last `len(s)`
+        Axes over which to compute the FFT.  If not given, the last ``len(s)``
         axes are used, or all axes if `s` is also not specified.
         Repeated indices in `axes` means that the transform over that axis is
         performed multiple times.
@@ -544,9 +556,9 @@ def fftn(a, s=None, axes=None):
     Raises
     ------
     ValueError
-        if `s` and `axes` have different length.
+        If `s` and `axes` have different length.
     IndexError
-        if an element of `axes` is larger than than the number of axes of `a`.
+        If an element of `axes` is larger than than the number of axes of `a`.
 
     See Also
     --------
@@ -556,11 +568,10 @@ def fftn(a, s=None, axes=None):
     fft : The one-dimensional FFT, with definitions and conventions used.
     rfftn : The *n*-dimensional FFT of real input.
     fft2 : The two-dimensional FFT.
-    fftshift : shifts zero-frequency terms to centre of array
+    fftshift : Shifts zero-frequency terms to centre of array
 
     Notes
     -----
-
     The output, analogously to `fft`, contains the term for zero frequency in
     the low-order corner of all axes, the positive frequency terms in the
     first half of all axes, the term for the Nyquist frequency in the middle
@@ -571,10 +582,8 @@ def fftn(a, s=None, axes=None):
 
     Examples
     --------
-    >>> from numpy import mgrid
-    >>> from numpy.fft import fftn
-    >>> a = mgrid[:3,:3,:3][0]
-    >>> fftn(a, axes=(1,2))
+    >>> a = mgrid[:3, :3, :3][0]
+    >>> np.fft.fftn(a, axes=(1, 2))
     array([[[  0.+0.j,   0.+0.j,   0.+0.j],
             [  0.+0.j,   0.+0.j,   0.+0.j],
             [  0.+0.j,   0.+0.j,   0.+0.j]],
@@ -584,19 +593,18 @@ def fftn(a, s=None, axes=None):
            [[ 18.+0.j,   0.+0.j,   0.+0.j],
             [  0.+0.j,   0.+0.j,   0.+0.j],
             [  0.+0.j,   0.+0.j,   0.+0.j]]])
-    >>> fftn(a, (2,2), axes=(0,1))
+    >>> np.fft.fftn(a, (2, 2), axes=(0, 1))
     array([[[ 2.+0.j,  2.+0.j,  2.+0.j],
             [ 0.+0.j,  0.+0.j,  0.+0.j]],
            [[-2.+0.j, -2.+0.j, -2.+0.j],
             [ 0.+0.j,  0.+0.j,  0.+0.j]]])
 
-    >>> from numpy import meshgrid, pi, arange, sin, cos, log, abs
-    >>> from numpy.fft import fftn, fftshift
     >>> import matplotlib.pyplot as plt
-    >>> [X, Y] = np.meshgrid(2*pi*arange(200)/12, 2*pi*arange(200)/34)
+    >>> [X, Y] = np.meshgrid(2 * np.pi * np.arange(200) / 12,
+                             2 * np.pi * np.arange(200) / 34)
     >>> S = np.sin(X) + np.cos(Y) + np.random.uniform(0, 1, X.shape)
     >>> FS = np.fft.fftn(S)
-    >>> plt.imshow(np.log(np.abs(fftshift(FS))**2))
+    >>> plt.imshow(np.log(np.abs(np.fft.fftshift(FS))**2))
     >>> plt.show()
 
     """
@@ -605,12 +613,12 @@ def fftn(a, s=None, axes=None):
 
 def ifftn(a, s=None, axes=None):
     """
-    Compute the N-dimensional inverse discrete Fourier Transform
+    Compute the N-dimensional inverse discrete Fourier Transform.
 
     This function computes the inverse of the N-dimensional discrete
     Fourier Transform over any number of axes in an M-dimensional array by
     means of the Fast Fourier Transform (FFT).  In other words,
-    `ifftn(fftn(a)) == a` to within numerical accuracy.
+    ``ifftn(fftn(a)) == a`` to within numerical accuracy.
     For a description of the definitions and conventions used, see `numpy.fft`.
 
