Signals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters

Size: px
Start display at page:

Download "Signals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters"

Transcription

1 Signals, Instruments, and Systems W5 Introduction to Signal Processing Sampling, Reconstruction, and Filters

2 Acknowledgments

3 Recapitulation of Key Concepts from the Last Lecture

4 Dirac delta function ( t) 0,, x x ( t) dt

5 Convolution f g ( t) f ( ) g( t ) d

6 Examples * =

7 From Fourier Series to Transform f(t) - an aperiodic signal - view it as the limit of a periodic signal as T For a periodic signal, the harmonic components are spaced ω 0 = 2π/T apart As T, ω 0 0, and harmonic components are spaced closer and closer in frequency

8 A Linear n-dimensional Transform F Time, space, etc. domain F -1 Frequency, spatial frequencies, etc. domain Idea: generalize to any n-dimensional signal so that a frequency representation can be found; we will focus on dimension 1 (and you will see dimension 2 in the next lab)

9 Fourier Transform i2 f ( ) f ( t) e dt t Fourier Transform i2 f ( t) f ( ) e d t Inverse Fourier Transform Note: typically in EE: ( i booked for current) 2 j i2

10 Simple Example Time [s] f ( t) cos(2 at) f a a Frequency [Hz] a a ( ) 2

11 Train of Dirac Deltas

12 Sampling

13 Sampling the Signal (t-domain)

14 Sampling the Signal (f-domain)

15 Sampling a Band-Limited Signal

16 Original Signal Time [s] f ( t) sin(2 t) 0.4sin(2 2t) 0.2sin(2 5t)

17 Too few samples (1Hz) Time [s] Data is lost

18 Too many samples (100 Hz) Redundant data Time [s] Increase of data size

19 Minimal Possible Sampling (> 10 Hz) Time [s]

20 Nyquist Shannon theorem If a function x(t) contains no frequencies higher than B Hz, it is completely determined by giving its coordinates at a series of points spaced 1/(2B) seconds apart. Sampling frequency must be at least two times greater than the maximal signal frequency

21 Sampling in Practice Sampling frequency two times greater than maximal frequency is the limit If possible, try to use a sampling frequency 10 times greater than the maximal frequency Audio CD, sampling at 44.1 khz Maximal hearable frequency: 20 khz

22 Sampling in Practice

23 Signal Reconstruction

24

25

26 Signal reconstruction Whittaker-Shannon Interpolation x( t) n x[ n] sinc t nt T s s x( t) n x[ n] ( t nt s ) sinc t T s Signal has to be band limited (i.e. Fourier transform for frequencies greater than B equal 0) The sampling rate must exceed twice the bandwidth, 2B 1 T s 2B

27

28

29 Aliasing

30 No Problems in Reconstruction

31 Reconstruction Problems Overlapping Alias

32 Harmonics Fundamental Frequency

33 Harmonics Time [s] 1Hz 2Hz 3Hz 4Hz

34 La-Tone (440 Hz) sampled at 44.1 khz (CD standard)

35 La-Tone (440 Hz) sampled at 4 khz without filtering

36 La-Tone (440Hz) sampled at 4 khz filtered at 2 khz

37 Aliasing example Original sound Aliases 4 khz Correct sampling 4 khz

38 Moiré Pattern

39 Various Transforms

40 Laplace transform F ( s) L f(t) e st f ( t) dt 0 s i

41 Fourier - Laplace F ( ) F{ f ( t)} L{ f ( t)} s i F ( s) s i e i t f ( t) dt Fourier is a special case of Laplace transform Fourier: frequency response (especially in signal processing) Laplace: impulse response (especially in control)

42 Z transform Corresponds to Laplace transform for timediscrete signals Transform signals from time-domain to frequency domain X ( z) Z{ x[ n]} x[ n] z n n z Ae j or z A(cos j sin )

43 Different Transforms Fourier Transform F( ) f ( t) e 2 i t dt Laplace Transform F( s) f ( t) e st dt, s i 0 Discrete- Time Fourier Transform X[ ] x[ n] e i n n Z Transform Z{ x[ n]} x[ n] z n, z Ae j Discrete Fourier Transform X[n] N 1 n x[ n] e 2 N i kn, k 0,, N 1 n 0 Fast Fourier Transform (FFT) is an algorithm to compute the Discrete Fourier Transform (DFT)

