2.161 Signal Processing: Continuous and Discrete Fall 2008
|
|
- Leonard Jacobs
- 5 years ago
- Views:
Transcription
1 IT OpenCourseWare Signal Processing: Continuous and Discrete all 2008 or information about citing these materials or our Terms of Use, visit:
2 assachusetts Institute of Technology Department of echanical Engineering Signal Processing - Continuous and Discrete all Term 2008 Lecture 15 1 Reading: Proakis & anolakis, Ch. 7 Oppenheim, Schafer & Buck. Ch. 10 Cartinhour, Chs. 6 & 9 1 requency Response and Poles and Zeros As we did for the continuous case, factor the discrete-time transfer functions into as set of poles and zeros; H(z) = b 0 z 0 + b 1 z b z i=1 (z z i) = K a 0 z 0 + a 1 z a N z N N i=1 (z p i) where Z i are the system zeros, the p i are the system poles, and K = b 0 /a 0 is the overall gain. We note, as in the continuous case that the polse and zeros must be either real, or appear in complex conjugate pairs. As in the continuous case, we can draw a set of vectors from the poles and zeros to a test point in the z-plane, and evaluate H(z) in terms of the lengths and angles of these vectors. In particular, we choose to evaluate H( e j ω ) on the unit circle, ( ) e j ω H( e j ω i=1 z i ) = K N i=1 ( ej ω p i ) and H( e j ω ) H( e j ω ) = i=1 e j ω zi i=1 q i = K N i=1 ej ω p i = K N N ( ) e j ω zi i=1 r i N ( ) e j ω pi = θi φ i i=1 i=1 i=1 i=1 where the q i and θ i are the lengths and angles of the vectors from the zeros to the point z = e j ω, the r i and φ i are the lengths and angles of the vectors from the poles to the point z = e j ω, as shown below: 1 copyright c D.Rowell
3 : : G G B B H H! H! B! G G : A ) J B H A G K A? O 0 A G G H H H! 0 A G G B B B! We can interpret the effect of pole and zero locations on the frequency response as follows: (a) A pole (or conjugate pole pair) on the unit circle will cause H( e j ω ) to become infinite at frequency ω. (b) A zero (or conjugate zero pair) on the unit circle will cause H( e j ω ) to become zero at frequency ω. (c) Poles near the unit circle will cause a peak in H( e j ω ) in the neighborhood of those poles. (c) Zeros near the unit circle will cause a dip, or notch, in H( e j ω ) in the neighborhood of those zeros. (d) Poles and zeros at the origin z = 0 have no effect upon H( e j ω ), but add a frequency dependent linear phase taper ( ω for a pole, +ω for a zero), which is equivalent to shift. (e) A pole or zero at z = 1 forces H( e j ω ) to be infinite (pole) or 0 (zero) at ω = 0. (f) A pole or zero at z = 1 forces H( e j ω ) to be infinite (pole) or 0 (zero) at ω = π, (the Nyquist frequency). 2 IR Low-Pass ilter Design The IR (finite impulse response) filter is an all-zero system with a difference equation y n = b k f n k k=0 which is clearly a convolution of the input sequence with an impulse response {h k } = {b k }, for k = 0,,. A direct-form causal implementation is 15 2
4 B B B B! B > > > >! > O The transfer function is and the frequency response is H(z) = b k z k = 1 z k=0 k=0 H( e j ω ) = b k e j kω. k=0 bk z k Note that there are + 1 terms in the impulse response but the order of the polynomials is. Example 1 ind the frequency response H( e j ω ) for a simple three-point moving average filter: 1 y n = (f n + f n 1 + f n 2 ). 3 Solution: so that and H(z) = 1 z z z H( e j ω ) = 1 ( 1 + e j ω 2j + e ω) 3 = 1 e j ω ( e j ω e j ω) 3 1 = (1 + 2 cos(ω)) e j ω 3 H( e j ω ) 1 = (1 + 2 cos(ω)) 3 H( e j ω ) = ω. 15 3
5 0 A The ideal IR low-pass filter has a response { H(e j ω 1 )= 0 ω ω c ω c < ω π 0 A 0 A 0 A???? The impulse response h n = Z 1 {H(z)}, and although we are not given H(z) explicitly, we can use the formal definition of the inverse z-transform (Lecture 14) as a contour integral in the z-plane, 1 Z 1 {H(z)} = H(z)z n 1 dz 2πj where the path is a ccw contour enclosing all of the poles of H(z), and for a stable filter choose the contour as the unit-circle. Let z = e j ω, so that dz = je j ω dω, and h n = Z 1 {H(z)} = 1 2π ωc ( ) 1. e j nω dω = ω c sin(ω c n) π ω c n The impulse response of the IR ideal low-pass filter is therefore ( ) ω c sin(ω c n) h n = π ωc n ω c 15 4