     The input, analogously to `ifft`, should be ordered in the same way as is
@@ -623,17 +631,17 @@ def ifftn(a, s=None, axes=None):
     Parameters
     ----------
     a : array_like
-        Input array, can be complex
+        Input array, can be complex.
     s : sequence of ints, optional
         Shape (length of each transformed axis) of the output
-        (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.).
-        This corresponds to `n` for `ifft(x, n)`.
+        (``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.).
+        This corresponds to ``n`` for ``ifft(x, n)``.
         Along any axis, if the given shape is smaller than that of the input,
         the input is cropped.  If it is larger, the input is padded with zeros.
         if `s` is not given, the shape of the input (along the axes specified
-        by `axes`) is used.  See notes for issue on ifft zero padding.
+        by `axes`) is used.  See notes for issue on `ifft` zero padding.
     axes : sequence of ints, optional
-        Axes over which to compute the IFFT.  If not given, the last `len(s)`
+        Axes over which to compute the IFFT.  If not given, the last ``len(s)``
         axes are used, or all axes if `s` is also not specified.
         Repeated indices in `axes` means that the inverse transform over that
         axis is performed multiple times.
@@ -648,9 +656,9 @@ def ifftn(a, s=None, axes=None):
     Raises
     ------
     ValueError
-        if `s` and `axes` have different length.
+        If `s` and `axes` have different length.
     IndexError
-        if an element of `axes` is larger than than the number of axes of `a`.
+        If an element of `axes` is larger than than the number of axes of `a`.
 
     See Also
     --------
@@ -659,12 +667,11 @@ def ifftn(a, s=None, axes=None):
     fftn : The forward *n*-dimensional FFT, of which `ifftn` is the inverse.
     ifft : The one-dimensional inverse FFT.
     ifft2 : The two-dimensional inverse FFT.
-    ifftshift : undoes `fftshift`, shifts zero-frequency terms to beginning
-        of array
+    ifftshift : Undoes `fftshift`, shifts zero-frequency terms to beginning
+        of array.
 
     Notes
     -----
-
     See `numpy.fft` for definitions and conventions used.
 
     Zero-padding, analogously with `ifft`, is performed by appending zeros to
@@ -674,27 +681,23 @@ def ifftn(a, s=None, axes=None):
 
     Examples
     --------
-    >>> from numpy import eye
-    >>> from numpy.fft import ifftn, fftn
-    >>> a = eye(4)
-    >>> ifftn(fftn(a, axes=(0,)),axes=(1,))
+    >>> a = np.eye(4)
+    >>> np.fft.ifftn(np.fft.fftn(a, axes=(0,)), axes=(1,))
     array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],
            [ 0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j],
            [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],
            [ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j]])
 
-    >>> from numpy import zeros, exp, pi
-    >>> from numpy.random import uniform
-    >>> from numpy.fft import ifftn
+
+    Create and plot an image with band-limited frequency content:
+
     >>> import matplotlib.pyplot as plt
     >>> n = np.zeros((200,200), dtype=complex)
-    >>> n[60:80,20:40] = exp(1j*uniform(0, 2*pi, (20,20)))
+    >>> n[60:80, 20:40] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20, 20)))
     >>> im = np.fft.ifftn(n).real
     >>> plt.imshow(im)
     >>> plt.show()
 
