Fourier Methods of Spectral Estimation
|
|
- Roderick Dean
- 5 years ago
- Views:
Transcription
1 Department of Electrical Engineering IIT Madras
2 Outline Definition of Power Spectrum Deterministic signal example Power Spectrum of a Random Process The Periodogram Estimator The Averaged Periodogram Blackman-Tukey Method Use of Data Windowing in Spectral Analysis Spectrogram: Speech Signal Example
3 What is Spectral Analysis? Spectral analysis is the estimation of the frequency content of a random process By frequency content we mean the distribution of power over frequency Also called Power Spectral Density, or simply spectrum What frequency components are present? What is the intensity of each component?
4 The Earliest Spectral Analyzer All colours are present with equal intensity
5 What is Frequency? Our notion of frequency comes from sin(2πf t) and cos(2πf t) both are called sinusoids Frequency Sinusoidal Frequency: f cycles/sec (Hz) exp(j2πf t) is the basis function needed for representing a component with frequency f for an arbitrary frequency component, it becomes exp(j2πft) As f varies from to, we get the Fourier basis set! f is sometimes called Fourier frequency
6 Spectral Analysis Expansion Using Fourier Basis Spectral analysis is nothing but expanding a signal x(t) using the Fourier basis X (f ) = x(t) exp( j2πft) dt inner product! X (f ) is the continuous-time Fourier transform of x(t) X (f 1 ) large dominant frequency component at f = f 1 X (f 1 ) = no frequency component at f = f 1 Plot of X (f ) 2 as a function of f is called the power spectrum
7 Deterministic Signal Example Gated sinusoid with f = 1 khz Magnitude Time (ms) Frequency (Hz) Log Magnitude Frequency (Hz)
8 What About PSD of a Random Process? Spectral analysis is the estimation of the frequency content of a random process An ensemble of sample waveforms constitute a random process X (f )? = x(t) exp( j2πft) dt Does it exist? Even if it does, is it meaningful? We ll focus on discrete-time random processes, i.e., ensemble of x[n], where n Z
9 PSD of a WSS Random Process Let x[n] be a complex wide-sense stationary process Its autocorrelation sequence (ACS) is defined as Wiener-Khinchine Theorem: P xx (f ) = r xx [k] = E{x [n] x[n + k]} r xx [k] exp( j2πfk) 1 2 f 1 2 That is, ACS DTFT PSD
10 An Alternative Definition for PSD If the ACS decays sufficiently rapidly, P xx (f ) = lim E 1 M 2 x[n] exp( j2πfn) M 2M + 1 M The so-called Direct Method is based on the above formula
11 Why is the Problem Difficult? ACS is not available Finite number of samples from one realization We are only given x[], x[1],..., x[n 1] No best spectral estimator exists Many practical signals, such as speech, are non-stationary P xx (f ) obtained from given data is a random variable Bias versus Variance trade-off
12 The Periodogram Estimator Recall P xx (f ) = lim E 1 M 2 x[n] exp( j2πfn) M 2M + 1 In practice we drop because data are finite lim M M the expectation operator E since we have only one realization The Periodogram estimator is defined as ˆP PER (f ) def = 1 N N 1 x[n] exp( j2πfn) n= Direct Method, since it deals with the data directly 2
13 Example: Two Sine Waves + Noise x[n] = 1 exp(j 2π.15n) + 2 exp(j 2π.2n) + z[n] z[n] complex N (, 1), N = Magnitude (db) Frequency
14 Periodogram is a Biased Estimator For Finite Data For finite N, periodogram is a biased estimator Bias is the difference between the true and expected values 5 4 averaged noiseless Magnitude (db) N = Frequency
15 Periodogram: Bias Decreases With Increasing N If data length is increased, bias decreases: } lim {ˆP E xx (f ) = P xx (f ) N Magnitude (db) N = Frequency
16 The More (Samples) the Merrier? For most estimators, bias and variance decrease with increasing N An estimator is said to be consistent if ( ) lim ˆθ Pr θ > ɛ = N where ˆθ is the estimate of θ This implies that, as N, bias variance
17 Is the Periodogram Consistent? Consider white noise sequence for various N True P xx (f ) = constant If the Periodogram estimator were consistent, ˆP xx (f ) constant as N increases Consider noise sequences of length 32, 64, 128, and 256 N = 32; % white noise sequence x = randn(n,1); % of length 32 Does ˆP xx (f ) tend to a constant as N increases?
18 White Noise Example 4 PSD of White Noise N= N= Frequency As N increases, variance of the estimate does not decrease Periodogram is an inconsistent estimator
19 What Went Wrong? In practice we drop lim because data are finite M the expectation operator E since we have only one realization For white noise, increasing the data length did not help What can be done to capture the benefits of E{ }?