6 The following figure shows the central region of the impulse response of an ideal IR filter with ω c =0.2π: D It is obvious that this impulse response has two problems: (a) It is infinite in extent, and (b) It is non-causal. To produce a causal, finite length filter (a) Truncate {h n } to include +1 central points ( +1 odd), that is select the points /2 n /2. Let this truncated filter be designated Ĥ(z). (b) Shift the truncated impulse response {ĥ n } to the right by /2 to form a causal sequence {h n}, where h n = ĥ (/2 n), for n = 0,... Take {h n} as the IR causal approximation to the ideal low-pass filter. Then H (z) = z ( 1)/2 Ĥ(z), that is the response is delayed by ( 1)/2 samples. The frequency response is H (e j ω )= e j( 1)ω/2 Ĥ(e j ω ), and because Ĥ(e j ω ) is real H (e j ω ) = Ĥ(e j ω ) H (e j ω ) = ( 1)ω/
7 Example 2 Design a five point causal IR low-pass filter with a cut-off frequency ω c =0.4π. Solution: The ideal filter has an impulse response ( ) ( ) ω c sin(ω c n) 1 sin(πn/2) h n = = π ω c n 2 πn/2 Select + 1 = 5, and select the five central components: n : h : n Shift to the right by /2 = 2, and form the causal impulse response {ĥ n }: with difference equation n : ĥ n : y n =0.0935f n f n f n f n f n 4 The causal impulse response and the frequency response magnitude are shown below: " D 0 A! % & $ " '! # " " $ & # # #! 2.1 The Effect of Truncation and Shifting (a) The Effect of Truncation The selection of the + 1 central components of the non-causal impulse response {h n } can be written as a product {h } = {h n r n } n where {r n } is an even rectangular window function { 1 n /2 r n = 0 otherwise. The following figure shows {r n } for =
8 The truncated frequency response is therefore or the window function H (e j ω )= 1 H(e j ω ) R(e j ω ). 2π /2 R(z) = z 1 k= /2 /2 /2 R(e j ω ) = e j kω =1+ 2 cos(kω) = D /2 (ω) k= /2 k=1 D /2 (ω) is known as the Dirichlet kernel, and is found in the study of truncated ourier series and convolution of periodic functions. It is easy to show (using the sum of a finite geometric series) that R(e j ω )= D /2 (ω) = sin(( +1)ω/2) sin(ω/2) 4 A! 15 7
9 Notice (1) The width of the main lobe decreases with, (2) the side lobes do not decay to zero as ω increases. Aside: The formal definition of the z-transform of the product of two sequences is given by the z-plane contour integral (z) = Z{x n y n } = 1 2πj X(ν)Y ( z ) ν 1 dν ν where X(z) = Z{x n }, Y (z) = Z{y n }, and the contour lies in the ROC of both X(z) and Y (z). In particular if the unit-circle lies within the ROC of both sequences, choose the unit-circle as the contour, ν = e j ω, then (e j ωo )= 1 π X(e j ω )Y (e j(ω0 ω) )dω 2π π which is the convolution of X(e j ω ) and Y (e j ω ). The frequency response of the truncated filter is or 1 H (e j ω ) = H(e j ω ) R(e j ω ) 2π 1 π = H(e j ν )R(e j(ν ω) )dν H (e j ω )= 2π π 1 ω c D /2 (ν ω)dν. 2π ω c which is shown below for filter with ω c =0.4π and lengths +1=11, 21, 31, and 41: 0 A! " ' " 15 8
10 In general: The amplitude of the ripple in the pass-band does not decrease with the filter order. Similarly, the stop-band attenuation is relatively unaffected by the filter order, and the amplitude of the first side-lobe is 0.091, so that the truncated filter has a stop-band attenuation of -21 db. The width of the transition-band decreases with increasing. (b) The Effect of the Right-Shift to orm a Causal ilter The truncated non-causal impulse response {h } is even and real, so that H ( e j ω ) is also real and even, that is H ( e j ω ) = 0. The right-shift of h by /2 samples to force causality imposes and therefore The phase response of the filter is which is a linear phase taper (lag). Ĥ(z) = z /2 H (z) Ĥ( e j ω ) = e j ω/2 H ( e j ω ). H ( e j ω ) = (/2)ω The effect of the right-shift on the impulse response of the ideal filter is to impose a phase lag that is proportional to frequency, with a slope of /
2.161 Signal Processing: Continuous and Discrete Fall 2008
IT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete all 2008 or information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. assachusetts
More information2.161 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. I Reading:
