Linear time invariant systems
|
|
- Elvin Dixon
- 6 years ago
- Views:
Transcription
1 Liear time ivariat systems Alejadro Ribeiro Dept. of Electrical ad Systems Egieerig Uiversity of Pesylvaia February 25, 2016 Sigal ad Iformatio Processig Samplig 1
2 Liear time ivariat systems Liear time ivariat systems Fiite impulse respose filter desig Sigal ad Iformatio Processig Samplig 2
3 Fourier trasform ad covolutio Fourier trasform eables sigal ad iformatio processig Patters ad properties easier to discer o frequecy domai Also eables aalysis ad deig of liear time ivariat (LTI) systems Not altogether urelated to patter disceribility Two properties of LTI systems Characterized by their (impulse) respose to a delta iput Resposes to other iputs are covolutios with impulse respose Equivalet properties i the frequecy domai Characterized by frequecy respose = F(impulse respose) Output spectrum = iput spectrum frequecy respose Sigal ad Iformatio Processig Samplig 3
4 Systems A system is characterized by a iput (x()) output (y()) relatio This relatio is betwee fuctios, ot values Each output value y() depeds o all iput values x() x() System y() x() y() We ca, alteratively, cosider cotiuous time systems. The same. Sigal ad Iformatio Processig Samplig 4
5 Time ivariat systems A system is time ivariat if a delayed iput yields a delayed output If iput x() yields output y() the iput x( k) yields y( k) Thik of output whe iput is applied k time uits later x( k) System y( k) x() y() x( k) y( k) Sigal ad Iformatio Processig Samplig 5
6 Liear systems I a liear system iput a liear combiatio of iputs Output the same liear combiatio of the respective outputs I.e., if iput x 1 () yields output y 1 () ad x 2 () yields y 2 () Iput a 1 x 1 () + a 2 x 2 () yields output a 1 y 1 () + a 2 y 2 () a 1x 1() + a 2x 2() System a 1y 1() + a 2y 2() x 1() y 1() x 2() y 2() a 1x 1() + a 2x 2() a 1y 1() + a 2y 2() Sigal ad Iformatio Processig Samplig 6
7 Liear time ivariat systems Liear + time ivariat system = liear time ivariat system (LTI) Also called a LTI filter, or a liear filter, or simply a filter The impulse respose is the output whe iput is a delta fuctio Iput is x() = δ() (discrete time, δ(0) = 1) Output is y() = h() = impulse respose δ() System h() δ() h() Sigal ad Iformatio Processig Samplig 7
8 Scale ad shifted impulse resposes Sice the system is time ivariat (shift) Iput δ( k) Iduces output respose h( k) Sice the system is liear (scale) iput x(k)δ( k) Output x(k)h( k) Sice the system is liear (sum) x(k 1)δ( k 1) + x(k 2)δ( k 2) x(k 1)h( k 1) + x(k 2)h( k 2) δ() System h() δ() h() Sigal ad Iformatio Processig Samplig 8
9 Scale ad shifted impulse resposes Sice the system is time ivariat (shift) Iput δ( k) Iduces output respose h( k) Sice the system is liear (scale) iput x(k)δ( k) Output x(k)h( k) Sice the system is liear (sum) x(k 1)δ( k 1) + x(k 2)δ( k 2) x(k 1)h( k 1) + x(k 2)h( k 2) δ( k) System h( k) δ( k) h( k) Sigal ad Iformatio Processig Samplig 9
10 Scale ad shifted impulse resposes Sice the system is time ivariat (shift) Iput δ( k) Iduces output respose h( k) Sice the system is liear (scale) iput x(k)δ( k) Output x(k)h( k) Sice the system is liear (sum) x(k 1)δ( k 1) + x(k 2)δ( k 2) x(k 1)h( k 1) + x(k 2)h( k 2) δ( k) System h( k) x(k)δ( k) x(k)h( k) Sigal ad Iformatio Processig Samplig 10
11 Scale ad shifted impulse resposes Sice the system is time ivariat (shift) Iput δ( k) Iduces output respose h( k) Sice the system is liear (scale) iput x(k)δ( k) Output x(k)h( k) Sice the system is liear (sum) x(k 1)δ( k 1) + x(k 2)δ( k 2) x(k 1)h( k 1) + x(k 2)h( k 2) δ( k) x(k 1)δ( k 1) + x(k 2)δ( k 2) System h( k) x(k 1)h( k 1) + x(k 2)h( k 2) Sigal ad Iformatio Processig Samplig 11
12 Output of a liear time ivariat system Shift, Scale, ad Sum Is this a Covolutio? Of course Ca write ay sigal x as x() = + k= x(k)δ( k) Thus, output of LTI with impulse respose h to iput x is give by y() = + k= x(k)h( k) The above sum is the covolutio of x ad h y = x h Sigal ad Iformatio Processig Samplig 12
