A Tutorial on Fourier Analysis
|
|
- Norah Haynes
- 7 years ago
- Views:
Transcription
1 A Tutorial on Fourier Analysis Douglas Eck University of Montreal NYU March 26
2 1.5 A fundamental and three odd harmonics (3,5,7) fund (freq 1) 3rd harm 5th harm 7th harmm Sum of odd harmonics approximate square wave fund (freq 1) fund+3rd harm fund+3rd+5th fund+3rd+5th+7th
3 1 Sum of odd harmonics from 1 to
4 Linear Combination In the interval [u 1, u 2 ] a function Θ(u) can be written as a linear combination: Θ(u) = α i ψ i (u) i= where functions ψ i (u) make up a set of simple elementary functions. If functions are orthogonal (roughly, perpindicular; inner product is zero)then coefficients α i are independant from one another.
5 Continuous Fourier Transform The most commonly used set of orthogonal functions is the Fourier series. Here is the analog version of the Fourier and Inverse Fourier: X (w) = x(t) = + + x(t)e ( 2πjwt) dt X (w)e (2πjwt) dw
6 Discrete Fourier and Inverse Fourier Transform X (n) = x(k) = 1 N N 1 k= N 1 n= x(k)e 2πjnk/N X (n)e 2πjnk/N
7 Taylor Series Expansion f (x) = f (x )+ f (x ) (x x )+ f (x ) (x x ) 2 + f (x ) (x x ) or more compactly as f (x) = n= f (n) (x ) (x x ) n n!
8 Taylor series expansion of e jθ Since e x is its own derivative, the Taylor series expansion for f (x) = e x is very simple: We can define: e x = e jθ == n= (jθ) n n= x n n! = 1 + x + x x 3 n! 3! +... = 1 + jθ Θ2 2 j Θ3 3! +...
9 Splitting out real and imaginary parts All even order terms are real; all odd-order terms are imaginary: ree jθ = 1 Θ 2 /2 + Θ 4 /4!... ime jθ = Θ Θ 3 /3! + Θ 5 /5!...
10 Fourier Transform as sum of sines and cosines Observe that: cos(θ) = n>=;n is even ( 1) n/2 n! Θ n sin(θ) = n>=;n is odd ( 1) (n 1)/2 Θ n Thus yielding Euler s formula: e jθ = cos(θ) + j sin(θ) n!
11
12 Fourier transform as kernel matrix
13 Example Sum of cosines with frequencies 12 and 9, sampling rate = signal real part two cosines (freqs=9, 12) imag part
14 Example FFT coefficients mapped onto unit circle 1 FFT projected onto unit circle
15 Impulse response impulse response signal magnitude phase
16 Impulse response delayed impulse response signal magnitude phase
17 A look at phase shifting Sinusoid frequency=5 phase shifted multiple times. sinusoid freq=5 phase shifted repeatedly magnitude number of points shifted 3 2 angle freq= number of points shifted
18 Sin period 1 + period freq (mag) phase component sinusoids reconstructed signal using component sinusoids vs original reconstructed signal using ifft vs original
19 Aliasing The useful range is the Nyquist frequency (fs/2) 1 cos(21) cos(21) sampled at 24 Hz cos(45) cos(45) sampled at 24 Hz cos( 3) cos( 3) sampled at 24 Hz
20 Leakage Even below Nyquist, when frequencies in the signal do not align well with sampling rate of signal, there can be leakage. First consider a well-aligned exampl (freq =.25 sampling rate) a) b) c) Amplitude Magnitude (db) Magnitude (Linear) Sinusoid at 1/4 the Sampling Rate Time (samples) Normalized Frequency (cycles per sample)) Normalized Frequency (cycles per sample))
21 Leakage Now consider a poorly-aligned example (freq = ( /N) * sampling rate) a) b) c) Amplitude Magnitude (Linear) Sinusoid NEAR 1/4 the Sampling Rate Time (samples) Normalized Frequency (cycles per sample)) 3 Magnitude (db) Normalized Frequency (cycles per sample))
22 Leakage Comparison: a) 1 Sinusoid at 1/4 the Sampling Rate a) 1 Sinusoid NEAR 1/4 the Sampling Rate Amplitude.5.5 Amplitude Time (samples) b) Time (samples) b) Magnitude (Linear) Normalized Frequency (cycles per sample)) c) Magnitude (Linear) Normalized Frequency (cycles per sample)) c) Magnitude (db) Normalized Frequency (cycles per sample)) Magnitude (db) Normalized Frequency (cycles per sample))
23 Windowing can help Can minimize effects by multiplying time series by a window that diminishes magnitude of points near signal edge: a) 1 Blackman Window Amplitude.5 b) c) Magnitude (db) Magnitude (db) Time (samples) Normalized Frequency (cycles per sample)) Normalized Frequency (cycles per sample))
