Exploring Anti-Aliasing Filters in Signal Conditioners for Mixed-Signal, Multimodal Sensor Conditioning by Arun T
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Detection and Estimation Theory Introduction to ECE 531 Mojtaba Soltanalian- UIC the Course
Detection and Estimation Theory Introduction to ECE 531 Mojtaba Soltanalian- UIC The course Lectures are given Tuesdays and Thursdays, 2:00-3:15pm Office hours: Thursdays 3:45-5:00pm, SEO 1031 Instructor: Prof. Mojtaba Soltanalian office: SEO 1031 email: [email protected] web: http://msol.people.uic.edu/ The course Course webpage: http://msol.people.uic.edu/ECE531 Textbook(s): * Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory, by Steven M. Kay, Prentice Hall, 1993, and (possibly) * Fundamentals of Statistical Signal Processing, Volume 2: Detection Theory, by Steven M. Kay, Prentice Hall 1998, available in hard copy form at the UIC Bookstore. The course Style: /Graduate Course with Active Participation/ Introduction Let’s start with a radar example! Introduction> Radar Example QUIZ Introduction> Radar Example You can actually explain it in ten seconds! Introduction> Radar Example Applications in Transportation, Defense, Medical Imaging, Life Sciences, Weather Prediction, Tracking & Localization Introduction> Radar Example The strongest signals leaking off our planet are radar transmissions, not television or radio. The most powerful radars, such as the one mounted on the Arecibo telescope (used to study the ionosphere and map asteroids) could be detected with a similarly sized antenna at a distance of nearly 1,000 light-years. - Seth Shostak, SETI Introduction> Estimation Traditionally discussed in STATISTICS. Estimation in Signal Processing: Digital Computers ADC/DAC (Sampling) Signal/Information Processing Introduction> Estimation The primary focus is on obtaining optimal estimation algorithms that may be implemented on a digital computer. We will work on digital signals/datasets which are typically samples of a continuous-time waveform. -
Z-Transformation - I
Digital signal processing: Lecture 5 z-transformation - I Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/1 Review of last lecture • Fourier transform & inverse Fourier transform: – Time domain & Frequency domain representations • Understand the “true face” (latent factor) of some physical phenomenon. – Key points: • Definition of Fourier transformation. • Definition of inverse Fourier transformation. • Condition for a signal to have FT. Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/2 Review of last lecture • The sampling theorem – If the highest frequency component contained in an analog signal is f0, the sampling frequency (the Nyquist rate) should be 2xf0 at least. • Frequency response of an LTI system: – The Fourier transformation of the impulse response – Physical meaning: frequency components to keep (~1) and those to remove (~0). – Theoretic foundation: Convolution theorem. Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/3 Topics of this lecture Chapter 6 of the textbook • The z-transformation. • z-変換 • Convergence region of z- • z-変換の収束領域 transform. -変換とフーリエ変換 • Relation between z- • z transform and Fourier • z-変換と差分方程式 transform. • 伝達関数 or システム • Relation between z- 関数 transform and difference equation. • Transfer function (system function). Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/4 z-transformation(z-変換) • Laplace transformation is important for analyzing analog signal/systems. • z-transformation is important for analyzing discrete signal/systems. • For a given signal x(n), its z-transformation is defined by ∞ Z[x(n)] = X (z) = ∑ x(n)z−n (6.3) n=0 where z is a complex variable. Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/5 One-sided z-transform and two-sided z-transform • The z-transform defined above is called one- sided z-transform (片側z-変換). -
Basics on Digital Signal Processing
Basics on Digital Signal Processing z - transform - Digital Filters Vassilis Anastassopoulos Electronics Laboratory, Physics Department, University of Patras Outline of the Lecture 1. The z-transform 2. Properties 3. Examples 4. Digital filters 5. Design of IIR and FIR filters 6. Linear phase 2/31 z-Transform 2 N 1 N 1 j nk n X (k) x(n) e N X (z) x(n) z n0 n0 Time to frequency Time to z-domain (complex) Transformation tool is the Transformation tool is complex wave z e j e j j j2k / N e e With amplitude ρ changing(?) with time With amplitude |ejω|=1 The z-transform is more general than the DFT 3/31 z-Transform For z=ejω i.e. ρ=1 we work on the N 1 unit circle X (z) x(n) z n And the z-transform degenerates n0 into the Fourier transform. I -z z=R+jI |z|=1 R The DFT is an expression of the z-transform on the unit circle. The quantity X(z) must exist with finite value on the unit circle i.e. must posses spectrum with which we can describe a signal or a system. 4/31 z-Transform convergence We are interested in those values of z for which X(z) converges. This region should contain the unit circle. I Why is it so? |a| N 1 N 1 X (z) x(n) zn x(n) (1/zn ) n0 n0 R At z=0, X(z) diverges ROC The values of z for which X(z) diverges are called poles of X(z). -
