FROM ANALOGUE TO DIGITAL
|
|
- Melinda Hensley
- 5 years ago
- Views:
Transcription
1 SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 7. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad FROM ANALOGUE TO DIGITAL To digitize signals it is necessary to define them by samples of discrete amplitude, taken at discrete instants in time. Each sample amplitude is quantized so that its voltage is represented by a number which has a minimum (typically 0) and a maximum value (depends on the number of bits allocated for representing the number 8 bits = 255 max, 16bits = max). Much signal processing today is done in the digital domain. Digital signals can be manipulated by computers which are intrinsically very flexible i.e. programmable. Because digital signals can be easily manipulated one can make digital systems which implement much more complicated processes than would be possible using analogue techniques. Digital signal processing is encountered in everyday devices like mobiles, CD,.mp3 Players DVD Players (Digital Video Disc.), soundblaster cards for the PC, web-cams, network cards etc. Even cars, microwaves... For digital processing to be any use we must be confident that representing a signal by a set of discrete samples still retains all the information of the original analogue signal. We must then be also confident that it is possible to reconstruct the analogue signal from the samples after the samples have been processed. We ll first look at the effects of sampling and then look at techniques for reconstructing the analogue signal from these samples. 1 This handout is based on the set of notes produced by Prof. Anil Kokaram 3C1 Signals and Systems 1
2 1 SAMPLING 1 Sampling C1 Signals and Systems 2
3 1.1 A note about representing spectra 1 SAMPLING 1.1 A note about representing spectra We will be deriving limits on sampling using spectal analysis. To do this we will be drawing a bunch of representations for signal spectra. Most naturally occuring (and interesting) signals have most of their spectral energy concentrated around the lower frequencies. In fact most natural signals (speech, music, pictures of natural scenes like landscapes) have spectra with energy that falls off with 1/f where f is frequency Frequency (Hz) Frequency (Hz) Frequency (Hz) So to visualise what is happening to signals in terms of frequency content, we tend to use pictures like below. Do not be alarmed by these pictures, they are just representations of signal spectra to allow us to visualise what would happen to the frequency content. In fact the most important thing about these pictures is that they allow us to visualise the bandwidth of signals without having to worry about a specific signal. 3C1 Signals and Systems 3
4 1.2 Sampling (Quantitative analysis) 1 SAMPLING 1.2 Sampling (Quantitative analysis) The goal is to develop an idealised mathematical model that describes the sampling process. Recall, the sifting property of the delta function x(t)δ(t u)dt = x(u) (1) In effect, the sifting property describes the sampling of x(t) at time t = u. Therefore we can model sampling over the entire signal by applying the sifting property to a series of different impulse functions located at regular intervals along the time axis. This sequence of impulses is called an impulse train. The impulse train p(t) selects the values of x(t) only at the desired sampling instants. We wish to examine the spectrum of the sampled signal and compare it to the spectrum of the original analogue signal. 3C1 Signals and Systems 4
5 1.2 Sampling (Quantitative analysis) 1 SAMPLING x(t) input signal Ideal sampler. p(t) sampling impulses x S (t) sampled signal Ideal Sampler x S (t) = x(t) p(t) (2) The sampling pulses p(t) are a periodic train of impulses of unit mean value and period T secs: p(t) = T δ(t mt ) (3) m= 3C1 Signals and Systems 5
6 1.2 Sampling (Quantitative analysis) 1 SAMPLING In order to make it easier to calculate the spectrum of x S (t), we use Fourier series analysis with ω 0 = 2π/T to express p(t) instead as a sum of its Fourier components: p(t) = a k e jkω 0t where a k = 1 T k= T/2 = e jkω 0. 0 T δ(t) e jkω 0t dt T/2 T/2 = 1 for all n. p(t) = n= e jnω 0t T/2 δ(t) dt 3C1 Signals and Systems 6
7 1.2 Sampling (Quantitative analysis) 1 SAMPLING An interesting point to note here is that, since the impulse train is a periodic function, the frequency spectrum of the impulse train is also a impulse train. In the frequency domain the interval between impulses is equal to the fundamental frequency which is the sampling frequency. The value of the fourier coefficients are uniform. Now we want to find X S (ω) so: x S (t) = x(t) p(t) = (4) Note that the sampling frequency, f S = 1 T Hz = ω 0 2π rad/sec. Procced to find X S (ω) by taking Fourier transforms and using the frequency shift theorem: HENCE X s (ω) = F(x s (t)) = = = = X S (ω) = k= k= n= x s (t)e jωt dt [ k= [ [ ] x(t)e jkω 0t e jωt dt ] x(t)e jkω0t e jωt dt ] x(t)e j(ω kω0)t dt X(ω nω 0 ) Alternatively we could have derived this result using the frequency shift and linearity properties of the fourier transform. 3C1 Signals and Systems 7
