Discrete Fourier Transform
|
|
- Ada Atkins
- 5 years ago
- Views:
Transcription
1 Discrete Fourier Transform Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno,
2 Diskrete Fourier transform (DFT) We have just one problem with DFS that needs to be solved. Infinite length of signal and finite length of computed spectrum. DFT transforms a sequence of length N to other sequence of length N we will see that it is a transform of one period of the input signal to one period in DFS. The procedure is the following:. periodize a sequence x[n] of length N : x[n] = x[ mod N (n)]. N. find DFS coefficients: X[k] = n= x[n]e j π N kn. Note, only one period of periodic signal x[ mod N (n)] is taken, therefore, we can work just with original sequence x[n]. Step. is taken to fulfill requirements for DFS computing. 3. resulting sequence is windowed again to the length N: X[k] = R N [k] N n= x[n]e j π N kn
3 Usually we find this formula with no windowing function as computing only through one period is assumed, X[k] for k = [, N ]: X[k] = N n= x[n]e j π N kn X[k] is a projection/image of DFT, denoted as x[n] DFT X[k]. Inverse DFT for samples n = [, N ] is obtained in the same manner ( periodization of DFT spectrum, inverse DFS application, windowing of the resulting periodic signal) : x[n] = N N k= X[k]e +j π N kn, we denote X[k] DFT x[n] 3
4 Frequency axis in DFT N samples of DFT are placed from approaching sampling frequency: sampling frequency is N. we have N samples placed from to N. thus for samples X[k] holds: normalized frequency k N to N N. normalized circle frequency π k to πn N N regular frequency k N F s to N N F s circle frequency k N πf s to N N πf s
5 Example : N = 6, shifted square signal of length 8, F s = Hz
6 8 norm. f 8 f x 8 norm. omega 8 omega x 5 6
7 Example : One period of a harmonic signal, N = 6, F s = Hz, ω = π 6 rad
8 8 norm. f 8 f x 8 norm. omega 8 omega x 5 8
9 Properties of DFT Image of a real sequence similarly to FS: X[k] = X [N k] X[] would be complex conjugate to X[N], but X[N] does not exist. Recall that according to DFS definition, X[] is a sum of discrete samples, that is a direct component. If N is even, then: X [ N ] = X [ N N is complex conjugate to itself, thus it must be real. Ilustration: see previous examples. ] = X [ N ]. 9
10 Linearity x [n] DFT X [k] x [n] DFT X [k] ax [n] + bx [n] DFT ax [k] + bx [k] Image of a circulary shifted sequence x[n] DFT X[k] R N x[ mod N (n m)] DFT X[k]e j π N mk
11
12 Image of circular convolution x [n] DFT X [k] x [n] DFT X [k] x [n] N x [n] DFT X [k]x [k] Similarly as for regular FT convolution of two signals corresponded to multiplication of their spectra in frequency, the DFT image of a circular convolution is a product of DFT coefficients of the convoluted signals.
13
14 Fast Fourier transform FFT Computing of DFT according to: X[k] = N n= x[n]e j π N kn requires N operations (multiplication or addition) with complex numbers. Cooley and Tukey invented an efficient algorithm for DFT and its inverse with N = k, where k is an integer: Fast Fourier transform FFT. The number of operations becomes only N log N. FFT recursively breaks the transform into two N/ transforms processing a pair of samples producing a pair of coefficients in each step. Example: pro N =, N = MOPS, N log N = kops FFT produces the same output as DFT!
