Processing Signals#

This section describes the signal processing features available in DataLab.

Axis transformation#

Linear calibration#

Create a new signal which is a linear calibration of each selected signal with respect to X or Y axis:

Parameter	Linear calibration
X-axis	\(x_{1} = a.x_{0} + b\)
Y-axis	\(y_{1} = a.y_{0} + b\)

Swap X/Y axes#

Create a new signal which is the result of swapping X/Y data.

Reverse X-axis#

Create a new signal which is the result of reversing X data.

Replace X by other signal’s Y#

Create a new signal by combining two signals, using Y values from the second signal as X coordinates for the first signal’s Y values.

This feature is particularly useful for calibration scenarios, such as:

Applying wavelength calibration to spectroscopy data
Using a calibrated reference signal to rescale measurement data
Combining signals where one contains calibration points

The two signals must have the same number of points. The operation directly uses the Y arrays without interpolation.

Input	Description
Signal 1	Provides the Y data for the output signal
Signal 2	Provides the Y data that becomes the X coordinates of the output signal

X-Y mode#

Create a new signal by simulating the X-Y mode of an oscilloscope.

This operation plots one signal as a function of another by:

Finding the overlapping X range between both signals
Resampling both signals onto a common X grid within that overlap
Using the resampled Y values as coordinates (first signal’s Y as X, second signal’s Y as Y)

Note

This is different from “Replace X by other signal’s Y” as it performs interpolation to handle signals with different X arrays.

Convert to Cartesian coordinates#

Create a new signal which is the result of converting polar coordinates to Cartesian coordinates.

This function assumes that the x-axis represents the radius and the y-axis the angle.

Warning

A radius cannot be negative. Any negative value is clipped to 0.

Convert to polar coordinates#

Create a new signal which is the result of converting Cartesian coordinates to polar coordinates.

Level adjustment#

Normalize#

Create a new signal which is the normalization of each selected signal by maximum, amplitude, sum, energy or RMS:

Parameter	Normalization
Maximum	\(y_{1}= \dfrac{y_{0}}{\max\left(y_{0}\right)}\)
Amplitude	\(y_{1}= \dfrac{y_{0}'}{\max\left(y_{0}'\right)}\) with \(y_{0}'=y_{0}-\min\left(y_{0}\right)\)
Area	\(y_{1}= \dfrac{y_{0}}{\sum_{n=0}^{N}y_{0}[n]}\)
Energy	\(y_{1}= \dfrac{y_{0}}{\sqrt{\sum_{n=0}^{N}\left\|y_{0}[n]\right\|^2}}\)
RMS	\(y_{1}= \dfrac{y_{0}}{\sqrt{\dfrac{1}{N}\sum_{n=0}^{N}\left\|y_{0}[n]\right\|^2}}\)

Clipping#

Create a new signal which is the result of clipping each selected signal.

Offset correction#

Create a new signal which is the result of offset correction of each selected signal. This operation is performed by subtracting the signal baseline which is estimated by the mean value of a user-defined range.

Noise addition#

Generate new signals by adding the same noise to each selected signal. The available noise types are:

Noise	Description
Gaussian	Normal distribution
Uniform	Uniform distribution
Poisson	Poisson distribution

Noise reduction#

Create a new signal which is the result of noise reduction of each selected signal.

The following filters are available:

Filter	Formula/implementation
Gaussian filter	scipy.ndimage.gaussian_filter
Moving average	scipy.ndimage.uniform_filter
Moving median	scipy.ndimage.median_filter
Wiener filter	scipy.signal.wiener

Fourier analysis#

Zero padding#

Create a new signal which is the result of zero padding of each selected signal.

The following parameters are available:

Parameter	Description
Strategy	Zero padding strategy (see below)
Number of points	Custom length (if strategy is “custom”)

Zero padding strategy refers to the method used to increase the length of the signal, and it can be one of the following:

Strategy	Description
next_pow2	Next power of 2 (e.g. 1024 → 2048)
double	Double the length (e.g. 1024 → 2048)
triple	Triple the length (e.g. 1024 → 3072)
custom	Custom length (user-defined)

FFT related functions#

Create a new signal which is the result of a Fourier analysis of each selected signal.

