"""

Numerical python functions written for compatability with MATLAB

commands with the same names.

MATLAB compatible functions

-------------------------------

:func:`cohere`

Coherence (normalized cross spectral density)

:func:`csd`

Cross spectral density uing Welch's average periodogram

:func:`detrend`

Remove the mean or best fit line from an array

:func:`find`

Return the indices where some condition is true;

numpy.nonzero is similar but more general.

:func:`griddata`

Interpolate irregularly distributed data to a

regular grid.

:func:`prctile`

Find the percentiles of a sequence

:func:`prepca`

Principal Component Analysis

:func:`psd`

Power spectral density uing Welch's average periodogram

:func:`rk4`

A 4th order runge kutta integrator for 1D or ND systems

:func:`specgram`

Spectrogram (spectrum over segments of time)

Miscellaneous functions

-------------------------

Functions that don't exist in MATLAB, but are useful anyway:

:func:`cohere_pairs`

Coherence over all pairs. This is not a MATLAB function, but we

compute coherence a lot in my lab, and we compute it for a lot of

pairs. This function is optimized to do this efficiently by

caching the direct FFTs.

:func:`rk4`

A 4th order Runge-Kutta ODE integrator in case you ever find

yourself stranded without scipy (and the far superior

scipy.integrate tools)

:func:`contiguous_regions`

Return the indices of the regions spanned by some logical mask

:func:`cross_from_below`

Return the indices where a 1D array crosses a threshold from below

:func:`cross_from_above`

Return the indices where a 1D array crosses a threshold from above

:func:`complex_spectrum`

Return the complex-valued frequency spectrum of a signal

:func:`magnitude_spectrum`

Return the magnitude of the frequency spectrum of a signal

:func:`angle_spectrum`

Return the angle (wrapped phase) of the frequency spectrum of a signal

:func:`phase_spectrum`

Return the phase (unwrapped angle) of the frequency spectrum of a signal

:func:`detrend_mean`

Remove the mean from a line.

:func:`demean`

Remove the mean from a line. This function is the same as as

:func:`detrend_mean` except for the default *axis*.

:func:`detrend_linear`

Remove the best fit line from a line.

:func:`detrend_none`

Return the original line.

:func:`stride_windows`

Get all windows in an array in a memory-efficient manner

:func:`stride_repeat`

Repeat an array in a memory-efficient manner

:func:`apply_window`

Apply a window along a given axis

record array helper functions

-------------------------------

A collection of helper methods for numpyrecord arrays

.. _htmlonly:

See :ref:`misc-examples-index`

:func:`rec2txt`

Pretty print a record array

:func:`rec2csv`

Store record array in CSV file

:func:`csv2rec`

Import record array from CSV file with type inspection

:func:`rec_append_fields`

Adds field(s)/array(s) to record array

:func:`rec_drop_fields`

Drop fields from record array

:func:`rec_join`

Join two record arrays on sequence of fields

:func:`recs_join`

A simple join of multiple recarrays using a single column as a key

:func:`rec_groupby`

Summarize data by groups (similar to SQL GROUP BY)

:func:`rec_summarize`

Helper code to filter rec array fields into new fields

For the rec viewer functions(e rec2csv), there are a bunch of Format

objects you can pass into the functions that will do things like color

negative values red, set percent formatting and scaling, etc.

Example usage::

r = csv2rec('somefile.csv', checkrows=0)

formatd = dict(

weight = FormatFloat(2),

change = FormatPercent(2),

cost = FormatThousands(2),

)

rec2excel(r, 'test.xls', formatd=formatd)

rec2csv(r, 'test.csv', formatd=formatd)

scroll = rec2gtk(r, formatd=formatd)

win = gtk.Window()

win.set_size_request(600,800)

win.add(scroll)

win.show_all()

gtk.main()

"""

from __future__ import (absolute_import, division, print_function,

unicode_literals)

import six

from six.moves import map, xrange, zip

import copy

import csv

import operator

import os

import warnings

import numpy as np

from matplotlib import verbose

import matplotlib.cbook as cbook

from matplotlib import docstring

from matplotlib.path import Path

import math

ma = np.ma

if six.PY3:

long = int

def logspace(xmin, xmax, N):

'''

Return N values logarithmically spaced between xmin and xmax.

