Signal Processing (`scipy.signal`)
==================================

.. sectionauthor:: Travis E. Oliphant

.. sectionauthor:: Pim Schellart

.. currentmodule:: scipy.signal

The signal processing toolbox currently contains some filtering
functions, a limited set of filter design tools, and a few B-spline
interpolation algorithms for one- and two-dimensional data. While the
B-spline algorithms could technically be placed under the
interpolation category, they are included here because they only work
with equally-spaced data and make heavy use of filter-theory and
transfer-function formalism to provide a fast B-spline transform. To
understand this section you will need to understand that a signal in
SciPy is an array of real or complex numbers.


B-splines
---------

A B-spline is an approximation of a continuous function over a finite-
domain in terms of B-spline coefficients and knot points. If the knot-
points are equally spaced with spacing :math:`\Delta x` , then the B-spline
approximation to a 1-dimensional function is the finite-basis expansion.

.. math::


    y\left(x\right)\approx\sum_{j}c_{j}\beta^{o}\left(\frac{x}{\Delta x}-j\right).

In two dimensions with knot-spacing :math:`\Delta x` and :math:`\Delta y` , the
function representation is

.. math::

    z\left(x,y\right)\approx\sum_{j}\sum_{k}c_{jk}\beta^{o}\left(\frac{x}{\Delta x}-j\right)\beta^{o}\left(\frac{y}{\Delta y}-k\right).

In these expressions, :math:`\beta^{o}\left(\cdot\right)` is the space-limited
B-spline basis function of order, :math:`o` . The requirement of equally-spaced
knot-points and equally-spaced data points, allows the development of fast
(inverse-filtering) algorithms for determining the coefficients, :math:`c_{j}`
, from sample-values, :math:`y_{n}` . Unlike the general spline interpolation
algorithms, these algorithms can quickly find the spline coefficients for large
images.

The advantage of representing a set of samples via B-spline basis
functions is that continuous-domain operators (derivatives, re-
sampling, integral, etc.) which assume that the data samples are drawn
from an underlying continuous function can be computed with relative
ease from the spline coefficients. For example, the second-derivative
of a spline is

.. math::

    y{}^{\prime\prime}\left(x\right)=\frac{1}{\Delta x^{2}}\sum_{j}c_{j}\beta^{o\prime\prime}\left(\frac{x}{\Delta x}-j\right).

Using the property of B-splines that

.. math::

    \frac{d^{2}\beta^{o}\left(w\right)}{dw^{2}}=\beta^{o-2}\left(w+1\right)-2\beta^{o-2}\left(w\right)+\beta^{o-2}\left(w-1\right)

it can be seen that

.. math::

    y^{\prime\prime}\left(x\right)=\frac{1}{\Delta x^{2}}\sum_{j}c_{j}\left[\beta^{o-2}\left(\frac{x}{\Delta x}-j+1\right)-2\beta^{o-2}\left(\frac{x}{\Delta x}-j\right)+\beta^{o-2}\left(\frac{x}{\Delta x}-j-1\right)\right].

If :math:`o=3` , then at the sample points,

.. math::
   :nowrap:

    \begin{eqnarray*} \Delta x^{2}\left.y^{\prime}\left(x\right)\right|_{x=n\Delta x} & = & \sum_{j}c_{j}\delta_{n-j+1}-2c_{j}\delta_{n-j}+c_{j}\delta_{n-j-1},\\  & = & c_{n+1}-2c_{n}+c_{n-1}.\end{eqnarray*}

Thus, the second-derivative signal can be easily calculated from the spline
fit. if desired, smoothing splines can be found to make the second-derivative
less sensitive to random-errors.

The savvy reader will have already noticed that the data samples are related
to the knot coefficients via a convolution operator, so that simple
convolution with the sampled B-spline function recovers the original data from
the spline coefficients. The output of convolutions can change depending on
how boundaries are handled (this becomes increasingly more important as the
number of dimensions in the data- set increases). The algorithms relating to
B-splines in the signal- processing sub package assume mirror-symmetric
boundary conditions. Thus, spline coefficients are computed based on that
assumption, and data-samples can be recovered exactly from the spline
coefficients by assuming them to be mirror-symmetric also.

Currently the package provides functions for determining second- and third-
order cubic spline coefficients from equally spaced samples in one- and two-
dimensions (:func:`qspline1d`, :func:`qspline2d`, :func:`cspline1d`,
:func:`cspline2d`). The package also supplies a function ( :func:`bspline` )
for evaluating the bspline basis function, :math:`\beta^{o}\left(x\right)` for
arbitrary order and :math:`x.` For large :math:`o` , the B-spline basis
function can be approximated well by a zero-mean Gaussian function with
standard-deviation equal to :math:`\sigma_{o}=\left(o+1\right)/12` :

.. math::

    \beta^{o}\left(x\right)\approx\frac{1}{\sqrt{2\pi\sigma_{o}^{2}}}\exp\left(-\frac{x^{2}}{2\sigma_{o}}\right).