13 Output of a liear time ivariat system Theorem A liear time ivariat system is completely determied by its impulse respose h. I particular, the respose to iput x is the sigal y = x h. Iocet lookig restrictios Liearity + time ivariace Iduce very strog structure (aythig but iocet) x() h() (x h)() = x(k)h(t k) Ca derive exact same result for cotiuous time systems Sigal ad Iformatio Processig Samplig 13
14 Frequecy respose Frequecy respose = trasform of impulse respose H = F(h) Corollary A liear time ivariat system is completely determied by its frequecy respose H. I particular, the respose to iput X is the sigal Y = HX. X (f ) H(f ) Y (f ) = H(f )X (f ) Desig i frequecy Implemet i time Have doe this already, but ow we kow its true for ay LTI Sigal ad Iformatio Processig Samplig 14
15 Causality A causal filter is oe with h() = 0 for all egative < 0 Otherwise, we would respod to spike before seeig spike I geeral y() = + x(k)h( k) = x(k)h( k) k= k= The value y() is oly affected by past iputs x(k), with k If filter is ot causal but h() = 0 for all < N Make it causal with a delay h() = h( N) Frequecy respose of delayed filter H(f ) = H(f )e j2πfn Qualitatively the same filter Sigal ad Iformatio Processig Samplig 15
16 Fiite impulse respose A causal fiite impulse respose filter (FIR) is oe for which h() = 0 for all N We say the filter is of legth N; oly N values i h() are ot ull Ca write output at time as y() = h(0)x() + h(1)x( 1) +... h(n 1)x( N + 1) Ruig iput vector x N () = [x(); x( 1);... ; x( N + 1)] FIR filter vector respose h = [h(0), h(1),..., h(n 1)] Ca the write output at time as y() = h T x N Sigal ad Iformatio Processig Samplig 16
17 Fiite impulse respose filter desig Liear time ivariat systems Fiite impulse respose filter desig Sigal ad Iformatio Processig Samplig 17
18 Filter desig ad implemetatio We wat to utilize a LTI system to process discrete time sigal x() E.g., to smooth out the sigal x() show below x() h() H(f ) y() x() y() All LTIs are completely determied by their impulse resposes h Desig h ad implemet filter as time covolutio y = x h All LTIs are completely determied by their frequecy resposes h Desig H ad implemet filter as spectral product Y = HX Sigal ad Iformatio Processig Samplig 18
19 Frequecy desig ad time implemetatio Time ad frequecy represetatios are equivalet x() h() y() = (x h)() F F 1 F F 1 F F 1 X (f ) H(f ) Y (f ) = H(f )X (f ) Idetify patter trasformatio i frequecy domai Desig H Use iverse DTFT to compute impulse respose h = F 1 (H) Implemet covolutio i time y() = (x h)() Sigal ad Iformatio Processig Samplig 19
20 Causality ad ifiite respose Impulse respose h = F 1 (H) is typically ot causal ad ifiite E.g., Low pass filter with cutoff freq. W /2 H(f ) = W (f ) h() = fs /2 f s /2 H(f )e j2πfts df = W sic(πwt s ) H(f ) = F (f ) 1 F 1 F h() W /2 W /2 f - 3 W - 2 W - 1 W 1 W 2 W 3 W t Multiply by widow (chop) for fiite respose with N ozero coeffs. Delay h() to obtai a causal filter with h() = 0 for 0 Sigal ad Iformatio Processig Samplig 20
21 FIR filter desig Trasform h() ito fiite impulse respose h w () = h()w() F h w () Widow w() = 0 for / [N mi, N max ] Filter legth N = N max N mi W - 2 W - 1 W 1 W 2 W 3 W t Trasform h w () ito causal respose h w () = h w ( N mi ) F h w ( N mi ) Choose borders N mi ad N max to retai highest values of h() Ofte, aroud = 0. But ot always - 3 W - 2 W - 1 W 1 W 2 W 3 W t Sigal ad Iformatio Processig Samplig 21
22 Spectral effects of widowig ad delayig Multiplicatio i time domai Covolutio i frequecy domai As a result, istead of filterig with H(f ), we filter with H w = H W Choose widows with spectrum W = F(w) close to delta fuctio Time delay Multiplicatio with complex expoetial i frequecy H w (f ) = H w (f )e j2πfn mit s Irrelevat, as it should, we just shifted the respose Sigal ad Iformatio Processig Samplig 22
23 FIR filter desig methodology Procedure to desig time coefficiets of a FIR filter (1) Spectral aalysis to determie filter frequecy respose H(f ) (2) Iverse DFT (ot DTFT) to determie impulse respose h() (3) Determie r. of coefficiets N ad coefficiet rage [N mi, N max ] (4) Select widow w() Alters spectrum to H w = H W (5) Shift impulse respose by N mi time steps to make filter causal How to we use FIR filter coefficiets h() to implemet the filter? Sigal ad Iformatio Processig Samplig 23