44 DAC ADC Transform Overview Continuous signal Time domain Fourier/Laplace Fourier Inverse Continuous signal Frequency domain Discrete signal Time domain Z/DTFT Inverse Z/DTFT Discrete signal Frequency domain

45 Filtering

46 Filtering noisy signals day/night cycle Solar radiation changing cloud cover low pass filter high pass filter

47 Decibel V1 V2 G db log 10 V V V V G 1( gain) db V V G 1( damping) db 2 1 Source of sound Sound pressure Sound pressure level pascal db re 20 μpa Jet engine at 30 m 630 Pa 150 db Rifle being fired at 1 m 200 Pa 140 db Threshold of pain 100 Pa 130 db Hearing damage (due to short-term exposure) 20 Pa approx. 120 db Jet at 100 m Pa db Jack hammer at 1 m 2 Pa approx. 100 db Hearing damage (due to long-term exposure) Pa approx. 85 db Major road at 10 m Pa db Passenger car at 10 m Pa db TV (set at home level) at 1 m Pa approx. 60 db Normal talking at 1 m Pa db Very calm room Pa db Leaves rustling, calm breathing Pa 10 db Auditory threshold at 1 khz Pa 0 db

48 Bode plot not to scale!

49 Filter design

50 Filters Analog Circuit Digital A/D Function y y f ( x x ) 1 n 1 n

51 Transfer Functions of Filters Analog Circuit Numerator Laplace Transf. Hs () v v c in 1 1 RCs Denominator Digital Function y y f ( x x ) 1 n 1 n z-transf. Numerator H( z) 1 b0 b1z bn z 1 1 a1z bm z N M Denominator

52 Bode Plot - Rules Zero (numerator = 0) Amplitude: 20 db/decade Phase: 90, 45 /decade, starting 1 decade before zero Pole (denominator = 0) Amplitude: -20 db/decade Phase: -90, -45 /decade, starting 1 decade before pole

53 Bode plot (magnitude) Zero (numerator = 0) Amplitude: 20 db/decade Pole (denominator = 0) Amplitude: -20 db/decade

54 Bode plot (phase) Zero (numerator = 0): 90, 45 /decade, starting 1 decade before zero Pole (denominator = 0): -90, -45 /decade, starting 1 decade before pole

55 Filter Order and Type 1 st order is equivalent to 20dB per decade Each successive order adds 20dB per decade Filter with a high order are closer to the ideal filter (rectangular function) Several filters exists and are defined by the polynomes at the numerator/denominator (Finite Impulse Response, Bessel, Butterworth, Tschebishev, etc.)

56 Analog Filter Order Filter order: 3 ++ faster cutoff -- more components -- higher power consumption Digital y[ n] a x[ n] 0 y[ n] a x[ n] a x[ n 1] 0 1 y[ n] a x[ n] a x[ n 1] a x[ n 2] Filter order: 1 Filter order: 2 Filter order: 3 ++ faster cutoff -- more computation -- higher power consumption

57 Bode plot First order Low Pass Filter

58 Low Pass Filter - RC circuit

59 High Pass Filter RC circuit

60 Bode plot first order high pass filter

61 Digital Low-Pass Filter

62 Digital High-Pass Filter

63 Digital Filter Transfer function: H( z) ( z 1) z 2z z z Y ( z) X( z) ( z )( z ) z z 1 z z Difference equation: N y[ n] a y[ n k] b x[ n k] k M k 1 k y[ n] x[ n] 2 x[ n 1] x[ n 2] y[ n 1] y[ n 2] 4 8 x General case

64 Conclusion

65 Take-Home Messages A number of n-dimensional linear transforms allow for moving from the original domain (e.g. time, space to) to the frequency domain and back Time-continuous and time-discrete versions exist A number of operations are easier to carry out and understand in the frequency domain Signals can be sampled and reconstructed properly if fundamental limits are respected Filters allow a number of operations (e.g., noise removal, contrast enhancement, etc.) and are often easier to design in the frequency domain

66 Additional Reading Books Ronald W. Schafer and James H. McClellan DSP First: A Multimedia Approach, 1998 A. Oppenheim and A. S. Willsky with S. Hamid, Signals and Systems, Prentice Hall, 1996.