Call signature::

logspace(xmin, xmax, N)

'''

return np.exp(np.linspace(np.log(xmin), np.log(xmax), N))

def _norm(x):

'''

Return sqrt(x dot x).

Call signature::

_norm(x)

'''

return np.sqrt(np.dot(x, x))

def window_hanning(x):

'''

Return x times the hanning window of len(x).

Call signature::

window_hanning(x)

.. seealso::

:func:`window_none`

:func:`window_none` is another window algorithm.

'''

return np.hanning(len(x))*x

def window_none(x):

'''

No window function; simply return x.

Call signature::

window_none(x)

.. seealso::

:func:`window_hanning`

:func:`window_hanning` is another window algorithm.

'''

return x

def apply_window(x, window, axis=0, return_window=None):

'''

Apply the given window to the given 1D or 2D array along the given axis.

Call signature::

apply_window(x, window, axis=0, return_window=False)

*x*: 1D or 2D array or sequence

Array or sequence containing the data.

*winodw*: function or array.

Either a function to generate a window or an array with length

*x*.shape[*axis*]

*axis*: integer

The axis over which to do the repetition.

Must be 0 or 1. The default is 0

*return_window*: bool

If true, also return the 1D values of the window that was applied

'''

x = np.asarray(x)

if x.ndim < 1 or x.ndim > 2:

raise ValueError('only 1D or 2D arrays can be used')

if axis+1 > x.ndim:

raise ValueError('axis(=%s) out of bounds' % axis)

xshape = list(x.shape)

xshapetarg = xshape.pop(axis)

if cbook.iterable(window):

if len(window) != xshapetarg:

raise ValueError('The len(window) must be the same as the shape '

'of x for the chosen axis')

windowVals = window

else:

windowVals = window(np.ones(xshapetarg, dtype=x.dtype))

if x.ndim == 1:

if return_window:

return windowVals * x, windowVals

else:

return windowVals * x

xshapeother = xshape.pop()

otheraxis = (axis+1) % 2

windowValsRep = stride_repeat(windowVals, xshapeother, axis=otheraxis)

if return_window:

return windowValsRep * x, windowVals

else:

return windowValsRep * x

def detrend(x, key=None, axis=None):

'''

Return x with its trend removed.

Call signature::

detrend(x, key='mean')

*x*: array or sequence

Array or sequence containing the data.

*key*: [ 'default' | 'constant' | 'mean' | 'linear' | 'none'] or function

Specifies the detrend algorithm to use. 'default' is 'mean',

which is the same as :func:`detrend_mean`. 'constant' is the same.

'linear' is the same as :func:`detrend_linear`. 'none' is the same

as :func:`detrend_none`. The default is 'mean'. See the

corresponding functions for more details regarding the algorithms.

Can also be a function that carries out the detrend operation.

*axis*: integer

The axis along which to do the detrending.

.. seealso::

:func:`detrend_mean`

:func:`detrend_mean` implements the 'mean' algorithm.

:func:`detrend_linear`

:func:`detrend_linear` implements the 'linear' algorithm.

:func:`detrend_none`

:func:`detrend_none` implements the 'none' algorithm.