24 FIR implemetatio The output y() of the FIR filter is give by the covolutio value y() = x(k)h( k) k= Sice h is fiite ad causal, oly N ozero terms. Make k = l N 1 y() = x(k)h( k)= h(l)x( l) k= (N 1) l=0 Easier to visualize whe writte i expaded form y() = h(0) x() + h(1) x( 1) h(n 1) x( N + 1) The expressio above ca be implemeted with a shift register Sigal ad Iformatio Processig Samplig 24
25 Shift registers Upo arrival of sigal value x() we compute output value y() by Delay (shift) uits to shift elemets of sigal x Product (scale) uits to multiply with filter coefficiets x() Sum uits to aggregate the products h(k)x( k) x() T s x( 1) T s x( 2) T s x( 3) T s x( N +1) h(0) h(1) h(2) h(3) h(n 1) h(0)x() h(1)x( 1) h(2)x( 2) h(3)x( 3) h(n)x( N +1) Shift register ca be implemeted i hardware (or software) Sigal ad Iformatio Processig Samplig 25
26 Voice recogitio Spectral desig For a give word to be recogized we compare the spectra X ad X X Average spectrum magitude of word to be recogized X Recorded spectrum durig executio time 0.5 Average spectrum of spoke word oe frequecy (KHz) Made copariso with ier product X T X Equivalet to usig X to filter X Y (f ) = H(f )X (f ) with H(f ) = X Sigal ad Iformatio Processig Samplig 26
27 Voice recogitio Filter desig (2) Impulse respose h() Iverse DFT of X (4) Widow to keep N = 1, 000 largest cosecutive taps Sigal ad Iformatio Processig Samplig 27
2D DSP Basics: 2D Systems
- Digital Image Processig ad Compressio D DSP Basics: D Systems D Systems T[ ] y = T [ ] Liearity Additivity: If T y = T [ ] The + T y = y + y Homogeeity: If The T y = T [ ] a T y = ay = at [ ] Liearity
More informationELEG3503 Introduction to Digital Signal Processing
ELEG3503 Itroductio to Digital Sigal Processig 1 Itroductio 2 Basics of Sigals ad Systems 3 Fourier aalysis 4 Samplig 5 Liear time-ivariat (LTI) systems 6 z-trasform 7 System Aalysis 8 System Realizatio
More informationSignal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform
Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1 1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the
More informationFinite-length Discrete Transforms. Chapter 5, Sections
Fiite-legth Discrete Trasforms Chapter 5, Sectios 5.2-50 5.0 Dr. Iyad djafar Outlie The Discrete Fourier Trasform (DFT) Matrix Represetatio of DFT Fiite-legth Sequeces Circular Covolutio DFT Symmetry Properties
More informationx[0] x[1] x[2] Figure 2.1 Graphical representation of a discrete-time signal.
x[ ] x[ ] x[] x[] x[] x[] 9 8 7 6 5 4 3 3 4 5 6 7 8 9 Figure. Graphical represetatio of a discrete-time sigal. From Discrete-Time Sigal Processig, e by Oppeheim, Schafer, ad Buck 999- Pretice Hall, Ic.
More informationFrequency Response of FIR Filters
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steady-state
More informationFall 2011, EE123 Digital Signal Processing
Lecture 5 Miki Lustig, UCB September 14, 211 Miki Lustig, UCB Motivatios for Discrete Fourier Trasform Sampled represetatio i time ad frequecy umerical Fourier aalysis requires a Fourier represetatio that
More informationAnalog and Digital Signals. Introduction to Digital Signal Processing. Discrete-time Sinusoids. Analog and Digital Signals
Itroductio to Digital Sigal Processig Chapter : Itroductio Aalog ad Digital Sigals aalog = cotiuous-time cotiuous amplitude digital = discrete-time discrete amplitude cotiuous amplitude discrete amplitude
More informationOlli Simula T / Chapter 1 3. Olli Simula T / Chapter 1 5
Sigals ad Systems Sigals ad Systems Sigals are variables that carry iformatio Systemstake sigals as iputs ad produce sigals as outputs The course deals with the passage of sigals through systems T-6.4
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationCh3 Discrete Time Fourier Transform
Ch3 Discrete Time Fourier Trasform 3. Show that the DTFT of [] is give by ( k). e k 3. Determie the DTFT of the two sided sigal y [ ],. 3.3 Determie the DTFT of the causal sequece x[ ] A cos( 0 ) [ ],
More informationDiscrete-Time Signals and Systems. Discrete-Time Signals and Systems. Signal Symmetry. Elementary Discrete-Time Signals.
