Sampling and Aliasing
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Sampling Signals on Graphs from Theory to Applications
1 Sampling Signals on Graphs From Theory to Applications Yuichi Tanaka, Yonina C. Eldar, Antonio Ortega, and Gene Cheung Abstract The study of sampling signals on graphs, with the goal of building an analog of sampling for standard signals in the time and spatial domains, has attracted considerable attention recently. Beyond adding to the growing theory on graph signal processing (GSP), sampling on graphs has various promising applications. In this article, we review current progress on sampling over graphs focusing on theory and potential applications. Although most methodologies used in graph signal sampling are designed to parallel those used in sampling for standard signals, sampling theory for graph signals significantly differs from the theory of Shannon–Nyquist and shift-invariant sampling. This is due in part to the fact that the definitions of several important properties, such as shift invariance and bandlimitedness, are different in GSP systems. Throughout this review, we discuss similarities and differences between standard and graph signal sampling and highlight open problems and challenges. I. INTRODUCTION Sampling is one of the fundamental tenets of digital signal processing (see [1] and references therein). As such, it has been studied extensively for decades and continues to draw considerable research efforts. Standard sampling theory relies on concepts of frequency domain analysis, shift invariant (SI) signals, and bandlimitedness [1]. Sampling of time and spatial domain signals in shift-invariant spaces is one of the most important building blocks of digital signal processing systems. However, in the big data era, the signals we need to process often have other types of connections and structure, such as network signals described by graphs. -
Lecture19.Pptx
Outline Foundations of Computer Graphics (Fall 2012) §. Basic ideas of sampling, reconstruction, aliasing CS 184, Lectures 19: Sampling and Reconstruction §. Signal processing and Fourier analysis §. http://inst.eecs.berkeley.edu /~cs184 Implementation of digital filters §. Section 14.10 of FvDFH (you really should read) §. Post-raytracing lectures more advanced topics §. No programming assignment §. But can be tested (at high level) in final Acknowledgements: Thomas Funkhouser and Pat Hanrahan Some slides courtesy Tom Funkhouser Sampling and Reconstruction Sampling and Reconstruction §. An image is a 2D array of samples §. Discrete samples from real-world continuous signal (Spatial) Aliasing (Spatial) Aliasing §. Jaggies probably biggest aliasing problem 1 Sampling and Aliasing Image Processing pipeline §. Artifacts due to undersampling or poor reconstruction §. Formally, high frequencies masquerading as low §. E.g. high frequency line as low freq jaggies Outline Motivation §. Basic ideas of sampling, reconstruction, aliasing §. Formal analysis of sampling and reconstruction §. Signal processing and Fourier analysis §. Important theory (signal-processing) for graphics §. Implementation of digital filters §. Also relevant in rendering, modeling, animation §. Section 14.10 of FvDFH Ideas Sampling Theory Analysis in the frequency (not spatial) domain §. Signal (function of time generally, here of space) §. Sum of sine waves, with possibly different offsets (phase) §. Each wave different frequency, amplitude §. Continuous: defined at all points; discrete: on a grid §. High frequency: rapid variation; Low Freq: slow variation §. Images are converting continuous to discrete. Do this sampling as best as possible. §. Signal processing theory tells us how best to do this §. Based on concept of frequency domain Fourier analysis 2 Fourier Transform Fourier Transform §. Tool for converting from spatial to frequency domain §. -
And Bandlimiting IMAHA, November 14, 2009, 11:00–11:40
Time- and bandlimiting IMAHA, November 14, 2009, 11:00{11:40 Joe Lakey (w Scott Izu and Jeff Hogan)1 November 16, 2009 Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting sampling theory and history Time- and bandlimiting, history, definitions and basic properties connecting sampling and time- and bandlimiting The multiband case Outline Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Time- and bandlimiting, history, definitions and basic properties connecting sampling and time- and bandlimiting The multiband case Outline sampling theory and history Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting connecting sampling and time- and bandlimiting The multiband case Outline sampling theory and history Time- and bandlimiting, history, definitions and basic properties Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting The multiband case Outline sampling theory and history Time- and bandlimiting, history, definitions and basic properties connecting sampling and time- and bandlimiting Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting Outline sampling theory and history Time- and bandlimiting, history, definitions and basic properties connecting sampling and time- and bandlimiting The multiband case Joe Lakey (w Scott Izu and Jeff Hogan) Time- and bandlimiting R −2πitξ Fourier transform: bf (ξ) = f (t) e dt R _ Bandlimiting: -
