Understanding the Lomb-Scargle Periodogram
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Spectrum Estimation Using Periodogram, Bartlett and Welch
Spectrum estimation using Periodogram, Bartlett and Welch Guido Schuster Slides follow closely chapter 8 in the book „Statistical Digital Signal Processing and Modeling“ by Monson H. Hayes and most of the figures and formulas are taken from there 1 Introduction • We want to estimate the power spectral density of a wide- sense stationary random process • Recall that the power spectrum is the Fourier transform of the autocorrelation sequence • For an ergodic process the following holds 2 Introduction • The main problem of power spectrum estimation is – The data x(n) is always finite! • Two basic approaches – Nonparametric (Periodogram, Bartlett and Welch) • These are the most common ones and will be presented in the next pages – Parametric approaches • not discussed here since they are less common 3 Nonparametric methods • These are the most commonly used ones • x(n) is only measured between n=0,..,N-1 • Ensures that the values of x(n) that fall outside the interval [0,N-1] are excluded, where for negative values of k we use conjugate symmetry 4 Periodogram • Taking the Fourier transform of this autocorrelation estimate results in an estimate of the power spectrum, known as the Periodogram • This can also be directly expressed in terms of the data x(n) using the rectangular windowed function xN(n) 5 Periodogram of white noise 32 samples 6 Performance of the Periodogram • If N goes to infinity, does the Periodogram converge towards the power spectrum in the mean squared sense? • Necessary conditions – asymptotically unbiased: – variance -
Configuring Spectrogram Views
Chapter 5: Spectrographic Analysis Configuring spectrographic views To create a new spectrogram, spectrogram slice, or selection spectrum view, click on the appropriate New View button in the view toolbar (Figure 5.3) or choose a view type from the View > New menu. Figure 5.3. New View buttons Figure 5.3. The New View buttons, in Raven’s view toolbar. 112 Raven Pro 1.4 User’s Manual Chapter 5: Spectrographic Analysis A dialog box appears, containing parameters for configuring the requested type of spectrographic view (Figure 5.4). The dialog boxes for configuring spectrogram and spectrogram slice views are identical, except for their titles. The dialog boxes are identical because both view types calculate a spectrogram of the entire sound; the only difference between spectrogram and spectrogram slice views is in how the data are displayed (see “How the spectrographic views are related” on page 110). The dialog box for configuring a selection spectrum view is the same, except that it lacks the Averaging parameter. The remainder of this section explains each of the parameters in the configuration dialog box. Figure 5.4 Configure Spectrogram dialog Figure 5.4. The Configure New Spectrogram dialog box. Window type Each data record is multiplied by a window function before its spectrum is calculated. Window functions are used to reduce the magnitude of spurious “sidelobe” energy that appears at frequencies flanking each analysis frequency in a spectrum. These sidelobes appear as a result of Raven Pro 1.4 User’s Manual 113 Chapter 5: Spectrographic Analysis analyzing a finite (truncated) portion of a signal. -
Observational Astrophysics II January 18, 2007
Observational Astrophysics II JOHAN BLEEKER SRON Netherlands Institute for Space Research Astronomical Institute Utrecht FRANK VERBUNT Astronomical Institute Utrecht January 18, 2007 Contents 1RadiationFields 4 1.1 Stochastic processes: distribution functions, mean and variance . 4 1.2Autocorrelationandautocovariance........................ 5 1.3Wide-sensestationaryandergodicsignals..................... 6 1.4Powerspectraldensity................................ 7 1.5 Intrinsic stochastic nature of a radiation beam: Application of Bose-Einstein statistics ....................................... 10 1.5.1 Intermezzo: Bose-Einstein statistics . ................... 10 1.5.2 Intermezzo: Fermi Dirac statistics . ................... 13 1.6 Stochastic description of a radiation field in the thermal limit . 15 1.7 Stochastic description of a radiation field in the quantum limit . 