'''

if key is None or key in ['constant', 'mean', 'default']:

return detrend(x, key=detrend_mean, axis=axis)

elif key == 'linear':

return detrend(x, key=detrend_linear, axis=axis)

elif key == 'none':

return detrend(x, key=detrend_none, axis=axis)

elif cbook.is_string_like(key):

raise ValueError("Unknown value for key %s, must be one of: "

"'default', 'constant', 'mean', "

"'linear', or a function" % key)

if not callable(key):

raise ValueError("Unknown value for key %s, must be one of: "

"'default', 'constant', 'mean', "

"'linear', or a function" % key)

x = np.asarray(x)

if axis is not None and axis+1 > x.ndim:

raise ValueError('axis(=%s) out of bounds' % axis)

if (axis is None and x.ndim == 0) or (not axis and x.ndim == 1):

return key(x)

# try to use the 'axis' argument if the function supports it,

# otherwise use apply_along_axis to do it

try:

return key(x, axis=axis)

except TypeError:

return np.apply_along_axis(key, axis=axis, arr=x)

def demean(x, axis=0):

'''

Return x minus its mean along the specified axis.

Call signature::

demean(x, axis=0)

*x*: array or sequence

Array or sequence containing the data

Can have any dimensionality

*axis*: integer

The axis along which to take the mean. See numpy.mean for a

description of this argument.

.. seealso::

:func:`delinear`

:func:`denone`

:func:`delinear` and :func:`denone` are other detrend algorithms.

:func:`detrend_mean`

This function is the same as as :func:`detrend_mean` except

for the default *axis*.

'''

return detrend_mean(x, axis=axis)

def detrend_mean(x, axis=None):

'''

Return x minus the mean(x).

Call signature::

detrend_mean(x, axis=None)

*x*: array or sequence

Array or sequence containing the data

Can have any dimensionality

*axis*: integer

The axis along which to take the mean. See numpy.mean for a

description of this argument.

.. seealso::

:func:`demean`

This function is the same as as :func:`demean` except

for the default *axis*.

:func:`detrend_linear`

:func:`detrend_none`

:func:`detrend_linear` and :func:`detrend_none` are other

detrend algorithms.

:func:`detrend`

:func:`detrend` is a wrapper around all the detrend algorithms.

'''

x = np.asarray(x)

if axis is not None and axis+1 > x.ndim:

raise ValueError('axis(=%s) out of bounds' % axis)

# short-circuit 0-D array.

if not x.ndim:

return np.array(0., dtype=x.dtype)

# short-circuit simple operations

if axis == 0 or axis is None or x.ndim <= 1:

return x - x.mean(axis)

ind = [slice(None)] * x.ndim

ind[axis] = np.newaxis

return x - x.mean(axis)[ind]

def detrend_none(x, axis=None):

'''

Return x: no detrending.

Call signature::

detrend_none(x, axis=None)

*x*: any object

An object containing the data

*axis*: integer

This parameter is ignored.

It is included for compatibility with detrend_mean

.. seealso::

:func:`denone`

This function is the same as as :func:`denone` except

for the default *axis*, which has no effect.

:func:`detrend_mean`

:func:`detrend_linear`

:func:`detrend_mean` and :func:`detrend_linear` are other

detrend algorithms.

:func:`detrend`

:func:`detrend` is a wrapper around all the detrend algorithms.

'''

return x

def detrend_linear(y):

'''

Return x minus best fit line; 'linear' detrending.

Call signature::

detrend_linear(y)

*y*: 0-D or 1-D array or sequence

Array or sequence containing the data

*axis*: integer

The axis along which to take the mean. See numpy.mean for a

description of this argument.

.. seealso::

:func:`delinear`

This function is the same as as :func:`delinear` except

for the default *axis*.

:func:`detrend_mean`

:func:`detrend_none`

:func:`detrend_mean` and :func:`detrend_none` are other

detrend algorithms.

:func:`detrend`

:func:`detrend` is a wrapper around all the detrend algorithms.

