A Tutorial on Fourier Analysis

Save this PDF as:

Size: px
Start display at page:

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

SGN-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:

FFT 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

Lab 4 Sampling, Aliasing, FIR Filtering

47 Lab 4 Sampling, Aliasing, FIR Filtering This is a software lab. In your report, please include all Matlab code, numerical results, plots, and your explanations of the theoretical questions. The due

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

Frequency 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

Fast Fourier Transforms and Power Spectra in LabVIEW

Application Note 4 Introduction Fast Fourier Transforms and Power Spectra in LabVIEW K. Fahy, E. Pérez Ph.D. The Fourier transform is one of the most powerful signal analysis tools, applicable to a wide

Polynomials and the Fast Fourier Transform (FFT) Battle Plan

Polynomials and the Fast Fourier Transform (FFT) Algorithm Design and Analysis (Wee 7) 1 Polynomials Battle Plan Algorithms to add, multiply and evaluate polynomials Coefficient and point-value representation

The 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

Purpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I

Part 3: Time Series I Purpose of Time Series Analysis (Figure from Panofsky and Brier 1968) Autocorrelation Function Harmonic Analysis Spectrum Analysis Data Window Significance Tests Some major purposes

ANALYZER 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

B3. 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

Fourier Transform and Image Filtering. CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010

Fourier Transform and Image Filtering CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010 The Fourier Transform Fourier Transform Forward, mapping to frequency domain: Backward, inverse mapping to time domain:

Design 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

The 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

Speech 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

Frequency Domain Characterization of Signals. Yao Wang Polytechnic University, Brooklyn, NY11201 http: //eeweb.poly.edu/~yao

Frequency Domain Characterization of Signals Yao Wang Polytechnic University, Brooklyn, NY1121 http: //eeweb.poly.edu/~yao Signal Representation What is a signal Time-domain description Waveform representation

L9: 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,

Correlation 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

Convolution, 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

Lecture 4: Analog Synthesizers

ELEN E4896 MUSIC SIGNAL PROCESSING Lecture 4: Analog Synthesizers 1. The Problem Of Electronic Synthesis 2. Oscillators 3. Envelopes 4. Filters Dan Ellis Dept. Electrical Engineering, Columbia University

IMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL

IMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL Saurabh Singh Rajput, Dr.S.S. Bhadauria Department of Electronics, Madhav Institute of Technology

UNIVERSITY 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

Time 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

Chapter 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

Auto-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

The 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

Adding 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

Sampling 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

5 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

Applications 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

University of Rhode Island Department of Electrical and Computer Engineering ELE 436: Communication Systems. FFT Tutorial

University of Rhode Island Department of Electrical and Computer Engineering ELE 436: Communication Systems FFT Tutorial 1 Getting to Know the FFT What is the FFT? FFT = Fast Fourier Transform. The FFT

Final 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

Short-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

1.4 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

Signaling is the way data is communicated. This type of signal used can be either analog or digital

3.1 Analog vs. Digital Signaling is the way data is communicated. This type of signal used can be either analog or digital 1 3.1 Analog vs. Digital 2 WCB/McGraw-Hill The McGraw-Hill Companies, Inc., 1998

Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34

Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in

Moving Average Filters

CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average

EE 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

SIGNAL 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

1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

Inverse Circular Function and Trigonometric Equation

Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse

Frequency Response and Continuous-time Fourier Transform

Frequency Response and Continuous-time Fourier Transform Goals Signals and Systems in the FD-part II I. (Finite-energy) signals in the Frequency Domain - The Fourier Transform of a signal - Classification

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;

Lab 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

MPEG, the MP3 Standard, and Audio Compression

MPEG, the MP3 Standard, and Audio Compression Mark ilgore and Jamie Wu Mathematics of the Information Age September 16, 23 Audio Compression Basic Audio Coding. Why beneficial to compress? Lossless versus

Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position

Sampling 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

Analog 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

Spectrum 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

Basic Electrical Theory

Basic Electrical Theory Mathematics Review PJM State & Member Training Dept. Objectives By the end of this presentation the Learner should be able to: Use the basics of trigonometry to calculate the different

