Digital Signal Processing Lecture 4

Size: px
Start display at page:

Download "Digital Signal Processing Lecture 4"

Transcription

1 Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I Povo, Trento, Italy Digital Signal Processing Lecture 4 Begüm Demir demir@disi.unitn.it Web page:

2 Linear Time-Invariant (LTI) Systems Impulse Response [ n] Discrete Time System hn [ ] yn [ ] xn [ k] k 0 hn [ ] [ n k] k 0 hn [ ] un [ ] yn [ ] xl [ ] n 0 hn [ ] 0 n 0 hn [ ] un [ ] 2 n l yn [ ] axn 1 [ ] axn 2 [ 1] axn 3 [ 2] hn [ ] a[ n] a[ n1] a[ n2]

3 Linear Time-Invariant (LTI) Systems [ n] h[ n] [ nk] h[ nk] x[ k] [ n k] x[ k] h[ nk] k k xn [ ] yn [ ] Convolution sum: yn [ ] xkhn [ ] [ k] k 3

4 Linear Time-Invariant Systems LTI Systems: Convolution

5 Example n xn [ ] (0.2) un [ ] n hn [ ] (0.6) un [ ] yn [ ] xn [ ] hn [ ]? y n u n n1 n1 [ ] 2.5( ) [ ] 5

6 Convolution Sum-Example The output of an LTI system can be obtained as the superposition of responses to individual samples of the input. This approach is shown to estimate y[n] in the case of x[n] and h[n] given in the following: y[n]?

7 Convolution Sum-Example-cont

8 Convolution Sum-Example-cont

9 Example y[n]? 9

10 Example-cont

11 Example-cont

12 LTI Systems-Convolution The output sequence y[n] can be also obtained by multiplying the input sequence (expressed as a function of k) by the sequence whose values are h[n k], <k< for any fixed value of n, and then summing all the values of the products x[k]h[n k], with k a counting index in the summation process. Therefore, the operation of convolving two sequences involves doing the computation specified by k y [ n] x[ k] h[ n k] for each value of n, thus generating the complete output sequence y[n], < n<.

13 LTI Systems-Convolution The process of computing the convolution between x[n] and h[n] involves the following steps: 1. Express x[n] and h[n] as a function of k and obtain x[k] and h[k] 2. Obtain h[-k] from h[k]. 3. Shift h[-k] by n 0 to the right if n is positive to obtain h[n 0 k] (shift h[- k] by n 0 to the left if n is negative). 4. Multiply x [k] by h[n 0 -k] to obtain the product sequence. 5. Sum all the values of the product sequence to obtain the value of the output at time n= n 0. The steps from 3 to 5 should be repeated for each values of n 0 (i.e., for any fixed value of n). 13

14 LTI Systems-Convolution k y [ n] x[ k] h[ n k] 14

15 LTI Systems-Convolution 15

16 LTI Systems-Convolution For each n value, estimate h[n-k], and then multiply with x[k] to estimate y[n]. 16

17 LTI Systems-Convolution 17

18 LTI Systems-Convolution yn [ ] xn [ ]* hn [ ] xkhn [ ] [ k] k x[k] k h[k] k h[0k] k

19 LTI Systems-Convolution How to compute y[n]? x[k] h[0k] h[1k] k compute y[0] k compute y[1] k For each n value, estimate h[n-k], and then multiply with x[k] to estimate y[n].

20 Example xn [ ] hn [ ] yn [ ]?

21 Example hn [ ] xn [ ] yn [ ]?

22 Convolution Features commutative: x [ n] x [ n] x [ n] x[ n] associative: x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] distributive: x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] x [ n] c [ n] x1[ n m] x2[ nk] c2[ nmk] x1[ n] [ n] x1[ n] if x [ n] has m samples and x [ n] k samples, 1 2 cn [ ] x[ n] x[ n] will have mk-1 samples. 1 2 x [ n] [ n] x [ n] 1 1

23 LTI Systems-Parallel Connection of Systems h 1 [n] x[n] y[n] x[n] h 1 [n]+h 2 [n] y[n] h 2 [n]

24 LTI Systems-Cascade Connection of Systems x[n] h 1 [n] h 2 [n] y[n] x[n] h 1 [n]*h 2 [n] y[n]

25 LTI Systems-Inverse Systems x[n] h[n] y[n] h'[n] x[n] hn [ ] h'[ n] [ n]

26 Example-Inverse Systems yn [ ] xn [ k] k0 hn [ ] [ nk] un [ ] k0 h'[ n]? h'[ n] [ nk] [ n] k0 k0 h'[ nk] [ n] h'[ n] h'[ n1] h'[ n2]... [ n] h'[0] 1 h'[1] h'[0] 0 h'[1] 1 h'[2] 0, h'[ n] 0, n1 h'[ n] [ n] [ n1]