More information2.161 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 informationDiscrete-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 informationLecture 3 - Design of Digital Filters
Lecture 3 - Design of Digital Filters 3.1 Simple filters In the previous lecture we considered the polynomial fit as a case example of designing a smoothing filter. The approximation to an ideal LPF can
More informationReview 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 informationMultimedia Signals and Systems - Audio and Video. Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2
Multimedia Signals and Systems - Audio and Video Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2 Kunio Takaya Electrical and Computer Engineering University of Saskatchewan December
More informationTransform Analysis of Linear Time-Invariant Systems
Transform Analysis of Linear Time-Invariant Systems Discrete-Time Signal Processing Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-Sen University Kaohsiung, Taiwan ROC Transform
More informationDigital Signal Processing Lecture 9 - Design of Digital Filters - FIR
Digital Signal Processing - Design of Digital Filters - FIR Electrical Engineering and Computer Science University of Tennessee, Knoxville November 3, 2015 Overview 1 2 3 4 Roadmap Introduction Discrete-time
More information-Digital Signal Processing- FIR Filter Design. Lecture May-16
-Digital Signal Processing- FIR Filter Design Lecture-17 24-May-16 FIR Filter Design! FIR filters can also be designed from a frequency response specification.! The equivalent sampled impulse response
More information(Refer Slide Time: 01:28 03:51 min)
Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture 40 FIR Design by Windowing This is the 40 th lecture and our topic for
More informationLike bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform.
Inversion of the z-transform Focus on rational z-transform of z 1. Apply partial fraction expansion. Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform. Let X(z)
More informationReview of Fundamentals of Digital Signal Processing
Solution Manual for Theory and Applications of Digital Speech Processing by Lawrence Rabiner and Ronald Schafer Click here to Purchase full Solution Manual at http://solutionmanuals.info Link download
More informationZ-TRANSFORMS. Solution: Using the definition (5.1.2), we find: for case (b). y(n)= h(n) x(n) Y(z)= H(z)X(z) (convolution) (5.1.
84 5. Z-TRANSFORMS 5 z-transforms Solution: Using the definition (5..2), we find: for case (a), and H(z) h 0 + h z + h 2 z 2 + h 3 z 3 2 + 3z + 5z 2 + 2z 3 H(z) h 0 + h z + h 2 z 2 + h 3 z 3 + h 4 z 4
More information2.004 Dynamics and Control II Spring 2008
MT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control Spring 2008 or information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Reading: ise: Chapter 8 Massachusetts
More informationSignal Processing. Lecture 10: FIR Filter Design. Ahmet Taha Koru, Ph. D. Yildiz Technical University Fall
Signal Processing Lecture 10: FIR Filter Design Ahmet Taha Koru, Ph. D. Yildiz Technical University 2017-2018 Fall ATK (YTU) Signal Processing 2017-2018 Fall 1 / 47 Introduction Introduction ATK (YTU)
More informationDiscrete-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 informationDigital Signal Processing. Midterm 1 Solution
EE 123 University of California, Berkeley Anant Sahai February 15, 27 Digital Signal Processing Instructions Midterm 1 Solution Total time allowed for the exam is 8 minutes Some useful formulas: Discrete
More informationYour solutions for time-domain waveforms should all be expressed as real-valued functions.
ECE-486 Test 2, Feb 23, 2017 2 Hours; Closed book; Allowed calculator models: (a) Casio fx-115 models (b) HP33s and HP 35s (c) TI-30X and TI-36X models. Calculators not included in this list are not permitted.
More informationOptimal Design of Real and Complex Minimum Phase Digital FIR Filters
Optimal Design of Real and Complex Minimum Phase Digital FIR Filters Niranjan Damera-Venkata and Brian L. Evans Embedded Signal Processing Laboratory Dept. of Electrical and Computer Engineering The University
More informationUNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.
UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 9th, 011 Examination hours: 14.30 18.30 This problem set
More informationEE123 Digital Signal Processing. M. Lustig, EECS UC Berkeley
EE123 Digital Signal Processing Today Last time: DTFT - Ch 2 Today: Continue DTFT Z-Transform Ch. 3 Properties of the DTFT cont. Time-Freq Shifting/modulation: M. Lustig, EE123 UCB M. Lustig, EE123 UCB
More informationDigital Signal Processing
COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #21 Friday, October 24, 2003 Types of causal FIR (generalized) linear-phase filters: Type I: Symmetric impulse response: with order M an even
More informationFilter Analysis and Design
Filter Analysis and Design Butterworth Filters Butterworth filters have a transfer function whose squared magnitude has the form H a ( jω ) 2 = 1 ( ) 2n. 1+ ω / ω c * M. J. Roberts - All Rights Reserved
More informationLecture 04: Discrete Frequency Domain Analysis (z-transform)
Lecture 04: Discrete Frequency Domain Analysis (z-transform) John Chiverton School of Information Technology Mae Fah Luang University 1st Semester 2009/ 2552 Outline Overview Lecture Contents Introduction
More informationSignal Analysis, Systems, Transforms
Michael J. Corinthios Signal Analysis, Systems, Transforms Engineering Book (English) August 29, 2007 Springer Contents Discrete-Time Signals and Systems......................... Introduction.............................................2
More informationDigital 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 information6.003: Signals and Systems
6.003: Signals and Systems Z Transform September 22, 2011 1 2 Concept Map: Discrete-Time Systems Multiple representations of DT systems. Delay R Block Diagram System Functional X + + Y Y Delay Delay X
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MI OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 008 For information about citing these materials or our erms of Use, visit: http://ocw.mit.edu/terms. Massachusetts
More informationChapter 13 Z Transform
Chapter 13 Z Transform 1. -transform 2. Inverse -transform 3. Properties of -transform 4. Solution to Difference Equation 5. Calculating output using -transform 6. DTFT and -transform 7. Stability Analysis
More informationDigital Signal Processing
Digital Signal Processing The -Transform and Its Application to the Analysis of LTI Systems Moslem Amiri, Václav Přenosil Embedded Systems Laboratory Faculty of Informatics, Masaryk University Brno, Cech
More informationEC 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 informationSolutions: Homework Set # 5
Signal Processing for Communications EPFL Winter Semester 2007/2008 Prof. Suhas Diggavi Handout # 22, Tuesday, November, 2007 Solutions: Homework Set # 5 Problem (a) Since h [n] = 0, we have (b) We can
More informationSignals & 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 information2.161 Signal Processing: Continuous and Discrete Fall 2008
MT OpenCourseWare http://ocw.mit.edu 2.6 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 informationLABORATORY 3 FINITE IMPULSE RESPONSE FILTERS
LABORATORY 3 FINITE IMPULSE RESPONSE FILTERS 3.. Introduction A digital filter is a discrete system, used with the purpose of changing the amplitude and/or phase spectrum of a signal. The systems (filters)
More informationSignals 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 informationECE 301 Division 1, Fall 2008 Instructor: Mimi Boutin Final Examination Instructions:
ECE 30 Division, all 2008 Instructor: Mimi Boutin inal Examination Instructions:. Wait for the BEGIN signal before opening this booklet. In the meantime, read the instructions below and fill out the requested
More information2.161 Signal Processing: Continuous and Discrete
MIT OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 8 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More informationSignals and Systems. Spring Room 324, Geology Palace, ,
Signals and Systems Spring 2013 Room 324, Geology Palace, 13756569051, zhukaiguang@jlu.edu.cn Chapter 10 The Z-Transform 1) Z-Transform 2) Properties of the ROC of the z-transform 3) Inverse z-transform
More information2.161 Signal Processing: Continuous and Discrete
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. M MASSACHUSETTS
More informationSEL4223 Digital Signal Processing. Inverse Z-Transform. Musa Mohd Mokji
SEL4223 Digital Signal Processing Inverse Z-Transform Musa Mohd Mokji Inverse Z-Transform Transform from z-domain to time-domain x n = 1 2πj c X z z n 1 dz Note that the mathematical operation for the
More informationEE 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 informationDIGITAL SIGNAL PROCESSING. Chapter 3 z-transform
DIGITAL SIGNAL PROCESSING Chapter 3 z-transform by Dr. Norizam Sulaiman Faculty of Electrical & Electronics Engineering norizam@ump.edu.my OER Digital Signal Processing by Dr. Norizam Sulaiman work is
More information2.004 Dynamics and Control II Spring 2008
MT OpenCourseWare http://ocw.mit.edu.004 Dynamics and Control Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts nstitute of Technology
More informationGrades will be determined by the correctness of your answers (explanations are not required).