A function to compute this Gaussian for arbitrary :math:`x` and :math:`o` is
also available ( :func:`gauss_spline` ). The following code and Figure uses
spline-filtering to compute an edge-image (the second-derivative of a smoothed
spline) of Lena's face which is an array returned by the command :func:`misc.lena`.
The command :func:`sepfir2d` was used to apply a separable two-dimensional FIR
filter with mirror- symmetric boundary conditions to the spline coefficients.
This function is ideally suited for reconstructing samples from spline
coefficients and is faster than :func:`convolve2d` which convolves arbitrary
two-dimensional filters and allows for choosing mirror-symmetric boundary
conditions.

.. plot::

   >>> from numpy import *
   >>> from scipy import signal, misc
   >>> import matplotlib.pyplot as plt

   >>> image = misc.lena().astype(float32)
   >>> derfilt = array([1.0,-2,1.0],float32)
   >>> ck = signal.cspline2d(image,8.0)
   >>> deriv = signal.sepfir2d(ck, derfilt, [1]) + \
   >>>         signal.sepfir2d(ck, [1], derfilt)

   Alternatively we could have done::

       laplacian = array([[0,1,0],[1,-4,1],[0,1,0]],float32)
       deriv2 = signal.convolve2d(ck,laplacian,mode='same',boundary='symm')

   >>> plt.figure()
   >>> plt.imshow(image)
   >>> plt.gray()
   >>> plt.title('Original image')
   >>> plt.show()

   >>> plt.figure()
   >>> plt.imshow(deriv)
   >>> plt.gray()
   >>> plt.title('Output of spline edge filter')
   >>> plt.show()

..   :caption: Example of using smoothing splines to filter images.


Filtering
---------

Filtering is a generic name for any system that modifies an input
signal in some way. In SciPy a signal can be thought of as a Numpy
array. There are different kinds of filters for different kinds of
operations. There are two broad kinds of filtering operations: linear
and non-linear. Linear filters can always be reduced to multiplication
of the flattened Numpy array by an appropriate matrix resulting in
another flattened Numpy array. Of course, this is not usually the best
way to compute the filter as the matrices and vectors involved may be
huge. For example filtering a :math:`512 \times 512` image with this
method would require multiplication of a :math:`512^2 \times 512^2`
matrix with a :math:`512^2` vector. Just trying to store the
:math:`512^2 \times 512^2` matrix using a standard Numpy array would
require :math:`68,719,476,736` elements. At 4 bytes per element this
would require :math:`256\textrm{GB}` of memory. In most applications
most of the elements of this matrix are zero and a different method
for computing the output of the filter is employed.


Convolution/Correlation
^^^^^^^^^^^^^^^^^^^^^^^

Many linear filters also have the property of shift-invariance. This
means that the filtering operation is the same at different locations
in the signal and it implies that the filtering matrix can be
constructed from knowledge of one row (or column) of the matrix alone.
In this case, the matrix multiplication can be accomplished using
Fourier transforms.

Let :math:`x\left[n\right]` define a one-dimensional signal indexed by the
integer :math:`n.` Full convolution of two one-dimensional signals can be
expressed as

.. math::

    y\left[n\right]=\sum_{k=-\infty}^{\infty}x\left[k\right]h\left[n-k\right].

This equation can only be implemented directly if we limit the
sequences to finite support sequences that can be stored in a
computer, choose :math:`n=0` to be the starting point of both
sequences, let :math:`K+1` be that value for which
:math:`y\left[n\right]=0` for all :math:`n>K+1` and :math:`M+1` be
that value for which :math:`x\left[n\right]=0` for all :math:`n>M+1` ,
then the discrete convolution expression is

.. math::

    y\left[n\right]=\sum_{k=\max\left(n-M,0\right)}^{\min\left(n,K\right)}x\left[k\right]h\left[n-k\right].

For convenience assume :math:`K\geq M.` Then, more explicitly the output of
this operation is

.. math::
   :nowrap:

    \begin{eqnarray*} y\left[0\right] & = & x\left[0\right]h\left[0\right]\\ y\left[1\right] & = & x\left[0\right]h\left[1\right]+x\left[1\right]h\left[0\right]\\ y\left[2\right] & = & x\left[0\right]h\left[2\right]+x\left[1\right]h\left[1\right]+x\left[2\right]h\left[0\right]\\ \vdots & \vdots & \vdots\\ y\left[M\right] & = & x\left[0\right]h\left[M\right]+x\left[1\right]h\left[M-1\right]+\cdots+x\left[M\right]h\left[0\right]\\ y\left[M+1\right] & = & x\left[1\right]h\left[M\right]+x\left[2\right]h\left[M-1\right]+\cdots+x\left[M+1\right]h\left[0\right]\\ \vdots & \vdots & \vdots\\ y\left[K\right] & = & x\left[K-M\right]h\left[M\right]+\cdots+x\left[K\right]h\left[0\right]\\ y\left[K+1\right] & = & x\left[K+1-M\right]h\left[M\right]+\cdots+x\left[K\right]h\left[1\right]\\ \vdots & \vdots & \vdots\\ y\left[K+M-1\right] & = & x\left[K-1\right]h\left[M\right]+x\left[K\right]h\left[M-1\right]\\ y\left[K+M\right] & = & x\left[K\right]h\left[M\right].\end{eqnarray*}