27 Stability of Systems All LTI systems are BIBO stable if and only if h[n] is absolutely summable: This is due to the fact that: k hk [ ] yn [ ] hkxn [ ] [ k] B, k y[ n] h[ k] x[ nk] h[ k] x[ nk] k if xn [ ] is bounded ( xn [ ] B),so that yn [ ] B hk [ ], hk [ ] x k k y k x

28 Stability of Systems-Example n hn [ ] un [ ] is stable? k hk [ ] k0 a k if 1 the system is stable if 1 the system is NOT stable

29 Stability of Systems-Example hn [ ] [ k]? 1 n 0 = 0 n<0 un [ ] n k The system is NOT stable

30 Stability of Systems-Example M 2 1 hn [ ] [ nk]? M M km 1 M n M = M1M2 1 0 otherwise The system is stable

31 Causality of Systems A LTI system is causal if an only if hn [ ] 0 for n 0

32 Causality of Systems yn [ ] xn [ n0] Is hn [ ] causal? hn [ ] [ nn] 0 Since h[n] includes only one sample at n=n 0, h[n] is causal. It is stable because it is absolutely summable. yn [ ] xn [ 1] xn [ ] Is hn [ ] causal? hn [ ] [ n1] [ n] Since h[n] includes sample at n=-1, h[n] is not causal, however it is stable because it is absolutely summable.

33 Causality of Systems N 1 1 yn [ ] xn [ k] N k0 Is hn [ ] causal? 1 N n N 1 hn [ ] [ n k] N N k 0 0 n 0, n N 1 Since h[n] does not include samples when at n<0, h[n] is causal. It is stable because it is absolutely summable.

34 Causality of Systems yn [ ] xk [ ] n k Is hn [ ] causal? n 1 n 0 hn [ ] [ k] k 0 n 0 hn [ ] un [ ] Since h[n] does not include samples when at n<0, h[n] is causal. It is NOT stable because it is NOT absolutely summable.

35 FIR and IIR Systems Finite Impulse Response (FIR) Systems: Systems with only a finite length of nonzero values in h[n] are called FIR systems. Examples: Ideal delay, moving average filter FIR systems are always STABLE Infinite Impulse Response (IIR) Systems: Systems with infinite length of nonzero values in h[n] are called IIR systems. Examples:Accumulator, filters IR systems can be STABLE or UNSTABLE

36 FIR and IIR Systems FIR Systems: IIR Systems: hn [ ] [ n n] 0 hn [ ] un [ ] hn [ ] [ n1] [ n] n hn [ ] aun [ ] N 1 1 hn [ ] [ n k] N k0

Digital Signal Processing Lecture 5

Digital Signal Processing Lecture 5 Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy Digital Signal Processing Lecture 5 Begüm Demir E-mail:

More information

ELEG 305: Digital Signal Processing

ELEG 305: Digital Signal Processing ELEG 305: Digital Signal Processing Lecture 1: Course Overview; Discrete-Time Signals & Systems Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 2008 K. E.

More information

ECE 308 Discrete-Time Signals and Systems

ECE 308 Discrete-Time Signals and Systems ECE 38-6 ECE 38 Discrete-Time Signals and Systems Z. Aliyazicioglu Electrical and Computer Engineering Department Cal Poly Pomona ECE 38-6 1 Intoduction Two basic methods for analyzing the response of

More information

Lecture 11 FIR Filters

Lecture 11 FIR Filters Lecture 11 FIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/4/12 1 The Unit Impulse Sequence Any sequence can be represented in this way. The equation is true if k ranges

More information

Analog vs. discrete signals

Analog vs. discrete signals Analog vs. discrete signals Continuous-time signals are also known as analog signals because their amplitude is analogous (i.e., proportional) to the physical quantity they represent. Discrete-time signals

More information

Lecture 19 IIR Filters

Lecture 19 IIR Filters Lecture 19 IIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/5/10 1 General IIR Difference Equation IIR system: infinite-impulse response system The most general class

More information

Digital Signal Processing

Digital Signal Processing Digital Signal Processing Discrete-Time Signals and Systems (2) Moslem Amiri, Václav Přenosil Embedded Systems Laboratory Faculty of Informatics, Masaryk University Brno, Czech Republic amiri@mail.muni.cz