Discrete-ime Sigals ad Systems Discrete-ime Sigals ad Systems Dr. Deepa Kudur Uiversity of oroto Referece: Sectios. -.5 of Joh G. Proakis ad Dimitris G. Maolakis, Digital Sigal Processig: Priciples, Algorithms,
More informationDiscrete-Time Signals and Systems. Signals and Systems. Digital Signals. Discrete-Time Signals. Operations on Sequences: Basic Operations
-6.3 Digital Sigal Processig ad Filterig..8 Discrete-ime Sigals ad Systems ime-domai Represetatios of Discrete-ime Sigals ad Systems ime-domai represetatio of a discrete-time sigal as a sequece of umbers
More informationFIR Filter Design: Part II
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we cosider how we might go about desigig FIR filters with arbitrary frequecy resposes, through compositio of multiple sigle-peak
More informationDIGITAL SIGNAL PROCESSING LECTURE 3
DIGITAL SIGNAL PROCESSING LECTURE 3 Fall 2 2K8-5 th Semester Tahir Muhammad tmuhammad_7@yahoo.com Cotet ad Figures are from Discrete-Time Sigal Processig, 2e by Oppeheim, Shafer, ad Buc, 999-2 Pretice
More informationCOMM 602: Digital Signal Processing
COMM 60: Digital Sigal Processig Lecture 4 -Properties of LTIS Usig Z-Trasform -Iverse Z-Trasform Properties of LTIS Usig Z-Trasform Properties of LTIS Usig Z-Trasform -ve +ve Properties of LTIS Usig Z-Trasform
More informationQuestion1 Multiple choices (circle the most appropriate one):
Philadelphia Uiversity Studet Name: Faculty of Egieerig Studet Number: Dept. of Computer Egieerig Fial Exam, First Semester: 2014/2015 Course Title: Digital Sigal Aalysis ad Processig Date: 01/02/2015
More informationChapter 8. DFT : The Discrete Fourier Transform
Chapter 8 DFT : The Discrete Fourier Trasform Roots of Uity Defiitio: A th root of uity is a complex umber x such that x The th roots of uity are: ω, ω,, ω - where ω e π /. Proof: (ω ) (e π / ) (e π )
More informationSignal Processing. Lecture 02: Discrete Time Signals and Systems. Ahmet Taha Koru, Ph. D. Yildiz Technical University.
Sigal Processig Lecture 02: Discrete Time Sigals ad Systems Ahmet Taha Koru, Ph. D. Yildiz Techical Uiversity 2017-2018 Fall ATK (YTU) Sigal Processig 2017-2018 Fall 1 / 51 Discrete Time Sigals Discrete
More informationADVANCED DIGITAL SIGNAL PROCESSING
ADVANCED DIGITAL SIGNAL PROCESSING PROF. S. C. CHAN (email : sccha@eee.hku.hk, Rm. CYC-702) DISCRETE-TIME SIGNALS AND SYSTEMS MULTI-DIMENSIONAL SIGNALS AND SYSTEMS RANDOM PROCESSES AND APPLICATIONS ADAPTIVE
More informationMAS160: Signals, Systems & Information for Media Technology. Problem Set 5. DUE: November 3, (a) Plot of u[n] (b) Plot of x[n]=(0.
MAS6: Sigals, Systems & Iformatio for Media Techology Problem Set 5 DUE: November 3, 3 Istructors: V. Michael Bove, Jr. ad Rosalid Picard T.A. Jim McBride Problem : Uit-step ad ruig average (DSP First
More informationLecture 2 Linear and Time Invariant Systems
EE3054 Sigals ad Systems Lecture 2 Liear ad Time Ivariat Systems Yao Wag Polytechic Uiversity Most of the slides icluded are extracted from lecture presetatios prepared by McClella ad Schafer Licese Ifo
More informationEE123 Digital Signal Processing
Aoucemets HW solutios posted -- self gradig due HW2 due Friday EE2 Digital Sigal Processig ham radio licesig lectures Tue 6:-8pm Cory 2 Lecture 6 based o slides by J.M. Kah SDR give after GSI Wedesday
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science. BACKGROUND EXAM September 30, 2004.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departmet of Electrical Egieerig ad Computer Sciece 6.34 Discrete Time Sigal Processig Fall 24 BACKGROUND EXAM September 3, 24. Full Name: Note: This exam is closed
More informationSignals and Systems. Problem Set: From Continuous-Time to Discrete-Time
Sigals ad Systems Problem Set: From Cotiuous-Time to Discrete-Time Updated: October 5, 2017 Problem Set Problem 1 - Liearity ad Time-Ivariace Cosider the followig systems ad determie whether liearity ad
More informationEECE 301 Signals & Systems
EECE 301 Sigals & Systems Prof. Mark Fowler Note Set #8 D-T Covolutio: The Tool for Fidig the Zero-State Respose Readig Assigmet: Sectio 2.1-2.2 of Kame ad Heck 1/14 Course Flow Diagram The arrows here
More informationA. Basics of Discrete Fourier Transform
A. Basics of Discrete Fourier Trasform A.1. Defiitio of Discrete Fourier Trasform (8.5) A.2. Properties of Discrete Fourier Trasform (8.6) A.3. Spectral Aalysis of Cotiuous-Time Sigals Usig Discrete Fourier
More informationExponential Moving Average Pieter P
Expoetial Movig Average Pieter P Differece equatio The Differece equatio of a expoetial movig average lter is very simple: y[] x[] + (1 )y[ 1] I this equatio, y[] is the curret output, y[ 1] is the previous
More informationChapter 2 Systems and Signals
Chapter 2 Systems ad Sigals 1 Itroductio Discrete-Time Sigals: Sequeces Discrete-Time Systems Properties of Liear Time-Ivariat Systems Liear Costat-Coefficiet Differece Equatios Frequecy-Domai Represetatio
More informationComputing the output response of LTI Systems.
