Block-by Block Convolution, FFT/IFFT, Digital Spectral Analysis
|
|
- Annabel Austin
- 6 years ago
- Views:
Transcription
1 Lecture 9 Outlie: Block-by Block Covolutio, FFT/IFFT, Digital Spectral Aalysis Aoucemets: Readig: 5: The Discrete Fourier Trasform pp. 3-5, 8, 9+block diagram at top of pg, pp. 7. HW 6 due today with free etesio to Thurs. 5pm Graded midterms ready for pickup (with TAs after class or from Julia) Eample of Liear Covolutio from Circular Block-by-block covolutio: Overlap/Add FFT/IFFT ad its Compleity Digital Spectral Aalysis
2 Midterm Postmortem Graded midterms/sols available (from TAs/Julia) Gradig Histogram?? Media: 65.5 Mea: Std. Dev: 4.46 Low: 4 High: 98 Regrade requests must be submitted i writig Describe gradig error that occurred; partial credit fied Course grade from all coursework: B+ class average Before etra credit; etra credit poits added to HW grade
3 Review of Last Lecture Computig circular covolutio: Liearly covolve ~ ad ~ Place sequeces o circle i opposite directios, sum up all pairs, rotate outer sequece clockwise each time icremet N N m ~ m~ m N otherwise Liear Covolutio usig Circular Covolutio zp Zero pad [] by appedig P- zeros: Zero pad h[] by appedig L- zeros: Both sequeces are of legth M=L+P-, same as y[] Take circular covolutio of zero padded sequeces This yields the liear covolutio: y h zp h M zp hzp * h L L L P P P L P M otherwise
4 Eample: Liear from Circular Liear Covolutio [ ] L=4 P= [] * [] Liear from Circular with Zero Paddig M=L+P-=9, zp 9, zp 3, zp , zp [ ] = =
5 Block Covolutio usig Overlap Methods Wat block-by-block liear covolutio for log sequeces Uses fied hardware. Has fied delay/compleity Goal: compute liear covolutio of [] ad y[] [] very log, h[] has legth P Wat to break [] ito shorter blocks ad compute portios of y[] block-by-block. Overlap-Add Method y * h mh m mhm m Breaks [] ito o-overlappig segmets of legth L: rl r L r Covolve each segmet with h[] ad sum: These covolutios computed usig DFT: r zp otherwise P m h N h FFT FFT * r,zp y r r * h * h FFTh r,zp zp r r
6
7 Overlap-Save (Not resposible for this topic; pp. 5-6) Breaks [] ito segmets of legth L>P, each segmet overlappig with previous oe at P- poits Perform L-poit circular covolutio of each segmet with zero-padded filter h[] (usig DFT): y L r p r hzp, Idetify portio of each circular covolutio that correspods to a liear covolutio, ad save it. First P poits are uusable, while the remaiig L P + poits correspod to a liear covolutio. Thus, we save L P + poits from each circular covolutio. yr, p P L y r otherwise Because first P poits are uusable, the iput segmets must overlap at P poits. y y rl P P r r
8 Blocklegth Choice I overlap methods, several factors affect the choice of the block legth L a shorter block legth miimizes latecy. a shorter block legth miimizes memory required for performig the FFTs, multiplicatio, ad iverse FFT. Give P (legth of h[]), there is a optimal block legth L that miimizes compleity. L too short, compleity icreased by overhead of adjacet block overlap L too log, compleity icreased because FFT compleity icreases with the block legth I practice, set block legth so that FFT blocklegth is a iteger power of (required for FFTs)
9 FFT ad IFFT Algorithms FFT computes the DFT of a sequece, IFFT N N k computes the iverse DFT: X k W N X k DFT as matri operatio: N comple multiplies (lect. 6) Compleity of FFT ad IFFT same: FFT/IFFT breaks dow a DFT with N comple multiplies ito may smaller DFTs with N multiplies Reduces compleity of computig N-poit DFT or IDFT from N comple multiplies to.5nlog N N N N log N N N log N , ,4,48,576 5, 4.8 8,9 67,8,864 53, N k DFT k W * X k DFT X k - I 994 Strag described the FFT as "the most importat umerical algorithm of our lifetime - Icluded i Top Algorithms of th Cetury by IEEE Joural of Computig i Sciece ad Egieerig N N *
10 Digital Spectral Aalysis (ppt slide oly) Ati-Aliasig Origial CT Sigal ct X c j t, tt Lowpass Filter sct Sc j j H aa Filtered CT Sigal Sampled Sigal t T A B Form Block of Legth L Block of L Sigal Samples, L C Widow of Legth L, L w Sampled, Widowed Sigal v w Aalog spectrum aalysis: D Zero-Pad To Legth N L Aalog sigal iput to a arrow BPF with tuable ceter frequecy. The ceter frequecy is swept over some rage Sigal recorded (amplitude ad phase) to obtai CTFT estimate Ca be used at ay frequecy rage: audio, radio ad microwave, optical, Digital spectrum aalysis is as show i the block diagram Sampler (ADC) techology limited to ~ GHz; sigals with BW>5 GHz distorted Widowig reduces distortio of FIR approimatio to IIR sigal Digital spectrum aalysis is cheaper, smaller, ad cosumes less power tha aalog Used by all small electroic devices that must do spectrum aalysis (e.g. WiFi) v L L N E N-Poit Block of N Spectral Samples V k, k N F DFT
