Review of Fourier Transform

Size: px
Start display at page:

Download "Review of Fourier Transform"

Transcription

1 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 with T Fourier series changes to Fourier transform, comple eponents are infinitesimally close in frequency Discrete spectrum becomes a continuous one, also known as spectral density (5)

2 Fourier Series -> Fourier Transform Periodic signal Its spectrum T = 4T T = 8T As T increases, spectral components are getting closer and closer, becoming the continuous spectrum at the limit T = 6T A.V. Oppenheim, A.S. Willsky, Signals and Systems, 997. (5)

3 Fourier Transform Fourier transform (spectrum): + j ( ) ( ) πft S f = t e dt Inverse Fourier transform: ( ) = + jπft ( ) ( ) ( ) t S f e df Dirichlet conditions (details on the net page): (t) is absolutely integrable, radial frequency jωt ( ) ( ) S ω = t e dt + jωt t = S ω e dω π (t) has a finite number of maima & minima within any finite interval, (t) has a finite number of discontinuities within any finite interval. These discontinuities must be finite. + radial frequency 3(5)

4 Convergence of Fourier Transform Dirichlet conditions: (t) must be absolutely integrable (finite energy) ( ) t dt < or ( t) dt < (t) has a finite number of maima & minima within any finite interval, (t) has a finite number of discontinuities within any finite interval. These discontinuities must be finite. Dirichlet conditions are only sufficient, but are not necessary. All physically-reasonable (practical) signals meet these conditions. 4(5)

5 Eample: Rectangular Pulse signal ( ) t, t < τ / = Π( t / τ ) = 0, t > τ / τ τ t sin πf τ S ( f ) = τ = τ sinc( f τ) πf τ spectrum τ S ( f ) τ τ f 5(5)

6 Eample: sinc(t) Shortening pulse widens its spectrum! f signal f f signal f spectrum S ( ) f spectrum f f S ( ) f f f f f 6(5)

7 Generalized (Singular) Functions Unit step function u( t), t > 0 = 0, t < 0 u(t) t Dirac delta function (unit impulse function) δ ( t) ( t) dt = (0), t = 0 δ ( t) = & δ ( t) dt = 0, t 0 δ( t) t 7(5)

8 Generalized Functions Dirac Delta Function as a limit: Π /, t < / ( t) = 0, t > δ ( t) = lim Π ( t) 0 Π ( t) t Useful properties of delta function Fourier transform? δ( t t ) ( t) dt = ( t ) 0 0 t δ ( t) dt = u( t) du( t) dt = δ( t) δ ( at) = δ ( t) δ( t) = δ ( t) ( t) δ ( t) = (0) δ( t) a 8(5)

9 Fourier Transform of Periodic Signal FT of a comple eponent: jω0 ( ) t = e πδ( ω ω ) = δ( f f ) Important property: ( ) FT of a periodic signal: jnω0 ( ) FT of? t + ± j π ft δ f = e dt FT + + t n n 0 n 0 n= n= n= t = c e π c δ( ω nω ) = c δ( f nf ) cos( ω t) 0 prove this property 9(5)

10 Properties of Fourier Transform * Very similar to those of Fourier series! Linearity: Time shifting: Time reversal: α ( t) + β ( t) α S ( f ) + βs ( f ) F ( ) ( ω) ( ) 0 ( ω) 0 jωt t S t t e S ( t) S ( ω) ( t) S ( ω) Time scaling: ω ( at) S a a Prove these properties. * properties are useful for evaluating Fourier transform in a simple way 0(5)

11 Properties of Fourier Transform Conjugation: Differentiation: Integration: Multiplication: * ( t) S ( ω) *( t) S ( ω) d( t) ( t) S( ω) jωs ( ω) dt t ( t) dt S( ω ) + πs(0) δ( ω) jω ( t) y( t) S ( ω ) Sy ( ω ω ) dω = π = S ( ω)* S ( ω) y Prove these properties Frequency shift (modulation): t e jω t ω ω 0 ( ) 0 S( ) (5)

12 Duality of Fourier Transform ( t) S ( ω) S ( t) π( ω ) ( t) S ( f ) S ( t) ( f ) A.V. Oppenheim, A.S. Willsky, Signals and Systems, 997. (5)

13 Convolution Property This property is of great importance y( t) = ( τ) h( t τ) dτ S ( ω) H ( ω ) = S ( ω) y (t) h(t) y(t) S ( ω ) H( ω ) S y ( ω) Cascade connection of LTI blocks A.V. Oppenheim, A.S. Willsky, Signals and Systems, 997. Eample: FT of a triangular pulse by convolution 3(5)

