sinc function T=1 sec T=2 sec angle(f(w)) angle(f(w))
|
|
- Audra Henderson
- 6 years ago
- Views:
Transcription
1 T=1 sec sinc function 3 angle(f(w)) T=2 sec angle(f(w)) 1
2 A quick script to plot mag & phase in MATLAB w=0:0.2:50; Real exponential func b=5; Fourier transform (filter) F=1.0./(b+j*w); subplot(211), plot(w, abs(f)); title( Real Exp *Step ); xlabel( frequency (rad/sec) ); ylabel( F(w) ); subplot(212), plot(w, angle(f)); title( Real Exp *Step ); xlabel( frequency (rad/sec) ); ylabel( phase angle ); 2
3 2D Fourier Transform (2DFT) Wave Equation (2D) 2 u(x,t) x 2 = 1 v 2 2 u(x,t) t 2 A particularly useful class of solutions: u(x,t) = Ae i(ωt ±kx) = A[cos(ωt ± kx) + isin(ωt ± kx)] k=wavenumber, v=speed, ω=angular frequency In fact, k could be a vector k in 3D, then we have several dimensions! v = w k (k expressed in radians/sec) This cons)tute a func)on of both wave number and frequency. To analyze the spectral characteris)cs we need to use a 2D Fourier Transform. Defini:on: The 2DFT of a func:on f(x,y) is defined as F(k x,k y ) = f (x, y) = 1 (2π) 2 Then inverse 2DFT is f (x, y)e j(k x x +k y y ) dxdy F(k x,k y )e j(k xx +k yy ) dk x dk y 3
4 Back to 1D seismic wave solution (assume propagating in +x direction): F(k,ω) = Ae j(ωt kx ) e j(ωt kx ) dxdt k=spatial frequency, ω=angular frequency=2π/t Other examples: Image compression or blurring/deblurring Image decompression: For a digital image, it is usually much simpler to save the coefs of number of discrete frequencies rather than the whole image. If need to reconstruct image, do Inv FT. As simple as doing these: (1) Load the image into a 2D matrix (2) Perform fft (2D) on this matrix (3) Multiplying frequencies with a function (of frequency), or simply remove small coefficients (4) Fft back to the space-time domain 4
5 2D FT seismic example: Velocity filter (I.e. F-k filter or dip filter) A seismic line contains 24 receivers, with 25 m spacing between two adjacent stations. The time shift is 15 ms/ trace. The total length of each seismogram is 1 sec, with sampling rate of sec. Find the seismic velocity (or propagation speed) of the medium using 12 Hz wavelet. f k = v = 12 cycles/s 7.2 cycles/km =1.67 km/sec (phase vel)
6 2D fft Numerical example (dip filter) 25 m (see program fkdip.m) 15 ms w = *2.0*12.0; % w=2*pi*freq Sample code: delta = 0.001; % sampling interval ss = ; % phase delay between adjacent seismographs Y=[]; time=(0:delta:1.0); m=length(time); % number of time points for i=1:24 Computes w*(t-phi) ss = ss+0.015; xx=w*time-ss*w; y = cos(xx); Fill matrix columns by 6 Y = [Y y']; sinusoids end
7 2D FFT Algorithm: 1D FT a column of Y Save the coef in the same column Once columns all become coefs, do 1D transform a row at a time, replace old coefs with new coefs Matlab function: fft2() Spatial wavenumber (cycles/km) Now, to compute velocity, do f/k = (12 cycles/sec) / (7.2 cycles/km) = 1.67 km/s = phase vel Why the artifact? 7
8 Discretization and Discrete Signals So far, we have talked about continuous functions and Fourier Series/Transforms for them. Realistically, observations are better stored and operated on as discrete sample points. An infinite discrete signal f s (t) is obtained by sampling f(t) every Δt seconds, f s (t) = f (t) k= δ(t kδt) Where Δt is called the sampling interval (or sample rate). Time series: Discrete time function discrete step function 8