6.00 (Fall 20) Final Examination December 9, 20 Name: Kerberos Username: Please circle your section number: Section Time 2 am pm 4 2 pm Grades will be determined by the correctness of your answers (explanations
More informationEEL3135: 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 informationLecture 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 informationDigital Filters Ying Sun
Digital Filters Ying Sun Digital filters Finite impulse response (FIR filter: h[n] has a finite numbers of terms. Infinite impulse response (IIR filter: h[n] has infinite numbers of terms. Causal filter:
More informationECE 410 DIGITAL SIGNAL PROCESSING D. Munson University of Illinois Chapter 12
. ECE 40 DIGITAL SIGNAL PROCESSING D. Munson University of Illinois Chapter IIR Filter Design ) Based on Analog Prototype a) Impulse invariant design b) Bilinear transformation ( ) ~ widely used ) Computer-Aided
More informationStability 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 informationThe Z transform (2) Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 1
The Z transform (2) Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 1 Outline Properties of the region of convergence (10.2) The inverse Z-transform (10.3) Definition Computational techniques Alexandra
More informationProblem Set 2: Solution Due on Wed. 25th Sept. Fall 2013
EE 561: Digital Control Systems Problem Set 2: Solution Due on Wed 25th Sept Fall 2013 Problem 1 Check the following for (internal) stability [Hint: Analyze the characteristic equation] (a) u k = 05u k
More informationTransform analysis of LTI systems Oppenheim and Schafer, Second edition pp For LTI systems we can write
Transform analysis of LTI systems Oppenheim and Schafer, Second edition pp. 4 9. For LTI systems we can write yœn D xœn hœn D X kd xœkhœn Alternatively, this relationship can be expressed in the z-transform
More informationPoles and Zeros in z-plane
M58 Mixed Signal Processors page of 6 Poles and Zeros in z-plane z-plane Response of discrete-time system (i.e. digital filter at a particular frequency ω is determined by the distance between its poles
More informationLINEAR-PHASE FIR FILTERS DESIGN
LINEAR-PHASE FIR FILTERS DESIGN Prof. Siripong Potisuk inimum-phase Filters A digital filter is a minimum-phase filter if and only if all of its zeros lie inside or on the unit circle; otherwise, it is
More informationA 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 informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture : Design of Digital IIR Filters (Part I) Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner (Univ.
More informationDigital Control & Digital Filters. Lectures 21 & 22
Digital Controls & Digital Filters Lectures 2 & 22, Professor Department of Electrical and Computer Engineering Colorado State University Spring 205 Review of Analog Filters-Cont. Types of Analog Filters:
More informationUniversity of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing
University of Illinois at Urbana-Champaign ECE 0: Digital Signal Processing Chandra Radhakrishnan PROBLEM SET : SOLUTIONS Peter Kairouz Problem. Hz z 7 z +/9, causal ROC z > contains the unit circle BIBO
More information( ) John A. Quinn Lecture. ESE 531: Digital Signal Processing. Lecture Outline. Frequency Response of LTI System. Example: Zero on Real Axis
John A. Quinn Lecture ESE 531: Digital Signal Processing Lec 15: March 21, 2017 Review, Generalized Linear Phase Systems Penn ESE 531 Spring 2017 Khanna Lecture Outline!!! 2 Frequency Response of LTI System
More informationGeneralizing the DTFT!