More information

2. CONVOLUTION. Convolution sum. Response of d.t. LTI systems at a certain input signal

2. CONVOLUTION. Convolution sum. Response of d.t. LTI systems at a certain input signal 2. CONVOLUTION Convolution sum. Response of d.t. LTI systems at a certain input signal Any signal multiplied by the unit impulse = the unit impulse weighted by the value of the signal in 0: xn [ ] δ [

More information

VU Signal and Image Processing

VU Signal and Image Processing 052600 VU Signal and Image Processing Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/18s/

More information

Digital Signal Processing Lecture 3 - Discrete-Time Systems

Digital Signal Processing Lecture 3 - Discrete-Time Systems Digital Signal Processing - Discrete-Time Systems Electrical Engineering and Computer Science University of Tennessee, Knoxville August 25, 2015 Overview 1 2 3 4 5 6 7 8 Introduction Three components of

More information

EEL3135: Homework #4

EEL3135: Homework #4 EEL335: Homework #4 Problem : For each of the systems below, determine whether or not the system is () linear, () time-invariant, and (3) causal: (a) (b) (c) xn [ ] cos( 04πn) (d) xn [ ] xn [ ] xn [ 5]

More information

Properties of LTI Systems

Properties of LTI Systems Properties of LTI Systems Properties of Continuous Time LTI Systems Systems with or without memory: A system is memory less if its output at any time depends only on the value of the input at that same

More information

ELEN E4810: Digital Signal Processing Topic 2: Time domain

ELEN E4810: Digital Signal Processing Topic 2: Time domain ELEN E4810: Digital Signal Processing Topic 2: Time domain 1. Discrete-time systems 2. Convolution 3. Linear Constant-Coefficient Difference Equations (LCCDEs) 4. Correlation 1 1. Discrete-time systems

More information

Chap 2. Discrete-Time Signals and Systems

Chap 2. Discrete-Time Signals and Systems Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,

More information

Digital Signal Processing, Homework 1, Spring 2013, Prof. C.D. Chung

Digital Signal Processing, Homework 1, Spring 2013, Prof. C.D. Chung Digital Signal Processing, Homework, Spring 203, Prof. C.D. Chung. (0.5%) Page 99, Problem 2.2 (a) The impulse response h [n] of an LTI system is known to be zero, except in the interval N 0 n N. The input

More information

Examples. 2-input, 1-output discrete-time systems: 1-input, 1-output discrete-time systems:

Examples. 2-input, 1-output discrete-time systems: 1-input, 1-output discrete-time systems: Discrete-Time s - I Time-Domain Representation CHAPTER 4 These lecture slides are based on "Digital Signal Processing: A Computer-Based Approach, 4th ed." textbook by S.K. Mitra and its instructor materials.

More information

Module 3. Convolution. Aim

Module 3. Convolution. Aim Module Convolution Digital Signal Processing. Slide 4. Aim How to perform convolution in real-time systems efficiently? Is convolution in time domain equivalent to multiplication of the transformed sequence?

More information

Lecture 2 Discrete-Time LTI Systems: Introduction

Lecture 2 Discrete-Time LTI Systems: Introduction Lecture 2 Discrete-Time LTI Systems: Introduction Outline 2.1 Classification of Systems.............................. 1 2.1.1 Memoryless................................. 1 2.1.2 Causal....................................

More information

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz Discrete Time Signals and Systems Time-frequency Analysis Gloria Menegaz Time-frequency Analysis Fourier transform (1D and 2D) Reference textbook: Discrete time signal processing, A.W. Oppenheim and R.W.

More information

Universiti Malaysia Perlis EKT430: DIGITAL SIGNAL PROCESSING LAB ASSIGNMENT 3: DISCRETE TIME SYSTEM IN TIME DOMAIN

Universiti Malaysia Perlis EKT430: DIGITAL SIGNAL PROCESSING LAB ASSIGNMENT 3: DISCRETE TIME SYSTEM IN TIME DOMAIN Universiti Malaysia Perlis EKT430: DIGITAL SIGNAL PROCESSING LAB ASSIGNMENT 3: DISCRETE TIME SYSTEM IN TIME DOMAIN Pusat Pengajian Kejuruteraan Komputer Dan Perhubungan Universiti Malaysia Perlis Discrete-Time

More information

Discrete-Time Systems

Discrete-Time Systems FIR Filters With this chapter we turn to systems as opposed to signals. The systems discussed in this chapter are finite impulse response (FIR) digital filters. The term digital filter arises because these

