# Miscellaneous¶

The Miscellaneous package provides various functions and methods which are used in the examples and in some of the other implemented routines.

The package contains the following modules:

## Tools for 2nd order systems¶

The PyDynamic.misc.SecondOrderSystem module is a collection of methods that are used throughout the whole package, specialized for second order dynamic systems, such as the ones used for high-class accelerometers.

This module contains the following functions:

PyDynamic.misc.SecondOrderSystem.sos_FreqResp(S, d, f0, freqs)[source]

Calculation of the system frequency response

The frequency response is calculated from the continuous physical model of a second order system given by

$$H(f) = \frac{4S\pi^2f_0^2}{(2\pi f_0)^2 + 2jd(2\pi f_0)f - f^2}$$

If the provided system parameters are vectors then $$H(f)$$ is calculated for each set of parameters. This is helpful for Monte Carlo simulations by using draws from the model parameters

Parameters: S (float or ndarray shape (K,)) – static gain d (float or ndarray shape (K,)) – damping parameter f0 (float or ndarray shape (K,)) – resonance frequency freqs (ndarray shape (N,)) – frequencies at which to calculate the freq response H – complex frequency response values ndarray shape (N,) or ndarray shape (N,K)
PyDynamic.misc.SecondOrderSystem.sos_phys2filter(S, d, f0)[source]

Calculation of continuous filter coefficients from physical parameters.

If the provided system parameters are vectors then the filter coefficients are calculated for each set of parameters. This is helpful for Monte Carlo simulations by using draws from the model parameters

Parameters: S (float) – static gain d (float) – damping parameter f0 (float) – resonance frequency b,a – analogue filter coefficients ndarray
PyDynamic.misc.SecondOrderSystem.sos_absphase(S, d, f0, uS, ud, uf0, f, runs=10000)[source]

Propagation of uncertainty from physical parameters to real and imaginary part of system’s transfer function using GUM S2 Monte Carlo.

Parameters: S (float) – static gain d (float) – damping f0 (float) – resonance frequency uS (float) – uncertainty associated with static gain ud (float) – uncertainty associated with damping uf0 (float) – uncertainty associated with resonance frequency f (ndarray, shape (N,)) – frequency values at which to calculate amplitue and phase Hmean (ndarray, shape (N,)) – best estimate of complex frequency response values Hcov (ndarray, shape (2N,2N)) – covariance matrix [ [u(abs,abs), u(abs,phase)], [u(phase,abs), u(phase,phase)] ]
PyDynamic.misc.SecondOrderSystem.sos_realimag(S, d, f0, uS, ud, uf0, f, runs=10000)[source]

Propagation of uncertainty from physical parameters to real and imaginary part of system’s transfer function using GUM S2 Monte Carlo.

Parameters: S (float) – static gain d (float) – damping f0 (float) – resonance frequency uS (float) – uncertainty associated with static gain ud (float) – uncertainty associated with damping uf0 (float) – uncertainty associated with resonance frequency f (ndarray, shape (N,)) – frequency values at which to calculate real and imaginary part Hmean (ndarray, shape (N,)) – best estimate of complex frequency response values Hcov (ndarray, shape (2N,2N)) – covariance matrix [ [u(real,real), u(real,imag)], [u(imag,real), u(imag,imag)] ]

## Tools for digital filters¶

The PyDynamic.misc.filterstuff module is a collection of methods which are related to filter design.

This module contains the following functions:

PyDynamic.misc.filterstuff.db(vals)[source]

Calculation of decibel values $$20\log_{10}(x)$$ for a vector of values

PyDynamic.misc.filterstuff.ua(vals)[source]

Shortcut for calculation of unwrapped angle of complex values

PyDynamic.misc.filterstuff.grpdelay(b, a, Fs, nfft=512)[source]

Calculation of the group delay of a digital filter

Parameters: b (ndarray) – IIR filter numerator coefficients a (ndarray) – IIR filter denominator coefficients Fs (float) – sampling frequency of the filter nfft (int) – number of FFT bins group_delay (np.ndarray) – group delay values frequencies (ndarray) – frequencies at which the group delay is calculated

References

PyDynamic.misc.filterstuff.mapinside(a)[source]

Maps the roots of polynomial to the unit circle.

Maps the roots of polynomial with coefficients $$a$$ to the unit circle.

Parameters: a (ndarray) – polynomial coefficients a – polynomial coefficients with all roots inside or on the unit circle ndarray
PyDynamic.misc.filterstuff.kaiser_lowpass(L, fcut, Fs, beta=8.0)[source]

Design of a FIR lowpass filter using the window technique with a Kaiser window.

This method uses a Kaiser window. Filters of that type are often used as FIR low-pass filters due to their linear phase.

Parameters: L (int) – filter order (window length) fcut (float) – desired cut-off frequency Fs (float) – sampling frequency beta (float) – scaling parameter for the Kaiser window blow (ndarray) – FIR filter coefficients shift (int) – delay of the filter (in samples)
PyDynamic.misc.filterstuff.isstable(b, a, ftype='digital')[source]

Determine whether IIR filter (b,a) is stable

Determine whether IIR filter (b,a) is stable by checking roots of the polynomial ´a´.