14 Parseval Theorem Total energy in time domain is the same as the total energy in frequency domain: ( ) ( ) ( ) π E = t dt = S f df = S ω dω E( f ) = S ( f ) - energy spectral density (ESD) of (t). Represents the amount of energy per Hz of bandwidth Counterpart of Parseval theorem for periodic signals Autocorrelation property: + ( τ ) = ( ) ( τ) ω R ( 0) R t t dt S ( ) = E 4(5)

15 Parseval Theorem: Eample + sin t t dt =? 5(5)

16 Fourier Transform of Real Signal if (t) is real, * { t } S S Im ( ) = 0 ( ω ) = ( ω) Fourier transform can be presented as 0 ( ) ( t) = S ( f ) cos π f + ϕ( f ) df, [ S f ] [ S f ] Im ( ) ϕ ( f ) = tan Re ( ) No negative frequencies! 6(5)

17 Signal Bandwidth & Negative Frequencies What negative frequency means? It means that there is term in signal spectrum Convenient mathematical tool. Do not eist in practice (i.e., cannot be measured on spectrum analyzer) What is the signal bandwidth? There are many definitions. Defined for positive frequencies only. Determines the frequency band over which a substantial part of the signal power/energy is concentrated. For band-limited signals j ft e π f = f f, f, f 0 ma min ma min 7(5)

18 Power and Energy Power P & energy E of signal (t) are: P = lim ( t) dt T T T T E = ( t) dt Energy-type signals: Power-type signals: E < 0 < P < Signal cannot be both energy & power type! Signal energy: if (t) is voltage or current, E is the energy dissipated in Ohm resistor Signal power: if (t) is voltage or current, P is the power dissipated in Ohm resistor. 8(5)

19 Energy-Type Signals (summary) Signal energy in time & frequency domains: ( ) ( ) ( ) π E = t dt = S f df = S ω dω Energy spectral density (energy per Hz of bandwidth): E ( f ) = S ( f ) ESD is FT of autocorrelation function: + ( τ ) = ( ) ( τ) ( ) R ( 0) R t t dt E f = E 9(5)

20 Power-Type Signals: PSD Definition of the power spectral density (PSD) (power per Hz of bandwidth): S ( ) lim T ( f P ) f = P P ( f ) df T T = < where ( t) is the truncated signal (to [-T/,T/]), T t ( t), T / t T / T ( t) = ( t) Π = T 0, otherwise and is its spectrum (FT), S T ( f ) { } S ( f ) = FT ( t) T T 0(5)

21 Power-Type Signals Time-average autocorrelation function: R T ( ) lim ( t ) τ = ( t ) dt T T τ T Power of the signal: Wiener-Khintchine theorem : P T = lim 0 T T t dt R = T ( ) ( ) { } P = P f df P f = FT R ( τ ) ( ) ( ) (5)

22 Power of a periodic signal: Autocorrelation function: Periodic Signals = = T n = T n= P ( t ) dt c R (0) jnω0 ( ) c e t + = n= n t jnω τ T n n= 0 ( ) ( ) ( ) R τ = t t τ T dt = c e Power spectral density (PSD): P ( f ) = FT { R ( τ )} = c δ f n n= n T Prove these properties! (5)

23 Relation Between Fourier Transform & Series consider a periodic signal (t)=(t+t) truncate it (one period only): find FT of the truncated signal T (t): ( t), T / < t T / ( t) = 0, otherwise Fourier series of the original periodic signal (t) is Continuous spectrum is the envelope of discrete spectrum (see slide )! T cn = S T ( nω0 ) T ( t) S ( ω) T T prove this 3(5)

24 Fourier Transform of Single Pulse ~ Envelope of Fourier Series of Pulse Train FT/S A.V. Oppenheim, A.S. Willsky, Signals and Systems, (5)

25 Summary Definition of Fourier Transform Properties of Fourier Transform Signal bandwidth Signal power & energy. Energy and power-type signals Fourier transform of periodic signals Relation between Fourier series & transform Reading: Couch,.-.6; Oppenheim & Willsky, Ch., 3 & 4. Study carefully all the eamples (including end-of-chapter study-aid eamples), make sure you understand and can solve them with the book closed. Do some end-of-chapter problems. Students solution manual provides solutions for many of them. 5(5)

Figure 3.1 Effect on frequency spectrum of increasing period T 0. Consider the amplitude spectrum of a periodic waveform as shown in Figure 3.2.

