Avoiding Aliasing in Allan Variance: an Application to Fiber Link Data Analysis
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Generalizing Sampling Theory for Time-Varying Nyquist Rates Using Self-Adjoint Extensions of Symmetric Operators with Deficiency Indices (1,1) in Hilbert Spaces
Generalizing Sampling Theory for Time-Varying Nyquist Rates using Self-Adjoint Extensions of Symmetric Operators with Deficiency Indices (1,1) in Hilbert Spaces by Yufang Hao A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Applied Mathematics Waterloo, Ontario, Canada, 2011 c Yufang Hao 2011 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Sampling theory studies the equivalence between continuous and discrete representa- tions of information. This equivalence is ubiquitously used in communication engineering and signal processing. For example, it allows engineers to store continuous signals as discrete data on digital media. The classical sampling theorem, also known as the theorem of Whittaker-Shannon- Kotel'nikov, enables one to perfectly and stably reconstruct continuous signals with a con- stant bandwidth from their discrete samples at a constant Nyquist rate. The Nyquist rate depends on the bandwidth of the signals, namely, the frequency upper bound. Intuitively, a signal's `information density' and ‘effective bandwidth' should vary in time. Adjusting the sampling rate accordingly should improve the sampling efficiency and information storage. While this old idea has been pursued in numerous publications, fundamental problems have remained: How can a reliable concept of time-varying bandwidth been defined? How can samples taken at a time-varying Nyquist rate lead to perfect and stable reconstruction of the continuous signals? This thesis develops a new non-Fourier generalized sampling theory which takes samples only as often as necessary at a time-varying Nyquist rate and maintains the ability to perfectly reconstruct the signals. -
Fourier Analysis and Sampling Theory
Reading Required: Shirley, Ch. 9 Recommended: Ron Bracewell, The Fourier Transform and Its Applications, McGraw-Hill. Fourier analysis and Don P. Mitchell and Arun N. Netravali, “Reconstruction Filters in Computer Computer sampling theory Graphics ,” Computer Graphics, (Proceedings of SIGGRAPH 88). 22 (4), pp. 221-228, 1988. Brian Curless CSE 557 Fall 2009 1 2 What is an image? Images as functions We can think of an image as a function, f, from R2 to R: f(x,y) gives the intensity of a channel at position (x,y) Realistically, we expect the image only to be defined over a rectangle, with a finite range: • f: [a,b]x[c,d] Æ [0,1] A color image is just three functions pasted together. We can write this as a “vector-valued” function: ⎡⎤rxy(, ) f (,xy )= ⎢⎥ gxy (, ) ⎢⎥ ⎣⎦⎢⎥bxy(, ) We’ll focus in grayscale (scalar-valued) images for now. 3 4 Digital images Motivation: filtering and resizing In computer graphics, we usually create or operate What if we now want to: on digital (discrete)images: smooth an image? Sample the space on a regular grid sharpen an image? Quantize each sample (round to nearest enlarge an image? integer) shrink an image? If our samples are Δ apart, we can write this as: In this lecture, we will explore the mathematical underpinnings of these operations. f [n ,m] = Quantize{ f (n Δ, m Δ) } 5 6 Convolution Convolution in 2D One of the most common methods for filtering a In two dimensions, convolution becomes: function, e.g., for smoothing or sharpening, is called convolution. -
Filter Effects and Filter Artifacts in the Analysis of Electrophysiological Data
GENERAL COMMENTARY published: 09 July 2012 doi: 10.3389/fpsyg.2012.00233 Filter effects and filter artifacts in the analysis of electrophysiological data Andreas Widmann* and Erich Schröger Institute of Psychology, University of Leipzig, Leipzig, Germany *Correspondence: [email protected] Edited by: Guillaume A. Rousselet, University of Glasgow, UK Reviewed by: Rufin VanRullen, Centre de Recherche Cerveau et Cognition, France Lucas C. Parra, City College of New York, USA A commentary on unity gain at DC (the step response never FILTER EFFECTS VS. FILTER ARTIFACTS returns to one). These artifacts are due to a We also recommend to distinguish between Four conceptual fallacies in mapping the known misconception in FIR filter design in filter effects, that is, the obligatory effects time course of recognition EEGLAB1. The artifacts are further ampli- any filter with equivalent properties – cutoff by VanRullen, R. (2011). Front. Psychol. fied by filtering twice, forward and back- frequency, roll-off, ripple, and attenuation 2:365. doi: 10.3389/fpsyg.2011.00365 ward, to achieve zero-phase. – would have on the data (e.g., smoothing With more appropriate filters the under- of transients as demonstrated by the filter’s Does filtering preclude us from studying estimation of signal onset latency due to step response), and filter artifacts, that is, ERP time-courses? the smoothing effect of low-pass filtering effects which can be minimized by selection by Rousselet, G. A. (2012). Front. Psychol. could be narrowed down to about 4–12 ms of filter type and parameters (e.g., ringing). 3:131. doi: 10.3389/fpsyg.2012.00131 in the simulated dataset (see Figure 1 and Appendix for a simulation), that is, about an CAUSAL FILTERING In a recent review, VanRullen (2011) con- order of magnitude smaller than assumed. -