Parameters: b (ndarray) – filter numerator coefficients a (ndarray) – filter denominator coefficients ftype (string) – type of filter (digital or analog) stable – whether filter is stable or not bool
PyDynamic.misc.filterstuff.savitzky_golay(y, window_size, order, deriv=0, delta=1.0)[source]

Smooth (and optionally differentiate) data with a Savitzky-Golay filter

The Savitzky-Golay filter removes high frequency noise from data. It has the advantage of preserving the original shape and features of the signal better than other types of filtering approaches, such as moving averages techniques.

Parameters: y (ndarray, shape (N,)) – the values of the time history of the signal window_size (int) – the length of the window. Must be an odd integer number order (int) – the order of the polynomial used in the filtering. Must be less then window_size - 1. deriv (int, optional) – The order of the derivative to compute. This must be a nonnegative integer. The default is 0, which means to filter the data without differentiating. delta (float, optional) – The spacing of the samples to which the filter will be applied. This is only used if deriv > 0. This includes a factor $$n! / h^n$$, where $$n$$ is represented by deriv and $$1/h$$ by delta. ys – the smoothed signal (or it’s n-th derivative). ndarray, shape (N,)

Notes

The Savitzky-Golay is a type of low-pass filter, particularly suited for smoothing noisy data. The main idea behind this approach is to make for each point a least-square fit with a polynomial of high order over a odd-sized window centered at the point.

References

## Test signals¶

The PyDynamic.misc.testsignals module is a collection of test signals which can be used to simulate dynamic measurements and test methods.

This module contains the following functions:

PyDynamic.misc.testsignals.shocklikeGaussian(time, t0, m0, sigma, noise=0.0)[source]

Generates a signal that resembles a shock excitation as a Gaussian followed by a smaller Gaussian of opposite sign.

Parameters: time (np.ndarray of shape (N,)) – time instants (equidistant) t0 (float) – time instant of signal maximum m0 (float) – signal maximum sigma (float) – std of main pulse noise (float, optional) – std of simulated signal noise x – signal amplitudes at time instants np.ndarray of shape (N,)
PyDynamic.misc.testsignals.GaussianPulse(time, t0, m0, sigma, noise=0.0)[source]

Generates a Gaussian pulse at t0 with height m0 and std sigma

Parameters: time (np.ndarray of shape (N,)) – time instants (equidistant) t0 (float) – time instant of signal maximum m0 (float) – signal maximum sigma (float) – std of pulse noise (float, optional) – std of simulated signal noise x – signal amplitudes at time instants np.ndarray of shape (N,)
PyDynamic.misc.testsignals.rect(time, t0, t1, height=1, noise=0.0)[source]

Rectangular signal of given height and width t1-t0

Parameters: time (np.ndarray of shape (N,)) – time instants (equidistant) t0 (float) – time instant of rect lhs t1 (float) – time instant of rect rhs height (float) – signal maximum noise (float or numpy.ndarray of shape (N,), optional) – float: standard deviation of additive white gaussian noise ndarray: user-defined additive noise x – signal amplitudes at time instants np.ndarray of shape (N,)
PyDynamic.misc.testsignals.squarepulse(time, height, numpulse=4, noise=0.0)[source]

Generates a series of rect functions to represent a square pulse signal

Parameters: time (np.ndarray of shape (N,)) – time instants height (float) – height of the rectangular pulses numpulse (int) – number of pulses noise (float, optional) – std of simulated signal noise x – signal amplitude at time instants np.ndarray of shape (N,)
class PyDynamic.misc.testsignals.corr_noise(w, sigma, seed=None)[source]

Base class for generation of a correlated noise process.

PyDynamic.misc.testsignals.sine(time, amp=1.0, freq=6.283185307179586, noise=0.0)[source]

Generate a sine signal

Parameters: time (np.ndarray of shape (N,)) – time instants amp (float, optional) – amplitude of the sine (default = 1.0) freq (float, optional) – frequency of the sine in Hz (default = $$2 * \pi$$) noise (float, optional) – std of simulated signal noise (default = 0.0) x – signal amplitude at time instants np.ndarray of shape (N,)

## Noise related functions¶

Collection of noise-signals

This module contains the following functions:

PyDynamic.misc.noise.get_alpha(color_value=0)[source]

Translate a color (given as string) into an exponent alpha or directly hand through a given numeric value of alpha.

Parameters: color_value (str, int or float) – if string -> check against known colornames -> return alpha if numeric -> directly return value alpha float
PyDynamic.misc.noise.power_law_noise(N=None, w=None, color_value='white', mean=0.0, std=1.0)[source]

Generate colored noise by * generate white gaussian noise * multiplying its Fourier-transform with f^(alpha/2) * inverse Fourier-transform to yield the colored/correlated noise * further adjustments to fit to specified mean/std

based on [Zhivomirov2018](A Method for Colored Noise Generation)

Parameters: N (int) – length of noise to be generated w (numpy.ndarray) – user-defined white noise if provided, N is ignored! color_value (str, int or float) – if string -> check against known colornames if numeric -> used as alpha to shape PSD mean (float) – mean of the output signal std (float) – standard deviation of the output signal w_filt filtered noise signal
PyDynamic.misc.noise.power_law_acf(N, color_value='white', std=1.0)[source]

Return the theoretic right-sided autocorrelation (Rww) of different colors of noise.