Figure 3.1 Effect on frequency spectrum of increasing period T 0. Consider the amplitude spectrum of a periodic waveform as shown in Figure 3.2. 3. Fourier ransorm From Fourier Series to Fourier ransorm [, 2] In communication systems, we oten deal with non-periodic signals. An extension o the time-requency relationship to a non-periodic signal

More information

Signals & Systems. Lecture 5 Continuous-Time Fourier Transform. Alp Ertürk

Signals & Systems. Lecture 5 Continuous-Time Fourier Transform. Alp Ertürk Signals & Systems Lecture 5 Continuous-Time Fourier Transform Alp Ertürk alp.erturk@kocaeli.edu.tr Fourier Series Representation of Continuous-Time Periodic Signals Synthesis equation: x t = a k e jkω

More information

ENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University

ENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University ENSC327 Communications Systems 2: Fourier Representations Jie Liang School of Engineering Science Simon Fraser University 1 Outline Chap 2.1 2.5: Signal Classifications Fourier Transform Dirac Delta Function

More information

Signals and Spectra - Review

Signals and Spectra - Review Signals and Spectra - Review SIGNALS DETERMINISTIC No uncertainty w.r.t. the value of a signal at any time Modeled by mathematical epressions RANDOM some degree of uncertainty before the signal occurs

More information

The Continuous-time Fourier

The Continuous-time Fourier The Continuous-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Representation of Aperiodic signals:

More information

LOPE3202: Communication Systems 10/18/2017 2

LOPE3202: Communication Systems 10/18/2017 2 By Lecturer Ahmed Wael Academic Year 2017-2018 LOPE3202: Communication Systems 10/18/2017 We need tools to build any communication system. Mathematics is our premium tool to do work with signals and systems.

More information

Chapter 4 The Fourier Series and Fourier Transform

Chapter 4 The Fourier Series and Fourier Transform Chapter 4 The Fourier Series and Fourier Transform Fourier Series Representation of Periodic Signals Let x(t) be a CT periodic signal with period T, i.e., xt ( + T) = xt ( ), t R Example: the rectangular

More information

Mechanical Vibrations Chapter 13

Mechanical Vibrations Chapter 13 Mechanical Vibrations Chapter 3 Peter Avitabile Mechanical Engineering Department University of Massachusetts Lowell Dr. Peter Avitabile Random Vibrations Up until now, everything analyzed was deterministic.

More information

Signals and Systems Spring 2004 Lecture #9

Signals and Systems Spring 2004 Lecture #9 Signals and Systems Spring 2004 Lecture #9 (3/4/04). The convolution Property of the CTFT 2. Frequency Response and LTI Systems Revisited 3. Multiplication Property and Parseval s Relation 4. The DT Fourier

More information

Lecture 8 ELE 301: Signals and Systems

Lecture 8 ELE 301: Signals and Systems Lecture 8 ELE 30: Signals and Systems Prof. Paul Cuff Princeton University Fall 20-2 Cuff (Lecture 7) ELE 30: Signals and Systems Fall 20-2 / 37 Properties of the Fourier Transform Properties of the Fourier

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

Continuous Time Signal Analysis: the Fourier Transform. Lathi Chapter 4

Continuous Time Signal Analysis: the Fourier Transform. Lathi Chapter 4 Continuous Time Signal Analysis: the Fourier Transform Lathi Chapter 4 Topics Aperiodic signal representation by the Fourier integral (CTFT) Continuous-time Fourier transform Transforms of some useful

More information

2.1 Basic Concepts Basic operations on signals Classication of signals

2.1 Basic Concepts Basic operations on signals Classication of signals Haberle³me Sistemlerine Giri³ (ELE 361) 9 Eylül 2017 TOBB Ekonomi ve Teknoloji Üniversitesi, Güz 2017-18 Dr. A. Melda Yüksel Turgut & Tolga Girici Lecture Notes Chapter 2 Signals and Linear Systems 2.1

More information

Signal Processing Signal and System Classifications. Chapter 13

Signal Processing Signal and System Classifications. Chapter 13 Chapter 3 Signal Processing 3.. Signal and System Classifications In general, electrical signals can represent either current or voltage, and may be classified into two main categories: energy signals

More information

Ch 4: The Continuous-Time Fourier Transform

Ch 4: The Continuous-Time Fourier Transform Ch 4: The Continuous-Time Fourier Transform Fourier Transform of x(t) Inverse Fourier Transform jt X ( j) x ( t ) e dt jt x ( t ) X ( j) e d 2 Ghulam Muhammad, King Saud University Continuous-time aperiodic