The Transform Generaliing the DTFT! The forward DTFT is defined by X e jω ( ) = x n e jωn in which n= Ω is discrete-time radian frequency, a real variable. The quantity e jωn is then a complex sinusoid
More informationLecture 14: Minimum Phase Systems and Linear Phase
EE518 Digital Signal Processing University of Washington Autumn 2001 Dept. of Electrical Engineering Lecture 14: Minimum Phase Systems and Linear Phase Nov 19, 2001 Prof: J. Bilmes
More informationSignals and Systems Lecture 8: Z Transform
Signals and Systems Lecture 8: Z Transform Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Winter 2012 Farzaneh Abdollahi Signal and Systems Lecture 8 1/29 Introduction
More informationFilter Design Problem
Filter Design Problem Design of frequency-selective filters usually starts with a specification of their frequency response function. Practical filters have passband and stopband ripples, while exhibiting
More informationEE 225D LECTURE ON DIGITAL FILTERS. University of California Berkeley
University of California Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences Professors : N.Morgan / B.Gold EE225D Digital Filters Spring,1999 Lecture 7 N.MORGAN
More informationLAB 6: FIR Filter Design Summer 2011
University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering ECE 311: Digital Signal Processing Lab Chandra Radhakrishnan Peter Kairouz LAB 6: FIR Filter Design Summer 011
More informationDSP-I DSP-I DSP-I DSP-I
DSP-I DSP-I DSP-I DSP-I Digital Signal Processing I (8-79) Fall Semester, 005 OTES FOR 8-79 LECTURE 9: PROPERTIES AD EXAPLES OF Z-TRASFORS Distributed: September 7, 005 otes: This handout contains in outline
More informationDigital Wideband Integrators with Matching Phase and Arbitrarily Accurate Magnitude Response (Extended Version)
Digital Wideband Integrators with Matching Phase and Arbitrarily Accurate Magnitude Response (Extended Version) Ça gatay Candan Department of Electrical Engineering, METU, Ankara, Turkey ccandan@metu.edu.tr
More informationGrades will be determined by the correctness of your answers (explanations are not required).
6.00 (Fall 2011) Final Examination December 19, 2011 Name: Kerberos Username: Please circle your section number: Section Time 2 11 am 1 pm 4 2 pm Grades will be determined by the correctness of your answers
More informationOn the Frequency-Domain Properties of Savitzky-Golay Filters
On the Frequency-Domain Properties of Savitzky-Golay Filters Ronald W Schafer HP Laboratories HPL-2-9 Keyword(s): Savitzky-Golay filter, least-squares polynomial approximation, smoothing Abstract: This
More informationDSP. Chapter-3 : Filter Design. Marc Moonen. Dept. E.E./ESAT-STADIUS, KU Leuven
DSP Chapter-3 : Filter Design Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be/stadius/ Filter Design Process Step-1 : Define filter specs Pass-band, stop-band,
More informationBasic Design Approaches
(Classic) IIR filter design: Basic Design Approaches. Convert the digital filter specifications into an analog prototype lowpass filter specifications. Determine the analog lowpass filter transfer function
More informationMassachusetts Institute of Technology
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing QUIZ 1, March 16, 2010 ANSWER BOOKLET
More informationDiscrete 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 informationGATE 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 informationVALLIAMMAI ENGINEERING COLLEGE. SRM Nagar, Kattankulathur DEPARTMENT OF INFORMATION TECHNOLOGY. Academic Year
VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur- 603 203 DEPARTMENT OF INFORMATION TECHNOLOGY Academic Year 2016-2017 QUESTION BANK-ODD SEMESTER NAME OF THE SUBJECT SUBJECT CODE SEMESTER YEAR
More information2. Typical Discrete-Time Systems All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5.
. Typical Discrete-Time Systems.1. All-Pass Systems (5.5).. Minimum-Phase Systems (5.6).3. Generalized Linear-Phase Systems (5.7) .1. All-Pass Systems An all-pass system is defined as a system which has
More informationSchool of Information Technology and Electrical Engineering EXAMINATION. ELEC3004 Signals, Systems & Control
This exam paper must not be removed from the venue Venue Seat Number Student Number Family Name First Name School of Information Technology and Electrical Engineering EXAMINATION Semester One Final Examinations,
More informationSIGNAL PROCESSING. B14 Option 4 lectures. Stephen Roberts
SIGNAL PROCESSING B14 Option 4 lectures Stephen Roberts Recommended texts Lynn. An introduction to the analysis and processing of signals. Macmillan. Oppenhein & Shafer. Digital signal processing. Prentice
More informationEE 3054: Signals, Systems, and Transforms Spring A causal discrete-time LTI system is described by the equation. y(n) = 1 4.