8 1.2 Sampling (Quantitative analysis) 1 SAMPLING Hence the spectrum of a sampled signal is the original spectrum X(ω), repeated at multiples of the sampling freq. See below Spectral representation of sampling. 3C1 Signals and Systems 8
9 1.2 Sampling (Quantitative analysis) 1 SAMPLING If the repeated spectra overlap, ALIASING occurs. (a) No Aliasing (b) Aliasing Signal recovery and the effect of aliasing. x(t) x S (t) Lowpass Filter y(t) recovered signal p(t) Lowpass filter recovers the original signal. If there is no aliasing, the original signal can be recovered perfectly from the sampled version by lowpass filtering, which removes all repeats of the original spectrum. 3C1 Signals and Systems 9
10 1.2 Sampling (Quantitative analysis) 1 SAMPLING If aliasing occurs, perfect recovery is not possible - f S is too low! X( ω+ω s ) X(ω) ω s - ω max X(ω ω s ) ω - ω s 0 ω max ω s As long as ω s ω max > ω max there is no aliasing. Therefore, minimum sampling frequency is when ω s ω max = ω max which implies ω s = 2ω max Same for Hz since ω s = 2πf S. The Sampling Theorem (Nyquist): If a signal has components with a maximum frequency of W Hz, then it is completely defined by samples which occur at intervals of 1/2W sec. The Sampling Theorem (Nyquist): If a signal has components with a maximum frequency of W Hz, then it is completely defined by samples that are created by sampling at 2W Hz. That is to say: if it is required to sample an analogue signal which has a spectrum containing frequency components up to W Hz, then the minimum sampling frequency required is 2W Hz. When this condition is satisfied it is possible to reconstruct the analogue signal exactly from the sampled data. This minimum sampling frequency required to prevent aliasing is called the Nyquist frequency. 3C1 Signals and Systems 10
11 2 QUANTIZATION 2 Quantization C1 Signals and Systems 11
12 2 QUANTIZATION Now we consider the effects of discrete quantization of the amplitude of our time samples. The figure below shows a digital coding system. x(t) Ideal Sampler x S (t) Quantiser x Q (t) Digital coding system. Quantization results in a transfer function similar to that shown in the figure below. A-D + D-A transfer function (ideal linear quantizer). 3C1 Signals and Systems 12
13 2 QUANTIZATION If the quantizer has a small step size and it always rounds to the nearest output value, a given input voltage results in a random quantizing error that has a uniform probability of being anywhere from δv/2 to +δv/2. Hence the quantizing errors may be modelled as additive random noise with a uniform probability density function (p.d.f.). Using these ideas we can estimate the average error caused by quantization and so work out how many bits we would need to represent a particular signal properly. This will not be discussed further here. x S (t) sampled signal quantising error e(t) + x Q (t) quantised signal Quantising noise model. p.d.f. of e(t) 1/δv δv/2 0 δv/2 e(t) Quantising noise probability density function (p.d.f.). 3C1 Signals and Systems 13
14 2 QUANTIZATION Quantising image samples Clockwise from top left: 8bit (256 Grey levels), 5bit (32 Grey Levels), 1bit (2 Grey levels), 2bit (4 Grey Levels) Quantisation. 3C1 Signals and Systems 14
15 3 Anti-aliasing/smoothing filters 3 ANTI-ALIASING/SMOOTHING FILTERS In most digitizing systems, 2 lowpass filters are required, as below. x Input Filter x F Sample & Hold x S A-D Conv. D-A Conv. y S Output Filter y Filters required for A-D and D-A conversion. Input filter (Anti-Aliasing Filter) prevents possible aliasing distortion by eliminating any input frequencies > 1 2 f S. A-D converter converts the sampled signal into a digital signal. This includes quantisation of the sampled signal. D-A converter converts the digital signal back into an analogue signal. Output filter removes all components of the sampled signal spectrum > 1 2 f S, to regenerate the original continuous signal. We want to examine the role of the input filter in the overal system. For now we will assume that the D-A converter is perfect and will ignore the effects of quantisation. Hence, we assume that x S (t) = y S (t). Next fig. shows the filter actions. 3C1 Signals and Systems 15
16 3 ANTI-ALIASING/SMOOTHING FILTERS 3C1 Signals and Systems 16
17 3 ANTI-ALIASING/SMOOTHING FILTERS Let the filters have a passband of bandwidth f P. Realisable filters must also have a transition band of finite width f T between the passband and the stopband. To make f T narrow, the filter complexity (no. of poles and zeroes) must be high. Passband Transition Band Stopband Typical filter response. 3C1 Signals and Systems 17