3F3 – Digital Signal Processing (DSP)
3F3 – Digital Signal Processing (DSP) Simon Godsill www-sigproc.eng.cam.ac.uk/~sjg/teaching/3F3 Course Overview • 11 Lectures • Topics: – Digital Signal Processing – DFT, FFT – Digital Filters – Filter Design – Filter Implementation – Random signals – Optimal Filtering – Signal Modelling • Books: – J.G. Proakis and D.G. Manolakis, Digital Signal Processing 3rd edition, Prentice-Hall. – Statistical digital signal processing and modeling -Monson H. Hayes –Wiley • Some material adapted from courses by Dr. Malcolm Macleod, Prof. Peter Rayner and Dr. Arnaud Doucet Digital Signal Processing - Introduction • Digital signal processing (DSP) is the generic term for techniques such as filtering or spectrum analysis applied to digitally sampled signals. • Recall from 1B Signal and Data Analysis that the procedure is as shown below: • is the sampling period • is the sampling frequency • Recall also that low-pass anti-aliasing filters must be applied before A/D and D/A conversion in order to remove distortion from frequency components higher than Hz (see later for revision of this). • Digital signals are signals which are sampled in time (“discrete time”) and quantised. • Mathematical analysis of inherently digital signals (e.g. sunspot data, tide data) was developed by Gauss (1800), Schuster (1896) and many others since. • Electronic digital signal processing (DSP) was first extensively applied in geophysics (for oil-exploration) then military applications, and is now fundamental to communications, mobile devices, broadcasting, and most applications of signal and image processing. There are many advantages in carrying out digital rather than analogue processing; among these are flexibility and repeatability. The flexibility stems from the fact that system parameters are simply numbers stored in the processor. -
Designing Filters Using the Digital Filter Design Toolkit Rahman Jamal, Mike Cerna, John Hanks
NATIONAL Application Note 097 INSTRUMENTS® The Software is the Instrument ® Designing Filters Using the Digital Filter Design Toolkit Rahman Jamal, Mike Cerna, John Hanks Introduction The importance of digital filters is well established. Digital filters, and more generally digital signal processing algorithms, are classified as discrete-time systems. They are commonly implemented on a general purpose computer or on a dedicated digital signal processing (DSP) chip. Due to their well-known advantages, digital filters are often replacing classical analog filters. In this application note, we introduce a new digital filter design and analysis tool implemented in LabVIEW with which developers can graphically design classical IIR and FIR filters, interactively review filter responses, and save filter coefficients. In addition, real-world filter testing can be performed within the digital filter design application using a plug-in data acquisition board. Digital Filter Design Process Digital filters are used in a wide variety of signal processing applications, such as spectrum analysis, digital image processing, and pattern recognition. Digital filters eliminate a number of problems associated with their classical analog counterparts and thus are preferably used in place of analog filters. Digital filters belong to the class of discrete-time LTI (linear time invariant) systems, which are characterized by the properties of causality, recursibility, and stability. They can be characterized in the time domain by their unit-impulse response, and in the transform domain by their transfer function. Obviously, the unit-impulse response sequence of a causal LTI system could be of either finite or infinite duration and this property determines their classification into either finite impulse response (FIR) or infinite impulse response (IIR) system. -
Finite Impulse Response (FIR) Digital Filters (II) Ideal Impulse Response Design Examples Yogananda Isukapalli