'''

# This is faster than an algorithm based on linalg.lstsq.

y = np.asarray(y)

if y.ndim > 1:

raise ValueError('y cannot have ndim > 1')

# short-circuit 0-D array.

if not y.ndim:

return np.array(0., dtype=y.dtype)

x = np.arange(y.size, dtype=np.float_)

C = np.cov(x, y, bias=1)

b = C[0, 1]/C[0, 0]

a = y.mean() - b*x.mean()

return y - (b*x + a)

def stride_windows(x, n, noverlap=None, axis=0):

'''

Get all windows of x with length n as a single array,

using strides to avoid data duplication.

.. warning::

It is not safe to write to the output array. Multiple

elements may point to the same piece of memory,

so modifying one value may change others.

Call signature::

stride_windows(x, n, noverlap=0)

*x*: 1D array or sequence

Array or sequence containing the data.

*n*: integer

The number of data points in each window.

*noverlap*: integer

The overlap between adjacent windows.

Default is 0 (no overlap)

*axis*: integer

The axis along which the windows will run.

Refs:

`stackoverflaw: Rolling window for 1D arrays in Numpy?

<http://stackoverflow.com/a/6811241>`_

`stackoverflaw: Using strides for an efficient moving average filter

<http://stackoverflow.com/a/4947453>`_

'''

if noverlap is None:

noverlap = 0

if noverlap >= n:

raise ValueError('noverlap must be less than n')

if n < 1:

raise ValueError('n cannot be less than 1')

x = np.asarray(x)

if x.ndim != 1:

raise ValueError('only 1-dimensional arrays can be used')

if n == 1 and noverlap == 0:

if axis == 0:

return x[np.newaxis]

else:

return x[np.newaxis].transpose()

if n > x.size:

raise ValueError('n cannot be greater than the length of x')

# np.lib.stride_tricks.as_strided easily leads to memory corruption for

# non integer shape and strides, i.e. noverlap or n. See #3845.

noverlap = int(noverlap)

n = int(n)

step = n - noverlap

if axis == 0:

shape = (n, (x.shape[-1]-noverlap)//step)

strides = (x.strides[0], step*x.strides[0])

else:

shape = ((x.shape[-1]-noverlap)//step, n)

strides = (step*x.strides[0], x.strides[0])

return np.lib.stride_tricks.as_strided(x, shape=shape, strides=strides)

def stride_repeat(x, n, axis=0):

'''

Repeat the values in an array in a memory-efficient manner. Array x is

stacked vertically n times.

.. warning::

It is not safe to write to the output array. Multiple

elements may point to the same piece of memory, so

modifying one value may change others.

Call signature::

stride_repeat(x, n, axis=0)

*x*: 1D array or sequence

Array or sequence containing the data.

*n*: integer

The number of time to repeat the array.

*axis*: integer

The axis along which the data will run.

Refs:

`stackoverflaw: Repeat NumPy array without replicating data?

<http://stackoverflow.com/a/5568169>`_

'''

if axis not in [0, 1]:

raise ValueError('axis must be 0 or 1')

x = np.asarray(x)

if x.ndim != 1:

raise ValueError('only 1-dimensional arrays can be used')

if n == 1:

if axis == 0:

return np.atleast_2d(x)

else:

return np.atleast_2d(x).T

if n < 1:

raise ValueError('n cannot be less than 1')

# np.lib.stride_tricks.as_strided easily leads to memory corruption for

# non integer shape and strides, i.e. n. See #3845.

n = int(n)

if axis == 0:

shape = (n, x.size)

strides = (0, x.strides[0])

else:

shape = (x.size, n)

strides = (x.strides[0], 0)

return np.lib.stride_tricks.as_strided(x, shape=shape, strides=strides)

def _spectral_helper(x, y=None, NFFT=None, Fs=None, detrend_func=None,

window=None, noverlap=None, pad_to=None,

sides=None, scale_by_freq=None, mode=None):

'''

This is a helper function that implements the commonality between the

psd, csd, spectrogram and complex, magnitude, angle, and phase spectrums.

It is *NOT* meant to be used outside of mlab and may change at any time.