24 Leakage Reduced Comparison: a) 1 Sinusoid NEAR 1/4 the Sampling Rate b) a) 1 Sinusoid at 1/4 the Sampling Rate Amplitude.5.5 Amplitude.5.5 b) Time (samples) Time (samples) c) Magnitude (Linear) Magnitude (db) Normalized Frequency (cycles per sample)) Normalized Frequency (cycles per sample)) c) Magnitude (Linear) Magnitude (db) Normalized Frequency (cycles per sample)) Normalized Frequency (cycles per sample))
25 Convolution theorem This can be understood in terms of the Convolution Theorem. Convolution in the time domain is multiplication in the frequency domain via the Fourier transform (F). F(f g) = F(f ) F(g)
26 Computing impulse response The impulse response h[n] is the response of a system to the unit impulse function.
27 Using the impulse response Once computed, the impulse response can be used to filter any signal x[n] yielding y[n].
28 Examples
29 Filtering using DFT Goal is to choose good impulse response h[n]
30 Filtering using DFT Goal is to choose good impulse response h[n] Transform signal into frequency domain
31 Filtering using DFT Goal is to choose good impulse response h[n] Transform signal into frequency domain Modify frequency properties of signal via multiplication
32 Filtering using DFT Goal is to choose good impulse response h[n] Transform signal into frequency domain Modify frequency properties of signal via multiplication Transform back into time domain
33 Difficulties (Why not a perfect filter?) You can have a perfect filter(!)
34 Difficulties (Why not a perfect filter?) You can have a perfect filter(!) Need long impulse response function in both directions
35 Difficulties (Why not a perfect filter?) You can have a perfect filter(!) Need long impulse response function in both directions Very non causal
36 Difficulties (Why not a perfect filter?) You can have a perfect filter(!) Need long impulse response function in both directions Very non causal In generating causal version, challenges arise
37 Gibbs Phenomenon ideal lopass filter in frequency domain ideal filter coeffs in time domain truncated causal filter Gibbs phenomenon
38 Spectral Analysis Often we want to see spectral energy as a signal evolves over time
39 Spectral Analysis Often we want to see spectral energy as a signal evolves over time Compute Fourier Transform over evenly-spaced frames of data
40 Spectral Analysis Often we want to see spectral energy as a signal evolves over time Compute Fourier Transform over evenly-spaced frames of data Apply window to minimize edge effects
41 Short-Timescale Fourier Transform (STFT) X (m, n) = N 1 k= x(k)w(k m)e 2πjnk/N Where w is some windowing function such as Hanning or gaussian centered around zero. The spectrogram is simply the squared magnitude of these STFT values
42 Trumpet (G4) 5 4 Frequency Time [Listen]
43 Violin (G4) 5 4 Frequency Time [Listen]
44 Flute (G4) 5 4 Frequency Time [Listen]
45 Piano (G4) 5 4 Frequency Time [Listen]
46 Voice Frequency Time [Listen]
47 C Major Scale (Piano) Frequency Time [Listen]
48 C Major Scale (Piano) Log Spectrogram (Constant-Q Transform) reveals low-frequency structure Frequency Time
49 Time-Space Tradeoff spoken "Steven Usma" Amp
50 Time-Space Tradeoff 4 Narrowband Spectrogram overlap=152 timepts= Frequency Time 4 Wideband Spectrogram overlap=3 timepts= Frequency Time
51 Auto-correlation and meter Autocorrelation long used to find meter in music (Brown 1993) Lag k auto-correlation a(k) is a special case of cross-correlation where a signal x is correlated with itself: a(k) = 1 N N 1 n=k x(n) x(n k)
52 Auto-correlation and meter Autocorrelation long used to find meter in music (Brown 1993) Lag k auto-correlation a(k) is a special case of cross-correlation where a signal x is correlated with itself: a(k) = 1 N N 1 n=k x(n) x(n k) Autocorrelation can also be defined in terms of Fourier analysis a = F 1 ( F(x) ) where F is the Fourier transform, F 1 is the inverse Fourier transform and indicates taking magnitude from a complex value.