The following functions are available:

Function	Description	Formula/implementation
FFT	Fast Fourier Transform	numpy.fft.fft
Inverse FFT	Inverse Fast Fourier Transform	numpy.fft.ifft
Magnitude spectrum	Optional: output in decibels (dB)	\(y_{1} = \left\|\FFT\left(y_{0}\right)\right\|\) or \(y_{1} = 20 \log_{10} \left(\left\|\FFT\left(y_{0}\right)\right\|\right)\) (dB)
Phase spectrum	Phase of the FFT expressed in degrees, using numpy.angle	\(y_{1} = \angle \FFT\left(y_{0}\right)\)
Power spectral density (PSD)	Optional: output in decibels (dB). PSD is estimated using Welch’s method (see scipy.signal.welch)	\(y_{1} = \PSD\left(y_{0}\right)\) or \(y_{1} = 10 \log_{10} \left(\PSD\left(y_{0}\right)\right)\) (dB)

Note

FFT and inverse FFT are performed using frequency shifting if the option is enabled in DataLab settings (see Settings).

Frequency filters#

Create a new signal which is the result of applying a frequency filter to each selected signal.

The following filters are available:

Filter	Description
Low-pass	Filter out high frequencies, above a cutoff frequency
High-pass	Filter out low frequencies, below a cutoff frequency
Band-pass	Filter out frequencies outside a range
Band-stop	Filter out frequencies inside a range

For each filter, the following methods are available:

Method	Description
Bessel	Bessel filter, using SciPy’s scipy.signal.bessel function
Brick wall	Ideal (rectangular) filter in the frequency domain
Butterworth	Butterworth filter, using SciPy’s scipy.signal.butter function
Chebyshev I	Chebyshev type I filter, using SciPy’s scipy.signal.cheby1 function
Chebyshev II	Chebyshev type II filter, using SciPy’s scipy.signal.cheby2 function
Elliptic	Elliptic filter, using SciPy’s scipy.signal.ellip function

Fitting#

Fit a model to each selected signal. This can be done automatically or through an interactive curve fitting dialog.

Model	Equation
Linear	\(y = c_{0} + c_{1} \cdot x\)
Polynomial	\(y = c_{0} + c_{1} \cdot x + c_{2} \cdot x^2 + ... + c_{n} \cdot x^n\)
Gaussian	\(y = y_{0} + \dfrac{A}{\sqrt{2 \pi} \sigma} \exp\left(-\dfrac{1}{2} \left(\dfrac{x - x_{0}}{\sigma}\right)^2\right)\)
Lorentzian	\(y = y_{0} + \dfrac{A}{\pi \sigma} \cdot \dfrac{1}{1+\left(\dfrac{x-x_{0}}{\sigma}\right)^2}\)
Voigt	\(y = y_{0} + A \cdot \dfrac{\Re\left(\exp(-z^2) \cdot \operatorname{erfc}(-j \cdot z)\right)}{\sqrt{2 \pi} \sigma}\) with \(z = \dfrac{x - x_{0} - j \cdot \sigma}{\sqrt{2} \sigma}\)
Multi-Gaussian	\(y = y_{0} + \sum_{i=0}^{N}\dfrac{A_{i}}{\sqrt{2 \pi} \sigma_{i}} \exp\left(-\dfrac{1}{2} \left(\dfrac{x - x_{0,i}}{\sigma_{i}}\right)^2\right)\)
Multi-Lorentzian	\(y = y_{0} + \sum_{i=0}^{N}\dfrac{A_{i}}{\pi \sigma_{i}} \cdot \dfrac{1}{1 + \left(\dfrac{x - x_{0,i}}{\sigma_{i}}\right)^2}\)
Planck	\(y = y_{0} + A\cdot \dfrac{2h \cdot c^2}{\lambda^5} \cdot \left(\exp\left(\dfrac{h \cdot c}{\lambda \cdot k \cdot T}\right)-1\right)^{-1}\)
Two half Gaussians	\(y = y_{0} + A \cdot \dfrac{1}{\sqrt{2 \pi} \sigma_{0}} \cdot \exp\left(-\dfrac{1}{2}\left(\dfrac{x - x_{0}}{\sigma_{0}}\right)^2\right)\) if \(x < x_{0}\) \(y = y_{0} + A \cdot \dfrac{1}{\sqrt{2 \pi} \sigma_{1}} \cdot \exp\left(-\dfrac{1}{2}\left(\dfrac{x - x_{1}}{\sigma_{1}}\right)^2\right)\) otherwise
Two back-to-back exponentials	\(y = y_{0} + A_{1} \cdot \exp\left(-x/\tau_{1}\right)\) if \(x < 0\) \(y = y_{0} + A_{2} \cdot \exp(-x/\tau_{2})\) otherwise
Exponential	\(y = y_{0} + A \exp\left(B \cdot x\right)\)
Sinusoidal	\(y = y_{0} + A \sin\left(2 \pi f \cdot x + \phi\right)\)
Cumulative Distribution Function (CDF)	\(y = y_{0} + A \erf\left(\dfrac{x - x_{0}}{\sqrt{2} \sigma}\right)\)