15 Computating FS and FT (with continuous time) using DFT We are interested how to compute a spectral representation (coefficients of FS or FT) just using DFT. first let us summarize what we compute using DFT: the signal is sampled, thus spectrum is periodic (eventhough we compute only one period of spectrum with N samples (, π, F s, πf s, according to the type of frequency). signal is periodic (by N samples) (eventhough we consider only one period for computating of DFT), spectrum is thus sampled (discrete). The step in spectrum is N, π N, F s N, πf s N according to the type of frequency. signal is windowed the spectrum of the window occurs also in DFT image, x(t)w(t) X(jω) W(jω). 5
16 Computation of coefficients FS using DFT To remind, for a continuous-time signal with period T, FS coefficients are: c k = T T x(t)e jkω t dt, If such signal is sampled with sampling period T, and T then contains N samples, we can aproximate the integral using: c k NT N n= π jk x(nt)e NT nt T = T NT N n= x(nt)e jk π N n = N N n= x[n]e jkn π N. This definition resembles the DFT formula with the only difference that we have to divide the c k by the number of samples N: c k = X[k] N. 6
17 The equation can be used only when the following restrictions are satisfied:. we can compute only coefficients c k for k < N one). (second half is mirrored to the first. sampling theorem must be satisfied: last non-zero coefficient of analog signal is for k max < N, otherwise aliasing ocures. We must to realize that N now corresponds to the sampling frequency, so the above equation is equivalent to: ω max < Ω s. 3. N samples must fit into exactly one period of the signal. When more periods m, we need to make a small modification: c k = S[mk] N 7
18 Example : signal with continuous time x(t) = cos(5πt + π/) sampled at khz. Compute coefficients of FS using DFT. Period T = π 5π =.6. Number of samples for computation is T T =.6/. = 6. Theoretic values of the coefficients are c = 5e jπ/, c = 5e jπ/
19 Example : signal with continuous time x(t) = cos(5πt) sampled at khz. Compute coefficients of FS using DFT. We don t know the period of the signal, we can choose N = 6. Theoretic values of coefficients are c = 5, c =
20 Example 3: signal with continuous time: periodic sequence of square impulses with D =, ϑ = 3 ms, T = 6 ms, sampled at khz. Compute coefficients of FS using DFT. Theoretic values of coefficients are c k = Dϑ T sinc( ϑ kω )
21 Computation of spectral function using DFT again let s remind X(jω) = + x(t)e jωt dt We will able to compute only FT of signal which is restricted from to T : if its is not, we cannot do anything. if it is, but elsewhere for example from t start to t start + T we will move it to [, T ], but we will remember it finally, just small fix of phase will be needed. If such signal is sampled with sampling period T, we get N samples. Integral can be aproximated, but only for some frequencies - that are multiples of Nth portion of the sampling frequency Ω s = π T : k Ω s N. Then: X(jk Ω N s N ) n= x(nt)e jk Ω s N nt T = T N n= π/t jk x(nt)e N nt = T N n= x[n]e jkn π N.
22 We again see the definition of DFT in the derived equation so for circular frequencies k Ω s N we can write: Again some restrictions: valid only for k < N. X(jk Ω s N ) = TX[k] sampling theorem must be satisfied: the maximum frequency ω max in the signal spectrum must be ω max < Ω s otherwise aliasing occurs. When we have a signal with ω max = (square,... ) we should use Ω s the highest possible so aliasing does not hurt.