18 3 ANTI-ALIASING/SMOOTHING FILTERS In both filters, the stopband must start at 1 2 f S to avoid aliasing and unwanted output components. Hencef P + f T 1 2 f S (5) If the required signal bandwidth = W, then f P W for negligible frequency distortion. Let us assume that f T = αf P for a given complexity of filter. Typically α = 0.1 (high complexity) to 1.0 (low complexity). Therefore the minimum sampling frequency f S is given by: f S = 2(f P + f T ) = 2W (1 + α) (6) Therefore if low filter complexity is important, we must choose f S to be significantly greater than 2W (e.g. 4W ). 3C1 Signals and Systems 18
19 3 ANTI-ALIASING/SMOOTHING FILTERS Sampling in images Top left: 4th Order butterworth filter (on rows then columns) for anti-aliasing Top right: No anti-aliasing filter Bottom: Analogue lenna 2 2 In fact, that s a lie, this picture is also digital, but the sampling rate used was 3 times higher than the top two images and the Nyquist limit is satisfied (in this case), even though an anti-aliasing filter was not used. 3C1 Signals and Systems 19
20 4 A PRACTICAL SYSTEM 4 A Practical system Although sampling signals using impulse trains is mathematically attractive it is not practical to implement. A more practical alternative is called a sample and hold filter. The idea is to sample the value at each interval and hold the value signal constant until the next sample occurs. This is illustrated below D-A converter hold action. Ignoring quantisation errors, the A-D + D-A system is equivalent to an ideal sampler followed by a filter which extends each sampling impulse into a pulse of width T = 1/f S, as below. 3C1 Signals and Systems 20
21 4 A PRACTICAL SYSTEM x(t) x S (t) Hold Filter h(t) = y S (t) p(t) Equivalent circuit for D-A hold action. If the filter impulse response is given by: then h(t) = 1 for 0 t < T, = 0 elsewhere, H(ω) = T ( ) ωt sinc = T 2 (π sinc f ) (7) f S See below for the effect of this filtering on the spectrum Y S (ω) of the D-A output. It is possible to modify the Output Filter response as shown so as to correct for the hold filter and obtain a flat response overall. 3C1 Signals and Systems 21
22 4 A PRACTICAL SYSTEM Spectra of signals modified by D-A hold action. 3C1 Signals and Systems 22
IB Paper 6: Signal and Data Analysis
IB Paper 6: Signal and Data Analysis Handout 5: Sampling Theory S Godsill Signal Processing and Communications Group, Engineering Department, Cambridge, UK Lent 2015 1 / 85 Sampling and Aliasing All of
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization
ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals 1. Sampling and Reconstruction 2. Quantization 1 1. Sampling & Reconstruction DSP must interact with an analog world: A to D D to A x(t)
More informationSIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 6a. Dr David Corrigan 1. Electronic and Electrical Engineering Dept.
SIGNALS AND SYSTEMS: PAPER 3C HANDOUT 6a. Dr David Corrigan. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.mee.tcd.ie/ corrigad FOURIER SERIES Have seen how the behaviour of systems can
More informationHomework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt
Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t
More information2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit dwm/courses/2tf
Time-Frequency Analysis II (HT 20) 2AH 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 20 For hints and answers visit www.robots.ox.ac.uk/ dwm/courses/2tf David Murray. A periodic
More information2A1H Time-Frequency Analysis II
2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 209 For any corrections see the course page DW Murray at www.robots.ox.ac.uk/ dwm/courses/2tf. (a) A signal g(t) with period
More informationMEDE2500 Tutorial Nov-7
(updated 2016-Nov-4,7:40pm) MEDE2500 (2016-2017) Tutorial 3 MEDE2500 Tutorial 3 2016-Nov-7 Content 1. The Dirac Delta Function, singularity functions, even and odd functions 2. The sampling process and
More informationChap 4. Sampling of Continuous-Time Signals
Digital Signal Processing Chap 4. Sampling of Continuous-Time Signals Chang-Su Kim Digital Processing of Continuous-Time Signals Digital processing of a CT signal involves three basic steps 1. Conversion
More informationSignals and Systems. Lecture 14 DR TANIA STATHAKI READER (ASSOCIATE PROFESSOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON
Signals and Systems Lecture 14 DR TAIA STATHAKI READER (ASSOCIATE PROFESSOR) I SIGAL PROCESSIG IMPERIAL COLLEGE LODO Introduction. Time sampling theorem resume. We wish to perform spectral analysis using
More informationSolutions to Problems in Chapter 4
Solutions to Problems in Chapter 4 Problems with Solutions Problem 4. Fourier Series of the Output Voltage of an Ideal Full-Wave Diode Bridge Rectifier he nonlinear circuit in Figure 4. is a full-wave