'''

if y is None:

# if y is None use x for y

same_data = True

else:

# The checks for if y is x are so that we can use the same function to

# implement the core of psd(), csd(), and spectrogram() without doing

# extra calculations. We return the unaveraged Pxy, freqs, and t.

same_data = y is x

if Fs is None:

Fs = 2

if noverlap is None:

noverlap = 0

if detrend_func is None:

detrend_func = detrend_none

if window is None:

window = window_hanning

# if NFFT is set to None use the whole signal

if NFFT is None:

NFFT = 256

if mode is None or mode == 'default':

mode = 'psd'

elif mode not in ['psd', 'complex', 'magnitude', 'angle', 'phase']:

raise ValueError("Unknown value for mode %s, must be one of: "

"'default', 'psd', 'complex', "

"'magnitude', 'angle', 'phase'" % mode)

if not same_data and mode != 'psd':

raise ValueError("x and y must be equal if mode is not 'psd'")

# Make sure we're dealing with a numpy array. If y and x were the same

# object to start with, keep them that way

x = np.asarray(x)

if not same_data:

y = np.asarray(y)

if sides is None or sides == 'default':

if np.iscomplexobj(x):

sides = 'twosided'

else:

sides = 'onesided'

elif sides not in ['onesided', 'twosided']:

raise ValueError("Unknown value for sides %s, must be one of: "

"'default', 'onesided', or 'twosided'" % sides)

# zero pad x and y up to NFFT if they are shorter than NFFT

if len(x) < NFFT:

n = len(x)

x = np.resize(x, (NFFT,))

x[n:] = 0

if not same_data and len(y) < NFFT:

n = len(y)

y = np.resize(y, (NFFT,))

y[n:] = 0

if pad_to is None:

pad_to = NFFT

if mode != 'psd':

scale_by_freq = False

elif scale_by_freq is None:

scale_by_freq = True

# For real x, ignore the negative frequencies unless told otherwise

if sides == 'twosided':

numFreqs = pad_to

if pad_to % 2:

freqcenter = (pad_to - 1)//2 + 1

else:

freqcenter = pad_to//2

scaling_factor = 1.

elif sides == 'onesided':

if pad_to % 2:

numFreqs = (pad_to + 1)//2

else:

numFreqs = pad_to//2 + 1

scaling_factor = 2.

result = stride_windows(x, NFFT, noverlap, axis=0)

result = detrend(result, detrend_func, axis=0)

result, windowVals = apply_window(result, window, axis=0,

return_window=True)

result = np.fft.fft(result, n=pad_to, axis=0)[:numFreqs, :]

freqs = np.fft.fftfreq(pad_to, 1/Fs)[:numFreqs]

if not same_data:

# if same_data is False, mode must be 'psd'

resultY = stride_windows(y, NFFT, noverlap)

resultY = apply_window(resultY, window, axis=0)

resultY = detrend(resultY, detrend_func, axis=0)

resultY = np.fft.fft(resultY, n=pad_to, axis=0)[:numFreqs, :]

result = np.conjugate(result) * resultY

elif mode == 'psd':

result = np.conjugate(result) * result

elif mode == 'magnitude':

result = np.absolute(result)

elif mode == 'angle' or mode == 'phase':

# we unwrap the phase later to handle the onesided vs. twosided case

result = np.angle(result)

elif mode == 'complex':

pass

if mode == 'psd':

# Scale the spectrum by the norm of the window to compensate for

# windowing loss; see Bendat & Piersol Sec 11.5.2.

result /= (np.abs(windowVals)**2).sum()

# Also include scaling factors for one-sided densities and dividing by

# the sampling frequency, if desired. Scale everything, except the DC

# component and the NFFT/2 component:

result[1:-1] *= scaling_factor

# MATLAB divides by the sampling frequency so that density function

# has units of dB/Hz and can be integrated by the plotted frequency

# values. Perform the same scaling here.

if scale_by_freq:

result /= Fs

t = np.arange(NFFT/2, len(x) - NFFT/2 + 1, NFFT - noverlap)/Fs

if sides == 'twosided':