53 time (seconds) time (seconds) lags (ms) Time series (top), envelope (middle) and autocorrelation (bottom) of excerpt from ISMIR 24 Tempo Induction contest (Albums-Cafe Paradiso-8.wav). A vertical line marks the actual tempo (484 msec, 124bpm). Stars mark the tempo and its integer multiples. Triangles mark levels in the metrical hierarchy.
54 Fast Fourier Transform Fourier Transform O(N 2 ) Fast version possible O(NlogN) Size must be a power of two Strategy is decimation in time or frequency Divide and conquer Rearrange the inputs in bit reversed order Output transformation (Decimation in Time)
Analysis/resynthesis with the short time Fourier transform
Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TU-Berlin IRCAM Analysis/Synthesis
More informationSGN-1158 Introduction to Signal Processing Test. Solutions
SGN-1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:
More informationFFT Algorithms. Chapter 6. Contents 6.1
Chapter 6 FFT Algorithms Contents Efficient computation of the DFT............................................ 6.2 Applications of FFT................................................... 6.6 Computing DFT
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More informationFrequency Response of FIR Filters
Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input
More informationANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 2-1
WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's
More informationB3. Short Time Fourier Transform (STFT)
B3. Short Time Fourier Transform (STFT) Objectives: Understand the concept of a time varying frequency spectrum and the spectrogram Understand the effect of different windows on the spectrogram; Understand
More informationThe Algorithms of Speech Recognition, Programming and Simulating in MATLAB
FACULTY OF ENGINEERING AND SUSTAINABLE DEVELOPMENT. The Algorithms of Speech Recognition, Programming and Simulating in MATLAB Tingxiao Yang January 2012 Bachelor s Thesis in Electronics Bachelor s Program
More informationDesign of FIR Filters
Design of FIR Filters Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 68 FIR as
More informationThe continuous and discrete Fourier transforms
FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1
More informationSpeech Signal Processing: An Overview
Speech Signal Processing: An Overview S. R. M. Prasanna Department of Electronics and Electrical Engineering Indian Institute of Technology Guwahati December, 2012 Prasanna (EMST Lab, EEE, IITG) Speech
More informationL9: Cepstral analysis
L9: Cepstral analysis The cepstrum Homomorphic filtering The cepstrum and voicing/pitch detection Linear prediction cepstral coefficients Mel frequency cepstral coefficients This lecture is based on [Taylor,
More informationCorrelation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs
Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in
More informationConvolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005
Convolution, Correlation, & Fourier Transforms James R. Graham 10/25/2005 Introduction A large class of signal processing techniques fall under the category of Fourier transform methods These methods fall
More informationUNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101 - Fall 2009 Linear Systems Fundamentals
UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101 - Fall 2009 Linear Systems Fundamentals MIDTERM EXAM You are allowed one 2-sided sheet of notes. No books, no other
More informationTime Series Analysis: Introduction to Signal Processing Concepts. Liam Kilmartin Discipline of Electrical & Electronic Engineering, NUI, Galway
Time Series Analysis: Introduction to Signal Processing Concepts Liam Kilmartin Discipline of Electrical & Electronic Engineering, NUI, Galway Aims of Course To introduce some of the basic concepts of