Discrete - Time Signals and Systems
Discrete - Time Signals and Systems Sampling – II Sampling theorem & Reconstruction Yogananda Isukapalli Sampling at diffe- -rent rates From these figures, it can be concluded that it is very important to sample the signal adequately to avoid problems in reconstruction, which leads us to Shannon’s sampling theorem 2 Fig:7.1 Claude Shannon: The man who started the digital revolution Shannon arrived at the revolutionary idea of digital representation by sampling the information source at an appropriate rate, and converting the samples to a bit stream Before Shannon, it was commonly believed that the only way of achieving arbitrarily small probability of error in a communication channel was to 1916-2001 reduce the transmission rate to zero. All this changed in 1948 with the publication of “A Mathematical Theory of Communication”—Shannon’s landmark work Shannon’s Sampling theorem A continuous signal xt( ) with frequencies no higher than fmax can be reconstructed exactly from its samples xn[ ]= xn [Ts ], if the samples are taken at a rate ffs ³ 2,max where fTss= 1 This simple theorem is one of the theoretical Pillars of digital communications, control and signal processing Shannon’s Sampling theorem, • States that reconstruction from the samples is possible, but it doesn’t specify any algorithm for reconstruction • It gives a minimum sampling rate that is dependent only on the frequency content of the continuous signal x(t) • The minimum sampling rate of 2fmax is called the “Nyquist rate” Example1: Sampling theorem-Nyquist rate x( t )= 2cos(20p t ), find the Nyquist frequency ? xt( )= 2cos(2p (10) t ) The only frequency in the continuous- time signal is 10 Hz \ fHzmax =10 Nyquist sampling rate Sampling rate, ffsnyq ==2max 20 Hz Continuous-time sinusoid of frequency 10Hz Fig:7.2 Sampled at Nyquist rate, so, the theorem states that 2 samples are enough per period. -
The Nyquist Sampling Rate for Spiraling Curves 11
THE NYQUIST SAMPLING RATE FOR SPIRALING CURVES PHILIPPE JAMING, FELIPE NEGREIRA & JOSE´ LUIS ROMERO Abstract. We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of under- sampling that compressible signals admit when sampled along spirals. More precisely, we derive a lower bound on the condition number for the reconstruction of functions of bounded variation, and for functions that are sparse in the Haar wavelet basis. 1. Introduction 1.1. The mobile sampling problem. In this article, we consider the reconstruction of a compactly supported function from samples of its Fourier transform taken along certain curves, that we call spiraling. This problem is relevant, for example, in magnetic resonance imaging (MRI), where the anatomy and physiology of a person are captured by moving sensors. The Fourier sampling problem is equivalent to the sampling problem for bandlimited functions - that is, functions whose Fourier transform are supported on a given compact set. The most classical setting concerns functions of one real variable with Fourier transform supported on the unit interval [ 1/2, 1/2], and sampled on a grid ηZ, with η > 0. The sampling rate η determines whether− every bandlimited function can be reconstructed from its samples: reconstruction fails if η > 1 and succeeds if η 6 1 [42]. The transition value η = 1 is known as the Nyquist sampling rate, and it is the benchmark for all sampling schemes: modern sampling strategies that exploit the particular structure of a certain class of signals are praised because they achieve sub-Nyquist sampling rates. -
2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing - Continuous and Discrete 1 Sampling and the Discrete Fourier Transform 1 Sampling Consider a continuous function f(t) that is limited in extent, T1 · t < T2. In order to process this function in the computer it must be sampled and represented by a ¯nite set of numbers. The most common sampling scheme is to use a ¯xed sampling interval ¢T and to form a sequence of length N: ffng (n = 0 : : : N ¡ 1), where fn = f(T1 + n¢T ): In subsequent processing the function f(t) is represented by the ¯nite sequence ffng and the sampling interval ¢T . In practice, sampling occurs in the time domain by the use of an analog-digital (A/D) converter. The mathematical operation of sampling (not to be confused with the physics of sampling) is most commonly described as a multiplicative operation, in which f(t) is multiplied by a Dirac comb sampling function s(t; ¢T ), consisting of a set of delayed Dirac delta functions: X1 s(t; ¢T ) = ±(t ¡ n¢T ): (1) n=¡1 ? We denote the sampled waveform f (t) as X1 ? f (t) = s(t; ¢T )f(t) = f(t)±(t ¡ n¢T ) (2) n=¡1 ? as shown in Fig. 1. Note that f (t) is a set of delayed weighted delta functions, and that the waveform must be interpreted in the distribution sense by the strength (or area) of each component impulse. -