11 Mai Poits For liear covolutio of log sequeces, computatio is doe i L-legth blocks usig overlap-add or overlap-save Methods are very similar, differ i where overlap is itroduced Choice of L optimizes tradeoff i latecy, memory, ad compleity The FFT ad IFFT drastically reduce the compleity of the DFT/IDFT computatio These algorithms are resposible for the widespread use of digital sigal processig i today s electroic devices Usig low-compleity FFTs ad IFFTs, spectral aalysis ca be doe with low-cost, low-power, small devices
EE123 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 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 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 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 information2D 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 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 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 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 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 informationVibratory Motion. Prof. Zheng-yi Feng NCHU SWC. National CHung Hsing University, Department of Soil and Water Conservation
Vibratory Motio Prof. Zheg-yi Feg NCHU SWC 1 Types of vibratory motio Periodic motio Noperiodic motio See Fig. A1, p.58 Harmoic motio Periodic motio Trasiet motio impact Trasiet motio earthquake A powerful
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 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 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 informationLecture 12: Spiral: Domain Specific HLS. James C. Hoe Department of ECE Carnegie Mellon University
8 643 Lecture : Spiral: Domai Specific HLS James C. Hoe Departmet of ECE Caregie Mello Uiversity 8 643 F7 L S, James C. Hoe, CMU/ECE/CALCM, 07 Houseeepig Your goal today: see a eample of really highlevel
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 informationLinear time invariant systems
Liear time ivariat systems Alejadro Ribeiro Dept. of Electrical ad Systems Egieerig Uiversity of Pesylvaia aribeiro@seas.upe.edu http://www.seas.upe.edu/users/~aribeiro/ February 25, 2016 Sigal ad Iformatio
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 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 informationChapter 9 Computation of the Discrete. Fourier Transform
Chapter 9 Coputatio of the Discrete Fourier Trasfor Itroductio Efficiet Coputatio of the Discrete Fourier Trasfor Goertzel Algorith Deciatio-I-Tie FFT Algoriths Deciatio-I-Frequecy FFT Algoriths Ipleetatio
More informationSpring 2014, EE123 Digital Signal Processing
Aoucemets EE3 Digital Sigal Processig Lecture 9 Lab part I ad II posted will post III today or tomorrow Lab-bash uesday -3pm Cory hree shorter Midterms: / i class / i class /3 (or BD) i class / or / (BD)
More informationWarped, Chirp Z-Transform: Radar Signal Processing
arped, Chirp Z-Trasform: Radar Sigal Processig by Garimella Ramamurthy Report o: IIIT/TR// Cetre for Commuicatios Iteratioal Istitute of Iformatio Techology Hyderabad - 5 3, IDIA Jauary ARPED, CHIRP Z
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 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 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 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 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 informationPolynomial Multiplication and Fast Fourier Transform
Polyomial Multiplicatio ad Fast Fourier Trasform Com S 477/577 Notes Ya-Bi Jia Sep 19, 2017 I this lecture we will describe the famous algorithm of fast Fourier trasform FFT, which has revolutioized digital
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 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 informationThe Discrete Fourier Transform
(7) The Discete Fouie Tasfom The Discete Fouie Tasfom hat is Discete Fouie Tasfom (DFT)? (ote: It s ot DTFT discete-time Fouie tasfom) A liea tasfomatio (mati) Samples of the Fouie tasfom (DTFT) of a apeiodic
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 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 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 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 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 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 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 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 informationSpectral Analysis. This week in lab. Next classes: 3/26 and 3/28. Your next experiment Homework is to prepare