ELEN 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 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 information

Analog 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 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 information

Signal Processing COS 323

Signal Processing COS 323 Signal Processing COS 323 Digital Signals D: functions of space or time e.g., sound 2D: often functions of 2 spatial dimensions e.g. images 3D: functions of 3 spatial dimensions CAT, MRI scans or 2 space,

More information

Chapter 5 Frequency Domain Analysis of Systems

Chapter 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 information

Chapter 5 Frequency Domain Analysis of Systems

Chapter 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 information

EE Homework 13 - Solutions

EE Homework 13 - Solutions EE3054 - Homework 3 - Solutions. (a) The Laplace transform of e t u(t) is s+. The pole of the Laplace transform is at which lies in the left half plane. Hence, the Fourier transform is simply the Laplace

More information

Tutorial Sheet #2 discrete vs. continuous functions, periodicity, sampling

Tutorial Sheet #2 discrete vs. continuous functions, periodicity, sampling 2.39 utorial Sheet #2 discrete vs. continuous functions, periodicity, sampling We will encounter two classes of signals in this class, continuous-signals and discrete-signals. he distinct mathematical

More information

E : Lecture 1 Introduction

E : Lecture 1 Introduction E85.2607: Lecture 1 Introduction 1 Administrivia 2 DSP review 3 Fun with Matlab E85.2607: Lecture 1 Introduction 2010-01-21 1 / 24 Course overview Advanced Digital Signal Theory Design, analysis, and implementation

More information

Fourier Transforms For additional information, see the classic book The Fourier Transform and its Applications by Ronald N. Bracewell (which is on the shelves of most radio astronomers) and the Wikipedia

More information

Topic 3: Fourier Series (FS)

Topic 3: Fourier Series (FS) ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties

More information

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010

DEPARTMENT 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 information

Review of Discrete-Time System

Review 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 information

IB Paper 6: Signal and Data Analysis

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 information

ECE 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 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 information

Sensors. Chapter Signal Conditioning

Sensors. 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 information

ETSF15 Analog/Digital. Stefan Höst

ETSF15 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 information

Experimental Fourier Transforms

Experimental Fourier Transforms Chapter 5 Experimental Fourier Transforms 5.1 Sampling and Aliasing Given x(t), we observe only sampled data x s (t) = x(t)s(t; T s ) (Fig. 5.1), where s is called sampling or comb function and can be

More information

Radar Systems Engineering Lecture 3 Review of Signals, Systems and Digital Signal Processing

Radar Systems Engineering Lecture 3 Review of Signals, Systems and Digital Signal Processing Radar Systems Engineering Lecture Review of Signals, Systems and Digital Signal Processing Dr. Robert M. O Donnell Guest Lecturer Radar Systems Course Review Signals, Systems & DSP // Block Diagram of

More information

CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals

CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 6, 2005 1 Sound Sound waves are longitudinal

More information

Introduction to Digital Signal Processing

Introduction to Digital Signal Processing Introduction to Digital Signal Processing 1.1 What is DSP? DSP is a technique of performing the mathematical operations on the signals in digital domain. As real time signals are analog in nature we need

More information

Grades will be determined by the correctness of your answers (explanations are not required).

Grades 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 information

SEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis

SEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some

More information

MEDE2500 Tutorial Nov-7

MEDE2500 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 information

Image Acquisition and Sampling Theory

Image Acquisition and Sampling Theory Image Acquisition and Sampling Theory Electromagnetic Spectrum The wavelength required to see an object must be the same size of smaller than the object 2 Image Sensors 3 Sensor Strips 4 Digital Image

More information

E2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)

E2.5 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 information

Lecture 5. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)

Lecture 5. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith) Lecture 5 The Digital Fourier Transform (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith) 1 -. 8 -. 6 -. 4 -. 2-1 -. 8 -. 6 -. 4 -. 2 -. 2. 4. 6. 8 1

More information

Grades will be determined by the correctness of your answers (explanations are not required).