Computig the output respose of LTI Systems. By breaig or decomposig ad represetig the iput sigal to the LTI system ito terms of a liear combiatio of a set of basic sigals. Usig the superpositio property
More informationIntroduction to Signals and Systems, Part V: Lecture Summary
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationPractical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement
Practical Spectral Aaysis (cotiue) (from Boaz Porat s book) Frequecy Measuremet Oe of the most importat applicatios of the DFT is the measuremet of frequecies of periodic sigals (eg., siusoidal sigals),
More informationLecture 3. Digital Signal Processing. Chapter 3. z-transforms. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev. 2016
Lecture 3 Digital Sigal Processig Chapter 3 z-trasforms Mikael Swartlig Nedelko Grbic Begt Madersso rev. 06 Departmet of Electrical ad Iformatio Techology Lud Uiversity z-trasforms We defie the z-trasform
More informationFIR Filters. Lecture #7 Chapter 5. BME 310 Biomedical Computing - J.Schesser
FIR Filters Lecture #7 Chapter 5 8 What Is this Course All About? To Gai a Appreciatio of the Various Types of Sigals ad Systems To Aalyze The Various Types of Systems To Lear the Skills ad Tools eeded
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More information6.003 Homework #12 Solutions
6.003 Homework # Solutios Problems. Which are rue? For each of the D sigals x [] through x 4 [] (below), determie whether the coditios listed i the followig table are satisfied, ad aswer for true or F
More informationELEG 4603/5173L Digital Signal Processing Ch. 1 Discrete-Time Signals and Systems
Departmet of Electrical Egieerig Uiversity of Arasas ELEG 4603/5173L Digital Sigal Processig Ch. 1 Discrete-Time Sigals ad Systems Dr. Jigxia Wu wuj@uar.edu OUTLINE 2 Classificatios of discrete-time sigals
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 06 Summer 07 Problem Set #5 Assiged: Jue 3, 07 Due Date: Jue 30, 07 Readig: Chapter 5 o FIR Filters. PROBLEM 5..* (The
More information6.003 Homework #12 Solutions
6.003 Homework # Solutios Problems. Which are rue? For each of the D sigals x [] through x 4 [] below), determie whether the coditios listed i the followig table are satisfied, ad aswer for true or F for
More informationFrequency Domain Filtering
Frequecy Domai Filterig Raga Rodrigo October 19, 2010 Outlie Cotets 1 Itroductio 1 2 Fourier Represetatio of Fiite-Duratio Sequeces: The Discrete Fourier Trasform 1 3 The 2-D Discrete Fourier Trasform
More informationThe Discrete Fourier Transform
The iscrete Fourier Trasform The discrete-time Fourier trasform (TFT) of a sequece is a cotiuous fuctio of!, ad repeats with period. I practice we usually wat to obtai the Fourier compoets usig digital
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 40 Digital Sigal Processig Prof. Mark Fowler Note Set #3 Covolutio & Impulse Respose Review Readig Assigmet: Sect. 2.3 of Proakis & Maolakis / Covolutio for LTI D-T systems We are tryig to fid y(t)
More informationExam. Notes: A single A4 sheet of paper (double sided; hand-written or computer typed)
Exam February 8th, 8 Sigals & Systems (5-575-) Prof. R. D Adrea Exam Exam Duratio: 5 Mi Number of Problems: 5 Number of Poits: 5 Permitted aids: Importat: Notes: A sigle A sheet of paper (double sided;
More informationECE4270 Fundamentals of DSP. Lecture 2 Discrete-Time Signals and Systems & Difference Equations. Overview of Lecture 2. More Discrete-Time Systems
ECE4270 Fudametals of DSP Lecture 2 Discrete-Time Sigals ad Systems & Differece Equatios School of ECE Ceter for Sigal ad Iformatio Processig Georgia Istitute of Techology Overview of Lecture 2 Aoucemet
More informationECE 308 Discrete-Time Signals and Systems
ECE 38-5 ECE 38 Discrete-Time Sigals ad Systems Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa ECE 38-5 1 Additio, Multiplicatio, ad Scalig of Sequeces Amplitude Scalig: (A Costat
More informationChapter 7: The z-transform. Chih-Wei Liu
Chapter 7: The -Trasform Chih-Wei Liu Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability
More informationEE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course
Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL
More informationTime-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
More informationModule 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation
Module 8 Discrete Time Sigals ad Z-Trasforms Objective:To uderstad represetig discrete time sigals, apply z trasform for aalyzigdiscrete time sigals ad to uderstad the relatio to Fourier trasform Itroductio
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationFilter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and
Filter bas Separately, the lowpass ad highpass filters are ot ivertible T removes the highest frequecy / ad removes the lowest frequecy Together these filters separate the sigal ito low-frequecy ad high-frequecy
More informationThe z-transform can be used to obtain compact transform-domain representations of signals and systems. It
3 4 5 6 7 8 9 10 CHAPTER 3 11 THE Z-TRANSFORM 31 INTRODUCTION The z-trasform ca be used to obtai compact trasform-domai represetatios of sigals ad systems It provides ituitio particularly i LTI system
More informationFIR Filter Design: Part I
EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I. Itroductio FIR Filter Desig: Part I I this set o otes, we cotiue our exploratio o the requecy respose o FIR ilters. First, we cosider some
More informationZ - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.