-    Creates and plots an image with band-limited frequency content
-
     """
 
     return _raw_fftnd(a, s, axes, ifft)
@@ -722,7 +725,7 @@ def fft2(a, s=None, axes=(-2,-1)):
         if `s` is not given, the shape of the input (along the axes specified
         by `axes`) is used.
     axes : sequence of ints, optional
-        Axes over which to compute the FFT.  If not given, the last 2
+        Axes over which to compute the FFT.  If not given, the last two
         axes are used.  A repeated index in `axes` means the transform over
         that axis is performed multiple times.  A one-element sequence means
         that a one-dimensional FFT is performed.
@@ -736,25 +739,24 @@ def fft2(a, s=None, axes=(-2,-1)):
     Raises
     ------
     ValueError
-        if `s` and `axes` have different length, or
-        `axes` not given and `len(s) != 2`
+        If `s` and `axes` have different length, or `axes` not given and
+        ``len(s) != 2``.
     IndexError
-        if an element of `axes` is larger than than the number of axes of `a`.
+        If an element of `axes` is larger than than the number of axes of `a`.
 
     See Also
     --------
     numpy.fft : Overall view of discrete Fourier transforms, with definitions
          and conventions used.
-    ifft2 : The inverse two-dimensional FFT
-    fft : The one-dimensional FFT
-    fftn : The *n*-dimensional FFT
-    fftshift : shifts zero-frequency terms to centre of array.
+    ifft2 : The inverse two-dimensional FFT.
+    fft : The one-dimensional FFT.
+    fftn : The *n*-dimensional FFT.
+    fftshift : Shifts zero-frequency terms to the center of the array.
         For two-dimensional input, swaps first and third quadrants, and second
         and fourth quadrants.
 
     Notes
     -----
-
     `fft2` is just `fftn` with a different default for `axes`.
 
     The output, analogously to `fft`, contains the term for zero frequency in
@@ -769,10 +771,8 @@ def fft2(a, s=None, axes=(-2,-1)):
 
     Examples
     --------
-    >>> from numpy import mgrid
-    >>> from numpy.fft import fft2
-    >>> a = mgrid[:5, :5][0]
-    >>> fft2(a)
+    >>> a = np.mgrid[:5, :5][0]
+    >>> np.fft.fft2(a)
     array([[  0.+0.j,   0.+0.j,   0.+0.j,   0.+0.j,   0.+0.j],
            [  5.+0.j,   0.+0.j,   0.+0.j,   0.+0.j,   0.+0.j],
            [ 10.+0.j,   0.+0.j,   0.+0.j,   0.+0.j,   0.+0.j],
@@ -786,11 +786,11 @@ def fft2(a, s=None, axes=(-2,-1)):
 
 def ifft2(a, s=None, axes=(-2,-1)):
     """
-    Compute the 2-dimensional inverse discrete Fourier Transform
+    Compute the 2-dimensional inverse discrete Fourier Transform.
 
     This function computes the inverse of the 2-dimensional discrete Fourier
     Transform over any number of axes in an M-dimensional array by means of
-    the Fast Fourier Transform (FFT).  In other words, `ifft2(fft2(a)) == a`
+    the Fast Fourier Transform (FFT).  In other words, ``ifft2(fft2(a)) == a``
     to within numerical accuracy.  By default, the inverse transform is
     computed over the last two axes of the input array.
 
@@ -804,16 +804,16 @@ def ifft2(a, s=None, axes=(-2,-1)):
     Parameters
     ----------
     a : array_like
-        Input array, can be complex
+        Input array, can be complex.
     s : sequence of ints, optional
-        Shape (length of each axis) of the output (`s[0]` refers to axis 0,
-        `s[1]` to axis 1, etc.).  This corresponds to `n` for `ifft(x, n)`.
+        Shape (length of each axis) of the output (``s[0]`` refers to axis 0,
+        ``s[1]`` to axis 1, etc.).  This corresponds to `n` for ``ifft(x, n)``.
         Along each axis, if the given shape is smaller than that of the input,
         the input is cropped.  If it is larger, the input is padded with zeros.
         if `s` is not given, the shape of the input (along the axes specified
-        by `axes`) is used.  See notes for issue on ifft zero padding.
+        by `axes`) is used.  See notes for issue on `ifft` zero padding.
     axes : sequence of ints, optional
-        Axes over which to compute the FFT.  If not given, the last 2
+        Axes over which to compute the FFT.  If not given, the last two
         axes are used.  A repeated index in `axes` means the transform over
         that axis is performed multiple times.  A one-element sequence means
         that a one-dimensional FFT is performed.
@@ -827,10 +827,10 @@ def ifft2(a, s=None, axes=(-2,-1)):
     Raises
     ------
     ValueError
-        if `s` and `axes` have different length, or
-        `axes` not given and `len(s) != 2`
+        If `s` and `axes` have different length, or `axes` not given and
+        ``len(s) != 2``.
     IndexError
-        if an element of `axes` is larger than than the number of axes of `a`.
+        If an element of `axes` is larger than than the number of axes of `a`.
 