20 2 Astronomical measuring process: information transfer 23 2.1Integralresponsefunctionforastronomicalmeasurements............ 23 2.2Timefiltering..................................... 24 2.2.1 Finiteexposureandtimeresolution.................... 24 2.2.2 Error assessment in sample average μT ................... 25 2.3 Data sampling in a bandlimited measuring system: Critical or Nyquist frequency 28 3 Indirect Imaging and Spectroscopy 31 3.1Coherence....................................... 31 3.1.1 The Visibility function . ................... 31 3.1.2 Young’sdualbeaminterferenceexperiment................ 31 3.1.3 Themutualcoherencefunction....................... 32 3.1.4 Interference -
An Information Theoretic Algorithm for Finding Periodicities in Stellar Light Curves Pablo Huijse, Student Member, IEEE, Pablo A
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 1, NO. 1, JANUARY 2012 1 An Information Theoretic Algorithm for Finding Periodicities in Stellar Light Curves Pablo Huijse, Student Member, IEEE, Pablo A. Estevez*,´ Senior Member, IEEE, Pavlos Protopapas, Pablo Zegers, Senior Member, IEEE, and Jose´ C. Pr´ıncipe, Fellow Member, IEEE Abstract—We propose a new information theoretic metric aligned to Earth. Periodic drops in brightness are observed for finding periodicities in stellar light curves. Light curves due to the mutual eclipses between the components of the are astronomical time series of brightness over time, and are system. Although most stars have at least some variation in characterized as being noisy and unevenly sampled. The proposed metric combines correntropy (generalized correlation) with a luminosity, current ground based survey estimations indicate periodic kernel to measure similarity among samples separated that 3% of the stars varying more than the sensitivity of the by a given period. The new metric provides a periodogram, called instruments and ∼1% are periodic [2]. Correntropy Kernelized Periodogram (CKP), whose peaks are associated with the fundamental frequencies present in the data. Detecting periodicity and estimating the period of stars is The CKP does not require any resampling, slotting or folding of high importance in astronomy. The period is a key feature scheme as it is computed directly from the available samples. for classifying variable stars [3], [4]; and estimating other CKP is the main part of a fully-automated pipeline for periodic light curve discrimination to be used in astronomical survey parameters such as mass and distance to Earth [5]. -
Windowing Techniques, the Welch Method for Improvement of Power Spectrum Estimation
Computers, Materials & Continua Tech Science Press DOI:10.32604/cmc.2021.014752 Article Windowing Techniques, the Welch Method for Improvement of Power Spectrum Estimation Dah-Jing Jwo1, *, Wei-Yeh Chang1 and I-Hua Wu2 1Department of Communications, Navigation and Control Engineering, National Taiwan Ocean University, Keelung, 202-24, Taiwan 2Innovative Navigation Technology Ltd., Kaohsiung, 801, Taiwan *Corresponding Author: Dah-Jing Jwo. Email: [email protected] Received: 01 October 2020; Accepted: 08 November 2020 Abstract: This paper revisits the characteristics of windowing techniques with various window functions involved, and successively investigates spectral leak- age mitigation utilizing the Welch method. The discrete Fourier transform (DFT) is ubiquitous in digital signal processing (DSP) for the spectrum anal- ysis and can be efciently realized by the fast Fourier transform (FFT). The sampling signal will result in distortion and thus may cause unpredictable spectral leakage in discrete spectrum when the DFT is employed. Windowing is implemented by multiplying the input signal with a window function and windowing amplitude modulates the input signal so that the spectral leakage is evened out. Therefore, windowing processing reduces the amplitude of the samples at the beginning and end of the window. In addition to selecting appropriate window functions, a pretreatment method, such as the Welch method, is effective to mitigate the spectral leakage. Due to the noise caused by imperfect, nite data, the noise reduction from Welch’s method is a desired treatment. The nonparametric Welch method is an improvement on the peri- odogram spectrum estimation method where the signal-to-noise ratio (SNR) is high and mitigates noise in the estimated power spectra in exchange for frequency resolution reduction. -