Signal Sampling
FYS3240 PC-based instrumentation and microcontrollers Signal sampling Spring 2017 – Lecture #5 Bekkeng, 30.01.2017 Content – Aliasing – Sampling – Analog to Digital Conversion (ADC) – Filtering – Oversampling – Triggering Analog Signal Information Three types of information: • Level • Shape • Frequency Sampling Considerations – An analog signal is continuous – A sampled signal is a series of discrete samples acquired at a specified sampling rate – The faster we sample the more our sampled signal will look like our actual signal Actual Signal – If not sampled fast enough a problem known as aliasing will occur Sampled Signal Aliasing Adequately Sampled SignalSignal Aliased Signal Bandwidth of a filter • The bandwidth B of a filter is defined to be between the -3 dB points Sampling & Nyquist’s Theorem • Nyquist’s sampling theorem: – The sample frequency should be at least twice the highest frequency contained in the signal Δf • Or, more correctly: The sample frequency fs should be at least twice the bandwidth Δf of your signal 0 f • In mathematical terms: fs ≥ 2 *Δf, where Δf = fhigh – flow • However, to accurately represent the shape of the ECG signal signal, or to determine peak maximum and peak locations, a higher sampling rate is required – Typically a sample rate of 10 times the bandwidth of the signal is required. Illustration from wikipedia Sampling Example Aliased Signal 100Hz Sine Wave Sampled at 100Hz Adequately Sampled for Frequency Only (Same # of cycles) 100Hz Sine Wave Sampled at 200Hz Adequately Sampled for Frequency and Shape 100Hz Sine Wave Sampled at 1kHz Hardware Filtering • Filtering – To remove unwanted signals from the signal that you are trying to measure • Analog anti-aliasing low-pass filtering before the A/D converter – To remove all signal frequencies that are higher than the input bandwidth of the device. -
Information Theory
Information Theory Professor John Daugman University of Cambridge Computer Science Tripos, Part II Michaelmas Term 2016/17 H(X,Y) I(X;Y) H(X|Y) H(Y|X) H(X) H(Y) 1 / 149 Outline of Lectures 1. Foundations: probability, uncertainty, information. 2. Entropies defined, and why they are measures of information. 3. Source coding theorem; prefix, variable-, and fixed-length codes. 4. Discrete channel properties, noise, and channel capacity. 5. Spectral properties of continuous-time signals and channels. 6. Continuous information; density; noisy channel coding theorem. 7. Signal coding and transmission schemes using Fourier theorems. 8. The quantised degrees-of-freedom in a continuous signal. 9. Gabor-Heisenberg-Weyl uncertainty relation. Optimal \Logons". 10. Data compression codes and protocols. 11. Kolmogorov complexity. Minimal description length. 12. Applications of information theory in other sciences. Reference book (*) Cover, T. & Thomas, J. Elements of Information Theory (second edition). Wiley-Interscience, 2006 2 / 149 Overview: what is information theory? Key idea: The movements and transformations of information, just like those of a fluid, are constrained by mathematical and physical laws. These laws have deep connections with: I probability theory, statistics, and combinatorics I thermodynamics (statistical physics) I spectral analysis, Fourier (and other) transforms I sampling theory, prediction, estimation theory I electrical engineering (bandwidth; signal-to-noise ratio) I complexity theory (minimal description length) I signal processing, representation, compressibility As such, information theory addresses and answers the two fundamental questions which limit all data encoding and communication systems: 1. What is the ultimate data compression? (answer: the entropy of the data, H, is its compression limit.) 2. -
Lecture 16 Bandlimiting and Nyquist Criterion