     See Also
     --------
@@ -838,12 +838,11 @@ def ifft2(a, s=None, axes=(-2,-1)):
          and conventions used.
     fft2 : The forward 2-dimensional FFT, of which `ifft2` is the inverse.
     ifftn : The inverse of the *n*-dimensional FFT.
-    fft : The one-dimensional FFT
+    fft : The one-dimensional FFT.
     ifft : The one-dimensional inverse FFT.
 
     Notes
     -----
-
     `ifft2` is just `ifftn` with a different default for `axes`.
 
     See `ifftn` for details and a plotting example, and `numpy.fft` for
@@ -856,10 +855,8 @@ def ifft2(a, s=None, axes=(-2,-1)):
 
     Examples
     --------
-    >>> from numpy import eye
-    >>> from numpy.fft import ifft2
-    >>> a = 4*eye(4)
-    >>> ifft2(a)
+    >>> a = 4 * np.eye(4)
+    >>> np.fft.ifft2(a)
     array([[ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],
            [ 0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j],
            [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],
@@ -883,18 +880,18 @@ def rfftn(a, s=None, axes=None):
     Parameters
     ----------
     a : array_like
-        Input array, taken to be real
+        Input array, taken to be real.
     s : sequence of ints, optional
         Shape (length along each transformed axis) to use from the input.
-        (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.).
-        The final element of `s` corresponds to `n` for `rfft(x, n)`, while
-        for the remaining axes, it corresponds to `n` for `fft(x, n)`.
+        (``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.).
+        The final element of `s` corresponds to `n` for ``rfft(x, n)``, while
+        for the remaining axes, it corresponds to `n` for ``fft(x, n)``.
         Along any axis, if the given shape is smaller than that of the input,
         the input is cropped.  If it is larger, the input is padded with zeros.
         if `s` is not given, the shape of the input (along the axes specified
         by `axes`) is used.
     axes : sequence of ints, optional
-        Axes over which to compute the FFT.  If not given, the last `len(s)`
+        Axes over which to compute the FFT.  If not given, the last ``len(s)``
         axes are used, or all axes if `s` is also not specified.
 
     Returns
@@ -903,16 +900,16 @@ def rfftn(a, s=None, axes=None):
         The truncated or zero-padded input, transformed along the axes
         indicated by `axes`, or by a combination of `s` and `a`,
         as explained in the parameters section above.
-        The length of the last axis transformed will be `s[-1]//2+1`,
+        The length of the last axis transformed will be ``s[-1]//2+1``,
         while the remaining transformed axes will have lengths according to
         `s`, or unchanged from the input.
 
     Raises
     ------
     ValueError
-        if `s` and `axes` have different length.
+        If `s` and `axes` have different length.
     IndexError
-        if an element of `axes` is larger than than the number of axes of `a`.
+        If an element of `axes` is larger than than the number of axes of `a`.
 
