DOCSLIB.ORG

Design of High-Speed, Low-Power, Nyquist Analog-To-Digital Converters

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Design of High-Speed, Low-Power, Nyquist Analog-To-Digital Converters Linköping Studies in Science and Technology Thesis No. 1423 Design of High‐Speed, Low‐Power, Nyquist Analog‐to‐Digital Converters Timmy Sundström LiU‐TEK‐LIC‐2009:31 Department of Electrical Engineering Linköping University, SE‐581 83 Linköping, Sweden Linköping 2009 ISBN 978‐91‐7393‐486‐2 ISSN 0280‐7971 ii Abstract The scaling of CMOS technologies has increased the performance of general purpose processors and DSPs while analog circuits designed in the same process have not been able to utilize the process scaling to the same extent, suffering from reduced voltage headroom and reduced analog gain. In order to design efficient analog ‐to‐digital converters in nanoscale CMOS there is a need to both understand the physical limitations as well as to develop new architectures and circuits that take full advantage of what the process has to offer. This thesis explores the power dissipation of Nyquist rate analog‐to‐digital converters and their lower bounds, set by both the thermal noise limit and the minimum device and feature sizes offered by the process. The use of digital error correction, which allows for low‐accuracy analog components leads to a power dissipation reduction. Developing the bounds for power dissipation based on this concept, it is seen that the power of low‐to‐medium resolution converters is reduced when going to more modern CMOS processes, something which is supported by published results. The design of comparators is studied in detail and a new topology is proposed which reduces the kickback by 6x compared to conventional topologies. This comparator is used in two flash ADCs, the first employing redundancy in the comparator array, allowing for the use of small sized, low‐ power, low‐accuracy comparators to achieve an overall low‐power solution. The flash ADC achieves 4 effective bits at 2.5 GS/s while dissipating 30 mW of power. iii iv The concept of low‐accuracy components is taken to its edge in the second ADC which does not include a reference network, instead relying on the process variations to generate the reference levels based on the mismatch induced comparator offsets. The reference‐free ADC achieves a resolution of 3.69 bits at 1.5 GS/s while dissipation 23 mW showing that process variations not necessarily must be seen as detrimental to circuit performance but rather can be seen as a source of diversity. Preface This licentiate thesis presents my research during the period March 2006 to July 2009 at the Electronic Devices group, Department of Electrical Engineering, Linköping University, Sweden. The following papers are included in the thesis: • Paper I – Timmy Sundström, Boris Murmann and Christer Svensson, “Power Dissipation Bounds for High‐Speed Nyquist Analog‐to‐Digital Converters,” in IEEE Transactions on Circuits and Systems—I: Regular Papers, Vol. 56, No. 3, pp. 509‐518, March 2009. • Paper II ‐ Timmy Sundström and Atila Alvandpour, ”A Kick‐back Reduced Comparator for a 4‐6‐bit 3‐GS/s Flash ADC in a 90nm CMOS Process, “ in Mixed Design of Integrated Circuits and Systems, MIXDES, Ciechocinek, Poland, 21‐23 June 2007. • Paper III ‐ Timmy Sundström and Atila Alvandpour, ”A 6‐bit 2.5‐GS/s Flash ADC using Comparator Redundancy for Low Power in 90nm CMOS, “ accepted for publication in Journal of Analog Integrated Circuits and Signal Processing, Springer, August 2009. • Paper IV ‐ Timmy Sundström and Atila Alvandpour, ”Utilizing Process Variations for Reference Generation in a Flash ADC, “ in IEEE Transactions on Circuits and Systems—II: Express Briefs, Vol. 56, No. 5, pp. 364‐368, May 2009. v vi The research has also generated the following peer‐reviewed publications which are not included in the thesis: • Timmy Sundström and Atila Alvandpour, “A comparative analysis of logic styles for secure IC's against DPA attacks, “ in Proceedings of the 23rd Norchip conference, Oulu, Finland, Nov 21‐22, 2005, pp 297 ‐ 300. • Timmy Sundström, Behzad Mesgarzadeh, Mattias Krysander, Markus Klein, Ingemar Söderquist, Anneli Crona, Torbjörn Fransson and Atila Alvandpour, “Prognostics of Electronic Systems through Power Supply Current Trends, “ in International Conference on Prognostics and Health Management 2008, PHM2008. • Timmy Sundström and Atila Alvandpour, “A 2.5‐GS/s 30‐mW 4‐bit Flash ADC in 90nm CMOS,” in Proceedings of the 26rd Norchip conference, Tallinn, Estonia, Nov 16‐17, 2008, pp 264 ‐ 267. Abbreviations ADC Analog‐to‐digital converter AGC Automatic gain control CMOS Complementary metal‐oxide‐semiconductor DNL Differential non‐linearity DR Dynamic range INL Integral non‐linearity LSB Least significant bit LUT Look‐up table MLSD Maximum‐likelihood sequence detection MMF Multimode fibre MSB Most significant bit PCM