Modeling of Atomic Clock Performance and Detection of Abnormal Clock Behavior
WITH) STATES APARTMENT OF NBS TECHNICAL NOTE 636 COMMERCE PUBLICATION Modeling of Atomic Clock Performance and Detection of Abnormal Clock Behavior U.S. APARTMENT OF COMMERCE National Rureau loo Js U51S3 i<?9 3 NATIONAL BUREAU OF STANDARDS 1 The National Bureau of Standards was established by an act of Congress March 3, 1 901 . The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measure- ment system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Center for Computer Sciences and Technology, and the Office for Information Programs. THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consistent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scien- tific community, industry, and commerce. The Institute consists of a Center for Radia- tion Research, an Office of Measurement Services and the following divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — -
Digital Signal Processing Tom Davinson
First SPES School on Experimental Techniques with Radioactive Beams INFN LNS Catania – November 2011 Experimental Challenges Lecture 4: Digital Signal Processing Tom Davinson School of Physics & Astronomy N I VE R U S I E T H Y T O H F G E D R I N B U Objectives & Outline Practical introduction to DSP concepts and techniques Emphasis on nuclear physics applications I intend to keep it simple … … even if it’s not … … I don’t intend to teach you VHDL! • Sampling Theorem • Aliasing • Filtering? Shaping? What’s the difference? … and why do we do it? • Digital signal processing • Digital filters semi-gaussian, moving window deconvolution • Hardware • To DSP or not to DSP? • Summary • Further reading Sampling Sampling Periodic measurement of analogue input signal by ADC Sampling Theorem An analogue input signal limited to a bandwidth fBW can be reproduced from its samples with no loss of information if it is regularly sampled at a frequency fs 2fBW The sampling frequency fs= 2fBW is called the Nyquist frequency (rate) Note: in practice the sampling frequency is usually >5x the signal bandwidth Aliasing: the problem Continuous, sinusoidal signal frequency f sampled at frequency fs (fs < f) Aliasing misrepresents the frequency as a lower frequency f < 0.5fs Aliasing: the solution Use low-pass filter to restrict bandwidth of input signal to satisfy Nyquist criterion fs 2fBW Digital Signal Processing D…igi wtahl aSt ignneaxtl? Processing Digital signal processing is the software controlled processing of sequential data derived from a digitised analogue signal. Some of the advantages of digital signal processing are: • functionality possible to implement functions which are difficult, impractical or impossible to achieve using hardware, e.g. -
Aliasing, Image Sampling and Reconstruction
https://youtu.be/yr3ngmRuGUc Recall: a pixel is a point… Aliasing, Image Sampling • It is NOT a box, disc or teeny wee light and Reconstruction • It has no dimension • It occupies no area • It can have a coordinate • More than a point, it is a SAMPLE Many of the slides are taken from Thomas Funkhouser course slides and the rest from various sources over the web… Image Sampling Imaging devices area sample. • An image is a 2D rectilinear array of samples • In video camera the CCD Quantization due to limited intensity resolution array is an area integral Sampling due to limited spatial and temporal resolution over a pixel. • The eye: photoreceptors Intensity, I Pixels are infinitely small point samples J. Liang, D. R. Williams and D. Miller, "Supernormal vision and high- resolution retinal imaging through adaptive optics," J. Opt. Soc. Am. A 14, 2884- 2892 (1997) 1 Aliasing Good Old Aliasing Reconstruction artefact Sampling and Reconstruction Sampling Reconstruction Slide © Rosalee Nerheim-Wolfe 2 Sources of Error Aliasing (in general) • Intensity quantization • In general: Artifacts due to under-sampling or poor reconstruction Not enough intensity resolution • Specifically, in graphics: Spatial aliasing • Spatial aliasing Temporal aliasing Not enough spatial resolution • Temporal aliasing Not enough temporal resolution Under-sampling Figure 14.17 FvDFH Sampling & Aliasing Spatial Aliasing • Artifacts due to limited spatial resolution • Real world is continuous • The computer world is discrete • Mapping a continuous function to a -
Allan Variance Method for Gyro Noise Analysis Using Weighted Least