Finite Impulse Response (FIR) Digital Filters (II) Ideal Impulse Response Design Examples Yogananda Isukapalli 1 • FIR Filter Design Problem Given H(z) or H(ejw), find filter coefficients {b0, b1, b2, ….. bN-1} which are equal to {h0, h1, h2, ….hN-1} in the case of FIR filters. 1 z-1 z-1 z-1 z-1 x[n] h0 h1 h2 h3 hN-2 hN-1 1 1 1 1 1 y[n] Consider a general (infinite impulse response) definition: ¥ H (z) = å h[n] z-n n=-¥ 2 From complex variable theory, the inverse transform is: 1 n -1 h[n] = ò H (z)z dz 2pj C Where C is a counterclockwise closed contour in the region of convergence of H(z) and encircling the origin of the z-plane • Evaluating H(z) on the unit circle ( z = ejw ) : ¥ H (e jw ) = åh[n]e- jnw n=-¥ 1 p h[n] = ò H (e jw )e jnwdw where dz = jejw dw 2p -p 3 • Design of an ideal low pass FIR digital filter H(ejw) K -2p -p -wc 0 wc p 2p w Find ideal low pass impulse response {h[n]} 1 p h [n] = H (e jw )e jnwdw LP ò 2p -p 1 wc = Ke jnwdw 2p ò -wc Hence K h [n] = sin(nw ) n = 0, ±1, ±2, …. ±¥ LP np c 4 Let K = 1, wc = p/4, n = 0, ±1, …, ±10 The impulse response coefficients are n = 0, h[n] = 0.25 n = ±4, h[n] = 0 = ±1, = 0.225 = ±5, = -0.043 = ±2, = 0.159 = ±6, = -0.053 = ±3, = 0.075 = ±7, = -0.032 n = ±8, h[n] = 0 = ±9, = 0.025 = ±10, = 0.032 5 Non Causal FIR Impulse Response We can make it causal if we shift hLP[n] by 10 units to the right: K h [n] = sin((n -10)w ) LP (n -10)p c n = 0, 1, 2, …. -
CONVOLUTION: Digital Signal Processing .R .Hamming, W
CONVOLUTION: Digital Signal Processing AN-237 National Semiconductor CONVOLUTION: Digital Application Note 237 Signal Processing January 1980 Introduction As digital signal processing continues to emerge as a major Decreasing the pulse width while increasing the pulse discipline in the field of electrical engineering, an even height to allow the area under the pulse to remain constant, greater demand has evolved to understand the basic theo- Figure 1c, shows from eq(1) and eq(2) the bandwidth or retical concepts involved in the development of varied and spectral-frequency content of the pulse to have increased, diverse signal processing systems. The most fundamental Figure 1d. concepts employed are (not necessarily listed in the order Further altering the pulse to that of Figure 1e provides for an [ ] of importance) the sampling theorem 1 , Fourier transforms even broader bandwidth, Figure 1f. If the pulse is finally al- [ ][] 2 3, convolution, covariance, etc. tered to the limit, i.e., the pulsewidth being infinitely narrow The intent of this article will be to address the concept of and its amplitude adjusted to still maintain an area of unity convolution and to present it in an introductory manner under the pulse, it is found in 1g and 1h the unit impulse hopefully easily understood by those entering the field of produces a constant, or ``flat'' spectrum equal to 1 at all digital signal processing. frequencies. Note that if ATe1 (unit area), we get, by defini- It may be appropriate to note that this article is Part II (Part I tion, the unit impulse function in time. -
The Scientist and Engineer's Guide to Digital Signal Processing Properties of Convolution
CHAPTER 7 Properties of Convolution A linear system's characteristics are completely specified by the system's impulse response, as governed by the mathematics of convolution. This is the basis of many signal processing techniques. For example: Digital filters are created by designing an appropriate impulse response. Enemy aircraft are detected with radar by analyzing a measured impulse response. Echo suppression in long distance telephone calls is accomplished by creating an impulse response that counteracts the impulse response of the reverberation. The list goes on and on. This chapter expands on the properties and usage of convolution in several areas. First, several common impulse responses are discussed. Second, methods are presented for dealing with cascade and parallel combinations of linear systems. Third, the technique of correlation is introduced. Fourth, a nasty problem with convolution is examined, the computation time can be unacceptably long using conventional algorithms and computers. Common Impulse Responses Delta Function The simplest impulse response is nothing more that a delta function, as shown in Fig. 7-1a. That is, an impulse on the input produces an identical impulse on the output. This means that all signals are passed through the system without change. Convolving any signal with a delta function results in exactly the same signal. Mathematically, this is written: EQUATION 7-1 The delta function is the identity for ( ' convolution. Any signal convolved with x[n] *[n] x[n] a delta function is left unchanged. This property makes the delta function the identity for convolution. This is analogous to zero being the identity for addition (a%0 ' a), and one being the identity for multiplication (a×1 ' a). -
Transformations for FIR and IIR Filters' Design