More informationChapter 8 - Power Density Spectrum
EE385 Class Notes 8/8/03 John Stensby Chapter 8 - Power Density Spectrum Let X(t) be a WSS random process. X(t) has an average power, given in watts, of E[X(t) ], a constant. his total average power is
More informationAuto-Tuning Using Fourier Coefficients
Auto-Tuning Using Fourier Coefficients Math 56 Tom Whalen May 20, 2013 The Fourier transform is an integral part of signal processing of any kind. To be able to analyze an input signal as a superposition
More informationThe Fourier Analysis Tool in Microsoft Excel
The Fourier Analysis Tool in Microsoft Excel Douglas A. Kerr Issue March 4, 2009 ABSTRACT AD ITRODUCTIO The spreadsheet application Microsoft Excel includes a tool that will calculate the discrete Fourier
More informationAdding Sinusoids of the Same Frequency. Additive Synthesis. Spectrum. Music 270a: Modulation
Adding Sinusoids of the Same Frequency Music 7a: Modulation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) February 9, 5 Recall, that adding sinusoids of
More informationSampling and Interpolation. Yao Wang Polytechnic University, Brooklyn, NY11201
Sampling and Interpolation Yao Wang Polytechnic University, Brooklyn, NY1121 http://eeweb.poly.edu/~yao Outline Basics of sampling and quantization A/D and D/A converters Sampling Nyquist sampling theorem
More information5 Signal Design for Bandlimited Channels
225 5 Signal Design for Bandlimited Channels So far, we have not imposed any bandwidth constraints on the transmitted passband signal, or equivalently, on the transmitted baseband signal s b (t) I[k]g
More informationFinal Year Project Progress Report. Frequency-Domain Adaptive Filtering. Myles Friel. Supervisor: Dr.Edward Jones
Final Year Project Progress Report Frequency-Domain Adaptive Filtering Myles Friel 01510401 Supervisor: Dr.Edward Jones Abstract The Final Year Project is an important part of the final year of the Electronic
More information1.4 Fast Fourier Transform (FFT) Algorithm
74 CHAPTER AALYSIS OF DISCRETE-TIME LIEAR TIME-IVARIAT SYSTEMS 4 Fast Fourier Transform (FFT Algorithm Fast Fourier Transform, or FFT, is any algorithm for computing the -point DFT with a computational
More informationShort-time FFT, Multi-taper analysis & Filtering in SPM12
Short-time FFT, Multi-taper analysis & Filtering in SPM12 Computational Psychiatry Seminar, FS 2015 Daniel Renz, Translational Neuromodeling Unit, ETHZ & UZH 20.03.2015 Overview Refresher Short-time Fourier
More informationApplications of the DFT
CHAPTER 9 Applications of the DFT The Discrete Fourier Transform (DFT) is one of the most important tools in Digital Signal Processing. This chapter discusses three common ways it is used. First, the DFT
More informationEE 179 April 21, 2014 Digital and Analog Communication Systems Handout #16 Homework #2 Solutions
EE 79 April, 04 Digital and Analog Communication Systems Handout #6 Homework # Solutions. Operations on signals (Lathi& Ding.3-3). For the signal g(t) shown below, sketch: a. g(t 4); b. g(t/.5); c. g(t
More informationLab 1. The Fourier Transform
Lab 1. The Fourier Transform Introduction In the Communication Labs you will be given the opportunity to apply the theory learned in Communication Systems. Since this is your first time to work in the
More informationSampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.
Sampling Theorem We will show that a band limited signal can be reconstructed exactly from its discrete time samples. Recall: That a time sampled signal is like taking a snap shot or picture of signal
More informationThis document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.