Z - Trasform The -trasform is a very importat tool i describig ad aalyig digital systems. It offers the techiques for digital filter desig ad frequecy aalysis of digital sigals. Defiitio of -trasform:
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationM2.The Z-Transform and its Properties
M2.The Z-Trasform ad its Properties Readig Material: Page 94-126 of chapter 3 3/22/2011 I. Discrete-Time Sigals ad Systems 1 What did we talk about i MM1? MM1 - Discrete-Time Sigal ad System 3/22/2011
More informationDiscrete-time signals and systems See Oppenheim and Schafer, Second Edition pages 8 93, or First Edition pages 8 79.
Discrete-time sigals ad systems See Oppeheim ad Schafer, Secod Editio pages 93, or First Editio pages 79. Discrete-time sigals A discrete-time sigal is represeted as a sequece of umbers: x D fxœg; <
More informationSolutions. Number of Problems: 4. None. Use only the prepared sheets for your solutions. Additional paper is available from the supervisors.
Quiz November 4th, 23 Sigals & Systems (5-575-) P. Reist & Prof. R. D Adrea Solutios Exam Duratio: 4 miutes Number of Problems: 4 Permitted aids: Noe. Use oly the prepared sheets for your solutios. Additioal
More informationSolution of EECS 315 Final Examination F09
Solutio of EECS 315 Fial Examiatio F9 1. Fid the umerical value of δ ( t + 4ramp( tdt. δ ( t + 4ramp( tdt. Fid the umerical sigal eergy of x E x = x[ ] = δ 3 = 11 = ( = ramp( ( 4 = ramp( 8 = 8 [ ] = (
More informationEE123 Digital Signal Processing
EE123 Digital Sigal Processig Lecture 20 Filter Desig Liear Filter Desig Used to be a art Now, lots of tools to desig optimal filters For DSP there are two commo classes Ifiite impulse respose IIR Fiite
More informationIntroduction to Digital Signal Processing
Fakultät Iformatik Istitut für Systemarchitektur Professur Recheretze Itroductio to Digital Sigal Processig Walteegus Dargie Walteegus Dargie TU Dresde Chair of Computer Networks I 45 Miutes Refereces
More informationSignals & Systems Chapter3
Sigals & Systems Chapter3 1.2 Discrete-Time (D-T) Sigals Electroic systems do most of the processig of a sigal usig a computer. A computer ca t directly process a C-T sigal but istead eeds a stream of
More informationImage pyramid example
Multiresolutio image processig Laplacia pyramids Discrete Wavelet Trasform (DWT) Quadrature mirror filters ad cojugate quadrature filters Liftig ad reversible wavelet trasform Wavelet theory Berd Girod:
More information(, ) (, ) (, ) ( ) ( )
PROBLEM ANSWER X Y x, x, rect, () X Y, otherwise D Fourier trasform is defied as ad i D case it ca be defied as We ca write give fuctio from Eq. () as It follows usig Eq. (3) it ( ) ( ) F f t e dt () i(
More informationDigital signal processing: Lecture 5. z-transformation - I. Produced by Qiangfu Zhao (Since 1995), All rights reserved
Digital sigal processig: Lecture 5 -trasformatio - I Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/ Review of last lecture Fourier trasform & iverse Fourier trasform: Time domai & Frequecy
More informationChapter 7 z-transform
Chapter 7 -Trasform Itroductio Trasform Uilateral Trasform Properties Uilateral Trasform Iversio of Uilateral Trasform Determiig the Frequecy Respose from Poles ad Zeros Itroductio Role i Discrete-Time
More informationChapter 4 : Laplace Transform
4. Itroductio Laplace trasform is a alterative to solve the differetial equatio by the complex frequecy domai ( s = σ + jω), istead of the usual time domai. The DE ca be easily trasformed ito a algebraic
More informationEE422G Homework #13 (12 points)
EE422G Homework #1 (12 poits) 1. (5 poits) I this problem, you are asked to explore a importat applicatio of FFT: efficiet computatio of covolutio. The impulse respose of a system is give by h(t) (.9),1,2,,1
More informationWritten exam Digital Signal Processing for BMT (8E070). Tuesday November 1, 2011, 09:00 12:00.