23 We compute values for some certain frequencies, but we are interested in all values of the spectral function. We must interpolate, or use zero-padding getting more samples in the spectrum. the phase need to be fixed if the signal s period was pushed to fit the interval [, T ]: X(jk Ω s N ) X(jkΩ s Ωs N )e jk N t start 3
24 Example: square impuls s D =, ϑ = 3 ms, sampled at khz. Theoretic spectral function is X(jω) = Dϑsinc( ϑ ω)
25 spectral function computed for N =
26 zero padded and spectral function computed for N =
27 good frequency axis (ω), scaling (multiplied by T) and corrected phase:
X. Chen More on Sampling
X. Chen More on Sampling 9 More on Sampling 9.1 Notations denotes the sampling time in second. Ω s = 2π/ and Ω s /2 are, respectively, the sampling frequency and Nyquist frequency in rad/sec. Ω and ω denote,
More 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 informationFourier series for continuous and discrete time signals
8-9 Signals and Systems Fall 5 Fourier series for continuous and discrete time signals The road to Fourier : Two weeks ago you saw that if we give a complex exponential as an input to a system, the output
More informationRandom signals II. ÚPGM FIT VUT Brno,
Random signals II. Jan Černocký ÚPGM FIT VUT Brno, cernocky@fit.vutbr.cz 1 Temporal estimate of autocorrelation coefficients for ergodic discrete-time random process. ˆR[k] = 1 N N 1 n=0 x[n]x[n + k],
More informationDiscrete Fourier transform (DFT)
Discrete Fourier transform (DFT) Signal Processing 2008/9 LEA Instituto Superior Técnico Signal Processing LEA (IST) Discrete Fourier transform 1 / 34 Periodic signals Consider a periodic signal x[n] with
More informationIn this Lecture. Frequency domain analysis
In this Lecture Frequency domain analysis Introduction In most cases we want to know the frequency content of our signal Why? Most popular analysis in frequency domain is based on work of Joseph Fourier
More informationChapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum
Chapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum CEN352, DR. Nassim Ammour, King Saud University 1 Fourier Transform History Born 21 March 1768 ( Auxerre ). Died 16 May 1830 ( Paris ) French
More informationLecture 19: Discrete Fourier Series
EE518 Digital Signal Processing University of Washington Autumn 2001 Dept. of Electrical Engineering Lecture 19: Discrete Fourier Series Dec 5, 2001 Prof: J. Bilmes TA: Mingzhou
More informationEEL3135: Homework #3 Solutions
EEL335: Homework #3 Solutions Problem : (a) Compute the CTFT for the following signal: xt () cos( πt) cos( 3t) + cos( 4πt). First, we use the trigonometric identity (easy to show by using the inverse Euler
More informationFast Fourier Transform Discrete-time windowing Discrete Fourier Transform Relationship to DTFT Relationship to DTFS Zero padding
Fast Fourier Transform Discrete-time windowing Discrete Fourier Transform Relationship to DTFT Relationship to DTFS Zero padding Fourier Series & Transform Summary x[n] = X[k] = 1 N k= n= X[k]e jkω
More informationx[n] = x a (nt ) x a (t)e jωt dt while the discrete time signal x[n] has the discrete-time Fourier transform x[n]e jωn
Sampling Let x a (t) be a continuous time signal. The signal is sampled by taking the signal value at intervals of time T to get The signal x(t) has a Fourier transform x[n] = x a (nt ) X a (Ω) = x a (t)e
More informationFast Fourier Transform Discrete-time windowing Discrete Fourier Transform Relationship to DTFT Relationship to DTFS Zero padding
Fast Fourier Transform Discrete-time windowing Discrete Fourier Transform Relationship to DTFT Relationship to DTFS Zero padding J. McNames Portland State University ECE 223 FFT Ver. 1.03 1 Fourier Series
More informationFourier transform representation of CT aperiodic signals Section 4.1
Fourier transform representation of CT aperiodic signals Section 4. A large class of aperiodic CT signals can be represented by the CT Fourier transform (CTFT). The (CT) Fourier transform (or spectrum)
More informationECE 301 Fall 2010 Division 2 Homework 10 Solutions. { 1, if 2n t < 2n + 1, for any integer n, x(t) = 0, if 2n 1 t < 2n, for any integer n.
ECE 3 Fall Division Homework Solutions Problem. Reconstruction of a continuous-time signal from its samples. Consider the following periodic signal, depicted below: {, if n t < n +, for any integer n,
More informationDiscrete Fourier Transform
Discrete Fourier Transform Virtually all practical signals have finite length (e.g., sensor data, audio records, digital images, stock values, etc). Rather than considering such signals to be zero-padded
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 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 information7.16 Discrete Fourier Transform
38 Signals, Systems, Transforms and Digital Signal Processing with MATLAB i.e. F ( e jω) = F [f[n]] is periodic with period 2π and its base period is given by Example 7.17 Let x[n] = 1. We have Π B (Ω)
More informationReview: Continuous Fourier Transform
Review: Continuous Fourier Transform Review: convolution x t h t = x τ h(t τ)dτ Convolution in time domain Derivation Convolution Property Interchange the order of integrals Let Convolution Property By
More information4.1 Introduction. 2πδ ω (4.2) Applications of Fourier Representations to Mixed Signal Classes = (4.1)
4.1 Introduction Two cases of mixed signals to be studied in this chapter: 1. Periodic and nonperiodic signals 2. Continuous- and discrete-time signals Other descriptions: Refer to pp. 341-342, textbook.