     See Also
     --------
@@ -925,7 +922,6 @@ def rfftn(a, s=None, axes=None):
 
     Notes
     -----
-
     The transform for real input is performed over the last transformation
     axis, as by `rfft`, then the transform over the remaining axes is
     performed as by `fftn`.  The order of the output is as for `rfft` for the
@@ -936,26 +932,18 @@ def rfftn(a, s=None, axes=None):
 
     Examples
     --------
-    >>> from numpy import ones
-    >>> from numpy.fft import rfftn
-    >>> a = ones((3,3,3))
-    >>> rfftn(a)
-    array([[[ 27.+0.j,   0.+0.j],
-            [  0.+0.j,   0.+0.j],
-            [  0.+0.j,   0.+0.j]],
-           [[  0.+0.j,   0.+0.j],
-            [  0.+0.j,   0.+0.j],
-            [  0.+0.j,   0.+0.j]],
-           [[  0.+0.j,   0.+0.j],
-            [  0.+0.j,   0.+0.j],
-            [  0.+0.j,   0.+0.j]]])
-    >>> rfftn(a, axes=(2,0))
-    array([[[ 9.+0.j,  0.+0.j,  0.+0.j],
-            [ 9.+0.j,  0.+0.j,  0.+0.j],
-            [ 9.+0.j,  0.+0.j,  0.+0.j]],
-           [[ 0.+0.j,  0.+0.j,  0.+0.j],
-            [ 0.+0.j,  0.+0.j,  0.+0.j],
-            [ 0.+0.j,  0.+0.j,  0.+0.j]]])
+    >>> a = np.ones((2, 2, 2))
+    >>> np.fft.rfftn(a)
+    array([[[ 8.+0.j,  0.+0.j],
+            [ 0.+0.j,  0.+0.j]],
+           [[ 0.+0.j,  0.+0.j],
+            [ 0.+0.j,  0.+0.j]]])
+
+    >>> np.fft.rfftn(a, axes=(2, 0))
+    array([[[ 4.+0.j,  0.+0.j],
+            [ 4.+0.j,  0.+0.j]],
+           [[ 0.+0.j,  0.+0.j],
+            [ 0.+0.j,  0.+0.j]]])
 
     """
 
@@ -968,21 +956,31 @@ def rfftn(a, s=None, axes=None):
 
 def rfft2(a, s=None, axes=(-2,-1)):
     """
-    Compute the 2-dimensional fft of a real array.
+    Compute the 2-dimensional FFT of a real array.
 
     Parameters
     ----------
-    a : array (real)
-        input array
-    s : sequence (int)
-        shape of the fft
-    axis : int
-        axis over which to compute the fft
+    a : array
+        Input array, taken to be real.
+    s : sequence of ints, optional
+        Shape of the FFT.
+    axes : sequence of ints, optional
+        Axes over which to compute the FFT.
+
+    Returns
+    -------
+    out : ndarray
+        The result of the real 2-D FFT.
+
+    See Also
+    --------
+    rfftn : Compute the N-dimensional discrete Fourier Transform for real
+            input.
 
     Notes
     -----
-    The 2-D fft of the real valued array a. This is really just rfftn with
-    different default behavior.
+    This is really just `rfftn` with different default behavior.
+    For more details see `rfftn`.
 
     """
 
@@ -995,9 +993,9 @@ def irfftn(a, s=None, axes=None):
     This function computes the inverse of the N-dimensional discrete
     Fourier Transform for real input over any number of axes in an
     M-dimensional array by means of the Fast Fourier Transform (FFT).  In
-    other words, `irfftn(rfftn(a), a.shape) == a` to within numerical accuracy.
-    (The `a.shape` is necessary like `len(a)` is for `irfft`, and for the same
-    reason.)
+    other words, ``irfftn(rfftn(a), a.shape) == a`` to within numerical
+    accuracy. (The ``a.shape`` is necessary like ``len(a)`` is for `irfft`,
+    and for the same reason.)
 