# center the frequency range at zero

freqs = np.concatenate((freqs[freqcenter:], freqs[:freqcenter]))

result = np.concatenate((result[freqcenter:, :],

result[:freqcenter, :]), 0)

elif not pad_to % 2:

# get the last value correctly, it is negative otherwise

freqs[-1] *= -1

# we unwrap the phase here to handle the onesided vs. twosided case

if mode == 'phase':

result = np.unwrap(result, axis=0)

return result, freqs, t

def _single_spectrum_helper(x, mode, Fs=None, window=None, pad_to=None,

sides=None):

'''

This is a helper function that implements the commonality between the

complex, magnitude, angle, and phase spectrums.

It is *NOT* meant to be used outside of mlab and may change at any time.

'''

if mode is None or mode == 'psd' or mode == 'default':

raise ValueError('_single_spectrum_helper does not work with %s mode'

% mode)

if pad_to is None:

pad_to = len(x)

spec, freqs, _ = _spectral_helper(x=x, y=None, NFFT=len(x), Fs=Fs,

detrend_func=detrend_none, window=window,

noverlap=0, pad_to=pad_to,

sides=sides,

scale_by_freq=False,

mode=mode)

if mode != 'complex':

spec = spec.real

if len(spec.shape) == 2 and spec.shape[1] == 1:

spec = spec[:, 0]

return spec, freqs

# Split out these keyword docs so that they can be used elsewhere

docstring.interpd.update(Spectral=cbook.dedent("""

Keyword arguments:

*Fs*: scalar

The sampling frequency (samples per time unit). It is used

to calculate the Fourier frequencies, freqs, in cycles per time

unit. The default value is 2.

*window*: callable or ndarray

A function or a vector of length *NFFT*. To create window

vectors see :func:`window_hanning`, :func:`window_none`,

:func:`numpy.blackman`, :func:`numpy.hamming`,

:func:`numpy.bartlett`, :func:`scipy.signal`,

:func:`scipy.signal.get_window`, etc. The default is

:func:`window_hanning`. If a function is passed as the

argument, it must take a data segment as an argument and

return the windowed version of the segment.

*sides*: [ 'default' | 'onesided' | 'twosided' ]

Specifies which sides of the spectrum to return. Default gives the

default behavior, which returns one-sided for real data and both

for complex data. 'onesided' forces the return of a one-sided

spectrum, while 'twosided' forces two-sided.

"""))

docstring.interpd.update(Single_Spectrum=cbook.dedent("""

*pad_to*: integer

The number of points to which the data segment is padded when

performing the FFT. While not increasing the actual resolution of

the spectrum (the minimum distance between resolvable peaks),

this can give more points in the plot, allowing for more

detail. This corresponds to the *n* parameter in the call to fft().

The default is None, which sets *pad_to* equal to the length of the

input signal (i.e. no padding).

"""))

docstring.interpd.update(PSD=cbook.dedent("""

*pad_to*: integer

The number of points to which the data segment is padded when

performing the FFT. This can be different from *NFFT*, which

specifies the number of data points used. While not increasing

the actual resolution of the spectrum (the minimum distance between

resolvable peaks), this can give more points in the plot,

allowing for more detail. This corresponds to the *n* parameter

in the call to fft(). The default is None, which sets *pad_to*

equal to *NFFT*

*NFFT*: integer

The number of data points used in each block for the FFT.

A power 2 is most efficient. The default value is 256.

This should *NOT* be used to get zero padding, or the scaling of the

result will be incorrect. Use *pad_to* for this instead.