Pulse‐coded modulation PCB Printedd circuit boar PLL Phase‐locked loop SAR Successive approximation register SINAD Signal to noise and distortion ratio vii viii SNDR Signal to noise and distortion ratio SNR Signal to noise ratio UWB Ultra‐wideband Acknowledgments I would especially like to thank the following persons for their support and encouragement. • My supervisor Professor Atila Alvandpour, for the support and guidance and for keeping me focused on that which is relevant. • Professor Christer Svensson for all valuable discussions and insight. • Dr. Martin Hansson for the great collaboration with tape‐outs and teaching and for being a great friend. • M.Sc. Jonas Fritzin for your great friendship and support. • Dr. Henrik Fredriksson and Dr. Stefan Andersson for all the valuable technical discussions. • Our secretary Anna Folkesson for helping with the non‐technical aspects of Dbeing a Ph. student. • Research Engineer Arta Alvandpour for his assistance in a wide variety of problems. • All the past and present members of the Electronic Devices research group especially, Ass. Prof. Jerzy Dabrowski, Ass. Prof. Behzad Mesgarzadeh, Adj. Prof. Aziz Ouacha, Adj. Prof. Ted Johansson, Dr. Peter ix x Caputa, Dr. Håkan Bengtsson, , Dr. Rashad Ramzan, Dr. Naveed Ahsan, Dr. Christer Jansson, Dr. Ingemar Söderquist, Dr. Sriram Vangal, M.Sc. Shakeel Ahmad, M.Sc. Ali Fazli, M.Sc. Dai Zhang and M.Sc. Amin Ojani. • All my friends for enriching my out‐of‐work life. • My family, especially my parents for all the love and support. • Finally I would like to thank Camilla for all the patience and love, and for always being there for me. Timmy Sundström Linköping, October 2009 Table of Contents Abstract iii Preface v Abbreviations vii Acknowledgments ix List of Figures xv Part I Design of High Speed ADCs 1 Chapter 1 Introduction 3 1.1 Introduction to Analog‐to‐Digital Converters .................................. 3 1.2 Brief History and Trends .................................................................. 3 1.3 Applications ..................................................................................... 4 1.3.1 Ultra‐Wideband Radio .................................................................................. 4 1.3.2 Serial‐Link Applications ................................................................................. 5 1.4 Motivation and Scope of the Thesis ................................................. 5 1.5 Organization of the Thesis ............................................................... 7 1.6 References ....................................................................................... 7 xi xii Chapter 2 Analog‐to‐Digital Conversion Basics 9 2.1 The Analog‐to‐Digital Converter ...................................................... 9 2.2 Quantization Error ........................................................................... 9 2.3 Static Errors ....................................................................................12 2.4 Dynamic Errors ...............................................................................13 2.5 References .....................................................................................14 Chapter 3 High‐Speed ADC Architectures 15 3.1 Flash ADCs ......................................................................................16 3.1.1 Inherent Sample and Hold .......................................................................... 17 3.1.2 Flash Decoders ............................................................................................ 18 3.1.3 Flash ADC Performance .............................................................................. 18 3.2 Interpolating and Folding ADCs .....................................................19 3.2.1 Folding ADC Performance ........................................................................... 21 3.3 Successive Approximation and Algorithmic ADCs ..........................21 3.3.1 Redundancy or Reduced Radix ................................................................... 23 3.3.2 Performance of Successive Approximation ADCs ....................................... 24 3.4 Pipeline ADCs .................................................................................24 3.4.1 Pipeline ADC Performance .......................................................................... 26 3.5 Interleaved ADCs............................................................................26 3.5.1 Performance of Interleaved ADCs .............................................................. 27 3.6 References .....................................................................................28 Chapter 4 The CMOS Process and its Limitations 33 4.1 Effects and Limitations due to Scaling ...........................................33 4.1.1 Reduced Supply Voltage ............................................................................