20 Averaging: The Poor Man s Expectation Operator Expectation operator can be approximated by averaging Averaged Periodogram: ˆP AVPER (f ) = 1 M M m=1 ˆP (m) PER (f ) where ˆP (m) PER (f ) is periodogram of m-th segment of length N For independent data records } var {ˆP AVPER (f ) = 1 } {ˆP M var PER (f )
21 Averaged Periodogram of White Noise Result of averaging 8 periodograms 4 Averaged Periodogram Magnitude (db) Frequency
22 Variance Decreases, But Bias Increases! Two Sines + Noise example 2 1 N=256, M=1 N=64, M=4 N=16, M=16 Averaged Periodogram for Two Sines + Noise Magnitude (db) Frequency
23 Welch s Method Overlapping blocks by 5% Reduces variance without worsening bias Block Length = 64 No. of blocks = 7 Overlap = 5% Welch s Method Magnitude (db) Frequency
24 Why Did The Periodogram Fail? Periodogram was defined as ˆP PER (f ) = 1 N N 1 n= x[n] exp( j2πfn) 2 Equivalent to where ˆr xx [k] = ˆP PER (f ) = N 1 k 1 N n= ˆr xx[ k] N 1 (N 1) ˆr xx [k] exp( j2πfk) x [n] x[n + k] k =, 1,..., N 1 Note that ˆr xx [N 1] = x []x[n 1]/N No averaging estimate with high variance! k = (N 1),..., 1
25 Blackman-Tukey Method Recall P xx (f ) = r xx [k] exp( j2πfk) 1 2 f 1 2 In practice: (a) replace r xx [k] by estimate ˆr xx [k], (b) truncate the summation, and (c) apply lag window M ˆP BT f ) = w[k] ˆr xx [k] exp( j2πfk) M where w[k] w[] = 1 w[k] = for k > M w[ k] = w[k] W (f ) Indirect Method, since it does not deal with the data directly
26 Example: Two Sine Waves + Noise Data length N = 1, Correlation Lag M = Magnitude (db) Frequency
27 Periodogram Vs. Blackman-Tukey Magnitude (db) Magnitude (db) Periodogram Blackman Tukey Frequency Blackman-Tukey method: reduction in variance comes at the expense of increased bias Speech Analysis
28 Data Windowing in Spectral Analysis Useful for data containing sinusoids + noise Sidelobes of a stronger sinusoid may mask the main lobe of a nearby weak sinusoid We multiply x[n] by data window w[n] before computing periodogram Weaker sinusoid becomes more visible Main lobe of each sinusoid broadens: two close peaks may merge into one
29 Example: How Many Sine Waves Are There? 6 How Many Sinusoids Are There? 4 Magnitude (db) Frequency
30 Example: Three Sine Waves 6 Three Sinusoids: Rectangular Window 4 Magnitude (db) Frequency
31 Example: Three Sine Waves 6 Three Sinusoids: Hanning Window 4 Magnitude (db) Frequency
32 Commonly Used Windows Name w[k] Fourier transform sin πf (2M + 1) Rectangular 1 W R (f ) = sin πf Bartlett 1 k ( ) 1 sin πfm 2 M M sin πf Hanning cos πk ( ).25 W R f 1 M ( 2M ) +.5 WR (f ) +.25 W R f + 1 Hamming cos πk M w[k] = for k > M 2M.23 W ( ) R f 1 ( 2M ) +.54 WR (f ) +.23 W R f + 1 2M
33 Hamming Vs. Hanning Magnitude (db) Magnitude (db) Frequency 2 4 Fourier Transforms of Hamming and Hanning Windows Hamming Hanning Frequency
34 Three Sine Waves 6 Rectangular Vs. Hamming Vs. Hanning 4 Magnitude (db) Frequency
35 How Can We Analyze Non-Stationary Signals? Consider a linear chirp, i.e., a signal whose frequency increases linearly from f 1 Hz to f 2 Hz over a time interval T What is its magnitude spectrum? 1 Amplitude Magnitdue (db) Time Frequency
36 Need a More Useful Representation In Fourier analysis, even if a signal is non-stationary, it is still represented using stationary sinusoids An unsatisfactory approach Power spectrum is identical to x( t), whose frequency decreases from f 2 to f 1 x(t) and x( t) differ only in the phase of the Fourier transform What we really want to know is how frequency varies with time Can it still be called frequency?