Thus, the full discrete convolution of two finite sequences of lengths
:math:`K+1` and :math:`M+1` respectively results in a finite sequence of length
:math:`K+M+1=\left(K+1\right)+\left(M+1\right)-1.`

One dimensional convolution is implemented in SciPy with the function
:func:`convolve`. This function takes as inputs the signals :math:`x,`
:math:`h` , and an optional flag and returns the signal :math:`y.` The
optional flag allows for specification of which part of the output signal to
return. The default value of 'full' returns the entire signal. If the flag has
a value of 'same' then only the middle :math:`K` values are returned starting
at :math:`y\left[\left\lfloor \frac{M-1}{2}\right\rfloor \right]` so that the
output has the same length as the largest input. If the flag has a value of
'valid' then only the middle :math:`K-M+1=\left(K+1\right)-\left(M+1\right)+1`
output values are returned where :math:`z` depends on all of the values of the
smallest input from :math:`h\left[0\right]` to :math:`h\left[M\right].` In
other words only the values :math:`y\left[M\right]` to :math:`y\left[K\right]`
inclusive are returned.

The code below shows a simple example for convolution of 2 sequences

>>> x = np.array([1.0, 2.0, 3.0])
>>> h = np.array([0.0, 1.0, 0.0, 0.0, 0.0])
>>> signal.convolve(x, h)
[ 0.  1.  2.  3.  0.  0.  0.]
>>> signal.convolve(x, h, 'same')
[ 2.  3.  0.]


This same function :func:`convolve` can actually take :math:`N` -dimensional
arrays as inputs and will return the :math:`N` -dimensional convolution of the
two arrays as is shown in the code example below. The same input flags are
available for that case as well.


>>> x = np.array([[1., 1., 0., 0.],[1., 1., 0., 0.],[0., 0., 0., 0.],[0., 0., 0., 0.]])
>>> h = np.array([[1., 0., 0., 0.],[0., 0., 0., 0.],[0., 0., 1., 0.],[0., 0., 0., 0.]])
>>> signal.convolve(x, h)
[[ 1.  1.  0.  0.  0.  0.  0.]
 [ 1.  1.  0.  0.  0.  0.  0.]
 [ 0.  0.  1.  1.  0.  0.  0.]
 [ 0.  0.  1.  1.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.]]

Correlation is very similar to convolution except for the minus sign
becomes a plus sign. Thus

.. math::

    w\left[n\right]=\sum_{k=-\infty}^{\infty}y\left[k\right]x\left[n+k\right]

is the (cross) correlation of the signals :math:`y` and :math:`x.` For
finite-length signals with :math:`y\left[n\right]=0` outside of the range
:math:`\left[0,K\right]` and :math:`x\left[n\right]=0` outside of the range
:math:`\left[0,M\right],` the summation can simplify to

.. math::

    w\left[n\right]=\sum_{k=\max\left(0,-n\right)}^{\min\left(K,M-n\right)}y\left[k\right]x\left[n+k\right].

Assuming again that :math:`K\geq M` this is

.. math::
   :nowrap:

    \begin{eqnarray*} w\left[-K\right] & = & y\left[K\right]x\left[0\right]\\ w\left[-K+1\right] & = & y\left[K-1\right]x\left[0\right]+y\left[K\right]x\left[1\right]\\ \vdots & \vdots & \vdots\\ w\left[M-K\right] & = & y\left[K-M\right]x\left[0\right]+y\left[K-M+1\right]x\left[1\right]+\cdots+y\left[K\right]x\left[M\right]\\ w\left[M-K+1\right] & = & y\left[K-M-1\right]x\left[0\right]+\cdots+y\left[K-1\right]x\left[M\right]\\ \vdots & \vdots & \vdots\\ w\left[-1\right] & = & y\left[1\right]x\left[0\right]+y\left[2\right]x\left[1\right]+\cdots+y\left[M+1\right]x\left[M\right]\\ w\left[0\right] & = & y\left[0\right]x\left[0\right]+y\left[1\right]x\left[1\right]+\cdots+y\left[M\right]x\left[M\right]\\ w\left[1\right] & = & y\left[0\right]x\left[1\right]+y\left[1\right]x\left[2\right]+\cdots+y\left[M-1\right]x\left[M\right]\\ w\left[2\right] & = & y\left[0\right]x\left[2\right]+y\left[1\right]x\left[3\right]+\cdots+y\left[M-2\right]x\left[M\right]\\ \vdots & \vdots & \vdots\\ w\left[M-1\right] & = & y\left[0\right]x\left[M-1\right]+y\left[1\right]x\left[M\right]\\ w\left[M\right] & = & y\left[0\right]x\left[M\right].\end{eqnarray*}


The SciPy function :func:`correlate` implements this operation. Equivalent
flags are available for this operation to return the full :math:`K+M+1` length
sequence ('full') or a sequence with the same size as the largest sequence
starting at :math:`w\left[-K+\left\lfloor \frac{M-1}{2}\right\rfloor \right]`
('same') or a sequence where the values depend on all the values of the
smallest sequence ('valid'). This final option returns the :math:`K-M+1`
values :math:`w\left[M-K\right]` to :math:`w\left[0\right]` inclusive.