Spectral Aalysis This week i lab Your ext experimet Homework is to prepare Next classes: 3/26 ad 3/28 Aero Testig, Fracture Toughess Testig Read the Experimets 5 ad 7 sectios of the course maual Spectral
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 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 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 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 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 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 informationThe Discrete Fourier Transform
The Discrete Fourier Trasform Complex Fourier Series Represetatio Recall that a Fourier series has the form a 0 + a k cos(kt) + k=1 b k si(kt) This represetatio seems a bit awkward, sice it ivolves two
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 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 informationReview of Discrete-time Signals. ELEC 635 Prof. Siripong Potisuk
Review of Discrete-time Sigals ELEC 635 Prof. Siripog Potisuk 1 Discrete-time Sigals Discrete-time, cotiuous-valued amplitude (sampled-data sigal) Discrete-time, discrete-valued amplitude (digital sigal)
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 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 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 informationI. Review of 1D continuous and discrete convolution. A. Continuous form: B. Discrete form: C. Example interface and review:
Lecture : Samplig Theorem ad Iterpolatio Learig Objectives: Review of cotiuous ad discrete covolutio Review of samplig with focus o sigal restoratio Applicatio of sigal iterpolatio I. Review of D cotiuous
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 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 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 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 informationLet A(x) and B(x) be two polynomials of degree n 1:
MI-EVY (2011/2012) J. Holub: 4. DFT, FFT ad Patter Matchig p. 2/42 Operatios o polyomials MI-EVY (2011/2012) J. Holub: 4. DFT, FFT ad Patter Matchig p. 4/42 Efficiet Patter Matchig (MI-EVY) 4. DFT, FFT
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 informationfrom definition we note that for sequences which are zero for n < 0, X[z] involves only negative powers of z.
We ote that for the past four examples we have expressed the -trasform both as a ratio of polyomials i ad as a ratio of polyomials i -. The questio is how does oe kow which oe to use? [] X ] from defiitio
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 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 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 informationMachine Learning Regression I Hamid R. Rabiee [Slides are based on Bishop Book] Spring
Machie Learig Regressio I Hamid R. Rabiee [Slides are based o Bishop Book] Sprig 015 http://ce.sharif.edu/courses/93-94//ce717-1 Liear Regressio Liear regressio: ivolves a respose variable ad a sigle predictor
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 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 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
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 informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
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 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 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 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 information2(25) Mean / average / expected value of a stochastic variable X: Variance of a stochastic variable X: 1(25)
Lecture 5: Codig of Aalog Sources Samplig ad Quatizatio Images ad souds are ot origially digital! The are cotiuous sigals i space/time as well as amplitude Typical model of a aalog source: A statioary
More informationApplications of Distributed Arithmetic to Digital Signal Processing: A Tutorial Review
pplicatios of Distriuted rithmetic to Digital Sigal Processig: Tutorial Review Ref: Staley. White, pplicatios of Distriuted rithmetic to Digital Sigal Processig: Tutorial Review, IEEE SSP Magazie, July,
More informationBIOINF 585: Machine Learning for Systems Biology & Clinical Informatics
BIOINF 585: Machie Learig for Systems Biology & Cliical Iformatics Lecture 14: Dimesio Reductio Jie Wag Departmet of Computatioal Medicie & Bioiformatics Uiversity of Michiga 1 Outlie What is feature reductio?