More informationHomework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, Signals & Systems Sampling P1
Homework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, 7.49 Signals & Systems Sampling P1 Undersampling & Aliasing Undersampling: insufficient sampling frequency ω s < 2ω M Perfect
More informationECE 301 Fall 2011 Division 1 Homework 10 Solutions. { 1, for 0.5 t 0.5 x(t) = 0, for 0.5 < t 1
ECE 3 Fall Division Homework Solutions Problem. Reconstruction of a continuous-time signal from its samples. Let x be a periodic continuous-time signal with period, such that {, for.5 t.5 x(t) =, for.5
More informationNAME: 11 December 2013 Digital Signal Processing I Final Exam Fall Cover Sheet
NAME: December Digital Signal Processing I Final Exam Fall Cover Sheet Test Duration: minutes. Open Book but Closed Notes. Three 8.5 x crib sheets allowed Calculators NOT allowed. This test contains four
More informationCITY UNIVERSITY LONDON. MSc in Information Engineering DIGITAL SIGNAL PROCESSING EPM746
No: CITY UNIVERSITY LONDON MSc in Information Engineering DIGITAL SIGNAL PROCESSING EPM746 Date: 19 May 2004 Time: 09:00-11:00 Attempt Three out of FIVE questions, at least One question from PART B PART
More informationSignals & Systems. Lecture 5 Continuous-Time Fourier Transform. Alp Ertürk
Signals & Systems Lecture 5 Continuous-Time Fourier Transform Alp Ertürk alp.erturk@kocaeli.edu.tr Fourier Series Representation of Continuous-Time Periodic Signals Synthesis equation: x t = a k e jkω
More informationSensors. Chapter Signal Conditioning
Chapter 2 Sensors his chapter, yet to be written, gives an overview of sensor technology with emphasis on how to model sensors. 2. Signal Conditioning Sensors convert physical measurements into data. Invariably,
More informationReview: Continuous Fourier Transform
Review: Continuous Fourier Transform Review: convolution x t h t = x τ h(t τ)dτ Convolution in time domain Derivation Convolution Property Interchange the order of integrals Let Convolution Property By
More informationDesign of IIR filters
Design of IIR filters Standard methods of design of digital infinite impulse response (IIR) filters usually consist of three steps, namely: 1 design of a continuous-time (CT) prototype low-pass filter;
More informationSignals & Systems. Chapter 7: Sampling. Adapted from: Lecture notes from MIT, Binghamton University, and Purdue. Dr. Hamid R.
Signals & Systems Chapter 7: Sampling Adapted from: Lecture notes from MIT, Binghamton University, and Purdue Dr. Hamid R. Rabiee Fall 2013 Outline 1. The Concept and Representation of Periodic Sampling
More informationAnalog Digital Sampling & Discrete Time Discrete Values & Noise Digital-to-Analog Conversion Analog-to-Digital Conversion
Analog Digital Sampling & Discrete Time Discrete Values & Noise Digital-to-Analog Conversion Analog-to-Digital Conversion 6.082 Fall 2006 Analog Digital, Slide Plan: Mixed Signal Architecture volts bits
More information6.003: Signals and Systems. CT Fourier Transform
6.003: Signals and Systems CT Fourier Transform April 8, 200 CT Fourier Transform Representing signals by their frequency content. X(jω)= x(t)e jωt dt ( analysis equation) x(t)= X(jω)e jωt dω ( synthesis
More informationSIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 6. Dr Anil Kokaram Electronic and Electrical Engineering Dept.
SIGNALS AND SYSTEMS: PAPER 3C HANDOUT 6. Dr Anil Kokaram Electronic and Electrical Engineering Dept. anil.kokaram@tcd.ie www.mee.tcd.ie/ sigmedia FOURIER ANALYSIS Have seen how the behaviour of systems
More informationSTABILITY. Have looked at modeling dynamic systems using differential equations. and used the Laplace transform to help find step and impulse
SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 4. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.sigmedia.tv STABILITY Have looked at modeling dynamic systems using differential
More informationReview of Discrete-Time System
Review of Discrete-Time System Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by Profs. K.J. Ray Liu and Min Wu.
More information6.003: Signals and Systems. Sampling and Quantization
6.003: Signals and Systems Sampling and Quantization December 1, 2009 Last Time: Sampling and Reconstruction Uniform sampling (sampling interval T ): x[n] = x(nt ) t n Impulse reconstruction: x p (t) =
More informationLecture 16: Filter Design: Impulse Invariance and Bilinear Transform
EE58 Digital Signal Processing University of Washington Autumn 2 Dept. of Electrical Engineering Lecture 6: Filter Design: Impulse Invariance and Bilinear Transform Nov 26, 2 Prof: J. Bilmes
More informationECE 301 Fall 2010 Division 2 Homework 10 Solutions. { 1, if 2n t < 2n + 1, for any integer n, x(t) = 0, if 2n 1 t < 2n, for any integer n.