Grades 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 information

Digital Image Processing

Digital Image Processing Digital Image Processing Part 3: Fourier Transform and Filtering in the Frequency Domain AASS Learning Systems Lab, Dep. Teknik Room T109 (Fr, 11-1 o'clock) achim.lilienthal@oru.se Course Book Chapter

More information

Review: Continuous Fourier Transform

Review: 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 information

Frequency Response and Continuous-time Fourier Series

Frequency Response and Continuous-time Fourier Series Frequency Response and Continuous-time Fourier Series Recall course objectives Main Course Objective: Fundamentals of systems/signals interaction (we d like to understand how systems transform or affect

More information

Homework 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 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 information

Chap 4. Sampling of Continuous-Time Signals

Chap 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 information

Lecture 8 - IIR Filters (II)

Lecture 8 - IIR Filters (II) Lecture 8 - IIR Filters (II) James Barnes (James.Barnes@colostate.edu) Spring 2009 Colorado State University Dept of Electrical and Computer Engineering ECE423 1 / 27 Lecture 8 Outline Introduction Digital

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Lecture 19 Practical ADC/DAC Ideal Anti-Aliasing ADC A/D x c (t) Analog Anti-Aliasing Filter HLP(jΩ) sampler t = nt x[n] =x c (nt ) Quantizer 1 X c (j ) and s < 2 1 T X

More information

EE482: Digital Signal Processing Applications

EE482: 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 information

Chapter 7: IIR Filter Design Techniques

Chapter 7: IIR Filter Design Techniques IUST-EE Chapter 7: IIR Filter Design Techniques Contents Performance Specifications Pole-Zero Placement Method Impulse Invariant Method Bilinear Transformation Classical Analog Filters DSP-Shokouhi Advantages

More information

ESS Finite Impulse Response Filters and the Z-transform

ESS Finite Impulse Response Filters and the Z-transform 9. Finite Impulse Response Filters and the Z-transform We are going to have two lectures on filters you can find much more material in Bob Crosson s notes. In the first lecture we will focus on some of

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE23 Digital Signal Processing Lecture 7B Sampling What is this Phenomena? https://www.youtube.com/watch?v=cxddi8m_mzk Sampling of Continuous ime Signals (Ch.4) Sampling: Conversion from C. (not quantized)

More information

Homework: 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, 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 information

EE 225D LECTURE ON DIGITAL FILTERS. University of California Berkeley

EE 225D LECTURE ON DIGITAL FILTERS. University of California Berkeley University of California Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences Professors : N.Morgan / B.Gold EE225D Digital Filters Spring,1999 Lecture 7 N.MORGAN

More information

Cast of Characters. Some Symbols, Functions, and Variables Used in the Book

Cast of Characters. Some Symbols, Functions, and Variables Used in the Book Page 1 of 6 Cast of Characters Some s, Functions, and Variables Used in the Book Digital Signal Processing and the Microcontroller by Dale Grover and John R. Deller ISBN 0-13-081348-6 Prentice Hall, 1998

More information

INTRODUCTION TO DELTA-SIGMA ADCS

INTRODUCTION TO DELTA-SIGMA ADCS ECE37 Advanced Analog Circuits INTRODUCTION TO DELTA-SIGMA ADCS Richard Schreier richard.schreier@analog.com NLCOTD: Level Translator VDD > VDD2, e.g. 3-V logic? -V logic VDD < VDD2, e.g. -V logic? 3-V

More information

CITY UNIVERSITY LONDON. MSc in Information Engineering DIGITAL SIGNAL PROCESSING EPM746

CITY 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 information

Filter Analysis and Design

Filter 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 information

Analog to Digital Converters (ADCs)

Analog 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

ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform

ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform Department of Electrical Engineering University of Arkansas ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Introduction Fourier Transform Properties of Fourier

More information

Poles and Zeros in z-plane

Poles and Zeros in z-plane M58 Mixed Signal Processors page of 6 Poles and Zeros in z-plane z-plane Response of discrete-time system (i.e. digital filter at a particular frequency ω is determined by the distance between its poles

More information

EE 521: Instrumentation and Measurements

EE 521: Instrumentation and Measurements Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA September 23, 2009 1 / 18 1 Sampling 2 Quantization 3 Digital-to-Analog Converter 4 Analog-to-Digital Converter