More information

06EC44-Signals and System Chapter Fourier Representation for four Signal Classes

06EC44-Signals and System Chapter Fourier Representation for four Signal Classes Chapter 5.1 Fourier Representation for four Signal Classes 5.1.1Mathematical Development of Fourier Transform If the period is stretched without limit, the periodic signal no longer remains periodic but

More information

ENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University

ENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University ENSC37 Communications Systems : Fourier Representations School o Engineering Science Simon Fraser University Outline Chap..5: Signal Classiications Fourier Transorm Dirac Delta Function Unit Impulse Fourier

More information

Signal and systems. Linear Systems. Luigi Palopoli. Signal and systems p. 1/5

Signal and systems. Linear Systems. Luigi Palopoli. Signal and systems p. 1/5 Signal and systems p. 1/5 Signal and systems Linear Systems Luigi Palopoli palopoli@dit.unitn.it Wrap-Up Signal and systems p. 2/5 Signal and systems p. 3/5 Fourier Series We have see that is a signal

More information

EE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal.

EE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal. EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuous-time LTI system that the input signal = 3 u(t) produces

More information

System Identification & Parameter Estimation

System Identification & Parameter Estimation System Identification & Parameter Estimation Wb3: SIPE lecture Correlation functions in time & frequency domain Alfred C. Schouten, Dept. of Biomechanical Engineering (BMechE), Fac. 3mE // Delft University

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

Chapter 4 The Fourier Series and Fourier Transform

Chapter 4 The Fourier Series and Fourier Transform Chapter 4 The Fourier Series and Fourier Transform Representation of Signals in Terms of Frequency Components Consider the CT signal defined by N xt () = Acos( ω t+ θ ), t k = 1 k k k The frequencies `present

More information

4 The Continuous Time Fourier Transform

4 The Continuous Time Fourier Transform 96 4 The Continuous Time ourier Transform ourier (or frequency domain) analysis turns out to be a tool of even greater usefulness Extension of ourier series representation to aperiodic signals oundation

More information

7. Find the Fourier transform of f (t)=2 cos(2π t)[u (t) u(t 1)]. 8. (a) Show that a periodic signal with exponential Fourier series f (t)= δ (ω nω 0

7. Find the Fourier transform of f (t)=2 cos(2π t)[u (t) u(t 1)]. 8. (a) Show that a periodic signal with exponential Fourier series f (t)= δ (ω nω 0 Fourier Transform Problems 1. Find the Fourier transform of the following signals: a) f 1 (t )=e 3 t sin(10 t)u (t) b) f 1 (t )=e 4 t cos(10 t)u (t) 2. Find the Fourier transform of the following signals:

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.6 Signal Processing: Continuous and Discrete Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS

More information

Sinc Functions. Continuous-Time Rectangular Pulse

Sinc Functions. Continuous-Time Rectangular Pulse Sinc Functions The Cooper Union Department of Electrical Engineering ECE114 Digital Signal Processing Lecture Notes: Sinc Functions and Sampling Theory October 7, 2011 A rectangular pulse in time/frequency

More information

Fourier transform representation of CT aperiodic signals Section 4.1

Fourier transform representation of CT aperiodic signals Section 4.1 Fourier transform representation of CT aperiodic signals Section 4. A large class of aperiodic CT signals can be represented by the CT Fourier transform (CTFT). The (CT) Fourier transform (or spectrum)

More information

Laplace Transforms and use in Automatic Control

Laplace Transforms and use in Automatic Control Laplace Transforms and use in Automatic Control P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: P.Santosh Krishna, SYSCON Recap Fourier series Fourier transform: aperiodic Convolution integral

More information

GATE EE Topic wise Questions SIGNALS & SYSTEMS

GATE EE Topic wise Questions SIGNALS & SYSTEMS www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)

More information

2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit dwm/courses/2tf

2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit   dwm/courses/2tf Time-Frequency Analysis II (HT 20) 2AH 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 20 For hints and answers visit www.robots.ox.ac.uk/ dwm/courses/2tf David Murray. A periodic

More information

Solutions to Problems in Chapter 4

Solutions to Problems in Chapter 4 Solutions to Problems in Chapter 4 Problems with Solutions Problem 4. Fourier Series of the Output Voltage of an Ideal Full-Wave Diode Bridge Rectifier he nonlinear circuit in Figure 4. is a full-wave

More information

ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52

ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52 1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n

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

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

Fourier transforms. Definition F(ω) = - should know these! f(t).e -jωt.dt. ω = 2πf. other definitions exist. f(t) = - F(ω).e jωt.