The function :func:`correlate` can also take arbitrary :math:`N`
-dimensional arrays as input and return the :math:`N` -dimensional
convolution of the two arrays on output.

When :math:`N=2,` :func:`correlate` and/or :func:`convolve` can be used
to construct arbitrary image filters to perform actions such as blurring,
enhancing, and edge-detection for an image.



.. plot::

   >>> import numpy as np
   >>> from scipy import signal, misc
   >>> import matplotlib.pyplot as plt

   >>> image = misc.lena()
   >>> w = np.zeros((50, 50))
   >>> w[0][0] = 1.0
   >>> w[49][25] = 1.0
   >>> image_new = signal.fftconvolve(image, w)

   >>> plt.figure()
   >>> plt.imshow(image)
   >>> plt.gray()
   >>> plt.title('Original image')
   >>> plt.show()

   >>> plt.figure()
   >>> plt.imshow(image_new)
   >>> plt.gray()
   >>> plt.title('Filtered image')
   >>> plt.show()


Using :func:`convolve` in the above example would take quite long to run.
Calculating the convolution in the time domain as above is mainly used for
filtering when one of the signals is much smaller than the other ( :math:`K\gg
M` ), otherwise linear filtering is more efficiently calculated in the
frequency domain provided by the function :func:`fftconvolve`.

If the filter function :math:`w[n,m]` can be factored according to

.. math::

  h[n, m] = h_1[n] h_2[m],

convolution can be calculated by means of the function :func:`sepfir2d`. As an
example we consider a Gaussian filter :func:`gaussian`

.. math::

  h[n, m] \propto e^{-x^2-y^2} = e^{-x^2} e^{-y^2}

which is often used for blurring.

.. plot::

   >>> import numpy as np
   >>> from scipy import signal, misc
   >>> import matplotlib.pyplot as plt

   >>> image = misc.lena()
   >>> w = signal.gaussian(50, 5.0)
   >>> image_new = signal.sepfir2d(image, w, w)

   >>> plt.figure()
   >>> plt.imshow(image)
   >>> plt.gray()
   >>> plt.title('Original image')
   >>> plt.show()

   >>> plt.figure()
   >>> plt.imshow(image_new)
   >>> plt.gray()
   >>> plt.title('Filtered image')
   >>> plt.show()



Difference-equation filtering
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

A general class of linear one-dimensional filters (that includes convolution
filters) are filters described by the difference equation

.. math::

    \sum_{k=0}^{N}a_{k}y\left[n-k\right]=\sum_{k=0}^{M}b_{k}x\left[n-k\right]

where :math:`x\left[n\right]` is the input sequence and
:math:`y\left[n\right]` is the output sequence. If we assume initial rest so
that :math:`y\left[n\right]=0` for :math:`n<0` , then this kind of filter can
be implemented using convolution.  However, the convolution filter sequence
:math:`h\left[n\right]` could be infinite if :math:`a_{k}\neq0` for
:math:`k\geq1.` In addition, this general class of linear filter allows
initial conditions to be placed on :math:`y\left[n\right]` for :math:`n<0`
resulting in a filter that cannot be expressed using convolution.

The difference equation filter can be thought of as finding
:math:`y\left[n\right]` recursively in terms of it's previous values

.. math::
    
    a_{0}y\left[n\right]=-a_{1}y\left[n-1\right]-\cdots-a_{N}y\left[n-N\right]+\cdots+b_{0}x\left[n\right]+\cdots+b_{M}x\left[n-M\right].

Often :math:`a_{0}=1` is chosen for normalization. The implementation in SciPy
of this general difference equation filter is a little more complicated then
would be implied by the previous equation. It is implemented so that only one
signal needs to be delayed. The actual implementation equations are (assuming
:math:`a_{0}=1` ).

.. math::
   :nowrap:

    \begin{eqnarray*} y\left[n\right] & = & b_{0}x\left[n\right]+z_{0}\left[n-1\right]\\ z_{0}\left[n\right] & = & b_{1}x\left[n\right]+z_{1}\left[n-1\right]-a_{1}y\left[n\right]\\ z_{1}\left[n\right] & = & b_{2}x\left[n\right]+z_{2}\left[n-1\right]-a_{2}y\left[n\right]\\ \vdots & \vdots & \vdots\\ z_{K-2}\left[n\right] & = & b_{K-1}x\left[n\right]+z_{K-1}\left[n-1\right]-a_{K-1}y\left[n\right]\\ z_{K-1}\left[n\right] & = & b_{K}x\left[n\right]-a_{K}y\left[n\right],\end{eqnarray*}

where :math:`K=\max\left(N,M\right).` Note that :math:`b_{K}=0` if :math:`K>M`
and :math:`a_{K}=0` if :math:`K>N.` In this way, the output at time :math:`n`
depends only on the input at time :math:`n` and the value of :math:`z_{0}` at
the previous time. This can always be calculated as long as the :math:`K`
values :math:`z_{0}\left[n-1\right]\ldots z_{K-1}\left[n-1\right]` are
computed and stored at each time step.