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 informationSignals & Systems. Lecture 5 Continuous-Time Fourier Transform. Alp Ertürk
Signals & Systems Lecture 5 Continuous-Time Fourier Transform Alp Ertürk alp.erturk@kocaeli.edu.tr Fourier Series Representation of Continuous-Time Periodic Signals Synthesis equation: x t = a k e jkω
More informationChapter 5. Fourier Analysis for Discrete-Time Signals and Systems Chapter
Chapter 5. Fourier Analysis for Discrete-Time Signals and Systems Chapter Objec@ves 1. Learn techniques for represen3ng discrete-)me periodic signals using orthogonal sets of periodic basis func3ons. 2.
More informationSummary of lecture 1. E x = E x =T. X T (e i!t ) which motivates us to define the energy spectrum Φ xx (!) = jx (i!)j 2 Z 1 Z =T. 2 d!
Summary of lecture I Continuous time: FS X FS [n] for periodic signals, FT X (i!) for non-periodic signals. II Discrete time: DTFT X T (e i!t ) III Poisson s summation formula: describes the relationship
More information[ ], [ ] [ ] [ ] = [ ] [ ] [ ]{ [ 1] [ 2]
4. he discrete Fourier transform (DF). Application goal We study the discrete Fourier transform (DF) and its applications: spectral analysis and linear operations as convolution and correlation. We use
More informationChapter 8 The Discrete Fourier Transform
Chapter 8 The Discrete Fourier Transform Introduction Representation of periodic sequences: the discrete Fourier series Properties of the DFS The Fourier transform of periodic signals Sampling the Fourier
More informationSignals & Systems. Lecture 4 Fourier Series Properties & Discrete-Time Fourier Series. Alp Ertürk
Signals & Systems Lecture 4 Fourier Series Properties & Discrete-Time Fourier Series Alp Ertürk alp.erturk@kocaeli.edu.tr Fourier Series Representation of Continuous-Time Periodic Signals Synthesis equation:
More informationLecture 10. Digital Signal Processing. Chapter 7. Discrete Fourier transform DFT. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev.
Lecture 10 Digital Signal Processing Chapter 7 Discrete Fourier transform DFT Mikael Swartling Nedelko Grbic Bengt Mandersson rev. 016 Department of Electrical and Information Technology Lund University
More information6.003 Signal Processing
6.003 Signal Processing Week 6, Lecture A: The Discrete Fourier Transform (DFT) Adam Hartz hz@mit.edu What is 6.003? What is a signal? Abstractly, a signal is a function that conveys information Signal
More informationECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name:
ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, 205 Name:. The quiz is closed book, except for one 2-sided sheet of handwritten notes. 2. Turn off
More information4.3 The Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT)
CHAPTER. TIME-FREQUECY AALYSIS: FOURIER TRASFORMS AD WAVELETS.3 The Discrete Fourier Transform (DFT and the Fast Fourier Transform (FFT.3.1 Introduction In this section, we discuss some of the mathematics
More informationReview 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 informationEE 224 Signals and Systems I Review 1/10
EE 224 Signals and Systems I Review 1/10 Class Contents Signals and Systems Continuous-Time and Discrete-Time Time-Domain and Frequency Domain (all these dimensions are tightly coupled) SIGNALS SYSTEMS
More informationECE 301 Fall 2011 Division 1 Homework 10 Solutions. { 1, for 0.5 t 0.5 x(t) = 0, for 0.5 < t 1
ECE 3 Fall Division Homework Solutions Problem. Reconstruction of a continuous-time signal from its samples. Let x be a periodic continuous-time signal with period, such that {, for.5 t.5 x(t) =, for.5
More informationChapter 6: Applications of Fourier Representation Houshou Chen
Chapter 6: Applications of Fourier Representation Houshou Chen Dept. of Electrical Engineering, National Chung Hsing University E-mail: houshou@ee.nchu.edu.tw H.S. Chen Chapter6: Applications of Fourier
More informationDFT and Matlab with some examples.