On the Duality of Regular and Local Functions
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 May 2017 doi:10.20944/preprints201705.0175.v1 mathematics ISSN 2227-7390 www.mdpi.com/journal/mathematics Article On the Duality of Regular and Local Functions Jens V. Fischer Institute of Mathematics, Ludwig Maximilians University Munich, 80333 Munich, Germany; E-Mail: jens.fi[email protected]; Tel.: +49-8196-934918 1 Abstract: In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty 2 principle. They both express Fourier dualities within the space of tempered distributions and 3 these dualities are furthermore the inverses of one another. While Poisson’s summation 4 formula expresses a duality between discretization and periodization, Heisenberg’s 5 uncertainty principle expresses a duality between regularization and localization. We define 6 regularization and localization on generalized functions and show that the Fourier transform 7 of regular functions are local functions and, vice versa, the Fourier transform of local 8 functions are regular functions. 9 Keywords: generalized functions; tempered distributions; regular functions; local functions; 10 regularization-localization duality; regularity; Heisenberg’s uncertainty principle 11 Classification: MSC 42B05, 46F10, 46F12 1 1.2 Introduction 13 Regularization is a popular trick in applied mathematics, see [1] for example. It is the technique 14 ”to approximate functions by more differentiable ones” [2]. Its terminology coincides moreover with 15 the terminology used in generalized function spaces. They contain two kinds of functions, ”regular 16 functions” and ”generalized functions”. While regular functions are being functions in the ordinary 17 functions sense which are infinitely differentiable in the ordinary functions sense, all other functions 18 become ”infinitely differentiable” in the ”generalized functions sense” [3]. -
Windowing Design and Performance Assessment for Mitigation of Spectrum Leakage
E3S Web of Conferences 94, 03001 (2019) https://doi.org/10.1051/e3sconf/20199403001 ISGNSS 2018 Windowing Design and Performance Assessment for Mitigation of Spectrum Leakage Dah-Jing Jwo 1,*, I-Hua Wu 2 and Yi Chang1 1 Department of Communications, Navigation and Control Engineering, National Taiwan Ocean University 2 Pei-Ning Rd., Keelung 202, Taiwan 2 Bison Electronics, Inc., Neihu District, Taipei 114, Taiwan Abstract. This paper investigates the windowing design and performance assessment for mitigation of spectral leakage. A pretreatment method to reduce the spectral leakage is developed. In addition to selecting appropriate window functions, the Welch method is introduced. Windowing is implemented by multiplying the input signal with a windowing function. The periodogram technique based on Welch method is capable of providing good resolution if data length samples are selected optimally. Windowing amplitude modulates the input signal so that the spectral leakage is evened out. Thus, windowing reduces the amplitude of the samples at the beginning and end of the window, altering leakage. The influence of various window functions on the Fourier transform spectrum of the signals was discussed, and the characteristics and functions of various window functions were explained. In addition, we compared the differences in the influence of different data lengths on spectral resolution and noise levels caused by the traditional power spectrum estimation and various window-function-based Welch power spectrum estimations. 1 Introduction knowledge. Due to computer capacity and limitations on processing time, only limited time samples can actually The spectrum analysis [1-4] is a critical concern in many be processed during spectrum analysis and data military and civilian applications, such as performance processing of signals, i.e., the original signals need to be test of electric power systems and communication cut off, which induces leakage errors. -
Algorithmic Framework for X-Ray Nanocrystallographic Reconstruction in the Presence of the Indexing Ambiguity