More informationMAXIMALLY FLAT FIR FILTERS
MAXIMALLY FLAT FIR FILTERS This sectio describes a family of maximally flat symmetric FIR filters first itroduced by Herrma [2]. The desig of these filters is particularly simple due to the availability
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 informationChapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers
Chapter 4 4-1 orth Seattle Commuity College BUS10 Busiess Statistics Chapter 4 Descriptive Statistics Summary Defiitios Cetral tedecy: The extet to which the data values group aroud a cetral value. Variatio:
More informationDigital Signal Processing, Fall 2010
Digital Sigal Processig, Fall 2 Lecture : Itroductio, Discrete-time sigals ad sstems Zheg-Hua Ta Departmet of Electroic Sstems alborg Uiversit, Demar zt@es.aau.d Digital Sigal Processig, I, Zheg-Hua Ta
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 information11. What are energy and power signals? (April/may 2011, Nov/Dec 2012) Energy signal: The energy of a discrete time signal x(n) is defined as
DHAALAKSHMI COLLEGE OF EGIEERIG, CHEAI DEPARTMET OF COMPUTER SCIECE AD EGIEERIG IT650 DIGITAL SIGAL PROCESSIG UIT - I : SIGALS AD SYSTEMS PART A MARKS. Defie Sigal ad Sigal Processig. A sigal is defied
More informationIntroduction to Computational Biology Homework 2 Solution
Itroductio to Computatioal Biology Homework 2 Solutio Problem 1: Cocave gap pealty fuctio Let γ be a gap pealty fuctio defied over o-egative itegers. The fuctio γ is called sub-additive iff it satisfies
More informationPipelined and Parallel Recursive and Adaptive Filters
VLSI Digital Sigal Processig Systems Pipelied ad Parallel Recursive ad Adaptive Filters La-Da Va 范倫達, Ph. D. Departmet of Computer Sciece Natioal Chiao ug Uiversity aiwa, R.O.C. Fall, 05 ldva@cs.ctu.edu.tw
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 informationParallel Vector Algorithms David A. Padua
Parallel Vector Algorithms 1 of 32 Itroductio Next, we study several algorithms where parallelism ca be easily expressed i terms of array operatios. We will use Fortra 90 to represet these algorithms.
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 informationApplications of Distributed Arithmetic to Digital Signal Processing: A Tutorial Review
pplicatios of Distriuted rithmetic to Digital Sigal Processig: Tutorial Review Ref: Staley. White, pplicatios of Distriuted rithmetic to Digital Sigal Processig: Tutorial Review, IEEE SSP Magazie, July,
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 informationModule 11: Applications : Linear prediction, Speech Analysis and Speech Enhancement Prof. Eliathamby Ambikairajah Dr. Tharmarajah Thiruvaran School
Module : Applicatios : Liear predictio, Speech Aalysis ad Speech Ehacemet Prof. Eliathamby Ambiairajah Dr. Tharmarajah Thiruvara School of Electrical Egieerig & Telecommuicatios The Uiversity of New South
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 informationCEMTool Tutorial. Fourier Analysis
CEMTool Tutorial Fourier Aalysis Overview This tutorial is part of the CEMWARE series. Each tutorial i this series will teach you a specific topic of commo applicatios by explaiig theoretical cocepts ad
More informationMath E-21b Spring 2018 Homework #2
Math E- Sprig 08 Homework # Prolems due Thursday, Feruary 8: Sectio : y = + 7 8 Fid the iverse of the liear trasformatio [That is, solve for, i terms of y, y ] y = + 0 Cosider the circular face i the accompayig
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationComplex Algorithms for Lattice Adaptive IIR Notch Filter
4th Iteratioal Coferece o Sigal Processig Systems (ICSPS ) IPCSIT vol. 58 () () IACSIT Press, Sigapore DOI:.7763/IPCSIT..V58. Complex Algorithms for Lattice Adaptive IIR Notch Filter Hog Liag +, Nig Jia
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 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 informationCS 179: GPU Programming. Lecture 9 / Homework 3
CS 179: GPU Programming Lecture 9 / Homework 3 Recap Some algorithms are less obviously parallelizable : Reduction Sorts FFT (and certain recursive algorithms) Parallel FFT structure (radix-2) Bit-reversed
More informationDigital Signal Processing
Digital Sigal Processig EC5 SUBJECT CODE : EC5 IA ARKS : 5 O. OF LECTURE HRS/WEEK : 4 EXA HOURS : 3 TOTAL O. OF LECTURE HRS. : 5 EXA ARKS : UIT - PART - A DISCRETE FOURIER TRASFORS DFT: FREQUECY DOAI SAPLIG
More information