 Lomb Scargle Periodogram - Maple Help

SignalProcessing

 LSPeriodogram
 plot the Lomb-Scargle periodogram of a signal Calling Sequence LSPeriodogram(t, f, options) Parameters

 t - rtable ; time f - rtable ; signal options - (optional) one or more of the options listed below Options

 • detrend : truefalse : indicates if the data should have any trend removed
 • frequencyscale : anything
 • powerscale : identical("dB", "absolute")
 • timeunit : anything
 • outputdata : posint : specify the number of the output data points
 • errors : rtable or realcons
 • precise : truefalse : indicates the methods to use
 • minimumfrequency : float : indicates minimum frequency
 • maximumfrequency : float : indicates maximum frequency
 • normalization : identical('standard', 'model', 'log', 'psd') : indicates the normalization methods
 • oversamplingfactor : numeric : indicates the oversampling factor
 • nyquistfactor : numeric : indicates the multiple of the average Nyquist frequency
 • centerdata : truefalse : indicates whether the data needs to subtract the weighted mean
 • generalized : truefalse : indicates the methods to use for the Lomb Scargle Fast Approximation Description

 • The LSPeriodogram(t, f) command plots the power spectrum for one signal. t and f must be a one-dimensional rtable.
 • The detrend option specifies if the data should have any linear trend removed before being plotted.
 • The frequencyscale option controls the scaling used on the frequency axis. The default is Hz.
 • The powerscale option controls the scaling used on the power axis and can be one of "dB" or "absolute". The default is "dB".
 • The timeunit option specifies the unit of the input time. The default value is s.
 • The errors option specifies the expected magnitude of the errors, which is used to get the normalized weight for generalized Lomb Scargle periodogram. It can only be a real number or an one-dimensional array. The default value is NULL.
 • The outputdata option specifies the number of output data points. These output data points correspond to equally spaced frequencies.
 • The minimumfrequency option specifies the minimum frequency.
 • The maximumfrequency option specifies the maximum frequency.
 • If one of maximumfrequency, minimumfrequency and outputdata is specified, then all of them must be provided. If all of them are not provided, then the frequencies used run from $\frac{1}{T\mathrm{oversamplingfactor}}$ to $\mathrm{nyquistfactor}\mathrm{Nf}$, where $T=\mathrm{max}\left(t\right)-\mathrm{min}\left(t\right)$ and $\mathrm{Nf}=\frac{n}{2T}$ where $n$ is the number of elements in t. Output frequencies are equally spaced.
 • The nyquistfactor option has a default value 1.
 • The normalization option controls the normalization used on the power and can be one of standard, model, log or psd. The default value is standard. Standard Normalization : ${P}_{\mathrm{standard}}\left(f\right)=\frac{{\mathrm{\chi }}_{\mathrm{ref}}^{2}-{\mathrm{\chi }\left(f\right)}^{2}}{{\mathrm{\chi }}_{\mathrm{ref}}^{2}}$. Model Normalization : ${P}_{\mathrm{model}}\left(f\right)=\frac{{\mathrm{\chi }}_{\mathrm{ref}}^{2}-{\mathrm{\chi }\left(f\right)}^{2}}{{\mathrm{\chi }\left(f\right)}^{2}}$. Logarithmic Normalization:  ${P}_{\mathrm{log}}\left(f\right)=\mathrm{log}\left(\frac{{\mathrm{\chi }}_{\mathrm{ref}}^{2}}{{\mathrm{\chi }\left(f\right)}^{2}}\right)$. PSD Normalization : ${P}_{\mathrm{psd}}\left(f\right)=\frac{{\mathrm{\chi }}_{\mathrm{ref}}^{2}}{2}-\frac{{\mathrm{\chi }\left(f\right)}^{2}}{2}$. In all normalization methods, let ${f}_{i}$ be the measurement in time ${t}_{i}$ with errors ${\mathrm{\sigma }}_{i}$, then ${\mathrm{\chi }\left(f\right)}^{2}=\sum _{i=1}^{n}\frac{{f}_{i}-{f\left({t}_{i}\right)}^{2}}{{\mathrm{\sigma }}_{i}^{2}}$ where $n$ is the number of elements in f, and ${\mathrm{\chi }}_{\mathrm{ref}}^{2}$ is the non-varying reference model which is the ${\mathrm{\chi }\left(f\right)}^{2}$ for weighted mean.
 • The oversamplingfactor option specifies the oversampling factor. The default value is 5.
 • If the centerdata=false option is provided, then the weighted mean will not be subtracted from signal data. The default value is true.
 • If the generalized=false option is provided, then the generalized Lomb Scargle Periodogram will not be used. The default value is true.
 • Additional plotting options as described on the plot/option help page may be included. Examples

 > $\mathrm{with}\left(\mathrm{SignalProcessing}\right):$

Generate a time vector

 > $t≔\mathrm{sort}\left(\mathrm{LinearAlgebra}:-\mathrm{RandomVector}\left({2}^{10},\mathrm{generator}=0..12\mathrm{\pi },\mathrm{datatype}=\mathrm{float}\left[8\right]\right)\right)$
 ${{\mathrm{_rtable}}}_{{36893628498096201844}}$ (1)

Assume the time unit is second and generate signal using frequencies of 1 Hz and 2 Hz:

 > $\mathrm{f1}≔1.0:$
 > $\mathrm{f2}≔2.0:$
 > $s≔\mathrm{Vector}\left({2}^{10},i↦\mathrm{sin}\left(2\cdot \mathrm{f1}\cdot \mathrm{\pi }\cdot t\left[i\right]\right)+1.5\cdot \mathrm{sin}\left(2\cdot \mathrm{f2}\cdot \mathrm{\pi }\cdot t\left[i\right]\right),\mathrm{datatype}=\mathrm{float}\left[8\right]\right)$
 ${{\mathrm{_rtable}}}_{{36893628498063077972}}$ (2)
 > $\mathrm{LSPeriodogram}\left(t,s,\mathrm{frequencyscale}="kHz"\right)$ > $\mathrm{LSPeriodogram}\left(t,s,\mathrm{frequencyscale}="Hz"\right)$ Assume the time unit is day and the frequencies are 1 cycle per day and 2 cycles per day, which are equivalent to 0.000011574 Hz and 0.000023148 Hz

 > $\mathrm{f3}≔\frac{1}{24.\cdot 60\cdot 60}$
 ${\mathrm{f3}}{≔}{0.00001157407407}$ (3)
 > $\mathrm{f4}≔\frac{2}{24.\cdot 60\cdot 60}$
 ${\mathrm{f4}}{≔}{0.00002314814815}$ (4)
 > $\mathrm{LSPeriodogram}\left(t,s,\mathrm{timeunit}=d,\mathrm{frequencyscale}=\mathrm{Hz}\right)$ > References

 Lomb, Nicholas R. "Least-Squares Frequency Analysis of Unequally Spaced Data." Astrophysics and Space Science. Vol. 39, 1976, pp. 447–462.
 Press, William H., and Rybicki, George B. "Fast Algorithm for Spectral Analysis of Unevenly Sampled Data." Astrophysical Journal. Vol. 338, 1989, pp. 277–280.
 Zechmeister, M., and Kürster, M. "The generalised Lomb-Scargle periodogram. A new formalism for the floating-mean and Keplerian periodograms." Astronomy & Astrophysics. Vol. 496, 2009, pp. 577-584 Compatibility

 • The SignalProcessing[LSPeriodogram] command was introduced in Maple 2019.