Algorithmic framework for X-ray nanocrystallographic reconstruction in the presence of the indexing ambiguity Jeffrey J. Donatellia,b and James A. Sethiana,b,1 aDepartment of Mathematics and bLawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720 Contributed by James A. Sethian, November 21, 2013 (sent for review September 23, 2013) X-ray nanocrystallography allows the structure of a macromolecule over multiple orientations. Although there has been some success in to be determined from a large ensemble of nanocrystals. How- determining structure from perfectly twinned data, reconstruction is ever, several parameters, including crystal sizes, orientations, and often infeasible without a good initial atomic model of the structure. incident photon flux densities, are initially unknown and images We present an algorithmic framework for X-ray nanocrystal- are highly corrupted with noise. Autoindexing techniques, com- lographic reconstruction which is based on directly reducing data monly used in conventional crystallography, can determine ori- variance and resolving the indexing ambiguity. First, we design entations using Bragg peak patterns, but only up to crystal lattice an autoindexing technique that uses both Bragg and non-Bragg symmetry. This limitation results in an ambiguity in the orienta- data to compute precise orientations, up to lattice symmetry. tions, known as the indexing ambiguity, when the diffraction Crystal sizes are then determined by performing a Fourier analysis pattern displays less symmetry than the lattice and leads to data around Bragg peak neighborhoods from a finely sampled low- that appear twinned if left unresolved. Furthermore, missing angle image, such as from a rear detector (Fig. 1). Next, we model phase information must be recovered to determine the imaged structure factor magnitudes for each reciprocal lattice point with object’s structure. -
Discrete-Time Fourier Transform
.@.$$-@@4&@$QEG'FC1. CHAPTER7 Discrete-Time Fourier Transform In Chapter 3 and Appendix C, we showed that interesting continuous-time waveforms x(t) can be synthesized by summing sinusoids, or complex exponential signals, having different frequencies fk and complex amplitudes ak. We also introduced the concept of the spectrum of a signal as the collection of information about the frequencies and corresponding complex amplitudes {fk,ak} of the complex exponential signals, and found it convenient to display the spectrum as a plot of spectrum lines versus frequency, each labeled with amplitude and phase. This spectrum plot is a frequency-domain representation that tells us at a glance “how much of each frequency is present in the signal.” In Chapter 4, we extended the spectrum concept from continuous-time signals x(t) to discrete-time signals x[n] obtained by sampling x(t). In the discrete-time case, the line spectrum is plotted as a function of normalized frequency ωˆ . In Chapter 6, we developed the frequency response H(ejωˆ ) which is the frequency-domain representation of an FIR filter. Since an FIR filter can also be characterized in the time domain by its impulse response signal h[n], it is not hard to imagine that the frequency response is the frequency-domain representation,orspectrum, of the sequence h[n]. 236 .@.$$-@@4&@$QEG'FC1. 7-1 DTFT: FOURIER TRANSFORM FOR DISCRETE-TIME SIGNALS 237 In this chapter, we take the next step by developing the discrete-time Fouriertransform (DTFT). The DTFT is a frequency-domain representation for a wide range of both finite- and infinite-length discrete-time signals x[n]. -
Optimal and Adaptive Subband Beamforming
Optimal and Adaptive Subband Beamforming Principles and Applications Nedelko Grbi´c Ronneby, June 2001 Department of Telecommunications and Signal Processing Blekinge Institute of Technology, S-372 25 Ronneby, Sweden c Nedelko Grbi´c ISBN 91-7295-002-1 ISSN 1650-2159 Published 2001 Printed by Kaserntryckeriet AB Karlskrona 2001 Sweden v Preface This doctoral thesis summarizes my work in the field of array signal processing. Frequency domain processing comprise a significant part of the material. The work is mainly aimed at speech enhancement in communication systems such as confer- ence telephony and handsfree mobile telephony. The work has been carried out at Department of Telecommunications and Signal Processing at Blekinge Institute of Technology. Some of the work has