The difference-equation filter is called using the command :func:`lfilter` in
SciPy. This command takes as inputs the vector :math:`b,` the vector,
:math:`a,` a signal :math:`x` and returns the vector :math:`y` (the same
length as :math:`x` ) computed using the equation given above. If :math:`x` is
:math:`N` -dimensional, then the filter is computed along the axis provided.
If, desired, initial conditions providing the values of
:math:`z_{0}\left[-1\right]` to :math:`z_{K-1}\left[-1\right]` can be provided
or else it will be assumed that they are all zero. If initial conditions are
provided, then the final conditions on the intermediate variables are also
returned. These could be used, for example, to restart the calculation in the
same state.

Sometimes it is more convenient to express the initial conditions in terms of
the signals :math:`x\left[n\right]` and :math:`y\left[n\right].` In other
words, perhaps you have the values of :math:`x\left[-M\right]` to
:math:`x\left[-1\right]` and the values of :math:`y\left[-N\right]` to
:math:`y\left[-1\right]` and would like to determine what values of
:math:`z_{m}\left[-1\right]` should be delivered as initial conditions to the
difference-equation filter. It is not difficult to show that for :math:`0\leq
m<K,`

.. math::

    z_{m}\left[n\right]=\sum_{p=0}^{K-m-1}\left(b_{m+p+1}x\left[n-p\right]-a_{m+p+1}y\left[n-p\right]\right).

Using this formula we can find the initial condition vector
:math:`z_{0}\left[-1\right]` to :math:`z_{K-1}\left[-1\right]` given initial
conditions on :math:`y` (and :math:`x` ). The command :func:`lfiltic` performs
this function.

As an example consider the following system:

.. math::

  y[n] = \frac{1}{2} x[n] + \frac{1}{4} x[n-1] + \frac{1}{3} y[n-1]

The code calculates the signal :math:`y[n]` for a given signal :math:`x[n]`;
first for initial condiditions :math:`y[-1] = 0` (default case), then for
:math:`y[-1] = 2` by means of :fun:`lfiltic`.

>>> import numpy as np
>>> from scipy import signal

>>> x = np.array([1., 0., 0., 0.])
>>> b = np.array([1.0/2, 1.0/4])
>>> a = np.array([1.0, -1.0/3])
>>> signal.lfilter(b, a, x)
[ 0.5         0.41666667  0.13888889  0.0462963 ]
>>> zi = signal.lfiltic(b, a, y=[2.])
>>> signal.lfilter(b, a, x, zi=zi)
[ 1.16666667,  0.63888889,  0.21296296,  0.07098765]

Note that the output signal :math:`y[n]` has the same length as the length as
the input signal :math:`x[n]`.


Analysis of Linear Systems
""""""""""""""""""""""""""

Linear system described a linear difference equation can be fully described by
the coefficient vectors a and b as was done above; an alternative
representation is to provide a factor :math:`k`, :math:`N_z` zeros :math:`z_k`
and :math:`N_p` poles :math:`p_k`, respectively, to describe the system by
means of its transfer function :math:`H(z)` according to

.. math::

   H(z) = k \frac{ (z-z_1)(z-z_2)...(z-z_{N_z})}{ (z-p_1)(z-p_2)...(z-p_{N_p})}

This alternative representation can be obtain wit hthe scipy function
:func:`tf2zpk`; the inverse is provided by :func:`zpk2tf`.

For the example from above we have

>>> b = np.array([1.0/2, 1.0/4])
>>> a = np.array([1.0, -1.0/3])
>>> signal.tf2zpk(b, a)
[-0.5] [ 0.33333333] 0.5

i.e. the system has a zero at :math:`z=-1/2` and a pole at :math:`z=1/3`. 

The scipy function :func:`freqz` allows calculation of the frequency response
of a system described by the coeffcients :math:`a_k` and :math:`b_k`. See the
help of the :func:`freqz` function of a comprehensive example.


Filter Design
^^^^^^^^^^^^^

Time-discrete filters can be classified into finite response (FIR) filters and
infinite response (IIR) filters. FIR filters provide a linear phase response,
whereas IIR filters do not exhibit this behaviour. Scipy provides functions
for designing both types of filters.

FIR Filter
""""""""""

The function :func:`firwin` designs filters according to the window method.
Depending on the provided arguments, the function returns different filter
types (e.g. low-pass, band-pass...).

The example below designs a low-pass and a band-stop filter, respectively.

.. plot::

   >>> import numpy as np
   >>> import scipy.signal as signal
   >>> import matplotlib.pyplot as plt

   >>> b1 = signal.firwin(40, 0.5)
   >>> b2 = signal.firwin(41, [0.3, 0.8])
   >>> w1, h1 = signal.freqz(b1)
   >>> w2, h2 = signal.freqz(b2)

   >>> plt.title('Digital filter frequency response')
   >>> plt.plot(w1, 20*np.log10(np.abs(h1)), 'b')
   >>> plt.plot(w2, 20*np.log10(np.abs(h2)), 'r')
   >>> plt.ylabel('Amplitude Response (dB)')
   >>> plt.xlabel('Frequency (rad/sample)')
   >>> plt.grid()
   >>> plt.show()

Note that :func:`firwin` uses per default a normalized frequency defined such
that the value :math:`1` corresponds to the Nyquist frequency, whereas the
function :func:`freqz` is defined such that the value :math:`\pi` corresponds
to the Nyquist frequency.