ECE 3 Fall Division Homework Solutions Problem. Reconstruction of a continuous-time signal from its samples. Consider the following periodic signal, depicted below: {, if n t < n +, for any integer n,
More information6.003: Signals and Systems. CT Fourier Transform
6.003: Signals and Systems CT Fourier Transform April 8, 200 CT Fourier Transform Representing signals by their frequency content. X(jω)= x(t)e jωt dt ( analysis equation) x(t)= 2π X(jω)e jωt dω ( synthesis
More informationDEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010
[E2.5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010 EEE/ISE PART II MEng. BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: 2:00 hours There are FOUR
More informationEE 224 Signals and Systems I Review 1/10
EE 224 Signals and Systems I Review 1/10 Class Contents Signals and Systems Continuous-Time and Discrete-Time Time-Domain and Frequency Domain (all these dimensions are tightly coupled) SIGNALS SYSTEMS
More informationPrinciples of Communications Lecture 8: Baseband Communication Systems. Chih-Wei Liu 劉志尉 National Chiao Tung University
Principles of Communications Lecture 8: Baseband Communication Systems Chih-Wei Liu 劉志尉 National Chiao Tung University cwliu@twins.ee.nctu.edu.tw Outlines Introduction Line codes Effects of filtering Pulse
More informationX. Chen More on Sampling
X. Chen More on Sampling 9 More on Sampling 9.1 Notations denotes the sampling time in second. Ω s = 2π/ and Ω s /2 are, respectively, the sampling frequency and Nyquist frequency in rad/sec. Ω and ω denote,
More informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Filtering in the Frequency Domain http://www.ee.unlv.edu/~b1morris/ecg782/ 2 Outline Background
More informationQuestion Paper Code : AEC11T02
Hall Ticket No Question Paper Code : AEC11T02 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B. Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)
More informationLecture 8: Signal Reconstruction, DT vs CT Processing. 8.1 Reconstruction of a Band-limited Signal from its Samples
EE518 Digital Signal Processing University of Washington Autumn 2001 Dept. of Electrical Engineering Lecture 8: Signal Reconstruction, D vs C Processing Oct 24, 2001 Prof: J. Bilmes
More informationFilter Analysis and Design
Filter Analysis and Design Butterworth Filters Butterworth filters have a transfer function whose squared magnitude has the form H a ( jω ) 2 = 1 ( ) 2n. 1+ ω / ω c * M. J. Roberts - All Rights Reserved
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels)
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30-Apr-14 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page
More informationLine Spectra and their Applications
In [ ]: cd matlab pwd Line Spectra and their Applications Scope and Background Reading This session concludes our introduction to Fourier Series. Last time (http://nbviewer.jupyter.org/github/cpjobling/eg-47-
More informationPrinciples of Communications
Principles of Communications Weiyao Lin, PhD Shanghai Jiao Tong University Chapter 4: Analog-to-Digital Conversion Textbook: 7.1 7.4 2010/2011 Meixia Tao @ SJTU 1 Outline Analog signal Sampling Quantization
More informationECE 350 Signals and Systems Spring 2011 Final Exam - Solutions. Three 8 ½ x 11 sheets of notes, and a calculator are allowed during the exam.
ECE 35 Spring - Final Exam 9 May ECE 35 Signals and Systems Spring Final Exam - Solutions Three 8 ½ x sheets of notes, and a calculator are allowed during the exam Write all answers neatly and show your
More informationE2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)
E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,
More informationOverview of Sampling Topics
Overview of Sampling Topics (Shannon) sampling theorem Impulse-train sampling Interpolation (continuous-time signal reconstruction) Aliasing Relationship of CTFT to DTFT DT processing of CT signals DT
More information1.1 SPECIAL FUNCTIONS USED IN SIGNAL PROCESSING. δ(t) = for t = 0, = 0 for t 0. δ(t)dt = 1. (1.1)
SIGNAL THEORY AND ANALYSIS A signal, in general, refers to an electrical waveform whose amplitude varies with time. Signals can be fully described in either the time or frequency domain. This chapter discusses
More information6.003: Signals and Systems. Applications of Fourier Transforms
6.003: Signals and Systems Applications of Fourier Transforms November 7, 20 Filtering Notion of a filter. LTI systems cannot create new frequencies. can only scale magnitudes and shift phases of existing
More informationGrades will be determined by the correctness of your answers (explanations are not required).