Fourier transforms. Definition F(ω) = - should know these! f(t).e -jωt.dt. ω = 2πf. other definitions exist. f(t) = - F(ω).e jωt. Fourier transforms This is intended to be a practical exposition, not fully mathematically rigorous ref The Fourier Transform and its Applications R. Bracewell (McGraw Hill) Definition F(ω) = - f(t).e

More information

LINEAR SYSTEMS. J. Elder PSYC 6256 Principles of Neural Coding

LINEAR SYSTEMS. J. Elder PSYC 6256 Principles of Neural Coding LINEAR SYSTEMS Linear Systems 2 Neural coding and cognitive neuroscience in general concerns input-output relationships. Inputs Light intensity Pre-synaptic action potentials Number of items in display

More information

Poularikas A. D. Fourier Transform The Handbook of Formulas and Tables for Signal Processing. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC,

Poularikas A. D. Fourier Transform The Handbook of Formulas and Tables for Signal Processing. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, Poularikas A. D. Fourier Transform The Handbook of Formulas and Tables for Signal Processing. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 999 3 Fourier Transform 3. One-Dimensional Fourier Transform

More information

Signals and Systems: Part 2

Signals and Systems: Part 2 Signals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms Some important Fourier transform theorems Convolution and Modulation Ideal filters Fourier transform definitions

More information

Fourier Analysis and Power Spectral Density

Fourier Analysis and Power Spectral Density Chapter 4 Fourier Analysis and Power Spectral Density 4. Fourier Series and ransforms Recall Fourier series for periodic functions for x(t + ) = x(t), where x(t) = 2 a + a = 2 a n = 2 b n = 2 n= a n cos

More information

ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name:

ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name: ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, 205 Name:. The quiz is closed book, except for one 2-sided sheet of handwritten notes. 2. Turn off

More information

Fourier Series and Transforms. Revision Lecture

Fourier Series and Transforms. Revision Lecture E. (5-6) : / 3 Periodic signals can be written as a sum of sine and cosine waves: u(t) u(t) = a + n= (a ncosπnft+b n sinπnft) T = + T/3 T/ T +.65sin(πFt) -.6sin(πFt) +.6sin(πFt) + -.3cos(πFt) + T/ Fundamental

More information

Continuous-Time Fourier Transform

Continuous-Time Fourier Transform Signals and Systems Continuous-Time Fourier Transform Chang-Su Kim continuous time discrete time periodic (series) CTFS DTFS aperiodic (transform) CTFT DTFT Lowpass Filtering Blurring or Smoothing Original

More information

Lecture 7 ELE 301: Signals and Systems

Lecture 7 ELE 301: Signals and Systems Lecture 7 ELE 30: Signals and Systems Prof. Paul Cuff Princeton University Fall 20-2 Cuff (Lecture 7) ELE 30: Signals and Systems Fall 20-2 / 22 Introduction to Fourier Transforms Fourier transform as

More information

Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, Prof. Young-Chai Ko

Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, Prof. Young-Chai Ko Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, 20 Prof. Young-Chai Ko koyc@korea.ac.kr Summary Fourier transform Properties Fourier transform of special function Fourier

More information

Laplace Transform Part 1: Introduction (I&N Chap 13)

Laplace Transform Part 1: Introduction (I&N Chap 13) Laplace Transform Part 1: Introduction (I&N Chap 13) Definition of the L.T. L.T. of Singularity Functions L.T. Pairs Properties of the L.T. Inverse L.T. Convolution IVT(initial value theorem) & FVT (final

More information

信號與系統 Signals and Systems

信號與系統 Signals and Systems Spring 2011 信號與系統 Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform Feng-Li Lian NTU-EE Feb11 Jun11 Figures and images used in these lecture notes are adopted from Signals & Systems

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

EC Signals and Systems

EC Signals and Systems UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J

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

7 The Waveform Channel

7 The Waveform Channel 7 The Waveform Channel The waveform transmitted by the digital demodulator will be corrupted by the channel before it reaches the digital demodulator in the receiver. One important part of the channel

More information

LTI Systems (Continuous & Discrete) - Basics

LTI Systems (Continuous & Discrete) - Basics LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying

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

LECTURE 12 Sections Introduction to the Fourier series of periodic signals

LECTURE 12 Sections Introduction to the Fourier series of periodic signals Signals and Systems I Wednesday, February 11, 29 LECURE 12 Sections 3.1-3.3 Introduction to the Fourier series of periodic signals Chapter 3: Fourier Series of periodic signals 3. Introduction 3.1 Historical

More information

Signals and Systems I Have Known and Loved. (Lecture notes for CSE 3451) Andrew W. Eckford

Signals and Systems I Have Known and Loved. (Lecture notes for CSE 3451) Andrew W. Eckford Signals and Systems I Have Known and Loved (Lecture notes for CSE 3451) Andrew W. Eckford Department of Electrical Engineering and Computer Science York University, Toronto, Ontario, Canada Version: December

More information

1 Signals and systems

1 Signals and systems 978--52-5688-4 - Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis Signals and systems In the first two chapters we will consider some basic concepts and ideas as

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

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

2 Frequency-Domain Analysis

2 Frequency-Domain Analysis 2 requency-domain Analysis Electrical engineers live in the two worlds, so to speak, of time and frequency. requency-domain analysis is an extremely valuable tool to the communications engineer, more so

More information

Functions not limited in time

Functions not limited in time Functions not limited in time We can segment an arbitrary L 2 function into segments of width T. The mth segment is u m (t) = u(t)rect(t/t m). We then have m 0 u(t) =l.i.m. m0 u m (t) m= m 0 This wors

More information

EA2.3 - Electronics 2 1

EA2.3 - Electronics 2 1 In the previous lecture, I talked about the idea of complex frequency s, where s = σ + jω. Using such concept of complex frequency allows us to analyse signals and systems with better generality. In this

More information

Review of Linear Time-Invariant Network Analysis

Review of Linear Time-Invariant Network Analysis D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x

More information

Fourier Series. Spectral Analysis of Periodic Signals

Fourier Series. Spectral Analysis of Periodic Signals Fourier Series. Spectral Analysis of Periodic Signals he response of continuous-time linear invariant systems to the complex exponential with unitary magnitude response of a continuous-time LI system at

More information

The Johns Hopkins University Department of Electrical and Computer Engineering Introduction to Linear Systems Fall 2002.

The Johns Hopkins University Department of Electrical and Computer Engineering Introduction to Linear Systems Fall 2002. The Johns Hopkins University Department of Electrical and Computer Engineering 505.460 Introduction to Linear Systems Fall 2002 Final exam Name: You are allowed to use: 1. Table 3.1 (page 206) & Table

More information

Ver 3808 E1.10 Fourier Series and Transforms (2014) E1.10 Fourier Series and Transforms. Problem Sheet 1 (Lecture 1)

Ver 3808 E1.10 Fourier Series and Transforms (2014) E1.10 Fourier Series and Transforms. Problem Sheet 1 (Lecture 1) Ver 88 E. Fourier Series and Transforms 4 Key: [A] easy... [E]hard Questions from RBH textbook: 4., 4.8. E. Fourier Series and Transforms Problem Sheet Lecture. [B] Using the geometric progression formula,

More information

Introduction to Fourier Transforms. Lecture 7 ELE 301: Signals and Systems. Fourier Series. Rect Example

Introduction to Fourier Transforms. Lecture 7 ELE 301: Signals and Systems. Fourier Series. Rect Example Introduction to Fourier ransforms Lecture 7 ELE 3: Signals and Systems Fourier transform as a limit of the Fourier series Inverse Fourier transform: he Fourier integral theorem Prof. Paul Cuff Princeton

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

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

e st f (t) dt = e st tf(t) dt = L {t f(t)} s

e st f (t) dt = e st tf(t) dt = L {t f(t)} s Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic

More information

A system that is both linear and time-invariant is called linear time-invariant (LTI).

A system that is both linear and time-invariant is called linear time-invariant (LTI). The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped

More information

be a deterministic function that satisfies x( t) dt. Then its Fourier

be a deterministic function that satisfies x( t) dt. Then its Fourier Lecture Fourier ransforms and Applications Definition Let ( t) ; t (, ) be a deterministic function that satisfies ( t) dt hen its Fourier it ransform is defined as X ( ) ( t) e dt ( )( ) heorem he inverse

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

Advanced Analog Building Blocks. Prof. Dr. Peter Fischer, Dr. Wei Shen, Dr. Albert Comerma, Dr. Johannes Schemmel, etc

