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 1- and 2-D 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 \(\Delta x\), then the B-spline approximation to a 1-D function is the finite-basis expansion.
In two dimensions with knot-spacing \(\Delta x\) and \(\Delta y\), the function representation is
In these expressions, \(\beta^{o}\left(\cdot\right)\) is the space-limited B-spline basis function of order \(o\). The requirement of equally-spaced knot-points and equally-spaced data points, allows the development of fast (inverse-filtering) algorithms for determining the coefficients, \(c_{j}\), from sample-values, \(y_{n}\). 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
Using the property of B-splines that
it can be seen that
If \(o=3\), 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 the boundaries are handled (this becomes increasingly more important as the number of dimensions in the dataset increases). The algorithms relating to B-splines in the signal-processing subpackage 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 B-spline basis function, \(\beta^{o}\left(x\right)\) for
arbitrary order and \(x.\) For large \(o\), the B-spline basis
function can be approximated well by a zero-mean Gaussian function with
standard-deviation equal to \(\sigma_{o}=\left(o+1\right)/12\) :
A function to compute this Gaussian for arbitrary \(x\) and \(o\) is
also available ( gauss_spline ). The following code and figure use
spline-filtering to compute an edge-image (the second derivative of a smoothed
spline) of a raccoon’s face, which is an array returned by the command scipy.misc.face.
The command sepfir2d was used to apply a separable 2-D 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
2-D filters and allows for choosing mirror-symmetric boundary
conditions.
>>> import numpy as np
>>> from scipy import signal, misc
>>> import matplotlib.pyplot as plt
>>> image = misc.face(gray=True).astype(np.float32)
>>> derfilt = np.array([1.0, -2, 1.0], dtype=np.float32)
>>> ck = signal.cspline2d(image, 8.0)
>>> deriv = (signal.sepfir2d(ck, derfilt, [1]) +
... signal.sepfir2d(ck, [1], derfilt))
Alternatively, we could have done:
laplacian = np.array([[0,1,0], [1,-4,1], [0,1,0]], dtype=np.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 \(512 \times 512\) image with this method would require multiplication of a \(512^2 \times 512^2\) matrix with a \(512^2\) vector. Just trying to store the \(512^2 \times 512^2\) matrix using a standard NumPy array would require \(68,719,476,736\) elements. At 4 bytes per element this would require \(256\textrm{GB}\) 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 \(x\left[n\right]\) define a 1-D signal indexed by the integer \(n.\) Full convolution of two 1-D 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 \(n=0\) to be the starting point of both sequences, let \(K+1\) be that value for which \(x\left[n\right]=0\) for all \(n\geq K+1\) and \(M+1\) be that value for which \(h\left[n\right]=0\) for all \(n\geq M+1\), then the discrete convolution expression is
For convenience, assume \(K\geq M.\) Then, more explicitly, the output of this operation is
Thus, the full discrete convolution of two finite sequences of lengths \(K+1\) and \(M+1\), respectively, results in a finite sequence of length \(K+M+1=\left(K+1\right)+\left(M+1\right)-1.\)
1-D convolution is implemented in SciPy with the function
convolve. This function takes as inputs the signals \(x,\)
\(h\), and two optional flags ‘mode’ and ‘method’, and returns the signal
\(y.\)
The first optional flag, ‘mode’, allows for the 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 \(K\) values are returned, starting at \(y\left[\left\lfloor \frac{M-1}{2}\right\rfloor \right]\), so that the output has the same length as the first input. If the flag has a value of ‘valid’, then only the middle \(K-M+1=\left(K+1\right)-\left(M+1\right)+1\) output values are returned, where \(z\) depends on all of the values of the smallest input from \(h\left[0\right]\) to \(h\left[M\right].\) In other words, only the values \(y\left[M\right]\) to \(y\left[K\right]\) inclusive are returned.
The second optional flag, ‘method’, determines how the convolution is computed,
either through the Fourier transform approach with fftconvolve or
through the direct method. By default, it selects the expected faster method.