More information

Discrete-time signals and systems

Discrete-time signals and systems Discrete-time signals and systems 1 DISCRETE-TIME DYNAMICAL SYSTEMS x(t) G y(t) Linear system: Output y(n) is a linear function of the inputs sequence: y(n) = k= h(k)x(n k) h(k): impulse response of the

More information

UNIT 1. SIGNALS AND SYSTEM

UNIT 1. SIGNALS AND SYSTEM Page no: 1 UNIT 1. SIGNALS AND SYSTEM INTRODUCTION A SIGNAL is defined as any physical quantity that changes with time, distance, speed, position, pressure, temperature or some other quantity. A SIGNAL

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Lecture 2 Discrete Time Systems Today Last time: Administration Overview Announcement: HW1 will be out today Lab 0 out webcast out Today: Ch. 2 - Discrete-Time Signals and

More information

ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM. Dr. Lim Chee Chin

ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM. Dr. Lim Chee Chin ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM Dr. Lim Chee Chin Outline Introduction Discrete Fourier Series Properties of Discrete Fourier Series Time domain aliasing due to frequency

More information

Linear Convolution Using FFT

Linear Convolution Using FFT Linear Convolution Using FFT Another useful property is that we can perform circular convolution and see how many points remain the same as those of linear convolution. When P < L and an L-point circular

More information

III. Time Domain Analysis of systems

III. Time Domain Analysis of systems 1 III. Time Domain Analysis of systems Here, we adapt properties of continuous time systems to discrete time systems Section 2.2-2.5, pp 17-39 System Notation y(n) = T[ x(n) ] A. Types of Systems Memoryless

More information

Let H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 )

Let H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 ) Review: Poles and Zeros of Fractional Form Let H() = P()/Q() be the system function of a rational form. Let us represent both P() and Q() as polynomials of (not - ) Then Poles: the roots of Q()=0 Zeros:

More information

Introduction to DSP Time Domain Representation of Signals and Systems

Introduction to DSP Time Domain Representation of Signals and Systems Introduction to DSP Time Domain Representation of Signals and Systems Dr. Waleed Al-Hanafy waleed alhanafy@yahoo.com Faculty of Electronic Engineering, Menoufia Univ., Egypt Digital Signal Processing (ECE407)

More information

University Question Paper Solution

University Question Paper Solution Unit 1: Introduction University Question Paper Solution 1. Determine whether the following systems are: i) Memoryless, ii) Stable iii) Causal iv) Linear and v) Time-invariant. i) y(n)= nx(n) ii) y(t)=

More information

Z Transform (Part - II)

Z Transform (Part - II) Z Transform (Part - II). The Z Transform of the following real exponential sequence x(nt) = a n, nt 0 = 0, nt < 0, a > 0 (a) ; z > (c) for all z z (b) ; z (d) ; z < a > a az az Soln. The given sequence

More information

Cosc 3451 Signals and Systems. What is a system? Systems Terminology and Properties of Systems

Cosc 3451 Signals and Systems. What is a system? Systems Terminology and Properties of Systems Cosc 3451 Signals and Systems Systems Terminology and Properties of Systems What is a system? an entity that manipulates one or more signals to yield new signals (often to accomplish a function) can be

More information

The Convolution Sum for Discrete-Time LTI Systems

The Convolution Sum for Discrete-Time LTI Systems The Convolution Sum for Discrete-Time LTI Systems Andrew W. H. House 01 June 004 1 The Basics of the Convolution Sum Consider a DT LTI system, L. x(n) L y(n) DT convolution is based on an earlier result

More information

Z-Transform. x (n) Sampler

Z-Transform. x (n) Sampler Chapter Two A- Discrete Time Signals: The discrete time signal x(n) is obtained by taking samples of the analog signal xa (t) every Ts seconds as shown in Figure below. Analog signal Discrete time signal

More information

Rui Wang, Assistant professor Dept. of Information and Communication Tongji University.

Rui Wang, Assistant professor Dept. of Information and Communication Tongji University. Linear Time Invariant (LTI) Systems Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Discrete-time LTI system: The convolution

More information

DIGITAL SIGNAL PROCESSING UNIT 1 SIGNALS AND SYSTEMS 1. What is a continuous and discrete time signal? Continuous time signal: A signal x(t) is said to be continuous if it is defined for all time t. Continuous

More information

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals. Z - Transform The z-transform is a very important tool in describing and analyzing digital systems. It offers the techniques for digital filter design and frequency analysis of digital signals. Definition

More information

Stability Condition in Terms of the Pole Locations

Stability Condition in Terms of the Pole Locations Stability Condition in Terms of the Pole Locations A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., 1 = S h [ n] < n= We now develop a stability

More information

Convolution. Define a mathematical operation on discrete-time signals called convolution, represented by *. Given two discrete-time signals x 1, x 2,

Convolution. Define a mathematical operation on discrete-time signals called convolution, represented by *. Given two discrete-time signals x 1, x 2, Filters Filters So far: Sound signals, connection to Fourier Series, Introduction to Fourier Series and Transforms, Introduction to the FFT Today Filters Filters: Keep part of the signal we are interested

More information

UNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z).

UNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z). Page no: 1 UNIT-II Z-TRANSFORM The Z-Transform The direct -transform, properties of the -transform, rational -transforms, inversion of the transform, analysis of linear time-invariant systems in the -

More information

Digital Signal Processing Lecture 10 - Discrete Fourier Transform

Digital Signal Processing Lecture 10 - Discrete Fourier Transform Digital Signal Processing - Discrete Fourier Transform Electrical Engineering and Computer Science University of Tennessee, Knoxville November 12, 2015 Overview 1 2 3 4 Review - 1 Introduction Discrete-time

More information

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Spring 2018 Exam #1

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Spring 2018 Exam #1 New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Spring 2018 Exam #1 Name: Prob. 1 Prob. 2 Prob. 3 Prob. 4 Total / 30 points / 20 points / 25 points /

More information

6.02 Fall 2012 Lecture #11

6.02 Fall 2012 Lecture #11 6.02 Fall 2012 Lecture #11 Eye diagrams Alternative ways to look at convolution 6.02 Fall 2012 Lecture 11, Slide #1 Eye Diagrams 000 100 010 110 001 101 011 111 Eye diagrams make it easy to find These

More information

ELEG 305: Digital Signal Processing

ELEG 305: Digital Signal Processing ELEG 305: Digital Signal Processing Lecture 18: Applications of FFT Algorithms & Linear Filtering DFT Computation; Implementation of Discrete Time Systems Kenneth E. Barner Department of Electrical and

More information

信號與系統 Signals and Systems

信號與系統 Signals and Systems Spring 2010 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb10 Jun10 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan

More information

Discrete-Time Signals & Systems

Discrete-Time Signals & Systems Chapter 2 Discrete-Time Signals & Systems 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 2-1-1 Discrete-Time Signals: Time-Domain Representation (1/10) Signals

More information

Module 1: Signals & System

Module 1: Signals & System Module 1: Signals & System Lecture 6: Basic Signals in Detail Basic Signals in detail We now introduce formally some of the basic signals namely 1) The Unit Impulse function. 2) The Unit Step function

More information

Discrete-Time Fourier Transform (DTFT)

Discrete-Time Fourier Transform (DTFT) Discrete-Time Fourier Transform (DTFT) 1 Preliminaries Definition: The Discrete-Time Fourier Transform (DTFT) of a signal x[n] is defined to be X(e jω ) x[n]e jωn. (1) In other words, the DTFT of x[n]

More information

6.02 Fall 2012 Lecture #10

6.02 Fall 2012 Lecture #10 6.02 Fall 2012 Lecture #10 Linear time-invariant (LTI) models Convolution 6.02 Fall 2012 Lecture 10, Slide #1 Modeling Channel Behavior codeword bits in generate x[n] 1001110101 digitized modulate DAC

More information

信號與系統 Signals and Systems

信號與系統 Signals and Systems Spring 2015 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb15 Jun15 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan

More information

Ch 2: Linear Time-Invariant System

Ch 2: Linear Time-Invariant System Ch 2: Linear Time-Invariant System A system is said to be Linear Time-Invariant (LTI) if it possesses the basic system properties of linearity and time-invariance. Consider a system with an output signal

More information

Chapter 7: The z-transform

Chapter 7: The z-transform Chapter 7: The -Transform ECE352 1 The -Transform - definition Continuous-time systems: e st H(s) y(t) = e st H(s) e st is an eigenfunction of the LTI system h(t), and H(s) is the corresponding eigenvalue.