This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. Title Transcription of polyphonic signals using fast filter bank( Accepted version ) Author(s) Foo, Say Wei;
More informationSpectrum Level and Band Level
Spectrum Level and Band Level ntensity, ntensity Level, and ntensity Spectrum Level As a review, earlier we talked about the intensity of a sound wave. We related the intensity of a sound wave to the acoustic
More informationAnalog and Digital Signals, Time and Frequency Representation of Signals
1 Analog and Digital Signals, Time and Frequency Representation of Signals Required reading: Garcia 3.1, 3.2 CSE 3213, Fall 2010 Instructor: N. Vlajic 2 Data vs. Signal Analog vs. Digital Analog Signals
More informationSIGNAL PROCESSING FOR EFFECTIVE VIBRATION ANALYSIS
SIGNAL PROCESSING FOR EFFECTIVE VIBRATION ANALYSIS Dennis H. Shreve IRD Mechanalysis, Inc Columbus, Ohio November 1995 ABSTRACT Effective vibration analysis first begins with acquiring an accurate time-varying
More informationAliasing, Image Sampling and Reconstruction
Aliasing, Image Sampling and Reconstruction Recall: a pixel is a point It is NOT a box, disc or teeny wee light It has no dimension It occupies no area It can have a coordinate More than a point, it is
More informationHow To Understand The Nyquist Sampling Theorem
Nyquist Sampling Theorem By: Arnold Evia Table of Contents What is the Nyquist Sampling Theorem? Bandwidth Sampling Impulse Response Train Fourier Transform of Impulse Response Train Sampling in the Fourier
More informationLinear Filtering Part II
Linear Filtering Part II Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr Fourier theory Jean Baptiste Joseph Fourier had a crazy idea: Any periodic function can
More informationThe Calculation of G rms
The Calculation of G rms QualMark Corp. Neill Doertenbach The metric of G rms is typically used to specify and compare the energy in repetitive shock vibration systems. However, the method of arriving
More informationPYKC Jan-7-10. Lecture 1 Slide 1
Aims and Objectives E 2.5 Signals & Linear Systems Peter Cheung Department of Electrical & Electronic Engineering Imperial College London! By the end of the course, you would have understood: Basic signal
More informationchapter Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction 1.2 Historical Perspective
Introduction to Digital Signal Processing and Digital Filtering chapter 1 Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction Digital signal processing (DSP) refers to anything
More informationConceptual similarity to linear algebra
Modern approach to packing more carrier frequencies within agivenfrequencyband orthogonal FDM Conceptual similarity to linear algebra 3-D space: Given two vectors x =(x 1,x 2,x 3 )andy = (y 1,y 2,y 3 ),
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationProbability and Random Variables. Generation of random variables (r.v.)
Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly
More informationMathematical Harmonies Mark Petersen
1 Mathematical Harmonies Mark Petersen What is music? When you hear a flutist, a signal is sent from her fingers to your ears. As the flute is played, it vibrates. The vibrations travel through the air
More informationAppendix D Digital Modulation and GMSK
D1 Appendix D Digital Modulation and GMSK A brief introduction to digital modulation schemes is given, showing the logical development of GMSK from simpler schemes. GMSK is of interest since it is used
More informationTime series analysis Matlab tutorial. Joachim Gross
Time series analysis Matlab tutorial Joachim Gross Outline Terminology Sampling theorem Plotting Baseline correction Detrending Smoothing Filtering Decimation Remarks Focus on practical aspects, exercises,
More informationA Sound Analysis and Synthesis System for Generating an Instrumental Piri Song
, pp.347-354 http://dx.doi.org/10.14257/ijmue.2014.9.8.32 A Sound Analysis and Synthesis System for Generating an Instrumental Piri Song Myeongsu Kang and Jong-Myon Kim School of Electrical Engineering,
More informationLecture 7 ELE 301: Signals and Systems
Lecture 7 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 22 Introduction to Fourier Transforms Fourier transform as a limit
More informationThe Effective Number of Bits (ENOB) of my R&S Digital Oscilloscope Technical Paper
The Effective Number of Bits (ENOB) of my R&S Digital Oscilloscope Technical Paper Products: R&S RTO1012 R&S RTO1014 R&S RTO1022 R&S RTO1024 This technical paper provides an introduction to the signal
More informationHow To Understand The Discrete Fourier Transform
The Fast Fourier Transform (FFT) and MATLAB Examples Learning Objectives Discrete Fourier transforms (DFTs) and their relationship to the Fourier transforms Implementation issues with the DFT via the FFT
More informationFAST Fourier Transform (FFT) and Digital Filtering Using LabVIEW
FAST Fourier Transform (FFT) and Digital Filtering Using LabVIEW Wei Lin Department of Biomedical Engineering Stony Brook University Instructor s Portion Summary This experiment requires the student to
More informationPerforming the Fast Fourier Transform with Microchip s dspic30f Series Digital Signal Controllers
Performing the Fast Fourier Transform with Microchip s dspic30f Series Digital Signal Controllers Application Note Michigan State University Dept. of Electrical & Computer Engineering Author: Nicholas
More informationIntroduction to IQ-demodulation of RF-data
Introduction to IQ-demodulation of RF-data by Johan Kirkhorn, IFBT, NTNU September 15, 1999 Table of Contents 1 INTRODUCTION...3 1.1 Abstract...3 1.2 Definitions/Abbreviations/Nomenclature...3 1.3 Referenced
More informationALFFT FAST FOURIER Transform Core Application Notes
ALFFT FAST FOURIER Transform Core Application Notes 6-20-2012 Table of Contents General Information... 3 Features... 3 Key features... 3 Design features... 3 Interface... 6 Symbol... 6 Signal description...