Aliasing, 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

Nyquist Sampling Theorem. By: Arnold Evia

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

Linear 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

The 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

PYKC 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

Original Lecture Notes developed by

Introduction to ADSL Modems Original Lecture Notes developed by Prof. Brian L. Evans Dept. of Electrical and Comp. Eng. The University of Texas at Austin http://signal.ece.utexas.edu Outline Broadband

chapter 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

Frequency Response MUMT 501 Digital Audio Signal Processing

Frequency Response MUMT 501 Digital Audio Signal Processing Gerald Lemay McGill University MUMT 501 Frequency Response 1 Table of Contents I Introduction... 3 I.1 Useful Math and Systems Concepts... 3

Probability 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

DTFT, Filter Design, Inverse Systems

EE3054 Signals and Systems DTFT, Filter Design, Inverse Systems Yao Wang Polytechnic University Discrete Time Fourier Transform Recall h[n] H(e^jw) = H(z) z=e^jw Can be applied to any discrete time

Exercises in Signals, Systems, and Transforms

Exercises in Signals, Systems, and Transforms Ivan W. Selesnick Last edit: October 7, 4 Contents Discrete-Time Signals and Systems 3. Signals...................................................... 3. System

Chapter 14. MPEG Audio Compression

Chapter 14 MPEG Audio Compression 14.1 Psychoacoustics 14.2 MPEG Audio 14.3 Other Commercial Audio Codecs 14.4 The Future: MPEG-7 and MPEG-21 14.5 Further Exploration 1 Li & Drew c Prentice Hall 2003 14.1

Conceptual 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 ),

1 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

Mathematical 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

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension

Trigonometry Lesson Objectives

Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

Appendix 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

Introduction 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

ENEE 425 Spring 2010 Assigned Homework Oppenheim and Shafer (3rd ed.) Instructor: S.A. Tretter

ENEE 425 Spring 2010 Assigned Homework Oppenheim and Shafer (3rd ed.) Instructor: S.A. Tretter Note: The dates shown are when the problems were assigned. Homework will be discussed in class at the beginning

Time 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,

ALFFT 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...

1.10 Using Figure 1.6, verify that equation (1.10) satisfies the initial velocity condition. t + ") # x (t) = A! n. t + ") # v(0) = A!

1.1 Using Figure 1.6, verify that equation (1.1) satisfies the initial velocity condition. Solution: Following the lead given in Example 1.1., write down the general expression of the velocity by differentiating

A 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,

Difference Equations

Difference Equations Andrew W H House 10 June 004 1 The Basics of Difference Equations Recall that in a previous section we saw that IIR systems cannot be evaluated using the convolution sum because it

Lecture 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

The 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

4.3 Analog-to-Digital Conversion

4.3 Analog-to-Digital Conversion overview including timing considerations block diagram of a device using a DAC and comparator example of a digitized spectrum number of data points required to describe

Introduction 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

Performing 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

FAST 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

The Fast Fourier Transform (FFT) and MATLAB Examples

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

Review 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

TCOM 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

Frequency response of a general purpose single-sided OpAmp amplifier

Frequency response of a general purpose single-sided OpAmp amplifier One configuration for a general purpose amplifier using an operational amplifier is the following. The circuit is characterized by:

Digital 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

Agilent 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

BX 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

Pulsed 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

MATH 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

CM0340 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

Using the TI-92 Plus: Some Examples

Liverpool John Moores University, 1-15 July 000 Using the TI-9 Plus: Some Examples Michel Beaudin École de technologie supérieure,canada mbeaudin@seg.etsmtl.ca 1. Introduction We incorporated the use of

(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,

7. 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

Interferometric Dispersion Measurements

Application Note 2004-022A Interferometric Dispersion Measurements Overview This application note describes how the dbm 2355 Dispersion Measurement Module operates. Summary There are two primary methods

USB 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,

Fourier Analysis and its applications

Fourier Analysis and its applications Fourier analysis originated from the study of heat conduction: Jean Baptiste Joseph Fourier (1768-1830) Fourier analysis enables a function (signal) to be decomposed