been conducted in collaboration with Ericsson Mobile Communications and Australian Telecommunications Research Institute. The thesis consists of seven stand alone parts: Part I Optimal and Adaptive Beamforming for Speech Signals Part II Structures and Performance Limits in Subband Beamforming Part III Blind Signal Separation using Overcomplete Subband Representation Part IV Neural Network based Adaptive Microphone Array System for Speech Enhancement Part V Design of Oversampled Uniform DFT Filter Banks with Delay Specifica- tion using Quadratic Optimization Part VI Design of Oversampled Uniform DFT Filter Banks with Reduced Inband Aliasing and Delay Constraints Part VII A New Pilot-Signal based Space-Time Adaptive Algorithm vii Acknowledgments I owe sincere gratitude to Professor Sven Nordholm with whom most of my work has been conducted. There have been enormous amount of inspiration and valuable discussions with him. Many thanks also goes to Professor Ingvar Claesson who has made every possible effort to make my study complete and rewarding. -
The Poisson Summation Formula, the Sampling Theorem, and Dirac Combs
The Poisson summation formula, the sampling theorem, and Dirac combs Jordan Bell [email protected] Department of Mathematics, University of Toronto April 3, 2014 1 Poisson summation formula Let S(R) be the set of all infinitely differentiable functions f on R such that for all nonnegative integers m and n, m (n) νm;n(f) = sup jxj jf (x)j x2R is finite. For each m and n, νm;n is a seminorm. This is a countable collection of seminorms, and S(R) is a Fréchet space. (One has to prove for vk 2 S(R) that if for each fixed m; n the sequence νm;n(fk) is a Cauchy sequence (of numbers), then there exists some v 2 S(R) such that for each m; n we have νm;n(f −fk) ! 0.) I mention that S(R) is a Fréchet space merely because it is satisfying to give a structure to a set with which we are working: if I call this set Schwartz space I would like to know what type of space it is, in the same sense that an Lp space is a Banach space. For f 2 S(R), define Z f^(ξ) = e−iξxf(x)dx; R and then 1 Z f(x) = eixξf^(ξ)dξ: 2π R Let T = R=2πZ. For φ 2 C1(T), define 1 Z 2π φ^(n) = e−intφ(t)dt; 2π 0 and then X φ(t) = φ^(n)eint: n2Z 1 For f 2 S(R) and λ 6= 0, define φλ : T ! C by X φλ(t) = 2π fλ(t + 2πj); j2Z 1 where fλ(x) = λf(λx). -
Airborne Radar Ground Clutter Suppression Using Multitaper Spectrum Estimation Comparison with Traditional Method
Airborne Radar Ground Clutter Suppression Using Multitaper Spectrum Estimation Comparison with Traditional Method Linus Ekvall Engineering Physics and Electrical Engineering, master's level 2018 Luleå University of Technology Department of Computer Science, Electrical and Space Engineering Airborne Radar Ground Clutter Suppression Using Multitaper Spectrum Estimation & Comparison with Traditional Method Linus C. Ekvall Lule˚aUniversity of Technology Dept. of Computer Science, Electrical and Space Engineering Div. Signals and Systems 20th September 2018 ABSTRACT During processing of data received by an airborne radar one of the issues is that the typical signal echo from the ground produces a large perturbation. Due to this perturbation it can be difficult to detect targets with low velocity or a low signal-to-noise ratio. Therefore, a filtering process is needed to separate the large perturbation from the target signal. The traditional method include a tapered Fourier transform that operates in parallel with a MTI filter to suppress the main spectral peak in order to produce a smoother spectral output. The difference between a typical signal echo produced from an object in the environment and the signal echo from the ground can be of a magnitude corresponding to more than a 60 dB difference. This thesis presents research of how the multitaper approach can be utilized in concurrence with the minimum variance estimation technique, to produce a spectral estimation that strives for a more effective clutter suppression. A simulation model of the ground clutter was constructed and also a number of simulations for the multitaper, minimum variance estimation technique was made. Compared to the traditional method defined in this thesis, there was a slight improve- ment of the improvement factor when using the multitaper approach.