6.00 (Fall 20) Final Examination December 9, 20 Name: Kerberos Username: Please circle your section number: Section Time 2 am pm 4 2 pm Grades will be determined by the correctness of your answers (explanations
More informationDigital Signal Processing IIR Filter Design via Bilinear Transform
Digital Signal Processing IIR Filter Design via Bilinear Transform D. Richard Brown III D. Richard Brown III 1 / 12 Basic Procedure We assume here that we ve already decided to use an IIR filter. The basic
More informationVID3: Sampling and Quantization
Video Transmission VID3: Sampling and Quantization By Prof. Gregory D. Durgin copyright 2009 all rights reserved Claude E. Shannon (1916-2001) Mathematician and Electrical Engineer Worked for Bell Labs
More informationJ. McNames Portland State University ECE 223 Sampling Ver
Overview of Sampling Topics (Shannon) sampling theorem Impulse-train sampling Interpolation (continuous-time signal reconstruction) Aliasing Relationship of CTFT to DTFT DT processing of CT signals DT
More informationSystems & Signals 315
1 / 15 Systems & Signals 315 Lecture 13: Signals and Linear Systems Dr. Herman A. Engelbrecht Stellenbosch University Dept. E & E Engineering 2 Maart 2009 Outline 2 / 15 1 Signal Transmission through a
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:30-15:45 CBC C222 Lecture 05 IIR Design 14/03/04 http://www.ee.unlv.edu/~b1morris/ee482/
More informationETSF15 Analog/Digital. Stefan Höst
ETSF15 Analog/Digital Stefan Höst Physical layer Analog vs digital Sampling, quantisation, reconstruction Modulation Represent digital data in a continuous world Disturbances Noise and distortion Synchronization
More informationEECE 2150 Circuits and Signals Final Exam Fall 2016 Dec 16
EECE 2150 Circuits and Signals Final Exam Fall 2016 Dec 16 Instructions: Write your name and section number on all pages Closed book, closed notes; Computers and cell phones are not allowed You can use
More informationDigital Signal Processing
COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #24 Tuesday, November 4, 2003 6.8 IIR Filter Design Properties of IIR Filters: IIR filters may be unstable Causal IIR filters with rational system
More informationLINEAR SYSTEMS. J. Elder PSYC 6256 Principles of Neural Coding
LINEAR SYSTEMS Linear Systems 2 Neural coding and cognitive neuroscience in general concerns input-output relationships. Inputs Light intensity Pre-synaptic action potentials Number of items in display
More information4.1 Introduction. 2πδ ω (4.2) Applications of Fourier Representations to Mixed Signal Classes = (4.1)
4.1 Introduction Two cases of mixed signals to be studied in this chapter: 1. Periodic and nonperiodic signals 2. Continuous- and discrete-time signals Other descriptions: Refer to pp. 341-342, textbook.
More informationDESIGN OF CMOS ANALOG INTEGRATED CIRCUITS
DESIGN OF CMOS ANALOG INEGRAED CIRCUIS Franco Maloberti Integrated Microsistems Laboratory University of Pavia Discrete ime Signal Processing F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete
More informationECE-700 Review. Phil Schniter. January 5, x c (t)e jωt dt, x[n]z n, Denoting a transform pair by x[n] X(z), some useful properties are
ECE-7 Review Phil Schniter January 5, 7 ransforms Using x c (t) to denote a continuous-time signal at time t R, Laplace ransform: X c (s) x c (t)e st dt, s C Continuous-ime Fourier ransform (CF): ote that:
More informationLine Codes and Pulse Shaping Review. Intersymbol interference (ISI) Pulse shaping to reduce ISI Embracing ISI
Line Codes and Pulse Shaping Review Line codes Pulse width and polarity Power spectral density Intersymbol interference (ISI) Pulse shaping to reduce ISI Embracing ISI Line Code Examples (review) on-off
More informationSampling اهمتسیس و اهلانگیس یرهطم لضفلاوبا دیس فیرش یتعنص هاگشناد رتویپماک هدکشناد
Sampling سیگنالها و سیستمها سید ابوالفضل مطهری دانشکده کامپیوتر دانشگاه صنعتی شریف Sampling Conversion of a continuous-time signal to discrete time. x(t) x[n] 0 2 4 6 8 10 t 0 2 4 6 8 10 n Sampling Applications
More informationSampling. Alejandro Ribeiro. February 8, 2018
Sampling Alejandro Ribeiro February 8, 2018 Signals exist in continuous time but it is not unusual for us to process them in discrete time. When we work in discrete time we say that we are doing discrete
More informationReview of Linear Time-Invariant Network Analysis
D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x
More informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 8: February 12th, 2019 Sampling and Reconstruction Lecture Outline! Review " Ideal sampling " Frequency response of sampled signal " Reconstruction " Anti-aliasing
More informationGrades will be determined by the correctness of your answers (explanations are not required).