Techische Uiversiteit Eidhove Fac. Biomedical Egieerig Writte exam Digital Sigal Processig for BMT (8E070). Tuesday November, 0, 09:00 :00. (oe page) ( problems) Problem. s Cosider a aalog filter with
More informationSpring 2014, EE123 Digital Signal Processing
Aoucemets EE3 Digital Sigal Processig Last time: FF oday: Frequecy aalysis with DF Widowig Effect of zero-paddig Lecture 9 based o slides by J.M. Kah Spectral Aalysis with the DF Spectral Aalysis with
More informationDigital Signal Processing, Fall 2006
Digital Sigal Processig, Fall 26 Lecture 1: Itroductio, Discrete-time sigals ad systems Zheg-Hua Ta Departmet of Electroic Systems Aalborg Uiversity, Demark zt@kom.aau.dk 1 Part I: Itroductio Itroductio
More informationRun-length & Entropy Coding. Redundancy Removal. Sampling. Quantization. Perform inverse operations at the receiver EEE
Geeral e Image Coder Structure Motio Video (s 1,s 2,t) or (s 1,s 2 ) Natural Image Samplig A form of data compressio; usually lossless, but ca be lossy Redudacy Removal Lossless compressio: predictive
More informationNotes 20 largely plagiarized by %khc
1 Notes 20 largely plagiarized by %khc 1 Warig This set of otes covers discrete time. However, i probably wo t be able to talk about everythig here; istead i will highlight importat properties or give
More informationDefinition of z-transform.
- Trasforms Frequecy domai represetatios of discretetime sigals ad LTI discrete-time systems are made possible with the use of DTFT. However ot all discrete-time sigals e.g. uit step sequece are guarateed
More informationChapter 3. z-transform
Chapter 3 -Trasform 3.0 Itroductio The -Trasform has the same role as that played by the Laplace Trasform i the cotiuous-time theorem. It is a liear operator that is useful for aalyig LTI systems such
More informationGeneralizing the DTFT. The z Transform. Complex Exponential Excitation. The Transfer Function. Systems Described by Difference Equations
Geeraliig the DTFT The Trasform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1 The forward DTFT is defied by X e jω = x e jω i which = Ω is discrete-time radia frequecy, a real variable.
More informationON THE EXISTENCE OF A GROUP ORTHONORMAL BASIS. Peter Zizler (Received 20 June, 2014)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 45 (2015, 45-52 ON THE EXISTENCE OF A GROUP ORTHONORMAL BASIS Peter Zizler (Received 20 Jue, 2014 Abstract. Let G be a fiite group ad let l 2 (G be a fiite dimesioal
More informationWavelet Transform and its relation to multirate filter banks
Wavelet Trasform ad its relatio to multirate filter bas Christia Walliger ASP Semiar th Jue 007 Graz Uiversity of Techology, Austria Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07
More informationSolution of Linear Constant-Coefficient Difference Equations
ECE 38-9 Solutio of Liear Costat-Coefficiet Differece Equatios Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa Solutio of Liear Costat-Coefficiet Differece Equatios Example: Determie
More informationUNIT-I. 2. A real valued sequence x(n) is anti symmetric if a) X(n)=x(-n) b) X(n)=-x(-n) c) A) and b) d) None Ans: b)
DIGITAL SIGNAL PROCESSING UNIT-I 1. The uit ramp sequece is Eergy sigal b) Power sigal c) Either Eergy or Power sigal d) Neither a Power sigal or a eergy sigal As: d) 2. A real valued sequece x() is ati
More informationEE123 Digital Signal Processing
Discrete Time Sigals Samples of a CT sigal: EE123 Digital Sigal Processig x[] =X a (T ) =1, 2, x[0] x[2] x[1] X a (t) T 2T 3T t Lecture 2 Or, iheretly discrete (Examples?) 1 2 Basic Sequeces Uit Impulse
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationMorphological Image Processing
Morphological Image Processig Biary dilatio ad erosio Set-theoretic iterpretatio Opeig, closig, morphological edge detectors Hit-miss filter Morphological filters for gray-level images Cascadig dilatios
More informationThe Z-Transform. Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, Prentice Hall Inc.
