# Signal Processing (`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 , then the B-spline approximation to a 1-dimensional function is the finite-basis expansion.

In two dimensions with knot-spacing and , the function representation is

In these expressions, is the space-limited B-spline basis function of order, . The requirement of equally-spaced knot-points and equally-spaced data points, allows the development of fast (inverse-filtering) algorithms for determining the coefficients, , from sample-values, . 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

Using the property of B-splines that

it can be seen that

If , then at the sample points,

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 (`qspline1d`, `qspline2d`, `cspline1d`,
`cspline2d`). The package also supplies a function ( `bspline` )
for evaluating the bspline basis function, for
arbitrary order and For large , the B-spline basis
function can be approximated well by a zero-mean Gaussian function with
standard-deviation equal to :

A function to compute this Gaussian for arbitrary and is
also available ( `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 `misc.lena`.
The command `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 `convolve2d` which convolves arbitrary
two-dimensional filters and allows for choosing mirror-symmetric boundary
conditions.

```
>>> 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()
```

## 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 image with this method would require multiplication of a matrix with a vector. Just trying to store the matrix using a standard Numpy array would require elements. At 4 bytes per element this would require 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 define a one-dimensional signal indexed by the integer Full convolution of two one-dimensional signals can be expressed as

This equation can only be implemented directly if we limit the sequences to finite support sequences that can be stored in a computer, choose to be the starting point of both sequences, let be that value for which for all and be that value for which for all , then the discrete convolution expression is

For convenience assume Then, more explicitly the output of this operation is

Thus, the full discrete convolution of two finite sequences of lengths and respectively results in a finite sequence of length

One dimensional convolution is implemented in SciPy with the function
`convolve`. This function takes as inputs the signals
, and an optional flag and returns the signal 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 values are returned starting
at so that the
output has the same length as the largest input. If the flag has a value of
‘valid’ then only the middle
output values are returned where depends on all of the values of the
smallest input from to In
other words only the values to
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 `convolve` can actually take -dimensional
arrays as inputs and will return the -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

is the (cross) correlation of the signals and For finite-length signals with outside of the range and outside of the range the summation can simplify to

Assuming again that this is

The SciPy function `correlate` implements this operation. Equivalent
flags are available for this operation to return the full length
sequence (‘full’) or a sequence with the same size as the largest sequence
starting at
(‘same’) or a sequence where the values depend on all the values of the
smallest sequence (‘valid’). This final option returns the
values to inclusive.

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

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

```
>>> 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 `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 ( ), otherwise linear filtering is more efficiently calculated in the
frequency domain provided by the function `fftconvolve`.

If the filter function can be factored according to

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

which is often used for blurring.

```
>>> 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

where is the input sequence and is the output sequence. If we assume initial rest so that for , then this kind of filter can be implemented using convolution. However, the convolution filter sequence could be infinite if for In addition, this general class of linear filter allows initial conditions to be placed on for resulting in a filter that cannot be expressed using convolution.

The difference equation filter can be thought of as finding recursively in terms of it’s previous values

Often 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 ).

where Note that if and if In this way, the output at time depends only on the input at time and the value of at the previous time. This can always be calculated as long as the values are computed and stored at each time step.

The difference-equation filter is called using the command `lfilter` in
SciPy. This command takes as inputs the vector the vector,
a signal and returns the vector (the same
length as ) computed using the equation given above. If is
-dimensional, then the filter is computed along the axis provided.
If, desired, initial conditions providing the values of
to 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 and In other words, perhaps you have the values of to and the values of to and would like to determine what values of should be delivered as initial conditions to the difference-equation filter. It is not difficult to show that for

Using this formula we can find the initial condition vector
to given initial
conditions on (and ). The command `lfiltic` performs
this function.

As an example consider the following system:

The code calculates the signal for a given signal ; first for initial condiditions (default case), then for 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 has the same length as the length as the input signal .

#### 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 , zeros and poles , respectively, to describe the system by means of its transfer function according to

This alternative representation can be obtain wit hthe scipy function
`tf2zpk`; the inverse is provided by `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 and a pole at .

The scipy function `freqz` allows calculation of the frequency response
of a system described by the coeffcients and . See the
help of the `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 `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.

```
>>> 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 `firwin` uses per default a normalized frequency defined such
that the value corresponds to the Nyquist frequency, whereas the
function `freqz` is defined such that the value corresponds
to the Nyquist frequency.

The function `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.

```
>>> 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 `firwin2` and `freqz` (as explained above).

#### IIR Filter¶

Scipy provides two functions to directly design IIR `iirdesign` and
`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. `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 dB.

```
>>> 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 `medfilt` . A specialized version that works
only for two-dimensional arrays is available as `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
`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 be the input signal, then the output is

where is the local estimate of the mean and is the local estimate of the variance. The window for these estimates is an optional input parameter (default is ). The parameter is a threshold noise parameter. If 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 then would return (except near the edges) In the frequency domain, the hilbert transform performs

where is 2 for positive frequencies, for negative frequencies and for zero-frequencies.

### Analog Filter Design¶

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

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

```
>>> 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 `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.

```
>>> 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 `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.

```
>>> 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 (`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 measurements sampled at times where , 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 is

Here, is the angular frequency. The frequency dependent time offset is given by

The `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:

and

Here,

while the sums are

This requires trigonometric function evaluations giving a factor of speed increase over the straightforward implementation.

## Detrend¶

Scipy provides the function `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.

```
>>> 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()
```

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 |