Lecture 8 ELE 301: Signals and Systems


 Brett Bridges
 3 years ago
 Views:
Transcription
1 Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 22 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 37 Properties of the Fourier Transform Properties of the Fourier Transform Linearity Timeshift Time Scaling Conjugation Duality Parseval Convolution and Modulation Periodic Signals ConstantCoefficient Differential Equations Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
2 Linearity Linear combination of two signals x (t) and x 2 (t) is a signal of the form ax (t) + bx 2 (t). Linearity Theorem: The Fourier transform is linear; that is, given two signals x (t) and x 2 (t) and two complex numbers a and b, then ax (t) + bx 2 (t) ax (jω) + bx 2 (jω). This follows from linearity of integrals: (ax (t) + bx 2 (t))e j2πft dt = a x (t)e j2πft dt + b x 2 (t)e j2πft dt = ax (f ) + bx 2 (f ) Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Finite Sums This easily extends to finite combinations. Given signals x k (t) with Fourier transforms X k (f ) and complex constants a k, k =, 2,... K, then K K a k x k (t) a k X k (f ). k= If you consider a system which has a signal x(t) as its input and the Fourier transform X (f ) as its output, the system is linear! k= Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
3 Linearity Example Find the Fourier transform of the signal This signal can be recognized as { x(t) = 2 2 t < t 2 x(t) = 2 rect ( t 2 ) + rect (t) 2 and hence from linearity we have ( ) X (f ) = 2 sinc(2f ) sinc(f ) = sinc(2f ) + 2 sinc(f ) Cuff (Lecture 7) ELE 3: Signals and Systems Fall / rect(t/2) + 2 rect(t).4.2!.2!2.5!2!.5!! sinc(ω/π) + 2 sinc(ω/(2π)).5!.5!!8!6!4! π 2π 2π ω 4π L Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
4 Scaling Theorem Stretch (Scaling) Theorem: Given a transform pair x(t) X (f ), and a realvalued nonzero constant a, x(at) ( ) f a X a Proof: Here consider only a >. (negative a left as an exercise) Change variables τ = at x(at)e j2πft j2πf τ/a dτ dt = x(τ)e a = ( ) f a X. a If a = time reversal theorem: X ( t) X ( f ) Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Scaling Examples We have already seen that rect(t/t ) T sinc(tf ) by brute force integration. The scaling theorem provides a shortcut proof given the simpler result rect(t) sinc(f ). This is a good point to illustrate a property of transform pairs. Consider this Fourier transform pair for a small T and large T, say T = and T = 5. The resulting transform pairs are shown below to a common horizontal scale: Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
5 Compress in time  Expand in frequency.2.8 rect(t) 6 4 sinc(ω/2π) !.2!2! t!2!!5 5 π 5π 5π π ω rect(t/5) !.2!2! t! 5sinc(5ω/2π)!2 π! 5π!5 5 5π π ω Cuff (Lecture 7) ELE 3: Signals and Narrower pulse means higher bandwidth. Systems Fall / 37 Scaling Example 2 As another example, find the transform of the timereversed exponential x(t) = e at u( t). This is the exponential signal y(t) = e at u(t) with time scaled by , so the Fourier transform is X (f ) = Y ( f ) = a j2πf. Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 37
6 Scaling Example 3 As a final example which brings two Fourier theorems into use, find the transform of x(t) = e a t. This signal can be written as e at u(t) + e at u( t). Linearity and timereversal yield X (f ) = = = Much easier than direct integration! a + j2πf + a j2πf 2a a 2 (j2πf ) 2 2a a 2 + (2πf ) 2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 37 Complex Conjugation Theorem Complex Conjugation Theorem: If x(t) X (f ), then x (t) X ( f ) Proof: The Fourier transform of x (t) is x (t)e j2πft dt = = ( ) x(t)e j2πft dt ( x(t)e dt) ( j2πf )t = X ( f ) Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
7 Duality Theorem We discussed duality in a previous lecture. Duality Theorem: If x(t) X (f ), then X (t) x( f ). This result effectively gives us two transform pairs for every transform we find. Exercise What signal x(t) has a Fourier transform e f? Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Shift Theorem The Shift Theorem: x(t τ) e j2πf τ X (f ) Proof: Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