The Z-Trasform Cotet ad Figures are from Discrete-Time Sigal Processig, e by Oppeheim, Shafer, ad Buck, 999- Pretice Hall Ic. The -Trasform Couterpart of the Laplace trasform for discrete-time sigals Geeraliatio
More informationAdvanced Training Course on FPGA Design and VHDL for Hardware Simulation and Synthesis
265-25 Advaced Traiig Course o FPGA esig ad VHL for Hardware Simulatio ad Sythesis 26 October - 2 ovember, 29 igital Sigal Processig The iscrete Fourier Trasform Massimiliao olich EEI Facolta' di Igegeria
More informationDigital Signal Processing
Digital Sigal Processig Z-trasform dftwave -Trasform Backgroud-Defiitio - Fourier trasform j ω j ω e x e extracts the essece of x but is limited i the sese that it ca hadle stable systems oly. jω e coverges
More informationT Signal Processing Systems Exercise material for autumn Solutions start from Page 16.
T-6.40 P (Problems&olutios, autum 003) Page / 9 T-6.40 P (Problems&olutios, autum 003) Page / 9 T-6.40 igal Processig ystems Exercise material for autum 003 - olutios start from Page 6.. Basics of complex
More informationELEC1200: A System View of Communications: from Signals to Packets Lecture 3
ELEC2: A System View of Commuicatios: from Sigals to Packets Lecture 3 Commuicatio chaels Discrete time Chael Modelig the chael Liear Time Ivariat Systems Step Respose Respose to sigle bit Respose to geeral
More informationSignal Processing in Mechatronics
Sigal Processig i Mechatroics Zhu K.P. AIS, UM. Lecture, Brief itroductio to Sigals ad Systems, Review of Liear Algebra ad Sigal Processig Related Mathematics . Brief Itroductio to Sigals What is sigal
More informationThe Z-Transform. (t-t 0 ) Figure 1: Simplified graph of an impulse function. For an impulse, it can be shown that (1)
The Z-Trasform Sampled Data The geeralied fuctio (t) (also kow as the impulse fuctio) is useful i the defiitio ad aalysis of sampled-data sigals. Figure below shows a simplified graph of a impulse. (t-t
More informationEE123 Digital Signal Processing
Today EE123 Digital Sigal Processig Lecture 2 Last time: Admiistratio Overview Today: Aother demo Ch. 2 - Discrete-Time Sigals ad Systems 1 2 Discrete Time Sigals Samples of a CT sigal: x[] =X a (T ) =1,
More informationLecture 3: Divide and Conquer: Fast Fourier Transform
Lecture 3: Divide ad Coquer: Fast Fourier Trasform Polyomial Operatios vs. Represetatios Divide ad Coquer Algorithm Collapsig Samples / Roots of Uity FFT, IFFT, ad Polyomial Multiplicatio Polyomial operatios
More informationEE Midterm Test 1 - Solutions
EE35 - Midterm Test - Solutios Total Poits: 5+ 6 Bous Poits Time: hour. ( poits) Cosider the parallel itercoectio of the two causal systems, System ad System 2, show below. System x[] + y[] System 2 The
More informationDiscrete-time Fourier transform (DTFT) of aperiodic and periodic signals
5 Discrete-time Fourier trasform (DTFT) of aperiodic ad periodic sigals We started with Fourier series which ca represet a periodic sigal usig siusoids. Fourier Trasform, a extesio of the Fourier series
More informationFIR Filter Design by Windowing
FIR Filter Desig by Widowig Take the low-pass filter as a eample of filter desig. FIR filters are almost etirely restricted to discretetime implemetatios. Passbad ad stopbad Magitude respose of a ideal
More informationFFTs in Graphics and Vision. The Fast Fourier Transform
FFTs i Graphics ad Visio The Fast Fourier Trasform 1 Outlie The FFT Algorithm Applicatios i 1D Multi-Dimesioal FFTs More Applicatios Real FFTs 2 Computatioal Complexity To compute the movig dot-product
More informationMathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1
Mathematical Descriptio of Discrete-Time Sigals 9/10/16 M. J. Roberts - All Rights Reserved 1 Samplig ad Discrete Time Samplig is the acquisitio of the values of a cotiuous-time sigal at discrete poits
More informationOptimum LMSE Discrete Transform
Image Trasformatio Two-dimesioal image trasforms are extremely importat areas of study i image processig. The image output i the trasformed space may be aalyzed, iterpreted, ad further processed for implemetig
More informationWeb Appendix O - Derivations of the Properties of the z Transform
M. J. Roberts - 2/18/07 Web Appedix O - Derivatios of the Properties of the z Trasform O.1 Liearity Let z = x + y where ad are costats. The ( z)= ( x + y )z = x z + y z ad the liearity property is O.2
More information