ELG3175 Introduction to Communication Systems Lecture 16 Bandlimiting and Nyquist Criterion Bandlimiting and ISI • Real systems are usually bandlimited. • When a signal is bandlimited in the frequency domain, it is usually smeared in the time domain. This smearing results in intersymbol interference (ISI). • The only way to avoid ISI is to satisfy the 1st Nyquist criterion. • For an impulse response this means at sampling instants having only one nonzero sample. Lecture 12 Band-limited Channels and Intersymbol Interference • Rectangular pulses are suitable for infinite-bandwidth channels (practically – wideband). • Practical channels are band-limited -> pulses spread in time and are smeared into adjacent slots. This is intersymbol interference (ISI). Input binary waveform Individual pulse response Received waveform Eye Diagram • Convenient way to observe the effect of ISI and channel noise on an oscilloscope. Eye Diagram • Oscilloscope presentations of a signal with multiple sweeps (triggered by a clock signal!), each is slightly larger than symbol interval. • Quality of a received signal may be estimated. • Normal operating conditions (no ISI, no noise) -> eye is open. • Large ISI or noise -> eye is closed. • Timing error allowed – width of the eye, called eye opening (preferred sampling time – at the largest vertical eye opening). • Sensitivity to timing error -> slope of the open eye evaluated at the zero crossing point. • Noise margin -> the height of the eye opening. Pulse shapes and bandwidth • For PAM: L L sPAM (t) = ∑ai p(t − iTs ) = p(t)*∑aiδ (t − iTs ) i=0 i=0 L Let ∑aiδ (t − iTs ) = y(t) i=0 Then SPAM ( f ) = P( f )Y( f ) BPAM = Bp. -
Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing
Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 1 News Homework #5 . Due this week . Submit via canvas Coding Problem #4 . Due this week . Submit via canvas Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 2 Exam 1 Grades The class did exceedingly well . Mean: 89.3 . Median: 91.5 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 3 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 4 Sampling Discrete-Time Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, -
Frames and Other Bases in Abstract and Function Spaces Novel Methods in Harmonic Analysis, Volume 1
Applied and Numerical Harmonic Analysis Isaac Pesenson Quoc Thong Le Gia Azita Mayeli Hrushikesh Mhaskar Ding-Xuan Zhou Editors Frames and Other Bases in Abstract and Function Spaces Novel Methods in Harmonic Analysis, Volume 1 Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland College Park, MD, USA Editorial Advisory Board Akram Aldroubi Gitta Kutyniok Vanderbilt University Technische Universität Berlin Nashville, TN, USA Berlin, Germany Douglas Cochran Mauro Maggioni Arizona State University Duke University Phoenix, AZ, USA Durham, NC, USA Hans G. Feichtinger Zuowei Shen University of Vienna National University of Singapore Vienna, Austria Singapore, Singapore Christopher Heil Thomas Strohmer Georgia Institute of Technology University of California Atlanta, GA, USA Davis, CA, USA Stéphane Jaffard Yang Wang University of Paris XII Michigan State University Paris, France East Lansing, MI, USA Jelena Kovaceviˇ c´ Carnegie Mellon University Pittsburgh, PA, USA More information about this series at http://www.springer.com/series/4968 Isaac Pesenson • Quoc Thong Le Gia Azita Mayeli • Hrushikesh Mhaskar Ding-Xuan Zhou Editors Frames and Other Bases in Abstract and Function Spaces Novel Methods in Harmonic Analysis, Volume 1 Editors Isaac Pesenson Quoc Thong Le Gia Department of Mathematics School of Mathematics and Statistics Temple University University of New South Wales Philadelphia, PA, USA Sydney, NSW, Australia Azita Mayeli Hrushikesh Mhaskar Department of Mathematics Institute of Mathematical -
Ideal C/D Converter
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341: Discrete-Time Signal Processing OpenCourseWare 2006 Lecture 4 DT Processing of CT Signals & CT Processing of DT Signals: Fractional Delay Reading: Sections 4.1 - 4.5 in Oppenheim, Schafer & Buck (OSB). A typical discrete-time system used to process a continuous-time signal is shown in OSB Figure 4.15. T is the sampling/reconstruction period for the C/D and the D/C converters. When the input signal is bandlimited, the effective system in Figure 4.15 (b) is equivalent to the continuous-time system shown in Figure 4.15 (a). Ideal C/D Converter The relationship between the input and output signals in an ideal C/D converter as depicted in OSB Figure 4.1 is: Time Domain: x[n] = xc(nT ) j! 1 P1 ! 2¼k Frequency Domain: X(e ) = T k=¡1 Xc(j( T ¡ T )); j! where X(e ) and Xc(jΩ) are the DTFT and CTFT of x[n], xc(t). These relationships are illustrated in the figures below. See Section 4.2 of OSB for their derivations and more detailed descriptions of C/D converters. Time/Frequency domain representation of C/D conversion 1 In the frequency domain, a C/D converter can be thought of in three stages: periodic 1 repetition of the spectrum at 2¼, normalization of frequency by , and scaling of the amplitude T 1 ! by . The CT frequency Ω is related to the DT frequency ! by Ω = . If x (t) is not T T c s s bandlimited to ¡ 2 < Ωmax < 2 , repetition at 2¼ will cause aliasing.