Colors of noise are defined to have a power spectral density (Sww) proportional to f^lpha. Sww and Rww form a Fourier-pair. Therefore Rww = ifft(Sww).

PyDynamic.misc.noise.ARMA(length, phi=0.0, theta=0.0, std=1.0)[source]

Generate time-series of a predefined ARMA-process based on this equation: $$\sum_{j=1}^{\min(p,n-1)} \phi_j \epsilon[n-j] + \sum_{j=1}^{\min(q,n-1)} \theta_j w[n-j]$$ where w is white gaussian noise. Equation and algorithm taken from [Eichst2012] .

Parameters: length (int) – how long the drawn sample will be phi (float, list or numpy.ndarray, shape (p, )) – AR-coefficients theta (float, list or numpy.ndarray) – MA-coefficients std (float) – std of the gaussian white noise that is feeded into the ARMA-model e – time-series of the predefined ARMA-process numpy.ndarray, shape (length, )

References

## Miscellaneous useful helper functions¶

The PyDynamic.misc.tools module is a collection of miscellaneous helper functions.

This module contains the following functions:

PyDynamic.misc.tools.print_mat(matrix, prec=5, vertical=False, retS=False)[source]

Print matrix (2D array) to the console or return as formatted string

Parameters: matrix ((M,N) array_like) – prec (int) – the precision of the output vertical (bool) – print out vertical or not retS (bool) – print or return string s – if retS is True str
PyDynamic.misc.tools.print_vec(vector, prec=5, retS=False, vertical=False)[source]

Print vector (1D array) to the console or return as formatted string

Parameters: vector ((M,) array_like) – prec (int) – the precision of the output vertical (bool) – print out vertical or not retS (bool) – print or return string s – if retS is True str
PyDynamic.misc.tools.make_semiposdef(matrix, maxiter=10, tol=1e-12, verbose=False)[source]

Make quadratic matrix positive semi-definite by increasing its eigenvalues

Parameters: matrix ((N,N) array_like) – the matrix to process maxiter (int) – the maximum number of iterations for increasing the eigenvalues tol (float) – tolerance for deciding if pos. semi-def. verbose (bool) – If True print smallest eigenvalue of the resulting matrix quadratic positive semi-definite matrix (N,N) array_like
PyDynamic.misc.tools.FreqResp2RealImag(Abs, Phase, Unc, MCruns=10000.0)[source]

Calculate real and imaginary parts from frequency response

Calculate real and imaginary parts from amplitude and phase with associated uncertainties.

Parameters: Abs ((N,) array_like) – amplitude values Phase ((N,) array_like) – phase values in rad Unc ((2N, 2N) or (2N,) array_like) – uncertainties MCruns (bool) – Iterations for Monte Carlo simulation Re, Im ((N,) array_like) – real and imaginary parts (best estimate) URI ((2N, 2N) array_like) – uncertainties assoc. with Re and Im
PyDynamic.misc.tools.make_equidistant(t, y, uy, dt=0.05, kind='linear')[source]

Interpolate non-equidistant time series to equidistant

Interpolate measurement values and propagate uncertainties accordingly.

Parameters: t ((N,) array_like) – timestamps (or frequencies) y ((N,) array_like) – corresponding measurement values uy ((N,) array_like) – corresponding measurement values’ standard uncertainties dt (float, optional) – desired interval length kind (str, optional) – Specifies the kind of interpolation for the measurement values as a string (‘previous’, ‘next’, ‘nearest’ or ‘linear’). t_new ((M,) array_like) – interpolation timestamps (or frequencies) y_new ((M,) array_like) – interpolated measurement values uy_new ((M,) array_like) – interpolated measurement values’ standard uncertainties

References

PyDynamic.misc.tools.trimOrPad(array, length, mode='constant')[source]

Trim or pad (with zeros) a vector to desired length

PyDynamic.misc.tools.progress_bar(count, count_max, width=30, prefix='', done_indicator='#', todo_indicator='.', fout=<_io.TextIOWrapper name='<stdout>' mode='w' encoding='UTF-8'>)[source]

A simple and reusable progress-bar

Parameters: count (int) – current status of iterations, assumed to be zero-based count_max (int) – total number of iterations width (int, optional) – width of the actual progressbar (actual printed line will be wider) prefix (str, optional) – some text that will be printed in front of the bar (i.e. “Progress of ABC:”) done_indicator (str, optional) – what character is used as “already-done”-indicator todo_indicator (str, optional) – what character is used as “not-done-yet”-indicator fout (file-object, optional) – where the progress-bar should be written/printed to