8 Example: square pulse Consider a causal square pulse p(t) = for t [, T ) and otherwise. We can write this as ( ) t T 2 p(t) = rect T From shift and scaling theorems P(f ) = Te jπft sinc(tf ). Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 The Derivative Theorem The Derivative Theorem: Given a signal x(t) that is differentiable almost everywhere with Fourier transform X (f ), x (t) j2πfx (f ) Similarly, if x(t) is n times differentiable, then d n x(t) dt n (j2πf ) n X (f ) Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
9 Dual Derivative Formula There is a dual to the derivative theorem, i.e., a result interchanging the role of t and f. Multiplying a signal by t is related to differentiating the spectrum with respect to f. ( j2πt)x(t) X (f ) Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 The Integral Theorem Recall that we can represent integration by a convolution with a unit step t x(τ)dτ = (x u)(t). Using the Fourier transform of the unit step function we can solve for the Fourier transform of the integral using the convolution theorem, [ t ] F x(τ)dτ = F [x(t)] F [u(t)] ( = X (f ) 2 δ(f ) + ) j2πf = X () X (f ) δ(f ) + 2 j2πf. Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
10 Fourier Transform of the Unit Step Function How do we know the derivative of the unit step function? The unit step function does not converge under the Fourier transform. But just as we use the delta function to accommodate periodic signals, we can handle the unit step function with some sleightofhand. Use the approximation that u(t) e at u(t) for small a. Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 A symmetric construction for approximating u(t) Example: Find the Fourier transform of the signum or sign signal t > f (t) = sgn(t) = t =. t < We can approximate f (t) by the signal as a. f a (t) = e at u(t) e at u( t) Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
11 This looks like.5.5 sgn(t) e t/5 e t!.5!!.5!2!.5!! t As a, f a (t) sgn(t). The Fourier transform of f a (t) is F a (f ) = F [f a (t)] = F [ e at u(t) e at u( t) ] = F [ e at u(t) ] F [ e at u( t) ] = a + j2πf a j2πf j4πf = a 2 + (2πf ) 2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Therefore, lim a(f ) a = j4πf lim a a 2 + (2πf ) 2 = j4πf (2πf ) 2 = jπf. This suggests we define the Fourier transform of sgn(t) as { 2 sgn(t) j2πf f f =. Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
12 With this, we can find the Fourier transform of the unit step, as can be seen from the plots u(t) = sgn(t) sgn(t) u(t) t t The Fourier transform of the unit step is then [ F [u(t)] = F 2 + ] 2 sgn(t) = 2 δ(f ) + ( ). 2 jπf Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 The transform pair is then u(t) 2 δ(f ) + j2πf. πδ(ω) + jω π jω ω Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
13 Parseval s Theorem (Parseval proved for Fourier series, Rayleigh for Fourier transforms. Also called Plancherel s theorem) Recall signal energy of x(t) is E x = x(t) 2 dt Interpretation: energy dissipated in a one ohm resistor if x(t) is a voltage. Can also be viewed as a measure of the size of a signal. Theorem: E x = x(t) 2 dt = X (f ) 2 df Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Example of Parseval s Theorem Parseval s theorem provides many simple integral evaluations. For example, evaluate sinc 2 (t) dt We have seen that sinc(t) rect(f ). Parseval s theorem yields sinc 2 (t) dt = rect 2 (f ) df /2 = df /2 =. Try to evaluate this integral directly and you will appreciate Parseval s shortcut. Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
14 The Convolution Theorem Convolution in the time domain multiplication in the frequency domain This can simplify evaluating convolutions, especially when cascaded. This is how most simulation programs (e.g., Matlab) compute convolutions, using the FFT. The Convolution Theorem: Given two signals x (t) and x 2 (t) with Fourier transforms X (f ) and X 2 (f ), (x x 2 )(t) X (f )X 2 (f ) Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Proof: The Fourier transform of (x x 2 )(t) is x (τ)x 2 (t τ) dτ e j2πft dt = x (τ) x 2 (t τ)e j2πft dt dτ. Using the shift theorem, this is = ( ) x (τ) e j2πf τ X 2 (f ) dτ = X 2 (f ) x (τ)e j2πf τ dτ = X 2 (f )X (f ). Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