The function :func:`firwin2` allows design of almost arbitrary frequency
responses by specifying an array of corner frequencies and corresponding
gains, respectively.

The example below designs a filter with such an arbitrary amplitude response.

.. plot::

   >>> import numpy as np
   >>> import scipy.signal as signal
   >>> import matplotlib.pyplot as plt

   >>> b = signal.firwin2(150, [0.0, 0.3, 0.6, 1.0], [1.0, 2.0, 0.5, 0.0])
   >>> w, h = signal.freqz(b)

   >>> plt.title('Digital filter frequency response')
   >>> plt.plot(w, np.abs(h))
   >>> plt.title('Digital filter frequency response')
   >>> plt.ylabel('Amplitude Response')
   >>> plt.xlabel('Frequency (rad/sample)')
   >>> plt.grid()
   >>> plt.show()

Note the linear scaling of the y-axis and the different definition of the
Nyquist frequency in :func:`firwin2` and :func:`freqz` (as explained above).


IIR Filter
""""""""""

Scipy provides two functions to directly design IIR :func:`iirdesign` and
:func:`iirfilter` where the filter type (e.g. elliptic) is passed as an
argument and several more filter design functions for specific filter types;
e.g. :func:`ellip`.

The example below designs an elliptic low-pass filter with defined passband
and stopband ripple, respectively. Note the much lower filter order (order 4)
compared with the FIR filters from the examples above in order to reach the same 
stop-band attenuation of :math:`\approx 60` dB.

.. plot::

   >>> import numpy as np
   >>> import scipy.signal as signal
   >>> import matplotlib.pyplot as plt

   >>> b, a = signal.iirfilter(4, Wn=0.2, rp=5, rs=60, btype='lowpass', ftype='ellip')
   >>> w, h = signal.freqz(b, a)

   >>> plt.title('Digital filter frequency response')
   >>> plt.plot(w, 20*np.log10(np.abs(h)))
   >>> plt.title('Digital filter frequency response')
   >>> plt.ylabel('Amplitude Response [dB]')
   >>> plt.xlabel('Frequency (rad/sample)')
   >>> plt.grid()
   >>> plt.show()


Other filters
^^^^^^^^^^^^^

The signal processing package provides many more filters as well.


Median Filter
"""""""""""""

A median filter is commonly applied when noise is markedly non-Gaussian or
when it is desired to preserve edges. The median filter works by sorting all
of the array pixel values in a rectangular region surrounding the point of
interest. The sample median of this list of neighborhood pixel values is used
as the value for the output array. The sample median is the middle array value
in a sorted list of neighborhood values. If there are an even number of
elements in the neighborhood, then the average of the middle two values is
used as the median. A general purpose median filter that works on
N-dimensional arrays is :func:`medfilt` . A specialized version that works
only for two-dimensional arrays is available as :func:`medfilt2d` .


Order Filter
""""""""""""

A median filter is a specific example of a more general class of filters
called order filters. To compute the output at a particular pixel, all order
filters use the array values in a region surrounding that pixel. These array
values are sorted and then one of them is selected as the output value. For
the median filter, the sample median of the list of array values is used as
the output. A general order filter allows the user to select which of the
sorted values will be used as the output. So, for example one could choose to
pick the maximum in the list or the minimum. The order filter takes an
additional argument besides the input array and the region mask that specifies
which of the elements in the sorted list of neighbor array values should be
used as the output. The command to perform an order filter is
:func:`order_filter`.


Wiener filter
"""""""""""""

The Wiener filter is a simple deblurring filter for denoising images. This is
not the Wiener filter commonly described in image reconstruction problems but
instead it is a simple, local-mean filter. Let :math:`x` be the input signal,
then the output is

.. math::

    y=\left\{ \begin{array}{cc} \frac{\sigma^{2}}{\sigma_{x}^{2}}m_{x}+\left(1-\frac{\sigma^{2}}{\sigma_{x}^{2}}\right)x & \sigma_{x}^{2}\geq\sigma^{2},\\ m_{x} & \sigma_{x}^{2}<\sigma^{2},\end{array}\right.

where :math:`m_{x}` is the local estimate of the mean and
:math:`\sigma_{x}^{2}` is the local estimate of the variance. The window for
these estimates is an optional input parameter (default is :math:`3\times3` ).
The parameter :math:`\sigma^{2}` is a threshold noise parameter. If
:math:`\sigma` is not given then it is estimated as the average of the local
variances.


Hilbert filter
""""""""""""""

The Hilbert transform constructs the complex-valued analytic signal
from a real signal. For example if :math:`x=\cos\omega n` then
:math:`y=\textrm{hilbert}\left(x\right)` would return (except near the
edges) :math:`y=\exp\left(j\omega n\right).` In the frequency domain,
the hilbert transform performs

.. math::

    Y=X\cdot H

where :math:`H` is 2 for positive frequencies, :math:`0` for negative
frequencies and :math:`1` for zero-frequencies.