37 Spectrogram Plot of power spectrum of short blocks of a signal as a function of time Over each short block, signal is considered to be stationary Speech is a classic example of a commonly occurring non-stationary signal Voiced sounds: /a/, /e/, /i/, /o/, /u/ (quasi-periodic) Unvoiced sounds: /s/, /sh/, /f/ (noise-like) Plosives: /p/, /t/, /k/ (transient sounds)
38 Spectrogram of Linear Chirp Frequency Time
39 Non-stationarity in Speech Signal.2 /k/ /ow/ /s/ Time
40 Application to Speech Analysis 1 Should We Chase Those Cowboys? Amplitude.5.5 Frequency Time
41 Application to Speech Analysis 1 Should We Chase Those Cowboys? Amplitude.5.5 Frequency Time
42 Application to Speech Analysis 1 Should We Chase Those Cowboys? Amplitude.5.5 Frequency Time
43 Application to Speech Analysis 1 Should We Chase Those Cowboys? Amplitude.5.5 Frequency Time
44 Summary Definition of Power Spectrum Deterministic signal example Power Spectrum of a Random Process The Periodogram Estimator The Averaged Periodogram Bias versus Variance Blackman-Tukey Method Use of Data Windowing in Spectral Analysis Spectrogram: Speech Signal Example
EE 451: Digital Signal Processing
EE 451: Digital Signal Processing Stochastic Processes and Spectral Estimation Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA November 29, 2011 Aly El-Osery (NMT)
More informationDigital Signal Processing
COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #29 Wednesday, November 19, 2003 Correlation-based methods of spectral estimation: In the periodogram methods of spectral estimation, a direct
More informationChapter 5 Window Functions. periodic with a period of N (number of samples). This is observed in table (3.1).
Chapter 5 Window Functions 5.1 Introduction As discussed in section (3.7.5), the DTFS assumes that the input waveform is periodic with a period of N (number of samples). This is observed in table (3.1).
More informationDesign of FIR Filter for Efficient Utilization of Speech Signal Akanksha. Raj 1 Arshiyanaz. Khateeb 2 Fakrunnisa.Balaganur 3
IJSRD - International Journal for Scientific Research & Development Vol. 3, Issue 03, 2015 ISSN (online): 2321-0613 Design of FIR Filter for Efficient Utilization of Speech Signal Akanksha. Raj 1 Arshiyanaz.
More informationBiomedical Signals. Signals and Images in Medicine Dr Nabeel Anwar
Biomedical Signals Signals and Images in Medicine Dr Nabeel Anwar Noise Removal: Time Domain Techniques 1. Synchronized Averaging (covered in lecture 1) 2. Moving Average Filters (today s topic) 3. Derivative
More informationEE 464 Short-Time Fourier Transform Fall and Spectrogram. Many signals of importance have spectral content that
EE 464 Short-Time Fourier Transform Fall 2018 Read Text, Chapter 4.9. and Spectrogram Many signals of importance have spectral content that changes with time. Let xx(nn), nn = 0, 1,, NN 1 1 be a discrete-time
More informationTopic 2. Signal Processing Review. (Some slides are adapted from Bryan Pardo s course slides on Machine Perception of Music)
Topic 2 Signal Processing Review (Some slides are adapted from Bryan Pardo s course slides on Machine Perception of Music) Recording Sound Mechanical Vibration Pressure Waves Motion->Voltage Transducer
More informationEE 451: Digital Signal Processing
EE 451: Digital Signal Processing Power Spectral Density Estimation Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA December 4, 2017 Aly El-Osery (NMT) EE 451:
More informationII. Random Processes Review
II. Random Processes Review - [p. 2] RP Definition - [p. 3] RP stationarity characteristics - [p. 7] Correlation & cross-correlation - [p. 9] Covariance and cross-covariance - [p. 10] WSS property - [p.