Advanced Analog Building Blocks. Prof. Dr. Peter Fischer, Dr. Wei Shen, Dr. Albert Comerma, Dr. Johannes Schemmel, etc Advanced Analog Building Blocks Prof. Dr. Peter Fischer, Dr. Wei Shen, Dr. Albert Comerma, Dr. Johannes Schemmel, etc 1 Topics 1. S domain and Laplace Transform Zeros and Poles 2. Basic and Advanced current

More information

Notes 07 largely plagiarized by %khc

Notes 07 largely plagiarized by %khc Notes 07 largely plagiarized by %khc Warning This set of notes covers the Fourier transform. However, i probably won t talk about everything here in section; instead i will highlight important properties

More information

ECE-700 Review. Phil Schniter. January 5, x c (t)e jωt dt, x[n]z n, Denoting a transform pair by x[n] X(z), some useful properties are

ECE-700 Review. Phil Schniter. January 5, x c (t)e jωt dt, x[n]z n, Denoting a transform pair by x[n] X(z), some useful properties are ECE-7 Review Phil Schniter January 5, 7 ransforms Using x c (t) to denote a continuous-time signal at time t R, Laplace ransform: X c (s) x c (t)e st dt, s C Continuous-ime Fourier ransform (CF): ote that:

More information

The Discrete-time Fourier Transform

The Discrete-time Fourier Transform The Discrete-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Representation of Aperiodic signals: The

More information

ENGIN 211, Engineering Math. Laplace Transforms

ENGIN 211, Engineering Math. Laplace Transforms ENGIN 211, Engineering Math Laplace Transforms 1 Why Laplace Transform? Laplace transform converts a function in the time domain to its frequency domain. It is a powerful, systematic method in solving

More information

Figure 1.1 (a) Model of a communication system, and (b) signal processing functions.

Figure 1.1 (a) Model of a communication system, and (b) signal processing functions. . Introduction to Signals and Operations Model of a Communication System [] Figure. (a) Model of a communication system, and (b) signal processing functions. Classification of Signals. Continuous-time

More information

2.1 Introduction 2.22 The Fourier Transform 2.3 Properties of The Fourier Transform 2.4 The Inverse Relationship between Time and Frequency 2.

2.1 Introduction 2.22 The Fourier Transform 2.3 Properties of The Fourier Transform 2.4 The Inverse Relationship between Time and Frequency 2. Chapter2 Fourier Theory and Communication Signals Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University Contents 2.1 Introduction 2.22

More information

Line Spectra and their Applications

Line Spectra and their Applications In [ ]: cd matlab pwd Line Spectra and their Applications Scope and Background Reading This session concludes our introduction to Fourier Series. Last time (http://nbviewer.jupyter.org/github/cpjobling/eg-47-

More information

Summary of Fourier Transform Properties

Summary of Fourier Transform Properties Summary of Fourier ransform Properties Frank R. Kschischang he Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of oronto January 7, 207 Definition and Some echnicalities

More information

The Laplace Transform

The Laplace Transform The Laplace Transform Generalizing the Fourier Transform The CTFT expresses a time-domain signal as a linear combination of complex sinusoids of the form e jωt. In the generalization of the CTFT to the

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

Review of Analog Signal Analysis

Review of Analog Signal Analysis Review of Analog Signal Analysis Chapter Intended Learning Outcomes: (i) Review of Fourier series which is used to analyze continuous-time periodic signals (ii) Review of Fourier transform which is used

More information

X. Chen More on Sampling

X. Chen More on Sampling X. Chen More on Sampling 9 More on Sampling 9.1 Notations denotes the sampling time in second. Ω s = 2π/ and Ω s /2 are, respectively, the sampling frequency and Nyquist frequency in rad/sec. Ω and ω denote,

More information

SYLLABUS. osmania university CHAPTER - 1 : TRANSIENT RESPONSE CHAPTER - 2 : LAPLACE TRANSFORM OF SIGNALS

SYLLABUS. osmania university CHAPTER - 1 : TRANSIENT RESPONSE CHAPTER - 2 : LAPLACE TRANSFORM OF SIGNALS i SYLLABUS osmania university UNIT - I CHAPTER - 1 : TRANSIENT RESPONSE Initial Conditions in Zero-Input Response of RC, RL and RLC Networks, Definitions of Unit Impulse, Unit Step and Ramp Functions,

More information

DSP-I DSP-I DSP-I DSP-I

DSP-I DSP-I DSP-I DSP-I NOTES FOR 8-79 LECTURES 3 and 4 Introduction to Discrete-Time Fourier Transforms (DTFTs Distributed: September 8, 2005 Notes: This handout contains in brief outline form the lecture notes used for 8-79