DFT and Matlab with some examples 1 www.icrf.nl Fourier transform Fourier transform is defined as: X + jωt = x( t e dt ( ω ) ) with ω= 2 π f Rad/s And x(t) a signal in the time domain. www.icrf.nl 2 Fourier
More informationDigital 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 informationTable 1: Properties of the Continuous-Time Fourier Series. Property Periodic Signal Fourier Series Coefficients
able : Properties of the Continuous-ime Fourier Series x(t = e jkω0t = = x(te jkω0t dt = e jk(/t x(te jk(/t dt Property Periodic Signal Fourier Series Coefficients x(t y(t } Periodic with period and fundamental
More information6.003 Signal Processing
6.003 Signal Processing Week 6, Lecture A: The Discrete Fourier Transform (DFT) Adam Hartz hz@mit.edu What is 6.003? What is a signal? Abstractly, a signal is a function that conveys information Signal
More informationLecture 20: Discrete Fourier Transform and FFT
EE518 Digital Signal Processing University of Washington Autumn 2001 Dept of Electrical Engineering Lecture 20: Discrete Fourier Transform and FFT Dec 10, 2001 Prof: J Bilmes TA:
More informationECE 301. Division 2, Fall 2006 Instructor: Mimi Boutin Midterm Examination 3
ECE 30 Division 2, Fall 2006 Instructor: Mimi Boutin Midterm Examination 3 Instructions:. Wait for the BEGIN signal before opening this booklet. In the meantime, read the instructions below and fill out
More informationELEN 4810 Midterm Exam
ELEN 4810 Midterm Exam Wednesday, October 26, 2016, 10:10-11:25 AM. One sheet of handwritten notes is allowed. No electronics of any kind are allowed. Please record your answers in the exam booklet. Raise
More informationSignals and Systems. Lecture 14 DR TANIA STATHAKI READER (ASSOCIATE PROFESSOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON
Signals and Systems Lecture 14 DR TAIA STATHAKI READER (ASSOCIATE PROFESSOR) I SIGAL PROCESSIG IMPERIAL COLLEGE LODO Introduction. Time sampling theorem resume. We wish to perform spectral analysis using
More informationFourier Representations of Signals & LTI Systems
3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n] 2. LTI system: LTI System Output = A weighted superposition of the system response to each complex
More informationThe Fourier Transform (and more )
The Fourier Transform (and more ) imrod Peleg ov. 5 Outline Introduce Fourier series and transforms Introduce Discrete Time Fourier Transforms, (DTFT) Introduce Discrete Fourier Transforms (DFT) Consider
More informationECE-700 Review. Phil Schniter. January 5, x c (t)e jωt dt, x[n]z n, Denoting a transform pair by x[n] X(z), some useful properties are
ECE-7 Review Phil Schniter January 5, 7 ransforms Using x c (t) to denote a continuous-time signal at time t R, Laplace ransform: X c (s) x c (t)e st dt, s C Continuous-ime Fourier ransform (CF): ote that:
More informationComplex symmetry Signals and Systems Fall 2015
18-90 Signals and Systems Fall 015 Complex symmetry 1. Complex symmetry This section deals with the complex symmetry property. As an example I will use the DTFT for a aperiodic discrete-time signal. The
More informationECGR4124 Digital Signal Processing Midterm Spring 2010
ECGR4124 Digital Signal Processing Midterm Spring 2010 Name: LAST 4 DIGITS of Student Number: Do NOT begin until told to do so Make sure that you have all pages before starting Open book, 1 sheet front/back
More informationDISCRETE FOURIER TRANSFORM
DISCRETE FOURIER TRANSFORM 1. Introduction The sampled discrete-time fourier transform (DTFT) of a finite length, discrete-time signal is known as the discrete Fourier transform (DFT). The DFT contains
More informationTable 1: Properties of the Continuous-Time Fourier Series. Property Periodic Signal Fourier Series Coefficients
able : Properties of the Continuous-ime Fourier Series x(t = a k e jkω0t = a k = x(te jkω0t dt = a k e jk(/t x(te jk(/t dt Property Periodic Signal Fourier Series Coefficients x(t y(t } Periodic with period
More information! Circular Convolution. " Linear convolution with circular convolution. ! Discrete Fourier Transform. " Linear convolution through circular
Previously ESE 531: Digital Signal Processing Lec 22: April 18, 2017 Fast Fourier Transform (con t)! Circular Convolution " Linear convolution with circular convolution! Discrete Fourier Transform " Linear
More informationHomework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt
Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t
More informationVII. Discrete Fourier Transform (DFT) Chapter-8. A. Modulo Arithmetic. (n) N is n modulo N, n is an integer variable.