More information

Various signal sampling and reconstruction methods

Various signal sampling and reconstruction methods Various signal sampling and reconstruction methods Rolands Shavelis, Modris Greitans 14 Dzerbenes str., Riga LV-1006, Latvia Contents Classical uniform sampling and reconstruction Advanced sampling and

More information

Correlator I. Basics. Chapter Introduction. 8.2 Digitization Sampling. D. Anish Roshi

Correlator 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 information

Multimedia Signals and Systems - Audio and Video. Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2

Multimedia Signals and Systems - Audio and Video. Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2 Multimedia Signals and Systems - Audio and Video Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2 Kunio Takaya Electrical and Computer Engineering University of Saskatchewan December

More information

Basic Electronics. Introductory Lecture Course for. Technology and Instrumentation in Particle Physics Chicago, Illinois June 9-14, 2011

Basic Electronics. Introductory Lecture Course for. Technology and Instrumentation in Particle Physics Chicago, Illinois June 9-14, 2011 Basic Electronics Introductory Lecture Course for Technology and Instrumentation in Particle Physics 2011 Chicago, Illinois June 9-14, 2011 Presented By Gary Drake Argonne National Laboratory Session 2

More information

6.003: Signals and Systems. Sampling and Quantization

6.003: 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 information

IIR digital filter design for low pass filter based on impulse invariance and bilinear transformation methods using butterworth analog filter

IIR digital filter design for low pass filter based on impulse invariance and bilinear transformation methods using butterworth analog filter IIR digital filter design for low pass filter based on impulse invariance and bilinear transformation methods using butterworth analog filter Nasser M. Abbasi May 5, 0 compiled on hursday January, 07 at

More information

2A1H Time-Frequency Analysis II

2A1H 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 information

Signals & 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. 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 information

DESIGN OF CMOS ANALOG INTEGRATED CIRCUITS

DESIGN 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 information

Sistemas 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 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 information

EE 224 Signals and Systems I Review 1/10

EE 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 information

SAMPLE EXAMINATION PAPER (with numerical answers)

SAMPLE EXAMINATION PAPER (with numerical answers) CID No: IMPERIAL COLLEGE LONDON Design Engineering MEng EXAMINATIONS For Internal Students of the Imperial College of Science, Technology and Medicine This paper is also taken for the relevant examination

More information

Each problem is worth 25 points, and you may solve the problems in any order.

Each 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 information

An Fir-Filter Example: Hanning Filter

An Fir-Filter Example: Hanning Filter An Fir-Filter Example: Hanning Filter Josef Goette Bern University of Applied Sciences, Biel Institute of Human Centered Engineering - microlab Josef.Goette@bfh.ch February 7, 2018 Contents 1 Mathematical

More information

Active Control? Contact : Website : Teaching

Active Control? Contact : Website :   Teaching Active Control? Contact : bmokrani@ulb.ac.be Website : http://scmero.ulb.ac.be Teaching Active Control? Disturbances System Measurement Control Controler. Regulator.,,, Aims of an Active Control Disturbances

More information

Fourier transform. Stefano Ferrari. Università degli Studi di Milano Methods for Image Processing. academic year

Fourier transform. Stefano Ferrari. Università degli Studi di Milano Methods for Image Processing. academic year Fourier transform Stefano Ferrari Università degli Studi di Milano stefano.ferrari@unimi.it Methods for Image Processing academic year 27 28 Function transforms Sometimes, operating on a class of functions

More information

The (Fast) Fourier Transform

The (Fast) Fourier Transform The (Fast) Fourier Transform The Fourier transform (FT) is the analog, for non-periodic functions, of the Fourier series for periodic functions can be considered as a Fourier series in the limit that the

More information

Chapter 2: Problem Solutions

Chapter 2: Problem Solutions Chapter 2: Problem Solutions Discrete Time Processing of Continuous Time Signals Sampling à Problem 2.1. Problem: Consider a sinusoidal signal and let us sample it at a frequency F s 2kHz. xt 3cos1000t

More information

DCSP-2: Fourier Transform

DCSP-2: Fourier Transform DCSP-2: Fourier Transform Jianfeng Feng Department of Computer Science Warwick Univ., UK Jianfeng.feng@warwick.ac.uk http://www.dcs.warwick.ac.uk/~feng/dcsp.html Data transmission Channel characteristics,