9 t=-4:6; % time axis, discrete at 1 sec per sample ft=[ ]; % make a function f(t) figure; % start figure environment subplot(2,2,1); % make a figure with 4 subfigs, % 2 rows, 2 columns, plot 1st. stem(t,ft, 'filled'); % make a stem plot with filled circles title('any discrete function'); xlabel('t(sec)'); ylabel('f(t)'); subplot(2,2,2); % set up second panel delta=[]; % declare delta to be an array for i=1: length(t) if(t(i)<0) discrete_sig.m delta(i)=0.0; else delta(i)=1; end end stem(t,delta, 'filled'); % make a stem plot with filled circles title('discrete delta function'); xlabel('t(sec)'); ylabel('f(t)'); subplot(2,2,3); % set up 3rd panel for i=1: length(t) if(t(i)<0) delta(i)=0.0; else delta(i)=t(i); end end stem(t,delta, 'filled'); % make a stem plot with filled circles title('discrete unit ramp function'); xlabel('t(sec)'); ylabel('f(t)'); subplot(2,2,4); % set up 4th panel delta=cos(t); 9 stem(t,delta, 'filled'); % make a stem plot with filled circles title('discrete unit cosine function'); xlabel('t(sec)'); ylabel('f(t)');
10 Idea of Nyquist Frequency: A continuous signal can be well restored by a discrete signal with sampling interval Δt --->0. For a given time series with sampling interval Δt, the highest frequency that can be restored is 1/(2Δt) which is the Nyquist Frequency of the time series. f Nyquist = 1 2Δt ; Δt = sampling interval = f s /2; f s = 1/Δt = sampling frequency To be authentic f Nyquist f max f max = max frequency of a bandlimitted signal f s 2 f max f s 2 f max i.e.: f s must be greater or equal to twice the f max of the signal! What happens otherwise? ALIASING! Aliasing Problem 10
11 See provided matlab code nyquist_spec.m P 0.1 sec sampling S Rayleigh 0.2 sec sampling 0.8 sec sampling 11
12 Folding Frequency and Aliasing Example of sampling without aliasing: (see adached Matlab code nyquist_gt_fmax_4panels.m) Experiment Setup: Sinosoidal, signal frequency = 25 Hz Fn =1/(2*0.002)=250 Hz Fn =125 Hz Example of sampling causing aliasing: (see adached Matlab code nyquist_150hz. m). Signal freq=150 Hz 12
13 Effect of Undersampling: (1) Bandlimit the spectrum of a con:nuous signal, with the highest frequency equals the Nyquist frequency. (2) Introduce aliasing, I.e., frequencies above the Nyquist are lost ater sampling, but reappears at frequencies below the Nyquist. The high frequencies are not lost per say, but actually folded back onto the spectrum as low frequencies. --> contamina:on of low frequencies. Computa:on of Alias frequency f a = 2mf N f s f a is the folding (aliasing) frequency, f s is the signal (or input) frequency, m is the smallest integer such that f a < f N Example: Suppose f s =65 Hz, f N =62.5 Hz, find alias frequency. Graphically, Guess m=0, f a = 65 Hz > f N, no Guess m=1, f a = 60 Hz < f N, yes f a =60 f n =62.5 f s =65 FREQ (Hz) Solu)on to undersampling: (1) Nothing can be done about effect (1) (2) People typically use some type of filter to remove high 13 frequency signals above Nyquist. Then do Fourier spectrum.
14 Non-Matlab issues: A few important points Input is a real function, so symmetric amp spectrum (why? As proved before F(-w)=F*(w), which yields same F(w) ) As demonstrated, nonzero average --> 0 frequency (this is one of the reasons that seismologists detrend the data before doing fft or filter, see nyquist_spec_const.m, see next page). Nonzero frequencies seem to shift to lower end (close to 0) when under-sampled due to Aliasing. F(k) = N Matlab issues: FFT(x) returns the discrete Fourier transform (DFT) of vector x. For length N input vector x, the DFT is a length N vector F, with elements x(n) e j*2π *(k 1)*(n 1)/ N where 1 k N n=1 n is t in the discrete sense, 2π(k-1)/N is related to frequency ω.! Examples: dft_simple.m, dft_simple_odd.m, dft_fftshift.m f (n) = 1 N N X(k) e j*2π *(k 1)*(n 1)/ N where 1 n N k=1 Do you see a potential problem in this formulation?