*detrend*: [ 'default' | 'constant' | 'mean' | 'linear' | 'none'] or

callable

The function applied to each segment before fft-ing,

designed to remove the mean or linear trend. Unlike in

MATLAB, where the *detrend* parameter is a vector, in

matplotlib is it a function. The :mod:`~matplotlib.pylab`

module defines :func:`~matplotlib.pylab.detrend_none`,

:func:`~matplotlib.pylab.detrend_mean`, and

:func:`~matplotlib.pylab.detrend_linear`, but you can use

a custom function as well. You can also use a string to choose

one of the functions. 'default', 'constant', and 'mean' call

:func:`~matplotlib.pylab.detrend_mean`. 'linear' calls

:func:`~matplotlib.pylab.detrend_linear`. 'none' calls

:func:`~matplotlib.pylab.detrend_none`.

*scale_by_freq*: boolean

Specifies whether the resulting density values should be scaled

by the scaling frequency, which gives density in units of Hz^-1.

This allows for integration over the returned frequency values.

The default is True for MATLAB compatibility.

"""))

@docstring.dedent_interpd

def psd(x, NFFT=None, Fs=None, detrend=None, window=None,

noverlap=None, pad_to=None, sides=None, scale_by_freq=None):

"""

Compute the power spectral density.

Call signature::

psd(x, NFFT=256, Fs=2, detrend=mlab.detrend_none,

window=mlab.window_hanning, noverlap=0, pad_to=None,

sides='default', scale_by_freq=None)

The power spectral density :math:`P_{xx}` by Welch's average

periodogram method. The vector *x* is divided into *NFFT* length

segments. Each segment is detrended by function *detrend* and

windowed by function *window*. *noverlap* gives the length of

the overlap between segments. The :math:`|\mathrm{fft}(i)|^2`

of each segment :math:`i` are averaged to compute :math:`P_{xx}`.

If len(*x*) < *NFFT*, it will be zero padded to *NFFT*.

*x*: 1-D array or sequence

Array or sequence containing the data

%(Spectral)s

%(PSD)s

*noverlap*: integer

The number of points of overlap between segments.

The default value is 0 (no overlap).

Returns the tuple (*Pxx*, *freqs*).

*Pxx*: 1-D array

The values for the power spectrum `P_{xx}` (real valued)

*freqs*: 1-D array

The frequencies corresponding to the elements in *Pxx*

Refs:

Bendat & Piersol -- Random Data: Analysis and Measurement

Procedures, John Wiley & Sons (1986)

.. seealso::

:func:`specgram`

:func:`specgram` differs in the default overlap; in not returning

the mean of the segment periodograms; and in returning the

times of the segments.

:func:`magnitude_spectrum`

:func:`magnitude_spectrum` returns the magnitude spectrum.

:func:`csd`

:func:`csd` returns the spectral density between two signals.

"""

Pxx, freqs = csd(x=x, y=None, NFFT=NFFT, Fs=Fs, detrend=detrend,

window=window, noverlap=noverlap, pad_to=pad_to,

sides=sides, scale_by_freq=scale_by_freq)

return Pxx.real, freqs

@docstring.dedent_interpd

def csd(x, y, NFFT=None, Fs=None, detrend=None, window=None,

noverlap=None, pad_to=None, sides=None, scale_by_freq=None):

"""

Compute the cross-spectral density.

Call signature::

csd(x, y, NFFT=256, Fs=2, detrend=mlab.detrend_none,

window=mlab.window_hanning, noverlap=0, pad_to=None,

sides='default', scale_by_freq=None)

The cross spectral density :math:`P_{xy}` by Welch's average

periodogram method. The vectors *x* and *y* are divided into

*NFFT* length segments. Each segment is detrended by function

*detrend* and windowed by function *window*. *noverlap* gives

the length of the overlap between segments. The product of

the direct FFTs of *x* and *y* are averaged over each segment

to compute :math:`P_{xy}`, with a scaling to correct for power

loss due to windowing.

If len(*x*) < *NFFT* or len(*y*) < *NFFT*, they will be zero

padded to *NFFT*.

*x*, *y*: 1-D arrays or sequences

Arrays or sequences containing the data