15 Examples of Convolution Theorem Unit Triangle Signal (t) { t if t < otherwise. Δ(t)  t Easy to show (t) = rect(t) rect(t). Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Since then rect(t) sinc(f ) (t) sinc 2 (f ) sinc 2 (ω/2π).5..5!!8!6!4! π 2π 2π 4π ω Transform Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
16 Multiplication Property If x (t) X (f ) and x 2 (t) X 2 (f ), x (t)x 2 (t) (X X 2 )(f ). This is the dual property of the convolution property. Note: If ω is used instead of f, then a /2π term must be included. Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Multiplication Example  Bandpass Filter A bandpass filter can be implemented using a lowpass filter and multiplication by a complex exponential. Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
17 Modulation The Modulation Theorem: Given a signal x(t) with spectrum x(f ), then x(t)e j2πft X (f f ), x(t) cos(2πf t) 2 (X (f f ) + X (f + f )), x(t) sin(2πf t) 2j (X (f f ) X (f + f )). Modulating a signal by an exponential shifts the spectrum in the frequency domain. This is a dual to the shift theorem. It results from interchanging the roles of t and f. Modulation by a cosine causes replicas of X (f ) to be placed at plus and minus the carrier frequency. Replicas are called sidebands. Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Amplitude Modulation (AM) Modulation of complex exponential (carrier) by signal x(t): x m (t) = x(t)e j2πft Variations: f c (t) = f (t) cos(ω t) (DSBSC) f s (t) = f (t) sin(ω t) (DSBSC) f a (t) = A[ + mf (t)] cos(ω t) (DSB, commercial AM radio) m is the modulation index Typically m and f (t) are chosen so that mf (t) < for all t Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
18 Examples of Modulation Theorem rect(t) sinc(ω/2π)!.2!2 2! 2.5!.5!!2 2! 2 t t rect(t)cos(πt)!.2!2! 2 2π π π 2π ω !.2!2! 2 2π π ( ) ω ω π 2 sinc 2π π 2π ) + 2 sinc ( ω + π 2π Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37 Periodic Signals Suppose x(t) is periodic with fundamental period T and frequency f = /T. Then the Fourier series representation is, x(t) = k= a k e j2πkft. Let s substitute in some δ functions using the sifting property: x(t) = = a k δ(f kf )e j2πft df, k= ( ) a k δ(f kf ) e j2πft df. k= This implies the Fourier transform: x(t) k= a kδ(f kf ). Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
19 ConstantCoefficient Differential Equations n k= a k d k y(t) dt k = M k= b k d k x(t) dt k. Find the Fourier Transform of the impulse response (the transfer function of the system, H(f )) in the frequency domain. Cuff (Lecture 7) ELE 3: Signals and Systems Fall / 37
EE 179 April 21, 2014 Digital and Analog Communication Systems Handout #16 Homework #2 Solutions
EE 79 April, 04 Digital and Analog Communication Systems Handout #6 Homework # Solutions. Operations on signals (Lathi& Ding.33). For the signal g(t) shown below, sketch: a. g(t 4); b. g(t/.5); c. g(t
More informationReview of Fourier series formulas. Representation of nonperiodic functions. ECE 3640 Lecture 5 Fourier Transforms and their properties
ECE 3640 Lecture 5 Fourier Transforms and their properties Objective: To learn about Fourier transforms, which are a representation of nonperiodic functions in terms of trigonometric functions. Also, to
More informationLecture 7 ELE 301: Signals and Systems
Lecture 7 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 22 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 22 Introduction to Fourier Transforms Fourier transform as a limit
More information9 Fourier Transform Properties
9 Fourier Transform Properties The Fourier transform is a major cornerstone in the analysis and representation of signals and linear, timeinvariant systems, and its elegance and importance cannot be overemphasized.