More information

Fourier Analysis Overview (0B)

Fourier Analysis Overview (0B) CTFS: Continuous Time Fourier Series CTFT: Continuous Time Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2009-2016 Young W. Lim. Permission

More information

Question Bank. UNIT 1 Part-A

Question Bank. UNIT 1 Part-A FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY Senkottai Village, Madurai Sivagangai Main Road, Madurai -625 020 An ISO 9001:2008 Certified Institution Question Bank DEPARTMENT OF ELECTRONICS AND COMMUNICATION

More information

!Sketch f(t) over one period. Show that the Fourier Series for f(t) is as given below. What is θ 1?

!Sketch f(t) over one period. Show that the Fourier Series for f(t) is as given below. What is θ 1? Second Year Engineering Mathematics Laboratory Michaelmas Term 998 -M L G Oldfield 30 September, 999 Exercise : Fourier Series & Transforms Revision 4 Answer all parts of Section A and B which are marked

More information

Bridge between continuous time and discrete time signals

Bridge 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 information

Chapter 3 Data Acquisition and Manipulation

Chapter 3 Data Acquisition and Manipulation 1 Chapter 3 Data Acquisition and Manipulation In this chapter we introduce z transf orm, or the discrete Laplace Transform, to solve linear recursions. Section 3.1 z-transform Given a data stream x {x

More information

ω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the

ω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the he ime-frequency Concept []. Review of Fourier Series Consider the following set of time functions {3A sin t, A sin t}. We can represent these functions in different ways by plotting the amplitude versus

More information

NAME: 11 December 2013 Digital Signal Processing I Final Exam Fall Cover Sheet

NAME: 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 information

Distortion Analysis T

Distortion Analysis T EE 435 Lecture 32 Spectral Performance Windowing Spectral Performance of Data Converters - Time Quantization - Amplitude Quantization Quantization Noise . Review from last lecture. Distortion Analysis

More information

Lecture 8 - IIR Filters (II)

Lecture 8 - IIR Filters (II) Lecture 8 - IIR Filters (II) James Barnes (James.Barnes@colostate.edu) Spring 24 Colorado State University Dept of Electrical and Computer Engineering ECE423 1 / 29 Lecture 8 Outline Introduction Digital

More information

2.161 Signal Processing: Continuous and Discrete Fall 2008

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

More information

ECE 413 Digital Signal Processing Midterm Exam, Spring Instructions:

ECE 413 Digital Signal Processing Midterm Exam, Spring Instructions: University of Waterloo Department of Electrical and Computer Engineering ECE 4 Digital Signal Processing Midterm Exam, Spring 00 June 0th, 00, 5:0-6:50 PM Instructor: Dr. Oleg Michailovich Student s name:

More information

Recursive Gaussian filters

Recursive Gaussian filters CWP-546 Recursive Gaussian filters Dave Hale Center for Wave Phenomena, Colorado School of Mines, Golden CO 80401, USA ABSTRACT Gaussian or Gaussian derivative filtering is in several ways optimal for

More information

FROM ANALOGUE TO DIGITAL

FROM ANALOGUE TO DIGITAL SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 7. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.mee.tcd.ie/ corrigad FROM ANALOGUE TO DIGITAL To digitize signals it is necessary

More information

A523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2011

A523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2011 A523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2011 Lecture 6 PDFs for Lecture 1-5 are on the web page Problem set 2 is on the web page Article on web page A Guided

More information

Digital Signal Processing. Lecture Notes and Exam Questions DRAFT

Digital 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 information

Chapter 1 Fundamental Concepts

Chapter 1 Fundamental Concepts Chapter 1 Fundamental Concepts Signals A signal is a pattern of variation of a physical quantity as a function of time, space, distance, position, temperature, pressure, etc. These quantities are usually

More information

2.161 Signal Processing: Continuous and Discrete Fall 2008

2.161 Signal Processing: Continuous and Discrete Fall 2008 MI OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 008 For information about citing these materials or our erms of Use, visit: http://ocw.mit.edu/terms. Massachusetts

More information

BME 50500: Image and Signal Processing in Biomedicine. Lecture 2: Discrete Fourier Transform CCNY