The Fourier transform method has order \(O(N\log N)\), while the direct
method has order \(O(N^2)\). Depending on the big O constant and the value
of \(N\), one of these two methods may be faster. The default value, ‘auto’,
performs a rough calculation and chooses the expected faster method, while the
values ‘direct’ and ‘fft’ force computation with the other two methods.
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)
array([ 0., 1., 2., 3., 0., 0., 0.])
>>> signal.convolve(x, h, 'same')
array([ 2., 3., 0.])
This same function convolve can actually take N-D
arrays as inputs and will return the N-D 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)
array([[ 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 that the minus sign becomes a plus sign. Thus,
is the (cross) correlation of the signals \(y\) and \(x.\) For finite-length signals with \(y\left[n\right]=0\) outside of the range \(\left[0,K\right]\) and \(x\left[n\right]=0\) outside of the range \(\left[0,M\right],\) the summation can simplify to
Assuming again that \(K\geq M\), this is
The SciPy function correlate implements this operation. Equivalent
flags are available for this operation to return the full \(K+M+1\) length
sequence (‘full’) or a sequence with the same size as the largest sequence
starting at \(w\left[-K+\left\lfloor \frac{M-1}{2}\right\rfloor \right]\)
(‘same’) or a sequence where the values depend on all the values of the
smallest sequence (‘valid’). This final option returns the \(K-M+1\)
values \(w\left[M-K\right]\) to \(w\left[0\right]\) inclusive.
The function correlate can also take arbitrary N-D arrays as input
and return the N-D convolution of the two arrays on output.
When \(N=2,\) 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.face(gray=True)
>>> 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()
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 ( \(K\gg
M\) ), otherwise linear filtering is more efficiently calculated in the
frequency domain provided by the function fftconvolve. By default,
convolve estimates the fastest method using choose_conv_method.
If the filter function \(w[n,m]\) 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.ascent()
>>> w = signal.windows.gaussian(51, 10.0)
>>> 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 1-D filters (that includes convolution filters) are filters described by the difference equation
where \(x\left[n\right]\) is the input sequence and \(y\left[n\right]\) is the output sequence. If we assume initial rest so that \(y\left[n\right]=0\) for \(n<0\), then this kind of filter can be implemented using convolution. However, the convolution filter sequence \(h\left[n\right]\) could be infinite if \(a_{k}\neq0\) for \(k\geq1.\) In addition, this general class of linear filter allows initial conditions to be placed on \(y\left[n\right]\) for \(n<0\) resulting in a filter that cannot be expressed using convolution.
The difference equation filter can be thought of as finding \(y\left[n\right]\) recursively in terms of its previous values
Often, \(a_{0}=1\) is chosen for normalization. The implementation in SciPy of this general difference equation filter is a little more complicated than 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 \(a_{0}=1\) ):
where \(K=\max\left(N,M\right).\) Note that \(b_{K}=0\) if \(K>M\) and \(a_{K}=0\) if \(K>N.\) In this way, the output at time \(n\) depends only on the input at time \(n\) and the value of \(z_{0}\) at the previous time. This can always be calculated as long as the \(K\) values \(z_{0}\left[n-1\right]\ldots z_{K-1}\left[n-1\right]\) are computed and stored at each time step.
The difference-equation filter is called using the command lfilter in
SciPy. This command takes as inputs the vector \(b,\) the vector,
\(a,\) a signal \(x\) and returns the vector \(y\) (the same
length as \(x\) ) computed using the equation given above. If \(x\) is
N-D, then the filter is computed along the axis provided.