Optik 126 (2015) 2529–2534 Contents lists available at ScienceDirect Optik jo urnal homepage: www.elsevier.de/ijleo Allan variance method for gyro noise analysis using weighted least square algorithm a a a,∗ b Pin Lv , Jianye Liu , Jizhou Lai , Kai Huang a Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China b AVIC Shanxi Baocheng Aviation Instrument Ltd., Baoji 721006, China a r a t i b s c t l e i n f o r a c t Article history: The Allan variance method is an effective way of analyzing gyro’s stochastic noises. In the traditional Received 3 May 2014 implementation, the ordinary least square algorithm is utilized to estimate the coefficients of gyro noises. Accepted 9 June 2015 However, the different accuracy of Allan variance values violates the prerequisite of the ordinary least square algorithm. In this study, a weighted least square algorithm is proposed to address this issue. The Keywords: new algorithm normalizes the accuracy of the Allan variance values by weighting them according to their Inertial navigation relative quantitative relationship. As a result, the problem associated with the traditional implementation Allan variance method can be solved. In order to demonstrate the effectiveness of the proposed algorithm, gyro simulations are Weighted least square algorithm carried out based on the various stochastic characteristics of SRS2000, VG951 and CRG20, which are Gyro stochastic noises three different-grade gyros. Different least square algorithms (traditional and this proposed method) are Gyro performance evaluation applied to estimate the coefficients of gyro noises. The estimation results demonstrate that the proposed algorithm outperforms the traditional algorithm, in terms of the accuracy and stability. -
Using the Allan Variance and Power Spectral Density to Characterize DC Nanovoltmeters Thomas J
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 2, APRIL 2001 445 Using the Allan Variance and Power Spectral Density to Characterize DC Nanovoltmeters Thomas J. Witt, Senior Member, IEEE Abstract—When analyzing nanovoltmeter measurements, sto- the noise is white, i.e., when the observations are independent chastic serial correlations are often ignored and the experimental and identically distributed, or, at least, over which the Allan vari- standard deviation of the mean is assumed to be the experimental ance decreases if the measurement time is extended. Such infor- standard deviation of a single observation divided by the square root of the number of observations. This is justified only for white mation is important when designing experiments. Furthermore, noise. This paper demonstrates the use of the power spectrum and knowledge of the minimum Allan variance achievable with a the Allan variance to analyze data, identify the regimes of white nanovoltmeter, and the time necessary to attain it, are useful noise, and characterize the performance of digital and analog dc specifications of the instrument. I nanovoltmeters. Limits imposed by temperature variations, Not surprisingly, it was found that, for low source resistances, noise and source resistance are investigated. the most important factor limiting precision is the stability of the Index Terms—Noise measurements, spectral analysis, time do- ambient temperature. This instigated an ancillary study of the main analysis, voltmeters, white noise. temperature coefficients of a number of nanovoltmeters. Sub- sequently the instruments were used in a temperature-stabilized I. INTRODUCTION enclosure, so that the next-greatest factor limiting the precision, HE experimental standard deviation of the mean is cor- noise, could be examined. -
Analyze Bus Goals & Constraints
16: Windowed-Sinc Filters * Acknowledgment: This material is derived and adapted from “The Scientist and Engineer's Guide to Digital Signal processing”, Steven W. Smith WOO-CCI504-SCI-UoN 1 Windowed-sinc filter . Windowed-sinc filters are used to separate one band of frequencies from another. For this, they are very stable, consistent, and can deliver high performance levels. The exceptional frequency domain characteristics are obtained, however, at the expense of poor performance in the time domain, including excessive ripple and overshoot in the step response. When carried out by standard convolution, windowed-sinc filters are easy to program, but slow to execute. the FFT can be used to dramatically improve the computational speed of these filters. WOO-CCI504-SCI-UoN 2 Strategy of the Windowed-Sinc WOO-CCI504-SCI-UoN 3 Strategy of the Windowed-Sinc . the idea behind the windowed-sinc filter. [see Figure 16-1] . The frequency response of the ideal low-pass filter: . All frequencies below the cutoff frequency, fC, are passed with unity amplitude, while all higher frequencies are blocked. The passband is perfectly flat, the attenuation in the stopband is infinite, and the transition between the two is infinitesimally small. Taking the Inverse Fourier Transform of this ideal frequency response produces the ideal filter kernel (impulse response) which is the sinc function, given by: WOO-CCI504-SCI-UoN 4 Strategy of the Windowed-Sinc Figure 16-1 WOO-CCI504-SCI-UoN 5 Strategy of the Windowed-Sinc . Convolving an input signal with this (sinc) filter kernel should provides a perfect low-pass filter. Problem: the sinc function continues to both negative and positive infinity without dropping to zero amplitude. -