EE : Signals, Systems, and Transforms Spring 7. A causal discrete-time LTI system is described by the equation Test y(n) = X x(n k) k= No notes, closed book. Show your work. Simplify your answers.. A discrete-time
More informationDHANALAKSHMI 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 informationUNIT - 7: FIR Filter Design
UNIT - 7: FIR Filter Design Dr. Manjunatha. P manjup.jnnce@gmail.com Professor Dept. of ECE J.N.N. College of Engineering, Shimoga October 5, 06 Unit 7 Syllabus Introduction FIR Filter Design:[,, 3, 4]
More informationHow 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/ (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 informationIIR digital filter design for low pass filter based on impulse invariance and bilinear transformation methods using butterworth analog filter
IIR digital filter design for low pass filter based on impulse invariance and bilinear transformation methods using butterworth analog filter Nasser M. Abbasi May 5, 0 compiled on hursday January, 07 at
More information2.161 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 information1. Design a 3rd order Butterworth low-pass filters having a dc gain of unity and a cutoff frequency, fc, of khz.
ECE 34 Experiment 6 Active Filter Design. Design a 3rd order Butterworth low-pass ilters having a dc gain o unity and a cuto requency, c, o.8 khz. c :=.8kHz K:= The transer unction is given on page 7 j
More informationChapter 5 THE APPLICATION OF THE Z TRANSFORM. 5.3 Stability
Chapter 5 THE APPLICATION OF THE Z TRANSFORM 5.3 Stability Copyright c 2005- Andreas Antoniou Victoria, BC, Canada Email: aantoniou@ieee.org February 13, 2008 Frame # 1 Slide # 1 A. Antoniou Digital Signal
More informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture 4: Inverse z Transforms & z Domain Analysis Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner
More informationZ-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 informationVery useful for designing and analyzing signal processing systems
z-transform z-transform The z-transform generalizes the Discrete-Time Fourier Transform (DTFT) for analyzing infinite-length signals and systems Very useful for designing and analyzing signal processing
More informationX (z) = n= 1. Ã! X (z) x [n] X (z) = Z fx [n]g x [n] = Z 1 fx (z)g. r n x [n] ª e jnω
3 The z-transform ² Two advantages with the z-transform:. The z-transform is a generalization of the Fourier transform for discrete-time signals; which encompasses a broader class of sequences. The z-transform
More informationLecture 9 Infinite Impulse Response Filters
Lecture 9 Infinite Impulse Response Filters Outline 9 Infinite Impulse Response Filters 9 First-Order Low-Pass Filter 93 IIR Filter Design 5 93 CT Butterworth filter design 5 93 Bilinear transform 7 9
More informationECE503: Digital Signal Processing Lecture 5
ECE53: Digital Signal Processing Lecture 5 D. Richard Brown III WPI 3-February-22 WPI D. Richard Brown III 3-February-22 / 32 Lecture 5 Topics. Magnitude and phase characterization of transfer functions
More informationUNIVERSITY OF OSLO. Faculty of mathematics and natural sciences. Forslag til fasit, versjon-01: Problem 1 Signals and systems.
UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 1th, 016 Examination hours: 14:30 18.30 This problem set
More informationTopic 4: The Z Transform
ELEN E480: Digital Signal Processing Topic 4: The Z Transform. The Z Transform 2. Inverse Z Transform . The Z Transform Powerful tool for analyzing & designing DT systems Generalization of the DTFT: G(z)
More informationThere are two main classes of digital lter FIR (Finite impulse response) and IIR (innite impulse reponse).
FIR Filters I There are two main classes of digital lter FIR (Finite impulse response) and IIR (innite impulse reponse). FIR Digital Filters These are described by dierence equations of the type: y[n]
More informationDiscrete-time Fourier transform (DTFT) representation of DT aperiodic signals Section The (DT) Fourier transform (or spectrum) of x[n] is
Discrete-time Fourier transform (DTFT) representation of DT aperiodic signals Section 5. 3 The (DT) Fourier transform (or spectrum) of x[n] is X ( e jω) = n= x[n]e jωn x[n] can be reconstructed from its
More informationSignals and Systems. Lecture 11 Wednesday 22 nd November 2017 DR TANIA STATHAKI
Signals and Systems Lecture 11 Wednesday 22 nd November 2017 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON Effect on poles and zeros on frequency response
More information