BME 50500: Image and Signal Processing in Biomedicine. Lecture 2: Discrete Fourier Transform CCNY 1 Lucas Parra, CCNY BME 50500: Image and Signal Processing in Biomedicine Lecture 2: Discrete Fourier Transform Lucas C. Parra Biomedical Engineering Department CCNY http://bme.ccny.cuny.edu/faculty/parra/teaching/signal-and-image/

More information

Overview of Sampling Topics

Overview 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 information

GEORGIA 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. 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 information

8/19/16. Fourier Analysis. Fourier analysis: the dial tone phone. Fourier analysis: the dial tone phone

8/19/16. Fourier Analysis. Fourier analysis: the dial tone phone. Fourier analysis: the dial tone phone Patrice Koehl Department of Biological Sciences National University of Singapore http://www.cs.ucdavis.edu/~koehl/teaching/bl5229 koehl@cs.ucdavis.edu Fourier analysis: the dial tone phone We use Fourier

More information

EE 435. Lecture 32. Spectral Performance Windowing

EE 435. Lecture 32. Spectral Performance Windowing EE 435 Lecture 32 Spectral Performance Windowing . Review from last lecture. Distortion Analysis T 0 T S THEOREM?: If N P is an integer and x(t) is band limited to f MAX, then 2 Am Χ mnp 1 0 m h N and

More information

200Pa 10million. Overview. Acoustics of Speech and Hearing. Loudness. Terms to describe sound. Matching Pressure to Loudness. Loudness vs.

200Pa 10million. Overview. Acoustics of Speech and Hearing. Loudness. Terms to describe sound. Matching Pressure to Loudness. Loudness vs. Overview Acoustics of Speech and Hearing Lecture 1-2 How is sound pressure and loudness related? How can we measure the size (quantity) of a sound? The scale Logarithmic scales in general Decibel scales

More information

Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: Student ID number... Signature...

Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: Student ID number... Signature... Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: 29..23 Given and family names......................solutions...................... Student ID number..........................

More information

Lecture 3 - Design of Digital Filters

Lecture 3 - Design of Digital Filters Lecture 3 - Design of Digital Filters 3.1 Simple filters In the previous lecture we considered the polynomial fit as a case example of designing a smoothing filter. The approximation to an ideal LPF can

More information

EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 20, Cover Sheet

EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 20, Cover Sheet NAME: NAME EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 20, 2017 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 5,7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet

More information

Fourier Analysis. David-Alexander Robinson ; Daniel Tanner; Jack Denning th October Abstract 2. 2 Introduction & Theory 2

Fourier Analysis. David-Alexander Robinson ; Daniel Tanner; Jack Denning th October Abstract 2. 2 Introduction & Theory 2 Fourier Analysis David-Alexander Robinson ; Daniel Tanner; Jack Denning 08332461 15th October 2009 Contents 1 Abstract 2 2 Introduction & Theory 2 3 Experimental Method 2 3.1 Experiment 1...........................

More information

Review of Fourier Transform

Review of Fourier Transform Review of Fourier Transform Fourier series works for periodic signals only. What s about aperiodic signals? This is very large & important class of signals Aperiodic signal can be considered as periodic

More information

Lecture Schedule: Week Date Lecture Title

Lecture Schedule: Week Date Lecture Title http://elec34.org Sampling and CONVOLUTION 24 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date Lecture Title 2-Mar Introduction 3-Mar

More information

The Laplace Transform

The Laplace Transform The Laplace Transform Syllabus ECE 316, Spring 2015 Final Grades Homework (6 problems per week): 25% Exams (midterm and final): 50% (25:25) Random Quiz: 25% Textbook M. Roberts, Signals and Systems, 2nd

More information

Fourier Series Representation of

Fourier Series Representation of Fourier Series Representation of Periodic Signals Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline The response of LIT system

More information

Digital Signal Processing

Digital Signal Processing Digital Signal Processing Introduction Moslem Amiri, Václav Přenosil Embedded Systems Laboratory Faculty of Informatics, Masaryk University Brno, Czech Republic amiri@mail.muni.cz prenosil@fi.muni.cz February

More information

J. McNames Portland State University ECE 223 Sampling Ver

J. 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 information