     The input should be ordered in the same way as is returned by `rfftn`,
     i.e. as for `irfft` for the final transformation axis, and as for `ifftn`
@@ -1009,38 +1007,38 @@ def irfftn(a, s=None, axes=None):
         Input array.
     s : sequence of ints, optional
         Shape (length of each transformed axis) of the output
-        (`s[0]` refers to axis 0, `s[1]` to axis 1, etc.).  `s` is also the
+        (``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.). `s` is also the
         number of input points used along this axis, except for the last axis,
-        where `s[-1]//2+1` points of the input are used.
+        where ``s[-1]//2+1`` points of the input are used.
         Along any axis, if the shape indicated by `s` is smaller than that of
         the input, the input is cropped.  If it is larger, the input is padded
-        with zeros.  if `s` is not given, the shape of the input (along the
+        with zeros. If `s` is not given, the shape of the input (along the
         axes specified by `axes`) is used.
     axes : sequence of ints, optional
-        Axes over which to compute the inverse FFT.  If not given, the last
+        Axes over which to compute the inverse FFT. If not given, the last
         `len(s)` axes are used, or all axes if `s` is also not specified.
         Repeated indices in `axes` means that the inverse transform over that
         axis is performed multiple times.
 
     Returns
     -------
-    out : real ndarray
+    out : ndarray
         The truncated or zero-padded input, transformed along the axes
         indicated by `axes`, or by a combination of `s` or `a`,
         as explained in the parameters section above.
         The length of each transformed axis is as given by the corresponding
         element of `s`, or the length of the input in every axis except for the
         last one if `s` is not given.  In the final transformed axis the length
-        of the output when `s` is not given is `2*(m-1)` where `m` is the
+        of the output when `s` is not given is ``2*(m-1)`` where `m` is the
         length of the final transformed axis of the input.  To get an odd
         number of output points in the final axis, `s` must be specified.
 
     Raises
     ------
     ValueError
-        if `s` and `axes` have different length.
+        If `s` and `axes` have different length.
     IndexError
-        if an element of `axes` is larger than than the number of axes of `a`.
+        If an element of `axes` is larger than than the number of axes of `a`.
 
     See Also
     --------
@@ -1052,33 +1050,21 @@ def irfftn(a, s=None, axes=None):
 
     Notes
     -----
-
     See `fft` for definitions and conventions used.
 
     See `rfft` for definitions and conventions used for real input.
 
     Examples
     --------
-    >>> from numpy import zeros
-    >>> from numpy.fft import irfftn, zeros
-    >>> a = zeros((4,4,3); a[0,0,0] = 64;
-    >>> irfftn(a)
-    array([[[ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.]],
-           [[ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.]],
-           [[ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.]],
-           [[ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.],
-            [ 1.,  1.,  1.,  1.]]])
+    >>> a = np.zeros((3, 2, 2))
+    >>> a[0, 0, 0] = 3 * 2 * 2
+    >>> np.fft.irfftn(a)
+    array([[[ 1.,  1.],
+            [ 1.,  1.]],
+           [[ 1.,  1.],
+            [ 1.,  1.]],
+           [[ 1.,  1.],
+            [ 1.,  1.]]])
 
     """
 
@@ -1091,20 +1077,31 @@ def irfftn(a, s=None, axes=None):
 
 def irfft2(a, s=None, axes=(-2,-1)):
     """
-    Compute the 2-dimensional inverse fft of a real array.
+    Compute the 2-dimensional inverse FFT of a real array.
 
     Parameters
     ----------
-    a : array (real)
-        input array
-    s : sequence (int)
-        shape of the inverse fft
-    axis : int
-        axis over which to compute the inverse fft
+    a : array_like
+        The input array
+    s : sequence of ints, optional
+        Shape of the inverse FFT.
+    axes : sequence of ints, optional
+        The axes over which to compute the inverse fft.
+        Default is the last two axes.
+
+    Returns
+    -------
+    out : ndarray
+        The result of the inverse real 2-D FFT.
+
+    See Also
+    --------
+    irfftn : Compute the inverse of the N-dimensional FFT of real input.
 
     Notes
     -----
-    This is really irfftn with different default.
+    This is really `irfftn` with different defaults.
+    For more details see `irfftn`.
 
     """