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Discrete Time Fourier Transform M. Lustig, EECS UC Berkeley A couple of things Read Ch 2 2.0-2.9 It s OK to use 2nd edition Class webcast in bcourses.berkeley.edu or linked

More information

Lecture 7 Discrete Systems

Lecture 7 Discrete Systems Lecture 7 Discrete Systems EE 52: Instrumentation and Measurements Lecture Notes Update on November, 29 Aly El-Osery, Electrical Engineering Dept., New Mexico Tech 7. Contents The z-transform 2 Linear

More information

DIGITAL SIGNAL PROCESSING LECTURE 1

DIGITAL SIGNAL PROCESSING LECTURE 1 DIGITAL SIGNAL PROCESSING LECTURE 1 Fall 2010 2K8-5 th Semester Tahir Muhammad tmuhammad_07@yahoo.com Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000

More information

Digital Filter Structures. Basic IIR Digital Filter Structures. of an LTI digital filter is given by the convolution sum or, by the linear constant

Digital Filter Structures. Basic IIR Digital Filter Structures. of an LTI digital filter is given by the convolution sum or, by the linear constant Digital Filter Chapter 8 Digital Filter Block Diagram Representation Equivalent Basic FIR Digital Filter Basic IIR Digital Filter. Block Diagram Representation In the time domain, the input-output relations

More information

/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by

/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by Code No: RR320402 Set No. 1 III B.Tech II Semester Regular Examinations, Apr/May 2006 DIGITAL SIGNAL PROCESSING ( Common to Electronics & Communication Engineering, Electronics & Instrumentation Engineering,

More information

Z-Transform. The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = g(t)e st dt. Z : G(z) =

Z-Transform. The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = g(t)e st dt. Z : G(z) = Z-Transform The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = Z : G(z) = It is Used in Digital Signal Processing n= g(t)e st dt g[n]z n Used to Define Frequency

More information

considered to be the elements of a column vector as follows 1.2 Discrete-time signals

considered to be the elements of a column vector as follows 1.2 Discrete-time signals Chapter 1 Signals and Systems 1.1 Introduction In this chapter we begin our study of digital signal processing by developing the notion of a discretetime signal and a discrete-time system. We will concentrate

More information

LECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13)

LECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) LECTURE NOTES ON DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) FACULTY : B.V.S.RENUKA DEVI (Asst.Prof) / Dr. K. SRINIVASA RAO (Assoc. Prof) DEPARTMENT OF ELECTRONICS AND COMMUNICATIONS

More information

Introduction to Signals and Systems Lecture #4 - Input-output Representation of LTI Systems Guillaume Drion Academic year

Introduction to Signals and Systems Lecture #4 - Input-output Representation of LTI Systems Guillaume Drion Academic year Introduction to Signals and Systems Lecture #4 - Input-output Representation of LTI Systems Guillaume Drion Academic year 2017-2018 1 Outline Systems modeling: input/output approach of LTI systems. Convolution

More information

R13 SET - 1

R13 SET - 1 R13 SET - 1 III B. Tech II Semester Regular Examinations, April - 2016 DIGITAL SIGNAL PROCESSING (Electronics and Communication Engineering) Time: 3 hours Maximum Marks: 70 Note: 1. Question Paper consists

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Lecture 2B D. T. Fourier Transform M. Lustig, EECS UC Berkeley Something Fun gotenna http://www.gotenna.com/# Text messaging radio Bluetooth phone interface MURS VHF radio

More information

Signals and Systems. Problem Set: The z-transform and DT Fourier Transform

Signals and Systems. Problem Set: The z-transform and DT Fourier Transform Signals and Systems Problem Set: The z-transform and DT Fourier Transform Updated: October 9, 7 Problem Set Problem - Transfer functions in MATLAB A discrete-time, causal LTI system is described by the

More information

Lecture 7 - IIR Filters

Lecture 7 - IIR Filters Lecture 7 - IIR Filters James Barnes (James.Barnes@colostate.edu) Spring 204 Colorado State University Dept of Electrical and Computer Engineering ECE423 / 2 Outline. IIR Filter Representations Difference

More information

Multidimensional digital signal processing

Multidimensional digital signal processing PSfrag replacements Two-dimensional discrete signals N 1 A 2-D discrete signal (also N called a sequence or array) is a function 2 defined over thex(n set 1 of, n 2 ordered ) pairs of integers: y(nx 1,

More information

Review of Fundamentals of Digital Signal Processing

Review of Fundamentals of Digital Signal Processing Chapter 2 Review of Fundamentals of Digital Signal Processing 2.1 (a) This system is not linear (the constant term makes it non linear) but is shift-invariant (b) This system is linear but not shift-invariant

More information

EE 210. Signals and Systems Solutions of homework 2

EE 210. Signals and Systems Solutions of homework 2 EE 2. Signals and Systems Solutions of homework 2 Spring 2 Exercise Due Date Week of 22 nd Feb. Problems Q Compute and sketch the output y[n] of each discrete-time LTI system below with impulse response