More informationTopic 4: ContinuousTime Fourier Transform (CTFT)
ELEC264: Signals And Systems Topic 4: ContinuousTime Fourier Transform (CTFT) Aishy Amer Concordia University Electrical and Computer Engineering o Introduction to Fourier Transform o Fourier transform
More informationFrequency Response and Continuoustime Fourier Transform
Frequency Response and Continuoustime Fourier Transform Goals Signals and Systems in the FDpart II I. (Finiteenergy) signals in the Frequency Domain  The Fourier Transform of a signal  Classification
More informationTCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS
TCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More informationL6: Shorttime Fourier analysis and synthesis
L6: Shorttime Fourier analysis and synthesis Overview Analysis: Fouriertransform view Analysis: filtering view Synthesis: filter bank summation (FBS) method Synthesis: overlapadd (OLA) method STFT magnitude
More information2 Background: Fourier Series Analysis and Synthesis
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Spring 2001 Lab #11: Design with Fourier Series Date: 3 6 April 2001 This is the official Lab #11 description. The
More informationAngle Modulation, II. Lecture topics FM bandwidth and Carson s rule. Spectral analysis of FM. Narrowband FM Modulation. Wideband FM Modulation
Angle Modulation, II EE 179, Lecture 12, Handout #19 Lecture topics FM bandwidth and Carson s rule Spectral analysis of FM Narrowband FM Modulation Wideband FM Modulation EE 179, April 25, 2014 Lecture
More information5 Signal Design for Bandlimited Channels
225 5 Signal Design for Bandlimited Channels So far, we have not imposed any bandwidth constraints on the transmitted passband signal, or equivalently, on the transmitted baseband signal s b (t) I[k]g
More informationChapter 8  Power Density Spectrum
EE385 Class Notes 8/8/03 John Stensby Chapter 8  Power Density Spectrum Let X(t) be a WSS random process. X(t) has an average power, given in watts, of E[X(t) ], a constant. his total average power is
More informationFourier Transform and Convolution
CHAPTER 3 Fourier Transform and Convolution 3.1 INTRODUCTION In this chapter, both the Fourier transform and Fourier series will be discussed. The properties of the Fourier transform will be presented
More informationFrequency Domain Characterization of Signals. Yao Wang Polytechnic University, Brooklyn, NY11201 http: //eeweb.poly.edu/~yao
Frequency Domain Characterization of Signals Yao Wang Polytechnic University, Brooklyn, NY1121 http: //eeweb.poly.edu/~yao Signal Representation What is a signal Timedomain description Waveform representation
More informationSampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.
Sampling Theorem We will show that a band limited signal can be reconstructed exactly from its discrete time samples. Recall: That a time sampled signal is like taking a snap shot or picture of signal
More informationFrequencyDomain Analysis: the Discrete Fourier Series and the Fourier Transform
FrequencyDomain Analysis: the Discrete Fourier Series and the Fourier Transform John Chiverton School of Information Technology Mae Fah Luang University 1st Semester 2009/ 2552 Outline Overview Lecture
More informationLecture 18: The TimeBandwidth Product
WAVELETS AND MULTIRATE DIGITAL SIGNAL PROCESSING Lecture 18: The TimeBandwih Product Prof.Prof.V.M.Gadre, EE, IIT Bombay 1 Introduction In this lecture, our aim is to define the time Bandwih Product,
More information1.1 DiscreteTime Fourier Transform
1.1 DiscreteTime Fourier Transform The discretetime Fourier transform has essentially the same properties as the continuoustime Fourier transform, and these properties play parallel roles in continuous
More informationThe University of Texas at Austin Dept. of Electrical and Computer Engineering Midterm #1. Spirou et Fantasio
The University of Texas at Austin Dept. of Electrical and Computer Engineering Midterm #1 Date: October 14, 2016 Course: EE 445S Evans Name: Spirou et Fantasio Last, First The exam is scheduled to last
More informationUNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2009 Linear Systems Fundamentals
UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2009 Linear Systems Fundamentals MIDTERM EXAM You are allowed one 2sided sheet of notes. No books, no other
More informationLaboratory Assignment 4. Fourier Sound Synthesis
Laboratory Assignment 4 Fourier Sound Synthesis PURPOSE This lab investigates how to use a computer to evaluate the Fourier series for periodic signals and to synthesize audio signals from Fourier series
More informationFrequency Response of FIR Filters
Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input
More informationApplications of Fourier transform. Convolution. Notes. Notes. Notes. Notes
Applications of Fourier transform So far, only considered Fourier transform as a way to obtain the frequency spectrum of a function/signal. However, there are other important applications: : Real physical
More informationCHAPTER 2 FOURIER SERIES
CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =
More informationDiscrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes
Discrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes (i) Understanding the relationships between the transform, discretetime Fourier transform (DTFT), discrete Fourier