1 VII. Discrete Fourier Transform (DFT) Chapter-8 A. Modulo Arithmetic (n) N is n modulo N, n is an integer variable. (n) N = n m N 0 n m N N-1, pick m Ex. (k) 4 W N = e -j2π/n 2 Note that W N k = 0 but
More informationSignals and Systems Profs. Byron Yu and Pulkit Grover Fall Midterm 2 Solutions
8-90 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 08 Midterm Solutions Name: Andrew ID: Problem Score Max 8 5 3 6 4 7 5 8 6 7 6 8 6 9 0 0 Total 00 Midterm Solutions. (8 points) Indicate whether
More informationContents. Digital Signal Processing, Part II: Power Spectrum Estimation
Contents Digital Signal Processing, Part II: Power Spectrum Estimation 5. Application of the FFT for 7. Parametric Spectrum Est. Filtering and Spectrum Estimation 7.1 ARMA-Models 5.1 Fast Convolution 7.2
More information6.003: Signal Processing
6.003: Signal Processing Discrete Fourier Transform Discrete Fourier Transform (DFT) Relations to Discrete-Time Fourier Transform (DTFT) Relations to Discrete-Time Fourier Series (DTFS) October 16, 2018
More informationL29: Fourier analysis
L29: Fourier analysis Introduction The discrete Fourier Transform (DFT) The DFT matrix The Fast Fourier Transform (FFT) The Short-time Fourier Transform (STFT) Fourier Descriptors CSCE 666 Pattern Analysis
More informationLecture 5. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)
Lecture 5 The Digital Fourier Transform (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith) 1 -. 8 -. 6 -. 4 -. 2-1 -. 8 -. 6 -. 4 -. 2 -. 2. 4. 6. 8 1
More informationINTRODUCTION TO THE DFS AND THE DFT
ITRODUCTIO TO THE DFS AD THE DFT otes: This brief handout contains in very brief outline form the lecture notes used for a video lecture in a previous year introducing the DFS and the DFT. This material
More informationDigital Signal Processing
Digital Signal Processing Introduction Moslem Amiri, Václav Přenosil Embedded Systems Laboratory Faculty of Informatics, Masaryk University Brno, Czech Republic amiri@mail.muni.cz prenosil@fi.muni.cz February
More informationHomework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, Signals & Systems Sampling P1
Homework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, 7.49 Signals & Systems Sampling P1 Undersampling & Aliasing Undersampling: insufficient sampling frequency ω s < 2ω M Perfect
More informationDiscrete Time Fourier Transform
Discrete Time Fourier Transform Recall that we wrote the sampled signal x s (t) = x(kt)δ(t kt). We calculate its Fourier Transform. We do the following: Ex. Find the Continuous Time Fourier Transform of
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 informationThe Discrete Fourier Transform
In [ ]: cd matlab pwd The Discrete Fourier Transform Scope and Background Reading This session introduces the z-transform which is used in the analysis of discrete time systems. As for the Fourier and
More informationDigital Signal Processing. Midterm 2 Solutions
EE 123 University of California, Berkeley Anant Sahai arch 15, 2007 Digital Signal Processing Instructions idterm 2 Solutions Total time allowed for the exam is 80 minutes Please write your name and SID
More informationE2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)
E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,
More informationDiscrete-Time David Johns and Ken Martin University of Toronto
Discrete-Time David Johns and Ken Martin University of Toronto (johns@eecg.toronto.edu) (martin@eecg.toronto.edu) University of Toronto 1 of 40 Overview of Some Signal Spectra x c () t st () x s () t xn
More informationRepresenting a Signal
The Fourier Series Representing a Signal The convolution method for finding the response of a system to an excitation takes advantage of the linearity and timeinvariance of the system and represents the