If desired, initial conditions providing the values of
\(z_{0}\left[-1\right]\) to \(z_{K-1}\left[-1\right]\) 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 \(x\left[n\right]\) and \(y\left[n\right].\) In other words, perhaps you have the values of \(x\left[-M\right]\) to \(x\left[-1\right]\) and the values of \(y\left[-N\right]\) to \(y\left[-1\right]\) and would like to determine what values of \(z_{m}\left[-1\right]\) should be delivered as initial conditions to the difference-equation filter. It is not difficult to show that, for \(0\leq m<K,\)
Using this formula, we can find the initial-condition vector
\(z_{0}\left[-1\right]\) to \(z_{K-1}\left[-1\right]\) given initial
conditions on \(y\) (and \(x\) ). The command lfiltic performs
this function.
As an example, consider the following system:
The code calculates the signal \(y[n]\) for a given signal \(x[n]\);
first for initial conditions \(y[-1] = 0\) (default case), then for
\(y[-1] = 2\) by means of 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)
array([0.5, 0.41666667, 0.13888889, 0.0462963])
>>> zi = signal.lfiltic(b, a, y=[2.])
>>> signal.lfilter(b, a, x, zi=zi)
(array([ 1.16666667, 0.63888889, 0.21296296, 0.07098765]), array([0.02366]))
Note that the output signal \(y[n]\) has the same length as the length as the input signal \(x[n]\).
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 \(k\), \(N_z\) zeros \(z_k\) and \(N_p\) poles \(p_k\), respectively, to describe the system by means of its transfer function \(H(z)\), according to
This alternative representation can be obtained with the scipy function
tf2zpk; the inverse is provided by zpk2tf.
For the above example we have
>>> b = np.array([1.0/2, 1.0/4])
>>> a = np.array([1.0, -1.0/3])
>>> signal.tf2zpk(b, a)
(array([-0.5]), array([ 0.33333333]), 0.5)
i.e., the system has a zero at \(z=-1/2\) and a pole at \(z=1/3\).
The scipy function freqz allows calculation of the frequency response
of a system described by the coefficients \(a_k\) and \(b_k\). See the
help of the freqz function for a comprehensive example.
Filter Design#
Time-discrete filters can be classified into finite response (FIR) filters and infinite response (IIR) filters. FIR filters can provide a linear phase response, whereas IIR filters cannot. 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 \(1\) corresponds to the Nyquist frequency, whereas the
function freqz is defined such that the value \(\pi\) 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 pass-band and stop-band 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 \(\approx 60\) 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()
Filter Coefficients#
Filter coefficients can be stored in several different formats:
‘ba’ or ‘tf’ = transfer function coefficients
‘zpk’ = zeros, poles, and overall gain
‘ss’ = state-space system representation
‘sos’ = transfer function coefficients of second-order sections
Functions, such as tf2zpk and zpk2ss, can convert between them.
Transfer function representation#
The ba or tf format is a 2-tuple (b, a) representing a transfer
function, where b is a length M+1 array of coefficients of the M-order
numerator polynomial, and a is a length N+1 array of coefficients of the
N-order denominator, as positive, descending powers of the transfer function
variable. So the tuple of \(b = [b_0, b_1, ..., b_M]\) and
\(a =[a_0, a_1, ..., a_N]\) can represent an analog filter of the form:
or a discrete-time filter of the form:
This “positive powers” form is found more commonly in controls engineering. If M and N are equal (which is true for all filters generated by the bilinear transform), then this happens to be equivalent to the “negative powers” discrete-time form preferred in DSP:
Although this is true for common filters, remember that this is not true in the general case. If M and N are not equal, the discrete-time transfer function coefficients must first be converted to the “positive powers” form before finding the poles and zeros.
This representation suffers from numerical error at higher orders, so other formats are preferred when possible.
Zeros and poles representation#
The zpk format is a 3-tuple (z, p, k), where z is an M-length
array of the complex zeros of the transfer function
\(z = [z_0, z_1, ..., z_{M-1}]\), p is an N-length array of the
complex poles of the transfer function \(p = [p_0, p_1, ..., p_{N-1}]\),
and k is a scalar gain. These represent the digital transfer function:
or the analog transfer function:
Although the sets of roots are stored as ordered NumPy arrays, their ordering
does not matter: ([-1, -2], [-3, -4], 1) is the same filter as
([-2, -1], [-4, -3], 1).