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Evaluating Oscilloscope Sample Rates Vs. Sampling Fidelity: How to Make the Most Accurate Digital Measurements
    AC 2011-2914: EVALUATING OSCILLOSCOPE SAMPLE RATES VS. SAM- PLING FIDELITY Johnnie Lynn Hancock, Agilent Technologies About the Author Johnnie Hancock is a Product Manager at Agilent Technologies Digital Test Division. He began his career with Hewlett-Packard in 1979 as an embedded hardware designer, and holds a patent for digital oscillo- scope amplifier calibration. Johnnie is currently responsible for worldwide application support activities that promote Agilent’s digitizing oscilloscopes and he regularly speaks at technical conferences world- wide. Johnnie graduated from the University of South Florida with a degree in electrical engineering. In his spare time, he enjoys spending time with his four grandchildren and restoring his century-old Victorian home located in Colorado Springs. Contact Information: Johnnie Hancock Agilent Technologies 1900 Garden of the Gods Rd Colorado Springs, CO 80907 USA +1 (719) 590-3183 johnnie [email protected] c American Society for Engineering Education, 2011 Evaluating Oscilloscope Sample Rates vs. Sampling Fidelity: How to Make the Most Accurate Digital Measurements Introduction Digital storage oscilloscopes (DSO) are the primary tools used today by digital designers to perform signal integrity measurements such as setup/hold times, rise/fall times, and eye margin tests. High performance oscilloscopes are also widely used in university research labs to accurately characterize high-speed digital devices and systems, as well as to perform high energy physics experiments such as pulsed laser testing. In addition, general-purpose oscilloscopes are used extensively by Electrical Engineering students in their various EE analog and digital circuits lab courses. The two key banner specifications that affect an oscilloscope’s signal integrity measurement accuracy are bandwidth and sample rate.
    [Show full text]
  • Enhancing ADC Resolution by Oversampling
    AVR121: Enhancing ADC resolution by oversampling 8-bit Features Microcontrollers • Increasing the resolution by oversampling • Averaging and decimation • Noise reduction by averaging samples Application Note 1 Introduction Atmel’s AVR controller offers an Analog to Digital Converter with 10-bit resolution. In most cases 10-bit resolution is sufficient, but in some cases higher accuracy is desired. Special signal processing techniques can be used to improve the resolution of the measurement. By using a method called ‘Oversampling and Decimation’ higher resolution might be achieved, without using an external ADC. This Application Note explains the method, and which conditions need to be fulfilled to make this method work properly. Figure 1-1. Enhancing the resolution. A/D A/D A/D 10-bit 11-bit 12-bit t t t Rev. 8003A-AVR-09/05 2 Theory of operation Before reading the rest of this Application Note, the reader is encouraged to read Application Note AVR120 - ‘Calibration of the ADC’, and the ADC section in the AVR datasheet. The following examples and numbers are calculated for Single Ended Input in a Free Running Mode. ADC Noise Reduction Mode is not used. This method is also valid in the other modes, though the numbers in the following examples will be different. The ADCs reference voltage and the ADCs resolution define the ADC step size. The ADC’s reference voltage, VREF, may be selected to AVCC, an internal 2.56V / 1.1V reference, or a reference voltage at the AREF pin. A lower VREF provides a higher voltage precision but minimizes the dynamic range of the input signal.