More informationReview of Fourier series formulas. Representation of nonperiodic functions. ECE 3640 Lecture 5 Fourier Transforms and their properties
ECE 3640 Lecture 5 Fourier Transforms and their properties Objective: To learn about Fourier transforms, which are a representation of nonperiodic functions in terms of trigonometric functions. Also, to
More informationTCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS
TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of
More informationTHE SPECTRAL MODELING TOOLBOX: A SOUND ANALYSIS/SYNTHESIS SYSTEM. A Thesis. Submitted to the Faculty
THE SPECTRAL MODELING TOOLBOX: A SOUND ANALYSIS/SYNTHESIS SYSTEM A Thesis Submitted to the Faculty in partial fulfillment of the requirements for the degree of Master of Arts in ELECTRO-ACOUSTIC MUSIC
More informationDigital Transmission of Analog Data: PCM and Delta Modulation
Digital Transmission of Analog Data: PCM and Delta Modulation Required reading: Garcia 3.3.2 and 3.3.3 CSE 323, Fall 200 Instructor: N. Vlajic Digital Transmission of Analog Data 2 Digitization process
More information(Refer Slide Time: 01:11-01:27)
Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 6 Digital systems (contd.); inverse systems, stability, FIR and IIR,
More informationAgilent Time Domain Analysis Using a Network Analyzer
Agilent Time Domain Analysis Using a Network Analyzer Application Note 1287-12 0.0 0.045 0.6 0.035 Cable S(1,1) 0.4 0.2 Cable S(1,1) 0.025 0.015 0.005 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Frequency (GHz) 0.005
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
More informationPulsed Fourier Transform NMR The rotating frame of reference. The NMR Experiment. The Rotating Frame of Reference.
Pulsed Fourier Transform NR The rotating frame of reference The NR Eperiment. The Rotating Frame of Reference. When we perform a NR eperiment we disturb the equilibrium state of the sstem and then monitor
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More informationUSB 3.0 CDR Model White Paper Revision 0.5
USB 3.0 CDR Model White Paper Revision 0.5 January 15, 2009 INTELLECTUAL PROPERTY DISCLAIMER THIS WHITE PAPER IS PROVIDED TO YOU AS IS WITH NO WARRANTIES WHATSOEVER, INCLUDING ANY WARRANTY OF MERCHANTABILITY,
More information7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )
34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using
More informationImproving A D Converter Performance Using Dither
Improving A D Converter Performance Using Dither 1 0 INTRODUCTION Many analog-to-digital converter applications require low distortion for a very wide dynamic range of signals Unfortunately the distortion
More informationClass Note for Signals and Systems. Stanley Chan University of California, San Diego
Class Note for Signals and Systems Stanley Chan University of California, San Diego 2 Acknowledgement This class note is prepared for ECE 101: Linear Systems Fundamentals at the University of California,
More informationPRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationAgilent Creating Multi-tone Signals With the N7509A Waveform Generation Toolbox. Application Note
Agilent Creating Multi-tone Signals With the N7509A Waveform Generation Toolbox Application Note Introduction Of all the signal engines in the N7509A, the most complex is the multi-tone engine. This application
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationCM0340 SOLNS. Do not turn this page over until instructed to do so by the Senior Invigilator.