State-space system representation#
The ss format is a 4-tuple of arrays (A, B, C, D) representing the
state-space of an N-order digital/discrete-time system of the form:
or a continuous/analog system of the form:
with P inputs, Q outputs and N state variables, where:
x is the state vector
y is the output vector of length Q
u is the input vector of length P
A is the state matrix, with shape
(N, N)B is the input matrix with shape
(N, P)C is the output matrix with shape
(Q, N)D is the feedthrough or feedforward matrix with shape
(Q, P). (In cases where the system does not have a direct feedthrough, all values in D are zero.)
State-space is the most general representation and the only one that allows
for multiple-input, multiple-output (MIMO) systems. There are multiple
state-space representations for a given transfer function. Specifically, the
“controllable canonical form” and “observable canonical form” have the same
coefficients as the tf representation, and, therefore, suffer from the same
numerical errors.
Second-order sections representation#
The sos format is a single 2-D array of shape (n_sections, 6),
representing a sequence of second-order transfer functions which, when
cascaded in series, realize a higher-order filter with minimal numerical
error. Each row corresponds to a second-order tf representation, with
the first three columns providing the numerator coefficients and the last
three providing the denominator coefficients:
The coefficients are typically normalized, such that \(a_0\) is always 1. The section order is usually not important with floating-point computation; the filter output will be the same, regardless of the order.
Filter transformations#
The IIR filter design functions first generate a prototype analog low-pass filter with a normalized cutoff frequency of 1 rad/sec. This is then transformed into other frequencies and band types using the following substitutions:
Type |
Transformation |
|---|---|
\(s \rightarrow \frac{s}{\omega_0}\) |
|
\(s \rightarrow \frac{\omega_0}{s}\) |
|
\(s \rightarrow \frac{s^2 + {\omega_0}^2}{s \cdot \mathrm{BW}}\) |
|
\(s \rightarrow \frac{s \cdot \mathrm{BW}}{s^2 + {\omega_0}^2}\) |
Here, \(\omega_0\) is the new cutoff or center frequency, and \(\mathrm{BW}\) is the bandwidth. These preserve symmetry on a logarithmic frequency axis.
To convert the transformed analog filter into a digital filter, the
bilinear transform is used, which makes the following substitution:
where T is the sampling time (the inverse of the sampling frequency).
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-D arrays is medfilt. A specialized version that works
only for 2-D 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 \(x\) be the input signal, then the output is
where \(m_{x}\) is the local estimate of the mean and \(\sigma_{x}^{2}\) is the local estimate of the variance. The window for these estimates is an optional input parameter (default is \(3\times3\) ). The parameter \(\sigma^{2}\) is a threshold noise parameter. If \(\sigma\) 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 \(x=\cos\omega n\), then \(y=\textrm{hilbert}\left(x\right)\) would return (except near the edges) \(y=\exp\left(j\omega n\right).\) In the frequency domain, the hilbert transform performs
where \(H\) is \(2\) for positive frequencies, \(0\) for negative frequencies, and \(1\) 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 the 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
\(\omega \approx 150\) and \(\omega \approx 300\) 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), 'ob', markerfacecolor='none')
>>> plt.plot(np.real(p), np.imag(p), 'xr')
>>> plt.legend(['Zeros', 'Poles'], loc=2)
>>> plt.title('Pole / Zero Plot')
>>> plt.xlabel('Real')
>>> plt.ylabel('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 spectrogram.
>>> 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) 1 2 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 \(N_{t}\) measurements \(X_{j}\equiv X(t_{j})\) sampled at times \(t_{j}\), where \((j = 1, \ldots, N_{t})\), 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 \(f\) is
Here, \(\omega \equiv 2\pi f\) is the angular frequency. The frequency-dependent time offset \(\tau\) 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 \(N_{f}(2N_{t}+3)\) trigonometric function evaluations giving a factor of \(\sim 2\) 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 second-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