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationNegative Feedback with Capacitors
Negative Feedback with Capacitors http://gaussmarkov.net January 8, 008 The web page OpAmps 4: Negative Feedback, describes several characteristics of the gain of a circuit like this one. This document
More informationSGN1158 Introduction to Signal Processing Test. Solutions
SGN1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:
More informationwhere T o defines the period of the signal. The period is related to the signal frequency according to
SIGNAL SPECTRA AND EMC One of the critical aspects of sound EMC analysis and design is a basic understanding of signal spectra. A knowledge of the approximate spectral content of common signal types provides
More informationThe continuous and discrete Fourier transforms
FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1
More informationEE 179 April 28, 2014 Digital and Analog Communication Systems Handout #21 Homework #3 Solutions
EE 179 April 28, 214 Digital and Analog Communication Systems Handout #21 Homework #3 Solutions 1. DSBSC modulator (Lathi& Ding 4.23). You are asked to design a DSBSC modulator to generate a modulated
More informationPYKC Jan710. Lecture 1 Slide 1
Aims and Objectives E 2.5 Signals & Linear Systems Peter Cheung Department of Electrical & Electronic Engineering Imperial College London! By the end of the course, you would have understood: Basic signal
More informationExperiment 3: Double Sideband Modulation (DSB)
Experiment 3: Double Sideband Modulation (DSB) This experiment examines the characteristics of the doublesideband (DSB) linear modulation process. The demodulation is performed coherently and its strict
More informationANALYSIS AND APPLICATIONS OF LAPLACE /FOURIER TRANSFORMATIONS IN ELECTRIC CIRCUIT
www.arpapress.com/volumes/vol12issue2/ijrras_12_2_22.pdf ANALYSIS AND APPLICATIONS OF LAPLACE /FOURIER TRANSFORMATIONS IN ELECTRIC CIRCUIT M. C. Anumaka Department of Electrical Electronics Engineering,
More information7: Fourier Transforms: Convolution and Parseval s Theorem
Convolution Parseval s E. Fourier Series and Transforms (245559) Fourier Transform  Parseval and Convolution: 7 / Multiplication of Signals Question: What is the Fourier transform ofw(t) = u(t)v(t)?
More informationNRZ Bandwidth  HF Cutoff vs. SNR
Application Note: HFAN09.0. Rev.2; 04/08 NRZ Bandwidth  HF Cutoff vs. SNR Functional Diagrams Pin Configurations appear at end of data sheet. Functional Diagrams continued at end of data sheet. UCSP
More informationPYKC 3Mar11. Lecture 15 Slide 1. Laplace transform. Fourier transform. Discrete Fourier transform. transform. L5.8 p560.
 derived from Laplace Lecture 15 DiscreteTime System Analysis using Transform (Lathi 5.1) Peter Cheung Department of Electrical & Electronic Engineering Imperial College London URL: www.ee.imperial.ac.uk/pcheung/teaching/ee2_signals
More informationPurpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I
Part 3: Time Series I Purpose of Time Series Analysis (Figure from Panofsky and Brier 1968) Autocorrelation Function Harmonic Analysis Spectrum Analysis Data Window Significance Tests Some major purposes
More information(Refer Slide Time: 01:1101:27)
Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture  6 Digital systems (contd.); inverse systems, stability, FIR and IIR,
More informationFast Fourier Transforms and Power Spectra in LabVIEW
Application Note 4 Introduction Fast Fourier Transforms and Power Spectra in LabVIEW K. Fahy, E. Pérez Ph.D. The Fourier transform is one of the most powerful signal analysis tools, applicable to a wide
More informationConvolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005
Convolution, Correlation, & Fourier Transforms James R. Graham 10/25/2005 Introduction A large class of signal processing techniques fall under the category of Fourier transform methods These methods fall
More informationClass Note for Signals and Systems. Stanley Chan University of California, San Diego
Class Note for Signals and Systems Stanley Chan University of California, San Diego 2 Acknowledgement This class note is prepared for ECE 101: Linear Systems Fundamentals at the University of California,
More informationOriginal Lecture Notes developed by
Introduction to ADSL Modems Original Lecture Notes developed by Prof. Brian L. Evans Dept. of Electrical and Comp. Eng. The University of Texas at Austin http://signal.ece.utexas.edu Outline Broadband
More informationAdvanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 28 Fourier Series (Contd.) Welcome back to the lecture on Fourier
More informationCONVERTERS. Filters Introduction to Digitization DigitaltoAnalog Converters AnalogtoDigital Converters
CONVERTERS Filters Introduction to Digitization DigitaltoAnalog Converters AnalogtoDigital Converters Filters Filters are used to remove unwanted bandwidths from a signal Filter classification according
More informationEELE445  Lab 2 Pulse Signals
EELE445  Lab 2 Pulse Signals PURPOSE The purpose of the lab is to examine the characteristics of some common pulsed waveforms in the time and frequency domain. The repetitive pulsed waveforms used are
More informationCurve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization.
Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization Subsections LeastSquares Regression Linear Regression General Linear LeastSquares Nonlinear
More informationSAMPLE SOLUTIONS DIGITAL SIGNAL PROCESSING: Signals, Systems, and Filters Andreas Antoniou
SAMPLE SOLUTIONS DIGITAL SIGNAL PROCESSING: Signals, Systems, and Filters Andreas Antoniou (Revision date: February 7, 7) SA. A periodic signal can be represented by the equation x(t) k A k sin(ω k t +
More informationFrom Fundamentals of Digital Communication Copyright by Upamanyu Madhow, 20032006
Chapter Introduction to Modulation From Fundamentals of Digital Communication Copyright by Upamanyu Madhow, 003006 Modulation refers to the representation of digital information in terms of analog waveforms
More informationchapter Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction 1.2 Historical Perspective
Introduction to Digital Signal Processing and Digital Filtering chapter 1 Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction Digital signal processing (DSP) refers to anything
More informationUNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2010 Linear Systems Fundamentals
UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2010 Linear Systems Fundamentals FINAL EXAM WITH SOLUTIONS (YOURS!) You are allowed one 2sided sheet of
More informationME 252 B. Computational Fluid Dynamics: Wavelet transforms and their applications to turbulence
ME 252 B Computational Fluid Dynamics: Wavelet transforms and their applications to turbulence Marie Farge 1 & Kai Schneider 2 Winter 2004 University of California, Santa Barbara 1 LMDCNRS, Ecole Normale
More informationDifference Equations
Difference Equations Andrew W H House 10 June 004 1 The Basics of Difference Equations Recall that in a previous section we saw that IIR systems cannot be evaluated using the convolution sum because it
More informationAdding Sinusoids of the Same Frequency. Additive Synthesis. Spectrum. Music 270a: Modulation
Adding Sinusoids of the Same Frequency Music 7a: Modulation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) February 9, 5 Recall, that adding sinusoids of
More informationProbability and Random Variables. Generation of random variables (r.v.)
Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly
More informationT = 10' s. p(t)= ( (tnt), T= 3. n=oo. Figure P16.2
16 Sampling Recommended Problems P16.1 The sequence x[n] = (1)' is obtained by sampling the continuoustime sinusoidal signal x(t) = cos oot at 1ms intervals, i.e., cos(oont) = (1)", Determine three
More informationLectures 56: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5: Taylor Series Weeks 5 Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationFIR Filter Design. FIR Filters and the zdomain. The zdomain model of a general FIR filter is shown in Figure 1. Figure 1
FIR Filters and the Domain FIR Filter Design The domain model of a general FIR filter is shown in Figure. Figure Each  box indicates a further delay of one sampling period. For example, the input to
More informationLecture 3: Quantization Effects
Lecture 3: Quantization Effects Reading: 6.76.8. We have so far discussed the design of discretetime filters, not digital filters. To understand the characteristics of digital filters, we need first
More informationAnalysis/resynthesis with the short time Fourier transform
Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TUBerlin IRCAM Analysis/Synthesis
More informationLecture 9, Multirate Signal Processing, zdomain Effects
Lecture 9, Multirate Signal Processing, zdomain Effects Last time we saw the effects of downsampling and upsampling. Observe that we can perfectly reconstruct the high pass signal in our example if we
More informationFrequency Domain Analysis
Exercise 4. Frequency Domain Analysis Required knowledge Fourierseries and Fouriertransform. Measurement and interpretation of transfer function of linear systems. Calculation of transfer function of
More informationCurrent Probes, More Useful Than You Think
Current Probes, More Useful Than You Think Training and design help in most areas of Electrical Engineering Copyright 1998 Institute of Electrical and Electronics Engineers. Reprinted from the IEEE 1998
More informationChapter 3 DiscreteTime Fourier Series. by the French mathematician Jean Baptiste Joseph Fourier in the early 1800 s. The
Chapter 3 DiscreteTime Fourier Series 3.1 Introduction The Fourier series and Fourier transforms are mathematical correlations between the time and frequency domains. They are the result of the heattransfer
More informationFirst and Second Order Filters
First and Second Order Filters These functions are useful for the design of simple filters or they can be cascaded to form highorder filter functions First Order Filters General first order bilinear transfer
More informationThe Fourier Transform
The Fourier Transorm Fourier Series Fourier Transorm The Basic Theorems and Applications Sampling Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGrawHill, 2. Eric W. Weisstein.