15 The code is different from the earlier version by this line: ff=load('regional_eq.v'); t=ff(:,1); ft=ff(:,2); ft=ft+2e-6; Adds a constant shift to the original time series. Gets back an average (actually, Matlab gives SUM, I divided by the number of data to get the average) at the zero freq! 15
16 Ins and Outs of Discrete Fourier Transform (DFT) 1. In mathematics, the Discrete Fourier Transform (DFT) is a specific kind of Fourier transform. 2. Requires a continuous function being Sampled at specific time/spatial points. 3. Only evaluates enough frequency samples to reconstruct the time-segmented finite samples 4. Assumes the limited time samples are from 1 period (hence the range is limited from 0à2 PI (or PI to PI) and this periodicity repeats forever. A window function is often added to remove high-freq contamination (recall: time limited signal cannot be band-limited)-à a little bit of artifacts Not too big since the frequencies at super high freq usually have small amps. 5. For the same reason as 4, inverse DFT (IFFT) will not be able to reconstruct the infinite time series from a non-infinite time series. 16
17 The DFT maps a discrete signal into the frequency domain where the limit of ω is [0, 2π). π N=8 Suppose: time series has N points, then the ω axis can be discretised as ω 0 2π ω k = k 2π N, k = 0, 1,..., N -1 N= length of Fourier Transform, and X(ω k ) = N 1 x n e jω kn, n= 0 k = 0, 1, 2,.., (N -1) So the DFT maps the N-point time series to N-point coefficients X k = X(ω k ) in frequency domain! Nyquist at ω=π complex coefficient ordering in Matlab. Matlab considers frequencies from 0 ---> 2π π/2 after fftshift, move 0 freq to center π 0 (-π) 0 π-df (-π) (2π) 1 N/2+1 N 3π/2 (-π/2) Check program dft_simple.m for unshifted frequency storage, useful for visualization (careful here!). 17 Questions: (1) What does it say about the coefficients (complex vs. real)? (2) What can I say about the time series?
18 Unshifted DFT storage in matlab >> type dft_simple.m % this little code finds DFT of a simple time series. z=[1, 2, 3, 4, 5, 6]; w=fft(z) >> dft_simple w = Columns 1 through i i Columns 4 through i i shifted DFT storage in matlab >> type dft_simple_shift.m % this little code finds shifted DFT of a simple time series. z=[1, 2, 3, 4, 5, 6]; w=fft(z); w=fftshift(w) >> dft_simple_shift w = Columns 1 through i i Columns 4 through i i 18
19 Symmetric function input/fourier symmetry (see code symmetry_fft.m) % this program demonstrates the input pattern for matlab, % as well as symmetry of Fourier transforms for real even and real odd functions z=[ ] fft(z) Output: (Not symmetric due to MATLAB corkiness, this input is actually regarded as a shifted symmetric func, which is multiplication of F(w) by a complex exponential i i i i Replace the input by z=[ ] fft(z) Output: (which is real symmetric spectrum!!) Real Odd func z=[ ] fft(z) Output: i i i i (Complex Odd Spectrum!!) 19
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 informationSEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis
SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some
More informationDigital Signal Processing: Signal Transforms
Digital Signal Processing: Signal Transforms Aishy Amer, Mohammed Ghazal January 19, 1 Instructions: 1. This tutorial introduces frequency analysis in Matlab using the Fourier and z transforms.. More Matlab
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 informationLAB # 5 HANDOUT. »»» The N-point DFT is converted into two DFTs each of N/2 points. N = -W N Then, the following formulas must be used. = k=0,...