Sigma-Delta Conversion Used for Motor Control
TECHNICAL ARTICLE SIGMA-DELTA CONVERSION USED Jens Sorensen Analog Devices FOR MOTOR CONTROL A - ADC has the lowest possible resolution of 1 bit, but through oversam- Share on Twitter Share on LinkedIn Email Σ Δ | | | pling, noise shaping, digital filtering, and decimation, very high signal quality can be achieved. The theory behind Σ-Δ ADCs and sinc filters is well under- stood and well documented,1, 2 so it will not be discussed in this article. Rather, Abstract the focus will be on how to get the best performance in a motor drive and how to utilize the performance in the control algorithms. Ʃ-Δ analog-to-digital converters are widely used in motor drives where high signal integrity and galvanic isolation are required. Phase Current Measurement with Σ-Δ ADCs While the Σ-Δ technology itself is well understood, the converters When a 3-phase motor is fed by a switching voltage source inverter, the are often used in ways that fail to unlock the full potential of the phase current can be seen as two components: an average component and technology. This article looks at Σ-Δ ADCs from an application a switching component, as seen in Figure 2. The top signal shows one phase point of view and discusses how to get the best performance in a current, the middle signal shows high-side PWM for the inverter phase-leg, motor drive. and the lower signal shows the sample synchronizing signal from the PWM timer, PWM_SYNC. PWM_SYNC is asserted at the beginning and the center Introduction of a PWM cycle and so it aligns with the midpoint of the current and voltage ripple waveforms. -
Analysis of Laser Frequency Stability Using Beat-Note Measurement V1.1 09 2016
Analysis of laser frequency stability using beat-note measurement V1.1 09 2016 The frequency of even the most stable ‘single frequency’ laser is not really constant in time, Frequency stability but fluctuates. measurement The laser’s frequency stability is often described Beat-note measurement in terms of its spectral linewidth. However, while For a detailed analysis of laser frequency stability, this term provides a simple figure for the most common measurement techniques are comparison of various lasers, it lacks detail spectral analysis of the laser transmission about the laser frequency noise which is often through a sensitive interferometer (filter) or a critical in laser applications. beat-note measurement with a reference laser. The beat-note measurement is a heterodyne For a detailed representation of laser frequency technique where the laser output is mixed with stability we use either the Allan variance of the the output of a reference laser and the combined frequency fluctuations or the frequency noise signal is measured using a photo detector. The spectrum. These representations are photo detector is a quadratic detector with a complimentary and originate respectively from a limited bandwidth and will only trace the time-domain description and a Fourier-domain difference frequency between the two lasers; the description of the frequency fluctuations. Both so-called beat frequency 휈퐵 (provided the beat the Allan variance and the frequency noise frequency is within the detector bandwidth). spectrum can be calculated from a beat-note measurement of the frequency fluctuations. laser under test Agilent 53230A frequency counter variable photo attenuator detector 3 dB coupler reference laser Fig. -
Design of a Low Power and Area Efficient Digital Down Converter and SINC Filter in CMOS 90-Nm Technology
Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2011 Design of a Low Power and Area Efficient Digital Down Converter and SINC Filter in CMOS 90-nm Technology Steven John Billman Wright State University Follow this and additional works at: https://corescholar.libraries.wright.edu/etd_all Part of the Electrical and Computer Engineering Commons Repository Citation Billman, Steven John, "Design of a Low Power and Area Efficient Digital Down Converter and SINC Filter in CMOS 90-nm Technology" (2011). Browse all Theses and Dissertations. 444. https://corescholar.libraries.wright.edu/etd_all/444 This Thesis is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. Design of a Low Power and Area Efficient Digital Down Converter and SINC Filter in CMOS 90-nm Technology A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Engineering By Steven John Billman B.S., Cedarville University, 2009 2011 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES Jun 10, 2011 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY Steven Billman ENTITLED Design of a Low Power and Area Efficient Digital Down Converter and SINC Filter In CMOS 90nm Technology BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in Engineering Saiyu Ren, Ph.D. Thesis Director Kefu Xue, Ph.D., Chair Department of Electrical Engineering College of Engineering and Computer Science Committee on Final Examination Saiyu Ren, Ph.D.