6.00 (Fall 2011) Final Examination December 19, 2011 Name: Kerberos Username: Please circle your section number: Section Time 2 11 am 1 pm 4 2 pm Grades will be determined by the correctness of your answers
More informationSignal and systems. Linear Systems. Luigi Palopoli. Signal and systems p. 1/5
Signal and systems p. 1/5 Signal and systems Linear Systems Luigi Palopoli palopoli@dit.unitn.it Wrap-Up Signal and systems p. 2/5 Signal and systems p. 3/5 Fourier Series We have see that is a signal
More informationLOPE3202: Communication Systems 10/18/2017 2
By Lecturer Ahmed Wael Academic Year 2017-2018 LOPE3202: Communication Systems 10/18/2017 We need tools to build any communication system. Mathematics is our premium tool to do work with signals and systems.
More informationUNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.
UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 9th, 011 Examination hours: 14.30 18.30 This problem set
More informationECGR4124 Digital Signal Processing Midterm Spring 2010
ECGR4124 Digital Signal Processing Midterm Spring 2010 Name: LAST 4 DIGITS of Student Number: Do NOT begin until told to do so Make sure that you have all pages before starting Open book, 1 sheet front/back
More informationEE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, Cover Sheet
EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, 2012 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 5,7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet Calculators
More informationFourier transform representation of CT aperiodic signals Section 4.1
Fourier transform representation of CT aperiodic signals Section 4. A large class of aperiodic CT signals can be represented by the CT Fourier transform (CTFT). The (CT) Fourier transform (or spectrum)
More informationDigital Signal Processing. Lecture Notes and Exam Questions DRAFT
Digital Signal Processing Lecture Notes and Exam Questions Convolution Sum January 31, 2006 Convolution Sum of Two Finite Sequences Consider convolution of h(n) and g(n) (M>N); y(n) = h(n), n =0... M 1
More informationMultirate signal processing
Multirate signal processing Discrete-time systems with different sampling rates at various parts of the system are called multirate systems. The need for such systems arises in many applications, including
More informationChapter 6 THE SAMPLING PROCESS 6.1 Introduction 6.2 Fourier Transform Revisited
Chapter 6 THE SAMPLING PROCESS 6.1 Introduction 6.2 Fourier Transform Revisited Copyright c 2005 Andreas Antoniou Victoria, BC, Canada Email: aantoniou@ieee.org July 14, 2018 Frame # 1 Slide # 1 A. Antoniou
More informationECE Unit 4. Realizable system used to approximate the ideal system is shown below: Figure 4.47 (b) Digital Processing of Analog Signals
ECE 8440 - Unit 4 Digital Processing of Analog Signals- - Non- Ideal Case (See sec8on 4.8) Before considering the non- ideal case, recall the ideal case: 1 Assump8ons involved in ideal case: - no aliasing
More informationTHE clock driving a digital-to-analog converter (DAC)
IEEE TRANSACTIONS ON CIRCUITS AND SYSTES II: EXPRESS BRIEFS, VOL. 57, NO. 1, JANUARY 2010 1 The Effects of Flying-Adder Clocks on Digital-to-Analog Converters Ping Gui, Senior ember, IEEE, Zheng Gao, Student
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING
GEORGIA INSIUE OF ECHNOLOGY SCHOOL of ELECRICAL and COMPUER ENGINEERING ECE 6250 Spring 207 Problem Set # his assignment is due at the beginning of class on Wednesday, January 25 Assigned: 6-Jan-7 Due
More informationMultirate Digital Signal Processing
Multirate Digital Signal Processing Basic Sampling Rate Alteration Devices Up-sampler - Used to increase the sampling rate by an integer factor Down-sampler - Used to decrease the sampling rate by an integer
More informationVII. Bandwidth Limited Time Series
VII. Bandwidth Limited Time Series To summarize the discussion up to this point: (1) In the general case of the aperiodic time series, which is infinite in time and frequency, both the time series and
More informationBridge between continuous time and discrete time signals
6 Sampling Bridge between continuous time and discrete time signals Sampling theorem complete representation of a continuous time signal by its samples Samplingandreconstruction implementcontinuous timesystems
More informationEach problem is worth 25 points, and you may solve the problems in any order.