More information

z-transforms Definition of the z-transform Chapter

z-transforms Definition of the z-transform Chapter z-transforms Chapter 7 In the study of discrete-time signal and systems, we have thus far considered the time-domain and the frequency domain. The z- domain gives us a third representation. All three domains

More information

EE 521: Instrumentation and Measurements

EE 521: Instrumentation and Measurements Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA November 1, 2009 1 / 27 1 The z-transform 2 Linear Time-Invariant System 3 Filter Design IIR Filters FIR Filters

More information

DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A

DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A Classification of systems : Continuous and Discrete

More information

Digital Signal Processing Module 1 Analysis of Discrete time Linear Time - Invariant Systems

Digital Signal Processing Module 1 Analysis of Discrete time Linear Time - Invariant Systems Digital Signal Processing Module 1 Analysis of Discrete time Linear Time - Invariant Systems Objective: 1. To understand the representation of Discrete time signals 2. To analyze the causality and stability

More information

LECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13)

LECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) LECTURE NOTES ON DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) FACULTY : B.V.S.RENUKA DEVI (Asst.Prof) / Dr. K. SRINIVASA RAO (Assoc. Prof) DEPARTMENT OF ELECTRONICS AND COMMUNICATIONS

More information

E : Lecture 1 Introduction

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

More information

Responses of Digital Filters Chapter Intended Learning Outcomes:

Responses of Digital Filters Chapter Intended Learning Outcomes: Responses of Digital Filters Chapter Intended Learning Outcomes: (i) Understanding the relationships between impulse response, frequency response, difference equation and transfer function in characterizing

More information

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Fall 2017 Exam #1

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Fall 2017 Exam #1 New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2017 Exam #1 Name: Prob. 1 Prob. 2 Prob. 3 Prob. 4 Total / 30 points / 20 points / 25 points / 25

More information

Frequency-Domain C/S of LTI Systems

Frequency-Domain C/S of LTI Systems Frequency-Domain C/S of LTI Systems x(n) LTI y(n) LTI: Linear Time-Invariant system h(n), the impulse response of an LTI systems describes the time domain c/s. H(ω), the frequency response describes the

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

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Lecture 2A D.T Systems D. T. Fourier Transform A couple of things Read Ch 2 2.0-2.9 It s OK to use 2nd edition My office hours: posted on-line W 4-5pm Cory 506 ham radio

More information

Lecture 18: Stability

Lecture 18: Stability Lecture 18: Stability ECE 401: Signal and Image Analysis University of Illinois 4/18/2017 1 Stability 2 Impulse Response 3 Z Transform Outline 1 Stability 2 Impulse Response 3 Z Transform BIBO Stability

More information

Digital Signal Processing:

Digital Signal Processing: Digital Signal Processing: Mathematical and algorithmic manipulation of discretized and quantized or naturally digital signals in order to extract the most relevant and pertinent information that is carried

More information

Use: Analysis of systems, simple convolution, shorthand for e jw, stability. Motivation easier to write. Or X(z) = Z {x(n)}

Use: Analysis of systems, simple convolution, shorthand for e jw, stability. Motivation easier to write. Or X(z) = Z {x(n)} 1 VI. Z Transform Ch 24 Use: Analysis of systems, simple convolution, shorthand for e jw, stability. A. Definition: X(z) = x(n) z z - transforms Motivation easier to write Or Note if X(z) = Z {x(n)} z

More information

Chapter 2 Time-Domain Representations of LTI Systems

Chapter 2 Time-Domain Representations of LTI Systems Chapter 2 Time-Domain Representations of LTI Systems 1 Introduction Impulse responses of LTI systems Linear constant-coefficients differential or difference equations of LTI systems Block diagram representations

More information

LTI H. the system H when xn [ ] is the input.

LTI H. the system H when xn [ ] is the input. REVIEW OF 1D LTI SYSTEMS LTI xn [ ] y[ n] H Operator notation: y[ n] = H{ x[ n] } In English, this is read: yn [ ] is the output of the system H when xn [ ] is the input. 2 THE MOST IMPORTANT PROPERTIES

More information

LTI Models and Convolution

LTI Models and Convolution MIT 6.02 DRAFT Lecture Notes Last update: November 4, 2012 CHAPTER 11 LTI Models and Convolution This chapter will help us understand what else besides noise (which we studied in Chapter 9) perturbs or

More information

Solution 10 July 2015 ECE301 Signals and Systems: Midterm. Cover Sheet

Solution 10 July 2015 ECE301 Signals and Systems: Midterm. Cover Sheet Solution 10 July 2015 ECE301 Signals and Systems: Midterm Cover Sheet Test Duration: 60 minutes Coverage: Chap. 1,2,3,4 One 8.5" x 11" crib sheet is allowed. Calculators, textbooks, notes are not allowed.