CARDIFF UNIVERSITY EXAMINATION PAPER Academic Year: 2008/2009 Examination Period: Examination Paper Number: Examination Paper Title: SOLUTIONS Duration: Autumn CM0340 SOLNS Multimedia 2 hours Do not turn
More informationCHAPTER 6 Frequency Response, Bode Plots, and Resonance
ELECTRICAL CHAPTER 6 Frequency Response, Bode Plots, and Resonance 1. State the fundamental concepts of Fourier analysis. 2. Determine the output of a filter for a given input consisting of sinusoidal
More informationIntroduction to Complex Fourier Series
Introduction to Complex Fourier Series Nathan Pflueger 1 December 2014 Fourier series come in two flavors. What we have studied so far are called real Fourier series: these decompose a given periodic function
More informationDoppler. Doppler. Doppler shift. Doppler Frequency. Doppler shift. Doppler shift. Chapter 19
Doppler Doppler Chapter 19 A moving train with a trumpet player holding the same tone for a very long time travels from your left to your right. The tone changes relative the motion of you (receiver) and
More informationIntroduction to Digital Filters
CHAPTER 14 Introduction to Digital Filters Digital filters are used for two general purposes: (1) separation of signals that have been combined, and (2) restoration of signals that have been distorted
More informationT = 1 f. Phase. Measure of relative position in time within a single period of a signal For a periodic signal f(t), phase is fractional part t p
Data Transmission Concepts and terminology Transmission terminology Transmission from transmitter to receiver goes over some transmission medium using electromagnetic waves Guided media. Waves are guided
More informationSampling Theory For Digital Audio By Dan Lavry, Lavry Engineering, Inc.
Sampling Theory Page Copyright Dan Lavry, Lavry Engineering, Inc, 24 Sampling Theory For Digital Audio By Dan Lavry, Lavry Engineering, Inc. Credit: Dr. Nyquist discovered the sampling theorem, one of
More informationMUSICAL INSTRUMENT FAMILY CLASSIFICATION
MUSICAL INSTRUMENT FAMILY CLASSIFICATION Ricardo A. Garcia Media Lab, Massachusetts Institute of Technology 0 Ames Street Room E5-40, Cambridge, MA 039 USA PH: 67-53-0 FAX: 67-58-664 e-mail: rago @ media.
More informationTMA4213/4215 Matematikk 4M/N Vår 2013
Norges teknisk naturvitenskapelige universitet Institutt for matematiske fag TMA43/45 Matematikk 4M/N Vår 3 Løsningsforslag Øving a) The Fourier series of the signal is f(x) =.4 cos ( 4 L x) +cos ( 5 L
More information4 Digital Video Signal According to ITU-BT.R.601 (CCIR 601) 43
Table of Contents 1 Introduction 1 2 Analog Television 7 3 The MPEG Data Stream 11 3.1 The Packetized Elementary Stream (PES) 13 3.2 The MPEG-2 Transport Stream Packet.. 17 3.3 Information for the Receiver
More informationSpike-Based Sensing and Processing: What are spikes good for? John G. Harris Electrical and Computer Engineering Dept
Spike-Based Sensing and Processing: What are spikes good for? John G. Harris Electrical and Computer Engineering Dept ONR NEURO-SILICON WORKSHOP, AUG 1-2, 2006 Take Home Messages Introduce integrate-and-fire
More informationWavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)
Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2π-periodic square-integrable function is generated by a superposition
More informationTTT4110 Information and Signal Theory Solution to exam
Norwegian University of Science and Technology Department of Electronics and Telecommunications TTT4 Information and Signal Theory Solution to exam Problem I (a The frequency response is found by taking
More informationAnalog & digital signals. Analog & digital signals
Analog & digital signals Analog & digital signals Analog Continuous function V of continuous variable t (time, space etc) : V(t). Digital Discrete function Vk of discrete sampling variable tk, with k =
More informationDesign of Efficient Digital Interpolation Filters for Integer Upsampling. Daniel B. Turek
Design of Efficient Digital Interpolation Filters for Integer Upsampling by Daniel B. Turek Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements
More informationComputing Fourier Series and Power Spectrum with MATLAB
Computing Fourier Series and Power Spectrum with MATLAB By Brian D. Storey. Introduction Fourier series provides an alternate way of representing data: instead of representing the signal amplitude as a
More informationPOWER SYSTEM HARMONICS. A Reference Guide to Causes, Effects and Corrective Measures AN ALLEN-BRADLEY SERIES OF ISSUES AND ANSWERS
A Reference Guide to Causes, Effects and Corrective Measures AN ALLEN-BRADLEY SERIES OF ISSUES AND ANSWERS By: Robert G. Ellis, P. Eng., Rockwell Automation Medium Voltage Business CONTENTS INTRODUCTION...
