Lecture 8 ELE 301: Signals and Systems


 Brett Bridges
 1 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationFourier Transform and Image Filtering. CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010
Fourier Transform and Image Filtering CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010 The Fourier Transform Fourier Transform Forward, mapping to frequency domain: Backward, inverse mapping to time domain:
More informationChapter 29 AlternatingCurrent Circuits
hapter 9 Alternatingurrent ircuits onceptual Problems A coil in an ac generator rotates at 6 Hz. How much time elapses between successive emf values of the coil? Determine the oncept Successive s are
More informationThe Fast Fourier Transform (FFT) and MATLAB Examples
The Fast Fourier Transform (FFT) and MATLAB Examples Learning Objectives Discrete Fourier transforms (DFTs) and their relationship to the Fourier transforms Implementation issues with the DFT via the FFT
More informationDesign of FIR Filters
Design of FIR Filters Elena Punskaya wwwsigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 68 FIR as
More informationThe Calculation of G rms
The Calculation of G rms QualMark Corp. Neill Doertenbach The metric of G rms is typically used to specify and compare the energy in repetitive shock vibration systems. However, the method of arriving
More informationELE745 Assignment and Lab Manual
ELE745 Assignment and Lab Manual August 22, 2010 CONTENTS 1. Assignment 1........................................ 1 1.1 Assignment 1 Problems................................ 1 1.2 Assignment 1 Solutions................................
More informationLecture 8: Signal Detection and Noise Assumption
ECE 83 Fall Statistical Signal Processing instructor: R. Nowak, scribe: Feng Ju Lecture 8: Signal Detection and Noise Assumption Signal Detection : X = W H : X = S + W where W N(, σ I n n and S = [s, s,...,
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationLecture 3 The Laplace transform
S. Boyd EE12 Lecture 3 The Laplace transform definition & examples properties & formulas linearity the inverse Laplace transform time scaling exponential scaling time delay derivative integral multiplication
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 informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
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 informationDigital Transmission (Line Coding)
Digital Transmission (Line Coding) Pulse Transmission Source Multiplexer Line Coder Line Coding: Output of the multiplexer (TDM) is coded into electrical pulses or waveforms for the purpose of transmission
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationMATRIX TECHNICAL NOTES
200 WOOD AVENUE, MIDDLESEX, NJ 08846 PHONE (732) 4699510 FAX (732) 4690418 MATRIX TECHNICAL NOTES MTN107 TEST SETUP FOR THE MEASUREMENT OF XMOD, CTB, AND CSO USING A MEAN SQUARE CIRCUIT AS A DETECTOR
More informationFFT Algorithms. Chapter 6. Contents 6.1
Chapter 6 FFT Algorithms Contents Efficient computation of the DFT............................................ 6.2 Applications of FFT................................................... 6.6 Computing DFT
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationFrequency Domain and Fourier Transforms
Chapter 4 Frequency Domain and Fourier Transforms Frequency domain analysis and Fourier transforms are a cornerstone of signal and system analysis. These ideas are also one of the conceptual pillars within
More informationHow to Graph Trigonometric Functions
How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationIntroduction to Receivers
Introduction to Receivers Purpose: translate RF signals to baseband Shift frequency Amplify Filter Demodulate Why is this a challenge? Interference (selectivity, images and distortion) Large dynamic range
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
More informationLezione 6 Communications Blockset
Corso di Tecniche CAD per le Telecomunicazioni A.A. 20072008 Lezione 6 Communications Blockset Ing. Marco GALEAZZI 1 What Is Communications Blockset? Communications Blockset extends Simulink with a comprehensive
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationFourier Analysis and its applications
Fourier Analysis and its applications Fourier analysis originated from the study of heat conduction: Jean Baptiste Joseph Fourier (17681830) Fourier analysis enables a function (signal) to be decomposed
More informationANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 21
WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's
More informationSolutions to Linear First Order ODE s
First Order Linear Equations In the previous session we learned that a first order linear inhomogeneous ODE for the unknown function x = x(t), has the standard form x + p(t)x = q(t) () (To be precise we
More informationTo define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions
Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationIMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL
IMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL Saurabh Singh Rajput, Dr.S.S. Bhadauria Department of Electronics, Madhav Institute of Technology
More informationFACTORING QUADRATIC EQUATIONS
FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods
More informationTables of Common Transform Pairs
ble of Common rnform Pir 0 by Mrc Ph. Stoecklin mrc toecklin.net http://www.toecklin.net/ 00 verion v.5.3 Engineer nd tudent in communiction nd mthemtic re confronted with tion uch the rnform, the ourier,
More informationBharathwaj Muthuswamy EE100 Active Filters
Bharathwaj Muthuswamy EE100 mbharat@cory.eecs.berkeley.edu 1. Introduction Active Filters In this chapter, we will deal with active filter circuits. Why even bother with active filters? Answer: Audio.
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationUsing the TI92 Plus: Some Examples
Liverpool John Moores University, 115 July 000 Using the TI9 Plus: Some Examples Michel Beaudin École de technologie supérieure,canada mbeaudin@seg.etsmtl.ca 1. Introduction We incorporated the use of
More informationf x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y
Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions:
More informationMODULATION Systems (part 1)
Technologies and Services on Digital Broadcasting (8) MODULATION Systems (part ) "Technologies and Services of Digital Broadcasting" (in Japanese, ISBN4339622) is published by CORONA publishing co.,
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MT OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 00 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More informationLinear Control Systems
Chapter 3 Linear Control Systems Topics : 1. Controllability 2. Observability 3. Linear Feedback 4. Realization Theory Copyright c Claudiu C. Remsing, 26. All rights reserved. 7 C.C. Remsing 71 Intuitively,
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationSecond Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
More informationElectronics Prof. D.C. Dube Department of Physics Indian Institute of Technology, Delhi
Electronics Prof. D.C. Dube Department of Physics Indian Institute of Technology, Delhi Module No. #06 Power Amplifiers Lecture No. #01 Power Amplifiers (Refer Slide Time: 00:44) We now move to the next
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More informationIntegrator Based Filters
Integrator Based Filters Main building block for this category of filters integrator By using signal flowgraph techniques conventional filter topologies can be converted to integrator based type filters
More informationUnderstanding CIC Compensation Filters
Understanding CIC Compensation Filters April 2007, ver. 1.0 Application Note 455 Introduction f The cascaded integratorcomb (CIC) filter is a class of hardwareefficient linear phase finite impulse response
More informationAgilent PN 8940013 Extending Vector Signal Analysis to 26.5 GHz with 20 MHz Information Bandwidth
Agilent PN 8940013 Extending Vector Signal Analysis to 26.5 GHz with 20 MHz Information Bandwidth Product Note The Agilent Technologies 89400 series vector signal analyzers provide unmatched signal analysis
More informationComputer Networks and Internets, 5e Chapter 6 Information Sources and Signals. Introduction
Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals Modified from the lecture slides of Lami Kaya (LKaya@ieee.org) for use CECS 474, Fall 2008. 2009 Pearson Education Inc., Upper
More informationDiscreteTime Signals and Systems
2 DiscreteTime Signals and Systems 2.0 INTRODUCTION The term signal is generally applied to something that conveys information. Signals may, for example, convey information about the state or behavior
More information