EEE4 Lab Handout. FAST FOURIER TRANSFORM LAB # 5 HANDOUT Data Sequence A = x, x, x, x3, x4, x5, x6, x7»»» The N-point DFT is converted into two DFTs each of N/ points. x, x, x4, x6 x, x3, x5, x7»»» N =e
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 informationV(t) = Total Power = Calculating the Power Spectral Density (PSD) in IDL. Thomas Ferree, Ph.D. August 23, 1999
Calculating the Power Spectral Density (PSD) in IDL Thomas Ferree, Ph.D. August 23, 1999 This note outlines the calculation of power spectra via the fast Fourier transform (FFT) algorithm. There are several
More informationDefinition. A signal is a sequence of numbers. sequence is also referred to as being in l 1 (Z), or just in l 1. A sequence {x(n)} satisfying
Signals and Systems. Definition. A signal is a sequence of numbers {x(n)} n Z satisfying n Z x(n)
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 informationPART 1. Review of DSP. f (t)e iωt dt. F(ω) = f (t) = 1 2π. F(ω)e iωt dω. f (t) F (ω) The Fourier Transform. Fourier Transform.
PART 1 Review of DSP Mauricio Sacchi University of Alberta, Edmonton, AB, Canada The Fourier Transform F() = f (t) = 1 2π f (t)e it dt F()e it d Fourier Transform Inverse Transform f (t) F () Part 1 Review
More informationFOURIER TRANSFORM AND
Version: 11 THE DISCRETE TEX d: Oct. 23, 2013 FOURIER TRANSFORM AND THE FFT PREVIEW Classical numerical analysis techniques depend largely on polynomial approximation of functions for differentiation,
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 informationMusic 270a: Complex Exponentials and Spectrum Representation
Music 270a: Complex Exponentials and Spectrum Representation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) October 24, 2016 1 Exponentials The exponential
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 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 information!Sketch f(t) over one period. Show that the Fourier Series for f(t) is as given below. What is θ 1?
Second Year Engineering Mathematics Laboratory Michaelmas Term 998 -M L G Oldfield 30 September, 999 Exercise : Fourier Series & Transforms Revision 4 Answer all parts of Section A and B which are marked
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 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 informationTutorial Sheet #2 discrete vs. continuous functions, periodicity, sampling
2.39 utorial Sheet #2 discrete vs. continuous functions, periodicity, sampling We will encounter two classes of signals in this class, continuous-signals and discrete-signals. he distinct mathematical
More informationLABORATORY 1 DISCRETE-TIME SIGNALS
LABORATORY DISCRETE-TIME SIGNALS.. Introduction A discrete-time signal is represented as a sequence of numbers, called samples. A sample value of a typical discrete-time signal or sequence is denoted as:
More informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 8: February 12th, 2019 Sampling and Reconstruction Lecture Outline! Review " Ideal sampling " Frequency response of sampled signal " Reconstruction " Anti-aliasing
More informationMath 56 Homework 5 Michael Downs
1. (a) Since f(x) = cos(6x) = ei6x 2 + e i6x 2, due to the orthogonality of each e inx, n Z, the only nonzero (complex) fourier coefficients are ˆf 6 and ˆf 6 and they re both 1 2 (which is also seen from
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 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 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 informationUp-Sampling (5B) Young Won Lim 11/15/12
Up-Sampling (5B) Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later version
More informationA523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2011
A523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2011 Lecture 6 PDFs for Lecture 1-5 are on the web page Problem set 2 is on the web page Article on web page A Guided
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 informationTTT4120 Digital Signal Processing Suggested Solutions for Problem Set 2
Norwegian University of Science and Technology Department of Electronics and Telecommunications TTT42 Digital Signal Processing Suggested Solutions for Problem Set 2 Problem (a) The spectrum X(ω) can be
More informationQUIZ #2 SOLUTION Version A
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING QUIZ #2 SOLUTION Version A DATE: 7-MAR-16 SOLUTION Version A COURSE: ECE 226A,B NAME: STUDENT #: LAST, FIRST 2 points 2 points
More informationCMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals
CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 6, 2005 1 Sound Sound waves are longitudinal
More informationA3. Frequency Representation of Continuous Time and Discrete Time Signals
A3. Frequency Representation of Continuous Time and Discrete Time Signals Objectives Define the magnitude and phase plots of continuous time sinusoidal signals Extend the magnitude and phase plots to discrete
More information1 1.27z z 2. 1 z H 2
E481 Digital Signal Processing Exam Date: Thursday -1-1 16:15 18:45 Final Exam - Solutions Dan Ellis 1. (a) In this direct-form II second-order-section filter, the first stage has
More informationVII. Bandwidth Limited Time Series
VII. Bandwidth Limited Time Series To summarize the discussion up to this point: (1) In the general case of the aperiodic time series, which is infinite in time and frequency, both the time series and
More informationLecture 3: Linear Filters
Lecture 3: Linear Filters Professor Fei Fei Li Stanford Vision Lab 1 What we will learn today? Images as functions Linear systems (filters) Convolution and correlation Discrete Fourier Transform (DFT)
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 informationInformation and Communications Security: Encryption and Information Hiding
Short Course on Information and Communications Security: Encryption and Information Hiding Tuesday, 10 March Friday, 13 March, 2015 Lecture 5: Signal Analysis Contents The complex exponential The complex
More informationFundamentals of the DFT (fft) Algorithms
Fundamentals of the DFT (fft) Algorithms D. Sundararajan November 6, 9 Contents 1 The PM DIF DFT Algorithm 1.1 Half-wave symmetry of periodic waveforms.............. 1. The DFT definition and the half-wave
More informationLecture 3: Linear Filters
Lecture 3: Linear Filters Professor Fei Fei Li Stanford Vision Lab 1 What we will learn today? Images as functions Linear systems (filters) Convolution and correlation Discrete Fourier Transform (DFT)
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 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 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 informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Filtering in the Frequency Domain http://www.ee.unlv.edu/~b1morris/ecg782/ 2 Outline Background
More informationFinal Exam January 31, Solutions
Final Exam January 31, 014 Signals & Systems (151-0575-01) Prof. R. D Andrea & P. Reist Solutions Exam Duration: Number of Problems: Total Points: Permitted aids: Important: 150 minutes 7 problems 50 points
More informationDiscrete Fourier Transform
Discrete Fourier Transform Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz Diskrete Fourier transform (DFT) We have just one problem with DFS that needs to be solved.
More informationChapter 4 The Fourier Series and Fourier Transform
Chapter 4 The Fourier Series and Fourier Transform Representation of Signals in Terms of Frequency Components Consider the CT signal defined by N xt () = Acos( ω t+ θ ), t k = 1 k k k The frequencies `present
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:305:45 CBC C222 Lecture 8 Frequency Analysis 14/02/18 http://www.ee.unlv.edu/~b1morris/ee482/
More information16.362: Signals and Systems: 1.0
16.362: Signals and Systems: 1.0 Prof. K. Chandra ECE, UMASS Lowell September 1, 2016 1 Background The pre-requisites for this course are Calculus II and Differential Equations. A basic understanding of
More informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 8: February 7th, 2017 Sampling and Reconstruction Lecture Outline! Review " Ideal sampling " Frequency response of sampled signal " Reconstruction " Anti-aliasing
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 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 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 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 informationCMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation
CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 23, 2006 1 Exponentials The exponential is
More information8/19/16. Fourier Analysis. Fourier analysis: the dial tone phone. Fourier analysis: the dial tone phone
Patrice Koehl Department of Biological Sciences National University of Singapore http://www.cs.ucdavis.edu/~koehl/teaching/bl5229 koehl@cs.ucdavis.edu Fourier analysis: the dial tone phone We use Fourier