More informationSOFTWARE FOR GENERATION OF SPECTRUM COMPATIBLE TIME HISTORY
3 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 24 Paper No. 296 SOFTWARE FOR GENERATION OF SPECTRUM COMPATIBLE TIME HISTORY ASHOK KUMAR SUMMARY One of the important
More informationFOURIER TRANSFORM BASED SIMPLE CHORD ANALYSIS. UIUC Physics 193 POM
FOURIER TRANSFORM BASED SIMPLE CHORD ANALYSIS Fanbo Xiang UIUC Physics 193 POM Professor Steven M. Errede Fall 2014 1 Introduction Chords, an essential part of music, have long been analyzed. Different
More informationAN-756 APPLICATION NOTE One Technology Way P.O. Box 9106 Norwood, MA 02062-9106 Tel: 781/329-4700 Fax: 781/326-8703 www.analog.com
APPLICATION NOTE One Technology Way P.O. Box 9106 Norwood, MA 02062-9106 Tel: 781/329-4700 Fax: 781/326-8703 www.analog.com Sampled Systems and the Effects of Clock Phase Noise and Jitter by Brad Brannon
More informationBasics of Digital Recording
Basics of Digital Recording CONVERTING SOUND INTO NUMBERS In a digital recording system, sound is stored and manipulated as a stream of discrete numbers, each number representing the air pressure at a
More informationExpression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds
Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative
More informationLecture 8 ELE 301: Signals and Systems
Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 37 Properties of the Fourier Transform Properties of the Fourier
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More information7. DYNAMIC LIGHT SCATTERING 7.1 First order temporal autocorrelation function.
7. DYNAMIC LIGHT SCATTERING 7. First order temporal autocorrelation function. Dynamic light scattering (DLS) studies the properties of inhomogeneous and dynamic media. A generic situation is illustrated
More informationEm bedded DSP : I ntroduction to Digital Filters
Embedded DSP : Introduction to Digital Filters 1 Em bedded DSP : I ntroduction to Digital Filters Digital filters are a important part of DSP. In fact their extraordinary performance is one of the keys
More informationPractical Design of Filter Banks for Automatic Music Transcription
Practical Design of Filter Banks for Automatic Music Transcription Filipe C. da C. B. Diniz, Luiz W. P. Biscainho, and Sergio L. Netto Federal University of Rio de Janeiro PEE-COPPE & DEL-Poli, POBox 6854,
More informationUnderstand the effects of clock jitter and phase noise on sampled systems A s higher resolution data converters that can
designfeature By Brad Brannon, Analog Devices Inc MUCH OF YOUR SYSTEM S PERFORMANCE DEPENDS ON JITTER SPECIFICATIONS, SO CAREFUL ASSESSMENT IS CRITICAL. Understand the effects of clock jitter and phase
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation
More informationFast Fourier Transform: Theory and Algorithms
Fast Fourier Transform: Theory and Algorithms Lecture Vladimir Stojanović 6.973 Communication System Design Spring 006 Massachusetts Institute of Technology Discrete Fourier Transform A review Definition
More information