# 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

Returns

H – complex frequency response values

Return type

ndarray shape (N,) or ndarray shape (N,K)

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

Returns

• 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_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

Returns

b,a – analogue filter coefficients

Return type

ndarray

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

Returns

• 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.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

Returns

• group_delay (np.ndarray) – group delay values

• frequencies (ndarray) – frequencies at which the group delay is calculated

References

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)

Returns

stable – whether filter is stable or not

Return type

bool

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

Returns

• blow (ndarray) – FIR filter coefficients

• shift (int) – delay of the filter (in samples)

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

Returns

a – polynomial coefficients with all roots inside or on the unit circle

Return type

ndarray

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.

Source obtained from scipy cookbook (online), downloaded 2013-09-13

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.

Returns

ys – the smoothed signal (or it’s n-th derivative).

Return type

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

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

Shortcut for calculation of unwrapped angle of complex values

## 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.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

Returns

x – signal amplitudes at time instants

Return type

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.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

Returns

x – signal amplitudes at time instants

Return type

np.ndarray of shape (N,)

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

Returns

x – signal amplitudes at time instants

Return type

np.ndarray of shape (N,)

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)

Returns

x – signal amplitude at time instants

Return type

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

Returns

x – signal amplitude at time instants

Return type

np.ndarray of shape (N,)

## Noise related functions¶

Collection of noise-signals

This module contains the following functions:

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

Returns

e – time-series of the predefined ARMA-process

Return type

numpy.ndarray, shape (length, )

References

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

Returns

alpha

Return type

float

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.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

Returns

w_filt

Return type

filtered noise signal

## 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.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

Returns

• 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_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

Returns

quadratic positive semi-definite matrix

Return type

(N,N) array_like

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

Returns

s – if retS is True

Return type

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

Returns

s – if retS is True

Return type

str

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

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

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