More informationIntroduction of Fourier Analysis and Timefrequency Analysis
Introduction of Fourier Analysis and Timefrequency Analysis March 1, 2016 Fourier Series Fourier transform Fourier analysis Mathematics compares the most diverse phenomena and discovers the secret analogies
More informationSECONDORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS
L SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A secondorder linear differential equation is one of the form d
More informationEngineering Mathematics II
PSUT Engineering Mathematics II Fourier Series and Transforms Dr. Mohammad Sababheh 4/14/2009 11.1 Fourier Series 2 Fourier Series and Transforms Contents 11.1 Fourier Series... 3 Periodic Functions...
More information1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution.
Name : Instructor: Marius Ionescu Instructions: 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution. 3. If you need extra pages
More informationRLC Resonant Circuits
C esonant Circuits Andrew McHutchon April 20, 203 Capacitors and Inductors There is a lot of inconsistency when it comes to dealing with reactances of complex components. The format followed in this document
More informationLecture 12 Basic Lyapunov theory
EE363 Winter 200809 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12
More informationFourier Transforms The Fourier Transform Properties of the Fourier Transform Some Special Fourier Transform Pairs 27
24 Contents Fourier Transforms 24.1 The Fourier Transform 2 24.2 Properties of the Fourier Transform 14 24.3 Some Special Fourier Transform Pairs 27 Learning outcomes In this Workbook you will learn about
More informationThe Membrane Equation
The Membrane Equation Professor David Heeger September 5, 2000 RC Circuits Figure 1A shows an RC (resistor, capacitor) equivalent circuit model for a patch of passive neural membrane. The capacitor represents
More informationNyquist Sampling Theorem. By: Arnold Evia
Nyquist Sampling Theorem By: Arnold Evia Table of Contents What is the Nyquist Sampling Theorem? Bandwidth Sampling Impulse Response Train Fourier Transform of Impulse Response Train Sampling in the Fourier
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationFiltering Data using Frequency Domain Filters
Filtering Data using Frequency Domain Filters Wouter J. Den Haan London School of Economics c Wouter J. Den Haan August 27, 2014 Overview Intro lag operator Why frequency domain? Fourier transform Data
More informationThe Discrete Fourier Transform
The Discrete Fourier Transform Introduction The discrete Fourier transform (DFT) is a fundamental transform in digital signal processing, with applications in frequency analysis, fast convolution, image
More informationNOTES ON PHASORS. where the three constants have the following meanings:
Chapter 1 NOTES ON PHASORS 1.1 TimeHarmonic Physical Quantities Timeharmonic analysis of physical systems is one of the most important skills for the electrical engineer to develop. Whether the application
More informationAn introduction to generalized vector spaces and Fourier analysis. by M. Croft
1 An introduction to generalized vector spaces and Fourier analysis. by M. Croft FOURIER ANALYSIS : Introduction Reading: Brophy p. 5863 This lab is u lab on Fourier analysis and consists of VI parts.
More informationGeneral Theory of Differential Equations Sections 2.8, 3.13.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.13.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationImplementation of Digital Signal Processing: Some Background on GFSK Modulation
Implementation of Digital Signal Processing: Some Background on GFSK Modulation Sabih H. Gerez University of Twente, Department of Electrical Engineering s.h.gerez@utwente.nl Version 4 (February 7, 2013)
More informationR U S S E L L L. H E R M A N
R U S S E L L L. H E R M A N A N I N T R O D U C T I O N T O F O U R I E R A N D C O M P L E X A N A LY S I S W I T H A P P L I C AT I O N S T O T H E S P E C T R A L A N A LY S I S O F S I G N A L S R.