Analog Filter Design
^^^^^^^^^^^^^^^^^^^^

The functions :func:`iirdesign`, :func:`iirfilter`, and the filter design
functions for specific filter types (e.g. :func:`ellip`) all have a flag
`analog` which allows design of analog filters as well.

The example below designs an analog (IIR) filter, obtains via :func:`tf2zpk`
the poles and zeros and plots them in the complex s-plane. The zeros at
:math:`\omega \approx 150` and :math:`\omega \approx 300` can be clearly seen
in the amplitude response.


.. plot::

   >>> import numpy as np
   >>> import scipy.signal as signal
   >>> import matplotlib.pyplot as plt

   >>> b, a = signal.iirdesign(wp=100, ws=200, gpass=2.0, gstop=40., analog=True)
   >>> w, h = signal.freqs(b, a)

   >>> plt.title('Analog filter frequency response')
   >>> plt.plot(w, 20*np.log10(np.abs(h)))
   >>> plt.ylabel('Amplitude Response [dB]')
   >>> plt.xlabel('Frequency')
   >>> plt.grid()
   >>> plt.show()


   >>> z, p, k = signal.tf2zpk(b, a)

   >>> plt.plot(np.real(z), np.imag(z), 'xb')
   >>> plt.plot(np.real(p), np.imag(p), 'or')
   >>> plt.legend(['Zeros', 'Poles'], loc=2)

   >>> plt.title('Pole / Zero Plot')
   >>> plt.ylabel('Real')
   >>> plt.xlabel('Imaginary')
   >>> plt.grid()
   >>> plt.show()



Spectral Analysis
------------------

Periodogram Measurements
^^^^^^^^^^^^^^^^^^^^^^^^

The scipy function :func:`periodogram` provides a method to estimate the
spectral density using the periodogram method.

The example below calculates the periodogram of a sine signal in white
Gaussian noise.

.. plot::

   >>> import numpy as np
   >>> import scipy.signal as signal
   >>> import matplotlib.pyplot as plt

   >>> fs = 10e3
   >>> N = 1e5
   >>> amp = 2*np.sqrt(2)
   >>> freq = 1270.0
   >>> noise_power = 0.001 * fs / 2
   >>> time = np.arange(N) / fs
   >>> x = amp*np.sin(2*np.pi*freq*time)
   >>> x += np.random.normal(scale=np.sqrt(noise_power), size=time.shape)

   >>> f, Pper_spec = signal.periodogram(x, fs, 'flattop', scaling='spectrum')

   >>> plt.semilogy(f, Pper_spec)
   >>> plt.xlabel('frequency [Hz]')
   >>> plt.ylabel('PSD')
   >>> plt.grid()
   >>> plt.show()



Spectral Analysis using Welch's Method
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

An improved method, especially with respect to noise immunity, is Welch's
method which is implemented by the scipy function :func:`welch`.

The example below estimates the spectrum using Welch's method and uses the
same parameters as the example above. Note the much smoother noise floor of
the spectogram.


.. plot::

   >>> import numpy as np
   >>> import scipy.signal as signal
   >>> import matplotlib.pyplot as plt

   >>> fs = 10e3
   >>> N = 1e5
   >>> amp = 2*np.sqrt(2)
   >>> freq = 1270.0
   >>> noise_power = 0.001 * fs / 2
   >>> time = np.arange(N) / fs
   >>> x = amp*np.sin(2*np.pi*freq*time)
   >>> x += np.random.normal(scale=np.sqrt(noise_power), size=time.shape)

   >>> f, Pwelch_spec = signal.welch(x, fs, scaling='spectrum')

   >>> plt.semilogy(f, Pwelch_spec)
   >>> plt.xlabel('frequency [Hz]')
   >>> plt.ylabel('PSD')
   >>> plt.grid()
   >>> plt.show()


Lomb-Scargle Periodograms (:func:`lombscargle`)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Least-squares spectral analysis (LSSA) is a method of estimating a frequency
spectrum, based on a least squares fit of sinusoids to data samples, similar
to Fourier analysis. Fourier analysis, the most used spectral method in
science, generally boosts long-periodic noise in long gapped records; LSSA
mitigates such problems.


The Lomb-Scargle method performs spectral analysis on unevenly sampled data and
is known to be a powerful way to find, and test the significance of, weak
periodic signals.

For a time series comprising :math:`N_{t}` measurements :math:`X_{j}\equiv
X(t_{j})` sampled at times :math:`t_{j}` where :math:`(j = 1, \ldots, N_{t})`,
assumed to have been scaled and shifted such that its mean is zero and its
variance is unity, the normalized Lomb-Scargle periodogram at frequency
:math:`f` is

.. math::

    P_{n}(f) \frac{1}{2}\left\{\frac{\left[\sum_{j}^{N_{t}}X_{j}\cos\omega(t_{j}-\tau)\right]^{2}}{\sum_{j}^{N_{t}}\cos^{2}\omega(t_{j}-\tau)}+\frac{\left[\sum_{j}^{N_{t}}X_{j}\sin\omega(t_{j}-\tau)\right]^{2}}{\sum_{j}^{N_{t}}\sin^{2}\omega(t_{j}-\tau)}\right\}.