    [Show full text]
  • 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,
    [Show full text]
  • Adaptive Sub-Nyquist Spectrum Sensing for Ultra-Wideband Communication Systems †
    S S symmetry Article Adaptive Sub-Nyquist Spectrum Sensing for Ultra-Wideband Communication Systems † Yong Lu , Shaohe Lv and Xiaodong Wang * Science and Technology on Parallel and Distributed Processing Laboratory, National University of Defense Technology, Changsha 410073, China; [email protected] (Y.L.); [email protected] (S.L.) * Correspondence: [email protected] † This paper is an extended version of our paper published in 2014 IEEE 17th International Conference on Computational Science and Engineering (CSE), Chengdu, China, 19–21 December 2014. Received: 14 January 2019; Accepted: 3 March 2019; Published: 7 March 2019 Abstract: With the ever-increasing demand for high-speed wireless data transmission, ultra-wideband spectrum sensing is critical to support the cognitive communication over an ultra-wide frequency band for ultra-wideband communication systems. However, it is challenging for the analog-to-digital converter design to fulfill the Nyquist rate for an ultra-wideband frequency band. Therefore, we explore the spectrum sensing mechanism based on the sub-Nyquist sampling and conduct extensive experiments to investigate the influence of sampling rate, bandwidth resolution and the signal-to-noise ratio on the accuracy of sub-Nyquist spectrum sensing. Afterward, an adaptive policy is proposed to determine the optimal sampling rate, and bandwidth resolution when the spectrum occupation or the strength of the existing signals is changed. The performance of the policy is verified by simulations. Keywords: ultra-wideband spectrum sensing; energy detection; cognitive radio; sub-Nyquist sensing; Chinese remainder theorem 1. Introduction After decades of rapid development, wireless technologies have been intensively applied to almost every field from daily communications to mobile medical and battlefield communications.
    [Show full text]
  • Above the Nyquist Rate, Modulo Folding Does Not Hurt
    1 Above the Nyquist Rate, Modulo Folding Does Not Hurt Elad Romanov and Or Ordentlich Abstract Q Q m −(R−1) —We consider the problem of recovering a continuous- choice will be xn . After all, xn xn < 2 2 , time bandlimited signal from the discrete-time signal obtained {res } | −m | −R · whereas xn xn can be as large as 2 (1 2 ). However, from sampling it every Ts seconds and reducing the result modulo somewhat| surprisingly,− | this letter shows that− the opposite is ∆, for some ∆ > 0. For ∆ = ∞ the celebrated Shannon- Nyquist sampling theorem guarantees that perfect recovery is true, provided that Ts < 1/2W . In particular, we improve possible provided that the sampling rate 1/Ts exceeds the so- upon the recent results of Bhandari et al. [1] and show called Nyquist rate. Recent work by Bhandari et al. has shown that whenever the sampling frequency exceeds the Shannon- ∆ 0 that for any > perfect reconstruction is still possible if the Nyquist frequency, for any R N, the signal xn can be sampling rate exceeds the Nyquist rate by a factor of πe. In perfectly reconstructed from x∈res . { } this letter we improve upon this result and show that for finite { n } energy signals, perfect recovery is possible for any ∆ > 0 and To be more concrete, the setup we consider is that of any sampling rate above the Nyquist rate. Thus, modulo folding sampling a modulo-reduced signal. For a given ∆ > 0, and does not degrade the signal, provided that the sampling rate a real number x R, we define x∗ = [x] mod ∆ as the exceeds the Nyquist rate.