More informationDiscrete Time Fourier Transform (DTFT) Digital Signal Processing, 2011 Robi Polikar, Rowan University
Discrete Time Fourier Transform (DTFT) Digital Signal Processing, 2 Robi Polikar, Rowan University Sinusoids & Exponentials Signals Phasors Frequency Impulse, step, rectangular Characterization Power /
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 informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 4 Digital Signal Processing Pro. Mark Fowler ote Set # Using the DFT or Spectral Analysis o Signals Reading Assignment: Sect. 7.4 o Proakis & Manolakis Ch. 6 o Porat s Book /9 Goal o Practical Spectral
More informationE : Lecture 1 Introduction
E85.2607: Lecture 1 Introduction 1 Administrivia 2 DSP review 3 Fun with Matlab E85.2607: Lecture 1 Introduction 2010-01-21 1 / 24 Course overview Advanced Digital Signal Theory Design, analysis, and implementation
More informationMultirate Digital Signal Processing
Multirate Digital Signal Processing Basic Sampling Rate Alteration Devices Up-sampler - Used to increase the sampling rate by an integer factor Down-sampler - Used to decrease the sampling rate by an integer
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 informationBME 50500: Image and Signal Processing in Biomedicine. Lecture 2: Discrete Fourier Transform CCNY
1 Lucas Parra, CCNY BME 50500: Image and Signal Processing in Biomedicine Lecture 2: Discrete Fourier Transform Lucas C. Parra Biomedical Engineering Department CCNY http://bme.ccny.cuny.edu/faculty/parra/teaching/signal-and-image/
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:30-15:45 CBC C222 Lecture 02 DSP Fundamentals 14/01/21 http://www.ee.unlv.edu/~b1morris/ee482/
More informationA Note on Upsampling by Integer Factors Using the DFT
A Note on Upsampling by Integer Factors Using the FT Mark Richards February 24, 202 Let x[n] be a sequence of length N with discrete-time Fourier transform (TFT) X(). Let x [n] be a decimated sequence
More informationEach of these functions represents a signal in terms of its spectral components in the frequency domain.
N INTRODUCTION TO SPECTRL FUNCTIONS Revision B By Tom Irvine Email: tomirvine@aol.com March 3, 000 INTRODUCTION This tutorial presents the Fourier transform. It also discusses the power spectral density
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 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 informationELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization
ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals 1. Sampling and Reconstruction 2. Quantization 1 1. Sampling & Reconstruction DSP must interact with an analog world: A to D D to A x(t)
More informationChirp Transform for FFT
Chirp Transform for FFT Since the FFT is an implementation of the DFT, it provides a frequency resolution of 2π/N, where N is the length of the input sequence. If this resolution is not sufficient in a
More informationComputational Methods for Astrophysics: Fourier Transforms
Computational Methods for Astrophysics: Fourier Transforms John T. Whelan (filling in for Joshua Faber) April 27, 2011 John T. Whelan April 27, 2011 Fourier Transforms 1/13 Fourier Analysis Outline: Fourier
More informationENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM. Dr. Lim Chee Chin
ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM Dr. Lim Chee Chin Outline Introduction Discrete Fourier Series Properties of Discrete Fourier Series Time domain aliasing due to frequency
More informationLecture 8: Signal Reconstruction, DT vs CT Processing. 8.1 Reconstruction of a Band-limited Signal from its Samples
EE518 Digital Signal Processing University of Washington Autumn 2001 Dept. of Electrical Engineering Lecture 8: Signal Reconstruction, D vs C Processing Oct 24, 2001 Prof: J. Bilmes
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 informationSubband Coding and Wavelets. National Chiao Tung University Chun-Jen Tsai 12/04/2014
Subband Coding and Wavelets National Chiao Tung Universit Chun-Jen Tsai /4/4 Concept of Subband Coding In transform coding, we use N (or N N) samples as the data transform unit Transform coefficients are
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationDCSP-2: Fourier Transform
DCSP-2: Fourier Transform Jianfeng Feng Department of Computer Science Warwick Univ., UK Jianfeng.feng@warwick.ac.uk http://www.dcs.warwick.ac.uk/~feng/dcsp.html Data transmission Channel characteristics,