EE 120: Signals & Systems Department of Electrical Engineering and Computer Sciences University of California, Berkeley Midterm Exam #2 April 11, 2016, 2:10-4:00pm Instructions: There are four questions
More informationSignals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters
Signals, Instruments, and Systems W5 Introduction to Signal Processing Sampling, Reconstruction, and Filters Acknowledgments Recapitulation of Key Concepts from the Last Lecture Dirac delta function (
More informationMassachusetts Institute of Technology
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing QUIZ 1, March 16, 2010 ANSWER BOOKLET
More informationpickup from external sources unwanted feedback RF interference from system or elsewhere, power supply fluctuations ground currents
Noise What is NOISE? A definition: Any unwanted signal obscuring signal to be observed two main origins EXTRINSIC NOISE examples... pickup from external sources unwanted feedback RF interference from system
More informationSistemas de Aquisição de Dados. Mestrado Integrado em Eng. Física Tecnológica 2016/17 Aula 3, 3rd September
Sistemas de Aquisição de Dados Mestrado Integrado em Eng. Física Tecnológica 2016/17 Aula 3, 3rd September The Data Converter Interface Analog Media and Transducers Signal Conditioning Signal Conditioning
More information2.1 Basic Concepts Basic operations on signals Classication of signals
Haberle³me Sistemlerine Giri³ (ELE 361) 9 Eylül 2017 TOBB Ekonomi ve Teknoloji Üniversitesi, Güz 2017-18 Dr. A. Melda Yüksel Turgut & Tolga Girici Lecture Notes Chapter 2 Signals and Linear Systems 2.1
More informationSignals and Systems Spring 2004 Lecture #9
Signals and Systems Spring 2004 Lecture #9 (3/4/04). The convolution Property of the CTFT 2. Frequency Response and LTI Systems Revisited 3. Multiplication Property and Parseval s Relation 4. The DT Fourier
More information16.362: Signals and Systems: 1.0
16.362: Signals and Systems: 1.0 Prof. K. Chandra ECE, UMASS Lowell September 1, 2016 1 Background The pre-requisites for this course are Calculus II and Differential Equations. A basic understanding of
More informationEE301 Signals and Systems Spring 2016 Exam 2 Thursday, Mar. 31, Cover Sheet
EE301 Signals and Systems Spring 2016 Exam 2 Thursday, Mar. 31, 2016 Cover Sheet Test Duration: 75 minutes. Coverage: Chapter 4, Hmwks 6-7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet Calculators
More informationThe Fourier Transform (and more )
The Fourier Transform (and more ) imrod Peleg ov. 5 Outline Introduce Fourier series and transforms Introduce Discrete Time Fourier Transforms, (DTFT) Introduce Discrete Fourier Transforms (DFT) Consider
More informationQuadrature-Mirror Filter Bank
Quadrature-Mirror Filter Bank In many applications, a discrete-time signal x[n] is split into a number of subband signals { v k [ n]} by means of an analysis filter bank The subband signals are then processed
More informationCHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME
CHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME Shri Mata Vaishno Devi University, (SMVDU), 2013 Page 13 CHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME When characterizing or modeling a random variable, estimates
More informationChapter 4 The Fourier Series and Fourier Transform
Chapter 4 The Fourier Series and Fourier Transform Fourier Series Representation of Periodic Signals Let x(t) be a CT periodic signal with period T, i.e., xt ( + T) = xt ( ), t R Example: the rectangular
More informationDigital Baseband Systems. Reference: Digital Communications John G. Proakis
Digital Baseband Systems Reference: Digital Communications John G. Proais Baseband Pulse Transmission Baseband digital signals - signals whose spectrum extend down to or near zero frequency. Model of the
More informationSlide Set Data Converters. Background Elements
0 Slide Set Data Converters Background Elements 1 Introduction Summary The Ideal Data Converter Sampling Amplitude Quantization Quantization Noise kt/c Noise Discrete and Fast Fourier Transforms The D/A
More informationMULTIRATE SYSTEMS. work load, lower filter order, lower coefficient sensitivity and noise,
MULIRAE SYSEMS ransfer signals between two systems that operate with different sample frequencies Implemented system functions more efficiently by using several sample rates (for example narrow-band filters).
More informationLecture 13: Discrete Time Fourier Transform (DTFT)
Lecture 13: Discrete Time Fourier Transform (DTFT) ECE 401: Signal and Image Analysis University of Illinois 3/9/2017 1 Sampled Systems Review 2 DTFT and Convolution 3 Inverse DTFT 4 Ideal Lowpass Filter
More informationCorrelator I. Basics. Chapter Introduction. 8.2 Digitization Sampling. D. Anish Roshi
Chapter 8 Correlator I. Basics D. Anish Roshi 8.1 Introduction A radio interferometer measures the mutual coherence function of the electric field due to a given source brightness distribution in the sky.
More informationDIGITAL SIGNAL PROCESSING. Chapter 6 IIR Filter Design
DIGITAL SIGNAL PROCESSING Chapter 6 IIR Filter Design OER Digital Signal Processing by Dr. Norizam Sulaiman work is under licensed Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
More informationAnalog to Digital Converters (ADCs)
Analog to Digital Converters (ADCs) Note: Figures are copyrighted Proakis & Manolakis, Digital Signal Processing, 4 th Edition, Pearson Publishers. Embedded System Design A Unified HW Approach, Vahid/Givargis,
More information