More informationIntroduction to IQdemodulation of RFdata
Introduction to IQdemodulation of RFdata by Johan Kirkhorn, IFBT, NTNU September 15, 1999 Table of Contents 1 INTRODUCTION...3 1.1 Abstract...3 1.2 Definitions/Abbreviations/Nomenclature...3 1.3 Referenced
More informationFourier Analysis Last Modified 9/5/06
Measurement Lab Fourier Analysis Last Modified 9/5/06 Any timevarying signal can be constructed by adding together sine waves of appropriate frequency, amplitude, and phase. Fourier analysis is a technique
More informationFILTER CIRCUITS. A filter is a circuit whose transfer function, that is the ratio of its output to its input, depends upon frequency.
FILTER CIRCUITS Introduction Circuits with a response that depends upon the frequency of the input voltage are known as filters. Filter circuits can be used to perform a number of important functions in
More informationConceptual similarity to linear algebra
Modern approach to packing more carrier frequencies within agivenfrequencyband orthogonal FDM Conceptual similarity to linear algebra 3D space: Given two vectors x =(x 1,x 2,x 3 )andy = (y 1,y 2,y 3 ),
More informationECE 5520: Digital Communications Lecture Notes Fall 2009
ECE 5520: Digital Communications Lecture Notes Fall 2009 Dr. Neal Patwari University of Utah Department of Electrical and Computer Engineering c 2006 ECE 5520 Fall 2009 2 Contents 1 Class Organization
More informationCHAPTER 6 Frequency Response, Bode Plots, and Resonance
ELECTRICAL CHAPTER 6 Frequency Response, Bode Plots, and Resonance 1. State the fundamental concepts of Fourier analysis. 2. Determine the output of a filter for a given input consisting of sinusoidal
More informationOSE801 Engineering System Identification Fall 2012 Lecture 5: Fourier Analysis: Introduction
OSE801 Engineering System Identification Fall 2012 Lecture 5: Fourier Analysis: Introduction Instructors: K. C. Park and I. K. Oh (Division of Ocean Systems Engineering) SystemIdentified State Space Model
More informationFourier Transform and Its Medical Application 서울의대의공학교실 김희찬
Fourier Transform and Its Medical Application 서울의대의공학교실 김희찬 강의내용 Fourier Transform 의수학적이해 Fourier Transform 과신호처리 Fourier Transform 과의학영상응용 Integral transform a particular kind of mathematical operator
More informationFourier Series for Periodic Functions. Lecture #8 5CT3,4,6,7. BME 333 Biomedical Signals and Systems  J.Schesser
Fourier Series for Periodic Functions Lecture #8 5C3,4,6,7 Fourier Series for Periodic Functions Up to now we have solved the problem of approximating a function f(t) by f a (t) within an interval. However,
More informationLaboratory Manual and Supplementary Notes. CoE 494: Communication Laboratory. Version 1.2
Laboratory Manual and Supplementary Notes CoE 494: Communication Laboratory Version 1.2 Dr. Joseph Frank Dr. Sol Rosenstark Department of Electrical and Computer Engineering New Jersey Institute of Technology
More informationWhat is a Filter? Output Signal. Input Signal Amplitude. Frequency. Low Pass Filter
What is a Filter? Input Signal Amplitude Output Signal Frequency Time Sequence Low Pass Filter Time Sequence What is a Filter Input Signal Amplitude Output Signal Frequency Signal Noise Signal Noise Frequency
More informationThe Fourier Analysis Tool in Microsoft Excel
The Fourier Analysis Tool in Microsoft Excel Douglas A. Kerr Issue March 4, 2009 ABSTRACT AD ITRODUCTIO The spreadsheet application Microsoft Excel includes a tool that will calculate the discrete Fourier
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More information7. FOURIER ANALYSIS AND DATA PROCESSING
7. FOURIER ANALYSIS AND DATA PROCESSING Fourier 1 analysis plays a dominant role in the treatment of vibrations of mechanical systems responding to deterministic or stochastic excitation, and, as has already
More informationRoots of quadratic equations
CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients
More information