More informationFrequency-domain representation of discrete-time signals
4 Frequency-domain representation of discrete-time signals So far we have been looing at signals as a function of time or an index in time. Just lie continuous-time signals, we can view a time signal as
More informationFourier Series and Transforms
Fourier Series and Transforms Website is now online at: http://www.unc.edu/courses/2008fall/comp/665/001/ 9/2/08 Comp 665 Real and Special Signals 1 Discrete Exponen8al Func8on Discrete Convolu?on: Convolu?on
More informationUniversity of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing
University of Illinois at Urbana-Champaign ECE 3: Digital Signal Processing Chandra Radhakrishnan PROBLEM SE 5: SOLUIONS Peter Kairouz Problem o derive x a (t (X a (Ω from X d (ω, we first need to get
More informationSection 3 Discrete-Time Signals EO 2402 Summer /05/2013 EO2402.SuFY13/MPF Section 3 1
Section 3 Discrete-Time Signals EO 2402 Summer 2013 07/05/2013 EO2402.SuFY13/MPF Section 3 1 [p. 3] Discrete-Time Signal Description Sampling, sampling theorem Discrete sinusoidal signal Discrete exponential
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 12 Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial,
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 informationProblem Value Score No/Wrong Rec
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING QUIZ #2 DATE: 14-Oct-11 COURSE: ECE-225 NAME: GT username: LAST, FIRST (ex: gpburdell3) 3 points 3 points 3 points Recitation
More informationG52IVG, School of Computer Science, University of Nottingham
Image Transforms Fourier Transform Basic idea 1 Image Transforms Fourier transform theory Let f(x) be a continuous function of a real variable x. The Fourier transform of f(x) is F ( u) f ( x)exp[ j2πux]
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 informationComputational Methods CMSC/AMSC/MAPL 460
Computational Methods CMSC/AMSC/MAPL 460 Fourier transform Balaji Vasan Srinivasan Dept of Computer Science Several slides from Prof Healy s course at UMD Last time: Fourier analysis F(t) = A 0 /2 + A
More informationDiscrete-Time Signals: Time-Domain Representation
Discrete-Time Signals: Time-Domain Representation 1 Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the
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 informationI. Signals & Sinusoids
I. Signals & Sinusoids [p. 3] Signal definition Sinusoidal signal Plotting a sinusoid [p. 12] Signal operations Time shifting Time scaling Time reversal Combining time shifting & scaling [p. 17] Trigonometric
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 informationDiscrete-Time Signals: Time-Domain Representation
Discrete-Time Signals: Time-Domain Representation 1 Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationMEDE2500 Tutorial Nov-7
(updated 2016-Nov-4,7:40pm) MEDE2500 (2016-2017) Tutorial 3 MEDE2500 Tutorial 3 2016-Nov-7 Content 1. The Dirac Delta Function, singularity functions, even and odd functions 2. The sampling process and
More informationEE Homework 13 - Solutions
EE3054 - Homework 3 - Solutions. (a) The Laplace transform of e t u(t) is s+. The pole of the Laplace transform is at which lies in the left half plane. Hence, the Fourier transform is simply the Laplace
More informationCorrelation, discrete Fourier transforms and the power spectral density
Correlation, discrete Fourier transforms and the power spectral density visuals to accompany lectures, notes and m-files by Tak Igusa tigusa@jhu.edu Department of Civil Engineering Johns Hopkins University
More informationWavenumber-Frequency Space
Wavenumber-Frequency Space ECE 6279: Spatial Array Processing Spring 2011 Lecture 3 Prof. Aaron D. Lanterman School of Electrical & Computer Engineering Georgia Institute of Technology AL: 404-385-2548
More informationIntroduction to Signal Analysis Parts I and II
41614 Dynamics of Machinery 23/03/2005 IFS Introduction to Signal Analysis Parts I and II Contents 1 Topics of the Lecture 11/03/2005 (Part I) 2 2 Fourier Analysis Fourier Series, Integral and Complex
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 informationImages have structure at various scales
Images have structure at various scales Frequency Frequency of a signal is how fast it changes Reflects scale of structure A combination of frequencies 0.1 X + 0.3 X + 0.5 X = Fourier transform Can we
More information