More informationSignals and Systems Spring 2004 Lecture #9
Signals and Systems Spring 2004 Lecture #9 (3/4/04). The convolution Property of the CTFT 2. Frequency Response and LTI Systems Revisited 3. Multiplication Property and Parseval s Relation 4. The DT Fourier
More informationLAB 2: DTFT, DFT, and DFT Spectral Analysis 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 2: DTFT, DFT, and DFT Spectral
More informationMAHALAKSHMI ENGINEERING COLLEGE-TRICHY
DIGITAL SIGNAL PROCESSING DEPT./SEM.: ECE&EEE /V DISCRETE FOURIER TRANFORM AND FFT PART-A 1. Define DFT of a discrete time sequence? AUC MAY 06 The DFT is used to convert a finite discrete time sequence
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 informationEA2.3 - Electronics 2 1
In the previous lecture, I talked about the idea of complex frequency s, where s = σ + jω. Using such concept of complex frequency allows us to analyse signals and systems with better generality. In this
More informationFourier transform. Alejandro Ribeiro. February 1, 2018
Fourier transform Alejandro Ribeiro February 1, 2018 The discrete Fourier transform (DFT) is a computational tool to work with signals that are defined on a discrete time support and contain a finite number
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 informationEE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 20, Cover Sheet
NAME: NAME EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 20, 2017 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 5,7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet
More information! Introduction. ! Discrete Time Signals & Systems. ! Z-Transform. ! Inverse Z-Transform. ! Sampling of Continuous Time Signals
ESE 531: Digital Signal Processing Lec 25: April 24, 2018 Review Course Content! Introduction! Discrete Time Signals & Systems! Discrete Time Fourier Transform! Z-Transform! Inverse Z-Transform! Sampling
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationANALOG AND DIGITAL SIGNAL PROCESSING ADSP - Chapter 8
ANALOG AND DIGITAL SIGNAL PROCESSING ADSP - Chapter 8 Fm n N fnt ( ) e j2mn N X() X() 2 X() X() 3 W Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 W W 3 W W x () x () x () 2 x ()
More informationEE 16B Final, December 13, Name: SID #:
EE 16B Final, December 13, 2016 Name: SID #: Important Instructions: Show your work. An answer without explanation is not acceptable and does not guarantee any credit. Only the front pages will be scanned
More informationEvaluating Fourier Transforms with MATLAB
ECE 460 Introduction to Communication Systems MATLAB Tutorial #2 Evaluating Fourier Transforms with MATLAB In class we study the analytic approach for determining the Fourier transform of a continuous
More informationDiscrete-time Fourier Series (DTFS)
Discrete-time Fourier Series (DTFS) Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 59 Opening remarks The Fourier series representation for discrete-time signals has some similarities with
More informationLECTURE 12 Sections Introduction to the Fourier series of periodic signals
Signals and Systems I Wednesday, February 11, 29 LECURE 12 Sections 3.1-3.3 Introduction to the Fourier series of periodic signals Chapter 3: Fourier Series of periodic signals 3. Introduction 3.1 Historical
More informationChapter 2: The Fourier Transform
EEE, EEE Part A : Digital Signal Processing Chapter Chapter : he Fourier ransform he Fourier ransform. Introduction he sampled Fourier transform of a periodic, discrete-time signal is nown as the discrete
More informationHST.582J / 6.555J / J Biomedical Signal and Image Processing Spring 2007