S S symmetry Article Transformations for FIR and IIR Filters’ Design V. N. Stavrou 1,*, I. G. Tsoulos 2 and Nikos E. Mastorakis 1,3 1 Hellenic Naval Academy, Department of Computer Science, Military Institutions of University Education, 18539 Piraeus, Greece 2 Department of Informatics and Telecommunications, University of Ioannina, 47150 Kostaki Artas, Greece; [email protected] 3 Department of Industrial Engineering, Technical University of Sofia, Bulevard Sveti Kliment Ohridski 8, 1000 Sofia, Bulgaria; mastor@tu-sofia.bg * Correspondence: [email protected] Abstract: In this paper, the transfer functions related to one-dimensional (1-D) and two-dimensional (2-D) filters have been theoretically and numerically investigated. The finite impulse response (FIR), as well as the infinite impulse response (IIR) are the main 2-D filters which have been investigated. More specifically, methods like the Windows method, the bilinear transformation method, the design of 2-D filters from appropriate 1-D functions and the design of 2-D filters using optimization techniques have been presented. Keywords: FIR filters; IIR filters; recursive filters; non-recursive filters; digital filters; constrained optimization; transfer functions Citation: Stavrou, V.N.; Tsoulos, I.G.; 1. Introduction Mastorakis, N.E. Transformations for There are two types of digital filters: the Finite Impulse Response (FIR) filters or Non- FIR and IIR Filters’ Design. Symmetry Recursive filters and the Infinite Impulse Response (IIR) filters or Recursive filters [1–5]. 2021, 13, 533. https://doi.org/ In the non-recursive filter structures the output depends only on the input, and in the 10.3390/sym13040533 recursive filter structures the output depends both on the input and on the previous outputs. -
Digital Signal Processing: Theory and Practice, Hardware and Software
AC 2009-959: DIGITAL SIGNAL PROCESSING: THEORY AND PRACTICE, HARDWARE AND SOFTWARE Wei PAN, Idaho State University Wei Pan is Assistant Professor and Director of VLSI Laboratory, Electrical Engineering Department, Idaho State University. She has several years of industrial experience including Siemens (project engineering/management.) Dr. Pan is an active member of ASEE and IEEE and serves on the membership committee of the IEEE Education Society. S. Hossein Mousavinezhad, Idaho State University S. Hossein Mousavinezhad is Professor and Chair, Electrical Engineering Department, Idaho State University. Dr. Mousavinezhad is active in ASEE and IEEE and is an ABET program evaluator. Hossein is the founding general chair of the IEEE International Conferences on Electro Information Technology. Kenyon Hart, Idaho State University Kenyon Hart is Specialist Engineer and Associate Lecturer, Electrical Engineering Department, Idaho State University, Pocatello, Idaho. Page 14.491.1 Page © American Society for Engineering Education, 2009 Digital Signal Processing, Theory/Practice, HW/SW Abstract Digital Signal Processing (DSP) is a course offered by many Electrical and Computer Engineering (ECE) programs. In our school we offer a senior-level, first-year graduate course with both lecture and laboratory sections. Our experience has shown that some students consider the subject matter to be too theoretical, relying heavily on mathematical concepts and abstraction. There are several visible applications of DSP including: cellular communication systems, digital image processing and biomedical signal processing. Authors have incorporated many examples utilizing software packages including MATLAB/MATHCAD in the course and also used classroom demonstrations to help students visualize some difficult (but important) concepts such as digital filters and their design, various signal transformations, convolution, difference equations modeling, signals/systems classifications and power spectral estimation as well as optimal filters. -
ELEG 5173L Digital Signal Processing Ch. 3 Discrete-Time Fourier Transform
Department of Electrical Engineering University of Arkansas ELEG 5173L Digital Signal Processing Ch. 3 Discrete-Time Fourier Transform Dr. Jingxian Wu [email protected] 2 OUTLINE • The Discrete-Time Fourier Transform (DTFT) • Properties • DTFT of Sampled Signals • Upsampling and downsampling 3 DTFT • Discrete-time Fourier Transform (DTFT) X () x(n)e jn n – (radians): digital frequency • Review: Z-transform: X (z) x(n)zn n0 j X () X (z) | j – Replace z with e . ze • Review: Fourier transform: X () x(t)e jt – (rads/sec): analog frequency 4 DTFT • Relationship between DTFT and Fourier Transform – Sample a continuous time signal x a ( t ) with a sampling period T xs (t) xa (t) (t nT ) xa (nT ) (t nT ) n n – The Fourier Transform of ys (t) jt jnT X s () xs (t)e dt xa (nT)e n – Define: T • : digital frequency (unit: radians) • : analog frequency (unit: radians/sec) – Let x(n) xa (nT) X () X s T 5 DTFT • Relationship between DTFT and Fourier Transform (Cont’d) – The DTFT can be considered as the scaled version of the Fourier transform of the sampled continuous-time signal jt jnT X s () xs (t)e dt xa (nT)e n x(n) x (nT) T a jn X () X s x(n)e T n 6 DTFT • Discrete Frequency – Unit: radians (the unit of continuous frequency is radians/sec) – X ( ) is a periodic function with period 2 j2 n jn j2n jn X ( 2 ) x(n)e x(n)e e x(n)e X () n n n – We only need to consider for • For Fourier transform, we need to consider 1 – f T 2 2T 1 – f T 2 2T 7 DTFT • Example: find the DTFT of the following signal – 1. -
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,