More information

Chapter 1 Fundamental Concepts

Chapter 1 Fundamental Concepts Chapter 1 Fundamental Concepts 1 Signals A signal is a pattern of variation of a physical quantity, often as a function of time (but also space, distance, position, etc). These quantities are usually the

More information

EE303: Communication Systems

EE303: Communication Systems EE303: Communication Systems Professor A. Manikas Chair of Communications and Array Processing Imperial College London Introductory Concepts Prof. A. Manikas (Imperial College) EE303: Introductory Concepts

More information

27. The Fourier Transform in optics, II

27. The Fourier Transform in optics, II 27. The Fourier Transform in optics, II Parseval s Theorem The Shift theorem Convolutions and the Convolution Theorem Autocorrelations and the Autocorrelation Theorem The Shah Function in optics The Fourier

More information

Then r (t) can be expanded into a linear combination of the complex exponential signals ( e j2π(kf 0)t ) k= as. c k e j2π(kf0)t + c k e j2π(kf 0)t

Then r (t) can be expanded into a linear combination of the complex exponential signals ( e j2π(kf 0)t ) k= as. c k e j2π(kf0)t + c k e j2π(kf 0)t .3 ourier Series Definition.37. Exponential ourier series: Let the real or complex signal r t be a periodic signal with period. Suppose the following Dirichlet conditions are satisfied: a r t is absolutely

More information

13.42 READING 6: SPECTRUM OF A RANDOM PROCESS 1. STATIONARY AND ERGODIC RANDOM PROCESSES

13.42 READING 6: SPECTRUM OF A RANDOM PROCESS 1. STATIONARY AND ERGODIC RANDOM PROCESSES 13.42 READING 6: SPECTRUM OF A RANDOM PROCESS SPRING 24 c A. H. TECHET & M.S. TRIANTAFYLLOU 1. STATIONARY AND ERGODIC RANDOM PROCESSES Given the random process y(ζ, t) we assume that the expected value

More information

Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n 1, Remarks:

Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n 1, Remarks: Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n, d n, t < δ n (t) = n, t 3 d3 d n, t > n. d t The Dirac delta generalized function

More information

Homework 7 Solution EE235, Spring Find the Fourier transform of the following signals using tables: te t u(t) h(t) = sin(2πt)e t u(t) (2)

Homework 7 Solution EE235, Spring Find the Fourier transform of the following signals using tables: te t u(t) h(t) = sin(2πt)e t u(t) (2) Homework 7 Solution EE35, Spring. Find the Fourier transform of the following signals using tables: (a) te t u(t) h(t) H(jω) te t u(t) ( + jω) (b) sin(πt)e t u(t) h(t) sin(πt)e t u(t) () h(t) ( ejπt e

More information

2.161 Signal Processing: Continuous and Discrete

2.161 Signal Processing: Continuous and Discrete MI OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 8 For information about citing these materials or our erms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSES INSIUE

More information

Frequency Domain Representations of Sampled and Wrapped Signals

Frequency Domain Representations of Sampled and Wrapped Signals Frequency Domain Representations of Sampled and Wrapped Signals Peter Kabal Department of Electrical & Computer Engineering McGill University Montreal, Canada v1.5 March 2011 c 2011 Peter Kabal 2011/03/11

More information

ANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE)

ANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE) 3. Linear System Response (general case) 3. INTRODUCTION In chapter 2, we determined that : a) If the system is linear (or operate in a linear domain) b) If the input signal can be assumed as periodic

More information

The Continuous Time Fourier Transform

The Continuous Time Fourier Transform COMM 401: Signals & Systems Theory Lecture 8 The Continuous Time Fourier Transform Fourier Transform Continuous time CT signals Discrete time DT signals Aperiodic signals nonperiodic periodic signals Aperiodic

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

06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1

06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1 IV. Continuous-Time Signals & LTI Systems [p. 3] Analog signal definition [p. 4] Periodic signal [p. 5] One-sided signal [p. 6] Finite length signal [p. 7] Impulse function [p. 9] Sampling property [p.11]

More information

Chapter 3 Convolution Representation

Chapter 3 Convolution Representation Chapter 3 Convolution Representation DT Unit-Impulse Response Consider the DT SISO system: xn [ ] System yn [ ] xn [ ] = δ[ n] If the input signal is and the system has no energy at n = 0, the output yn

More information