MIT OpenCourseare http://ocw.mit.edu HST.58J / 6.555J / 16.56J Biomedical Signal and Image Processing Spring 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationDESIGN OF CMOS ANALOG INTEGRATED CIRCUITS
DESIGN OF CMOS ANALOG INEGRAED CIRCUIS Franco Maloberti Integrated Microsistems Laboratory University of Pavia Discrete ime Signal Processing F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete
More informationChap 4. Sampling of Continuous-Time Signals
Digital Signal Processing Chap 4. Sampling of Continuous-Time Signals Chang-Su Kim Digital Processing of Continuous-Time Signals Digital processing of a CT signal involves three basic steps 1. Conversion
More informationRadar Systems Engineering Lecture 3 Review of Signals, Systems and Digital Signal Processing
Radar Systems Engineering Lecture Review of Signals, Systems and Digital Signal Processing Dr. Robert M. O Donnell Guest Lecturer Radar Systems Course Review Signals, Systems & DSP // Block Diagram of
More informationModule 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 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 informationOverview of Sampling Topics
Overview of Sampling Topics (Shannon) sampling theorem Impulse-train sampling Interpolation (continuous-time signal reconstruction) Aliasing Relationship of CTFT to DTFT DT processing of CT signals DT
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationEE 435. Lecture 30. Data Converters. Spectral Performance
EE 435 Lecture 30 Data Converters Spectral Performance . Review from last lecture. INL Often Not a Good Measure of Linearity Four identical INL with dramatically different linearity X OUT X OUT X REF X
More informationCorrelator I. Basics. Chapter Introduction. 8.2 Digitization Sampling. D. Anish Roshi
Chapter 8 Correlator I. Basics D. Anish Roshi 8.1 Introduction A radio interferometer measures the mutual coherence function of the electric field due to a given source brightness distribution in the sky.
More informationIntroduction to Digital Signal Processing
Introduction to Digital Signal Processing 1.1 What is DSP? DSP is a technique of performing the mathematical operations on the signals in digital domain. As real time signals are analog in nature we need
More information8 The Discrete Fourier Transform (DFT)
8 The Discrete Fourier Transform (DFT) ² Discrete-Time Fourier Transform and Z-transform are de ned over in niteduration sequence. Both transforms are functions of continuous variables (ω and z). For nite-duration
More informationCHAPTER 7. The Discrete Fourier Transform 436
CHAPTER 7. The Discrete Fourier Transform 36 hfa figconfg( P7a, long ); subplot() stem(k,abs(ck), filled );hold on stem(k,abs(ck_approx), filled, color, red ); xlabel( k, fontsize,lfs) title( Magnitude
More informationsummable Necessary and sufficient for BIBO stability of an LTI system. Also see poles.
EECS 206 DSP GLOSSARY c Andrew E. Yagle Fall 2005 absolutely impulse response: h[n] is finite. EX: n=0 ( 3 4 )n = 1 = 4 but 1 3 n=1 1 n. 4 summable Necessary and sufficient for BIBO stability of an LI
More informationUniversity of Connecticut Lecture Notes for ME5507 Fall 2014 Engineering Analysis I Part III: Fourier Analysis
University of Connecticut Lecture Notes for ME557 Fall 24 Engineering Analysis I Part III: Fourier Analysis Xu Chen Assistant Professor United Technologies Engineering Build, Rm. 382 Department of Mechanical
More informationThe Fast Fourier Transform: A Brief Overview. with Applications. Petros Kondylis. Petros Kondylis. December 4, 2014
December 4, 2014 Timeline Researcher Date Length of Sequence Application CF Gauss 1805 Any Composite Integer Interpolation of orbits of celestial bodies F Carlini 1828 12 Harmonic Analysis of Barometric
More information