Windowing#

Create a new signal which is the result of applying a window function to each selected signal.

The following window functions are available:

Window function	Reference
Barthann	`scipy.signal.windows.barthann()`
Bartlett	`numpy.bartlett()`
Blackman	`scipy.signal.windows.blackman()`
Blackman-Harris	`scipy.signal.windows.blackmanharris()`
Bohman	`scipy.signal.windows.bohman()`
Boxcar (rectangular)	`scipy.signal.windows.boxcar()`
Cosine	`scipy.signal.windows.cosine()`
Exponential	`scipy.signal.windows.exponential()`
Flat top	`scipy.signal.windows.flattop()`
Gaussian	`scipy.signal.windows.gaussian()`
Hamming	`numpy.hamming()`
Hann	`numpy.hanning()`
Kaiser	`scipy.signal.windows.kaiser()`
Lanczos	`scipy.signal.windows.lanczos()`
Nuttall	`scipy.signal.windows.nuttall()`
Parzen	`scipy.signal.windows.parzen()`
Taylor	`scipy.signal.windows.taylor()`
Tukey	`scipy.signal.windows.tukey()`

Detrending#

Create a new signal which is the detrending of each selected signal. This features is based on SciPy’s scipy.signal.detrend function.

The following parameters are available:

Parameter	Description
Method	Detrending method: ‘linear’ or ‘constant’. See SciPy’s scipy.signal.detrend function.

Interpolation#

Create a new signal which is the interpolation of each selected signal with respect to a second signal X-axis (which might be the same as one of the selected signals).

The following interpolation methods are available:

Method	Description
Linear	Linear interpolation, using using NumPy’s interp function
Spline	Cubic spline interpolation, using using SciPy’s scipy.interpolate.splev function
Quadratic	Quadratic interpolation, using using NumPy’s polyval function
Cubic	Cubic interpolation, using using SciPy’s Akima1DInterpolator class
Barycentric	Barycentric interpolation, using using SciPy’s BarycentricInterpolator class
PCHIP	Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) interpolation, using using SciPy’s PchipInterpolator class

Resampling#

Create a new signal which is the resampling of each selected signal.

The following parameters are available:

Parameter	Description
Method	Interpolation method (see previous section)
Fill value	Interpolation fill value (see previous section)
Xmin	Minimum X value
Xmax	Maximum X value
Mode	Resampling mode: step size or number of points
Step size	Resampling step size
Number of points	Resampling number of points

Stability analysis#

Create a new signal which is the result of a stability analysis of each selected signal.

The following stability analysis methods are available:

Function	Description
Allan variance	Measure of the stability of a signal: defined as the variance of the difference between two successive measurements as a function of the time interval between them.
Allan deviation	Square root of the Allan variance.
Overlapping Allan deviation	A more robust version of the Allan variance that overlaps successive segments to improve statistical confidence.
Modified Allan variance	A variation of the Allan variance that accounts for phase noise by introducing a filtering operation.
Hadamard variance	An alternative to Allan variance, more robust to linear frequency drift in the signal
Total variance	Extends the Allan variance concept to cover all possible averaging intervals.
Time deviation	Derived from Allan deviation, quantifies stability in terms of time rather than frequency.

Note

The “All stability features” option allows to compute all stability analysis methods at once.