Here, :math:`\omega \equiv 2\pi f` is the angular frequency. The frequency
dependent time offset :math:`\tau` is given by

.. math::

    \tan 2\omega\tau = \frac{\sum_{j}^{N_{t}}\sin 2\omega t_{j}}{\sum_{j}^{N_{t}}\cos 2\omega t_{j}}.

The :func:`lombscargle` function calculates the periodogram using a slightly
modified algorithm due to Townsend [3]_ which allows the periodogram to be
calculated using only a single pass through the input arrays for each
frequency.

The equation is refactored as:

.. math::

    P_{n}(f) = \frac{1}{2}\left[\frac{(c_{\tau}XC + s_{\tau}XS)^{2}}{c_{\tau}^{2}CC + 2c_{\tau}s_{\tau}CS + s_{\tau}^{2}SS} + \frac{(c_{\tau}XS - s_{\tau}XC)^{2}}{c_{\tau}^{2}SS - 2c_{\tau}s_{\tau}CS + s_{\tau}^{2}CC}\right]

and

.. math::

    \tan 2\omega\tau = \frac{2CS}{CC-SS}.

Here,

.. math::

    c_{\tau} = \cos\omega\tau,\qquad s_{\tau} = \sin\omega\tau

while the sums are

.. math::

    XC &= \sum_{j}^{N_{t}} X_{j}\cos\omega t_{j}\\
    XS &= \sum_{j}^{N_{t}} X_{j}\sin\omega t_{j}\\
    CC &= \sum_{j}^{N_{t}} \cos^{2}\omega t_{j}\\
    SS &= \sum_{j}^{N_{t}} \sin^{2}\omega t_{j}\\
    CS &= \sum_{j}^{N_{t}} \cos\omega t_{j}\sin\omega t_{j}.

This requires :math:`N_{f}(2N_{t}+3)` trigonometric function evaluations
giving a factor of :math:`\sim 2` speed increase over the straightforward
implementation.


Detrend
-------

Scipy provides the function :func:`detrend` to remove a constant or linear
trend in a data series in order to see effect of higher order.

The example below removes the constant and linear trend of a 2-nd order
polynomial time series and plots the remaining signal components.

.. plot::

   >>> import numpy as np
   >>> import scipy.signal as signal
   >>> import matplotlib.pyplot as plt

   >>> t = np.linspace(-10, 10, 20)
   >>> y = 1 + t + 0.01*t**2
   >>> yconst = signal.detrend(y, type='constant')
   >>> ylin = signal.detrend(y, type='linear')

   >>> plt.plot(t, y, '-rx')
   >>> plt.plot(t, yconst, '-bo')
   >>> plt.plot(t, ylin, '-k+')
   >>> plt.grid()
   >>> plt.legend(['signal', 'const. detrend', 'linear detrend'])
   >>> plt.show()




..
.. Filter design
.. -------------
..
..
.. Finite-impulse response design
.. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
..
..
.. Inifinite-impulse response design
.. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
..
..
.. Analog filter frequency response
.. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
..
..
.. Digital filter frequency response
.. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
..
..
.. Linear Time-Invariant Systems
.. -----------------------------
..
..
.. LTI Object
.. ^^^^^^^^^^
..
..
.. Continuous-Time Simulation
.. ^^^^^^^^^^^^^^^^^^^^^^^^^^
..
..
.. Step response
.. ^^^^^^^^^^^^^
..
..
.. Impulse response
.. ^^^^^^^^^^^^^^^^
..
..
.. Input/Output
.. ============
..
..
.. Binary
.. ------
..
..
.. Arbitrary binary input and output (fopen)
.. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
..
..
.. Read and write Matlab .mat files
.. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
..
..
.. Saving workspace
.. ^^^^^^^^^^^^^^^^
..
..
.. Text-file
.. ---------
..
..
.. Read text-files (read_array)
.. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
..
..
.. Write a text-file (write_array)
.. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
..
..
.. Fourier Transforms
.. ==================
..
..
.. One-dimensional
.. ---------------
..
..
.. Two-dimensional
.. ---------------
..
..
.. N-dimensional
.. -------------
..
..
.. Shifting
.. --------
..
..
.. Sample frequencies
.. ------------------
..
..
.. Hilbert transform
.. -----------------
..
..
.. Tilbert transform
.. -----------------


.. rubric:: References

Some further reading and related software:

.. [1] N.R. Lomb "Least-squares frequency analysis of unequally spaced
       data", Astrophysics and Space Science, vol 39, pp. 447-462, 1976

.. [2] J.D. Scargle "Studies in astronomical time series analysis. II - 
       Statistical aspects of spectral analysis of unevenly spaced data",
       The Astrophysical Journal, vol 263, pp. 835-853, 1982

.. [3] R.H.D. Townsend, "Fast calculation of the Lomb-Scargle
       periodogram using graphics processing units.", The Astrophysical
       Journal Supplement Series, vol 191, pp. 247-253, 2010