More information

Discrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections

Discrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections Discrete-Time Signals and Systems The z-transform and Its Application Dr. Deepa Kundur University of Toronto Reference: Sections 3. - 3.4 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:

More information

How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches

How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches Difference Equations (an LTI system) x[n]: input, y[n]: output That is, building a system that maes use of the current and previous

More information

Roll No. :... Invigilator s Signature :.. CS/B.Tech (EE-N)/SEM-6/EC-611/ DIGITAL SIGNAL PROCESSING. Time Allotted : 3 Hours Full Marks : 70

Roll No. :... Invigilator s Signature :.. CS/B.Tech (EE-N)/SEM-6/EC-611/ DIGITAL SIGNAL PROCESSING. Time Allotted : 3 Hours Full Marks : 70 Name : Roll No. :.... Invigilator s Signature :.. CS/B.Tech (EE-N)/SEM-6/EC-611/2011 2011 DIGITAL SIGNAL PROCESSING Time Allotted : 3 Hours Full Marks : 70 The figures in the margin indicate full marks.

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Processing Prof. Mark Fowler Note Set #21 Using the DFT to Implement FIR Filters Reading Assignment: Sect. 7.3 of Proakis & Manolakis Motivation: DTFT View of Filtering There are

More information

Shift Property of z-transform. Lecture 16. More z-transform (Lathi 5.2, ) More Properties of z-transform. Convolution property of z-transform

Shift Property of z-transform. Lecture 16. More z-transform (Lathi 5.2, ) More Properties of z-transform. Convolution property of z-transform Shift Property of -Transform If Lecture 6 More -Transform (Lathi 5.2,5.4-5.5) then which is delay causal signal by sample period. If we delay x[n] first: Peter Cheung Department of Electrical & Electronic

More information

Discrete Time Systems

Discrete Time Systems Discrete Time Systems Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz 1 LTI systems In this course, we work only with linear and time-invariant systems. We talked about

More information

Signals & Systems Handout #4

Signals & Systems Handout #4 Signals & Systems Handout #4 H-4. Elementary Discrete-Domain Functions (Sequences): Discrete-domain functions are defined for n Z. H-4.. Sequence Notation: We use the following notation to indicate the

More information

Signals and Systems Chapter 2

Signals and Systems Chapter 2 Signals and Systems Chapter 2 Continuous-Time Systems Prof. Yasser Mostafa Kadah Overview of Chapter 2 Systems and their classification Linear time-invariant systems System Concept Mathematical transformation

More information

DFT-Based FIR Filtering. See Porat s Book: 4.7, 5.6

DFT-Based FIR Filtering. See Porat s Book: 4.7, 5.6 DFT-Based FIR Filtering See Porat s Book: 4.7, 5.6 1 Motivation: DTFT View of Filtering There are two views of filtering: * Time Domain * Frequency Domain x[ X f ( θ ) h[ H f ( θ ) Y y[ = h[ * x[ f ( θ

More information

Analog LTI system Digital LTI system

Analog LTI system Digital LTI system Sampling Decimation Seismometer Amplifier AAA filter DAA filter Analog LTI system Digital LTI system Filtering (Digital Systems) input output filter xn [ ] X ~ [ k] Convolution of Sequences hn [ ] yn [

More information

ECE4270 Fundamentals of DSP Lecture 20. Fixed-Point Arithmetic in FIR and IIR Filters (part I) Overview of Lecture. Overflow. FIR Digital Filter

ECE4270 Fundamentals of DSP Lecture 20. Fixed-Point Arithmetic in FIR and IIR Filters (part I) Overview of Lecture. Overflow. FIR Digital Filter ECE4270 Fundamentals of DSP Lecture 20 Fixed-Point Arithmetic in FIR and IIR Filters (part I) School of ECE Center for Signal and Information Processing Georgia Institute of Technology Overview of Lecture

More information

z-transform Chapter 6

z-transform Chapter 6 z-transform Chapter 6 Dr. Iyad djafar Outline 2 Definition Relation Between z-transform and DTFT Region of Convergence Common z-transform Pairs The Rational z-transform The Inverse z-transform z-transform

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

Discrete-Time Signals and Systems

Discrete-Time Signals and Systems ECE 46 Lec Viewgraph of 35 Discrete-Time Signals and Systems Sequences: x { x[ n] }, < n

More information