    [Show full text]
  • MULTIRATE SIGNAL PROCESSING Multirate Signal Processing
    MULTIRATE SIGNAL PROCESSING Multirate Signal Processing • Definition: Signal processing which uses more than one sampling rate to perform operations • Upsampling increases the sampling rate • Downsampling reduces the sampling rate • Reference: Digital Signal Processing, DeFatta, Lucas, and Hodgkiss B. Baas, EEC 281 431 Multirate Signal Processing • Advantages of lower sample rates –May require less processing –Likely to reduce power dissipation, P = CV2 f, where f is frequently directly proportional to the sample rate –Likely to require less storage • Advantages of higher sample rates –May simplify computation –May simplify surrounding analog and RF circuitry • Remember that information up to a frequency f requires a sampling rate of at least 2f. This is the Nyquist sampling rate. –Or we can equivalently say the Nyquist sampling rate is ½ the sampling frequency, fs B. Baas, EEC 281 432 Upsampling Upsampling or Interpolation •For an upsampling by a factor of I, add I‐1 zeros between samples in the original sequence •An upsampling by a factor I is commonly written I For example, upsampling by two: 2 • Obviously the number of samples will approximately double after 2 •Note that if the sampling frequency doubles after an upsampling by two, that t the original sample sequence will occur at the same points in time t B. Baas, EEC 281 434 Original Signal Spectrum •Example signal with most energy near DC •Notice 5 spectral “bumps” between large signal “bumps” B. Baas, EEC 281 π 2π435 Upsampled Signal (Time) •One zero is inserted between the original samples for 2x upsampling B. Baas, EEC 281 436 Upsampled Signal Spectrum (Frequency) • Spectrum of 2x upsampled signal •Notice the location of the (now somewhat compressed) five “bumps” on each side of π B.
    [Show full text]
  • Introduction Simulation of Signal Averaging
    Signal Processing Naureen Ghani December 9, 2017 Introduction Signal processing is used to enhance signal components in noisy measurements. It is especially important in analyzing time-series data in neuroscience. Applications of signal processing include data compression and predictive algorithms. Data analysis techniques are often subdivided into operations in the spatial domain and frequency domain. For one-dimensional time series data, we begin by signal averaging in the spatial domain. Signal averaging is a technique that allows us to uncover small amplitude signals in the noisy data. It makes the following assumptions: 1. Signal and noise are uncorrelated. 2. The timing of the signal is known. 3. A consistent signal component exists when performing repeated measurements. 4. The noise is truly random with zero mean. In reality, all these assumptions may be violated to some degree. This technique is still useful and robust in extracting signals. Simulation of Signal Averaging To simulate signal averaging, we will generate a measurement x that consists of a signal s and a noise component n. This is repeated over N trials. For each digitized trial, the kth sample point in the jth trial can be written as xj(k) = sj(k) + nj(k) Here is the code to simulate signal averaging: % Signal Averaging Simulation % Generate 256 noisy trials trials = 256; noise_trials = randn(256); % Generate sine signal sz = 1:trials; sz = sz/(trials/2); S = sin(2*pi*sz); % Add noise to 256 sine signals for i = 1:trials noise_trials(i,:) = noise_trials(i,:) + S;
    [Show full text]
  • Can Compressed Sensing Beat the Nyquist Sampling Rate?
    Can compressed sensing beat the Nyquist sampling rate? Leonid P. Yaroslavsky, Dept. of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv 699789, Israel Data saving capability of “Compressed sensing (sampling)” in signal discretization is disputed and found to be far below the theoretical upper bound defined by the signal sparsity. On a simple and intuitive example, it is demonstrated that, in a realistic scenario for signals that are believed to be sparse, one can achieve a substantially larger saving than compressing sensing can. It is also shown that frequent assertions in the literature that “Compressed sensing” can beat the Nyquist sampling approach are misleading substitution of terms and are rooted in misinterpretation of the sampling theory. Keywords: sampling; sampling theorem; compressed sensing. Paper 150246C received Feb. 26, 2015; accepted for publication Jun. 26, 2015. This letter is motivated by recent OPN publications1,2 that Therefore, the inverse of the signal sparsity SPRS=K/N is the advertise wide use in optical sensing of “Compressed sensing” theoretical upper bound of signal dimensionality reduction (CS), a new method of digital image formation that has achievable by means of signal sparse approximation. This obtained considerable attention after publications3-7. This relationship is plotted in Fig. 1 (solid line). attention is driven by such assertions in numerous The upper bound of signal dimensionality reduction publications as “CS theory asserts that one can recover CSDRF of signals with sparsity SSP achievable by CS can be certain signals and images from far fewer samples or evaluated from the relationship: measurements than traditional methods use”6, “Beating 1/CSDRF>C(2SSPlogCSDRF) provided in Ref.
    [Show full text]