More informationDigital Signal Processing
COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #27 Tuesday, November 11, 23 6. SPECTRAL ANALYSIS AND ESTIMATION 6.1 Introduction to Spectral Analysis and Estimation The discrete-time Fourier
More informationDiscrete Fourier Transform (DFT)
Amplitude Amplitude Discrete Fourier Transform (DFT) DFT transforms the time domain signal samples to the frequency domain components. DFT Signal Spectrum Time Frequency DFT is often used to do frequency
More informationObjectives. Presentation Outline. Digital Modulation Lecture 03
Digital Modulation Lecture 03 Inter-Symbol Interference Power Spectral Density Richard Harris Objectives To be able to discuss Inter-Symbol Interference (ISI), its causes and possible remedies. To be able
More informationComplex Sounds. Reading: Yost Ch. 4
Complex Sounds Reading: Yost Ch. 4 Natural Sounds Most sounds in our everyday lives are not simple sinusoidal sounds, but are complex sounds, consisting of a sum of many sinusoids. The amplitude and frequency
More informationNoise estimation and power spectrum analysis using different window techniques
IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE) e-issn: 78-1676,p-ISSN: 30-3331, Volume 11, Issue 3 Ver. II (May. Jun. 016), PP 33-39 www.iosrjournals.org Noise estimation and power
More informationOutline. Introduction to Biosignal Processing. Overview of Signals. Measurement Systems. -Filtering -Acquisition Systems (Quantisation and Sampling)
Outline Overview of Signals Measurement Systems -Filtering -Acquisition Systems (Quantisation and Sampling) Digital Filtering Design Frequency Domain Characterisations - Fourier Analysis - Power Spectral
More informationChapter 7. Frequency-Domain Representations 语音信号的频域表征
Chapter 7 Frequency-Domain Representations 语音信号的频域表征 1 General Discrete-Time Model of Speech Production Voiced Speech: A V P(z)G(z)V(z)R(z) Unvoiced Speech: A N N(z)V(z)R(z) 2 DTFT and DFT of Speech The
More informationEE123 Digital Signal Processing
EE123 Digital Signal Processing Lecture 5A Time-Frequency Tiling Subtleties in filtering/processing with DFT x[n] H(e j! ) y[n] System is implemented by overlap-and-save Filtering using DFT H[k] π 2π Subtleties
More informationDCSP-10: DFT and PSD. Jianfeng Feng. Department of Computer Science Warwick Univ., UK
DCSP-10: DFT and PSD Jianfeng Feng Department of Computer Science Warwick Univ., UK Jianfeng.feng@warwick.ac.uk http://www.dcs.warwick.ac.uk/~feng/dcsp.html DFT Definition: The discrete Fourier transform
More information(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters
FIR Filter Design Chapter Intended Learning Outcomes: (i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters (ii) Ability to design linear-phase FIR filters according
More informationME scope Application Note 01 The FFT, Leakage, and Windowing
INTRODUCTION ME scope Application Note 01 The FFT, Leakage, and Windowing NOTE: The steps in this Application Note can be duplicated using any Package that includes the VES-3600 Advanced Signal Processing
More informationSpectrogram Review The Sampling Problem: 2π Ambiguity Fourier Series. Lecture 6: Sampling. ECE 401: Signal and Image Analysis. University of Illinois
Lecture 6: Sampling ECE 401: Signal and Image Analysis University of Illinois 2/7/2017 1 Spectrogram Review 2 The Sampling Problem: 2π Ambiguity 3 Fourier Series Outline 1 Spectrogram Review 2 The Sampling
More informationECE 201: Introduction to Signal Analysis
ECE 201: Introduction to Signal Analysis Prof. Paris Last updated: October 9, 2007 Part I Spectrum Representation of Signals Lecture: Sums of Sinusoids (of different frequency) Introduction Sum of Sinusoidal
More information(i) Understanding the basic concepts of signal modeling, correlation, maximum likelihood estimation, least squares and iterative numerical methods
Tools and Applications Chapter Intended Learning Outcomes: (i) Understanding the basic concepts of signal modeling, correlation, maximum likelihood estimation, least squares and iterative numerical methods
More informationHarmonic Analysis. Purpose of Time Series Analysis. What Does Each Harmonic Mean? Part 3: Time Series I
Part 3: Time Series I Harmonic Analysis Spectrum Analysis Autocorrelation Function Degree of Freedom Data Window (Figure from Panofsky and Brier 1968) Significance Tests Harmonic Analysis Harmonic analysis
More informationChapter 2: Signal Representation
Chapter 2: Signal Representation Aveek Dutta Assistant Professor Department of Electrical and Computer Engineering University at Albany Spring 2018 Images and equations adopted from: Digital Communications
More information(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters
FIR Filter Design Chapter Intended Learning Outcomes: (i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters (ii) Ability to design linear-phase FIR filters according
More informationTime Series/Data Processing and Analysis (MATH 587/GEOP 505)
Time Series/Data Processing and Analysis (MATH 587/GEOP 55) Rick Aster and Brian Borchers October 7, 28 Plotting Spectra Using the FFT Plotting the spectrum of a signal from its FFT is a very common activity.
More informationTheory of Telecommunications Networks
Theory of Telecommunications Networks Anton Čižmár Ján Papaj Department of electronics and multimedia telecommunications CONTENTS Preface... 5 1 Introduction... 6 1.1 Mathematical models for communication
More informationCHAPTER 3 DIGITAL SPECTRAL ANALYSIS
CHAPTER 3 DIGITAL SPECTRAL ANALYSIS Shri Mata Vaishno Devi University, (SMVDU), 2013 Page 22 CHAPTER 3 DIGITAL SPECTRAL ANALYSIS 3.1 Introduction The transformation of data from the time domain to the
More informationInternational Journal of Modern Trends in Engineering and Research e-issn No.: , Date: 2-4 July, 2015
International Journal of Modern Trends in Engineering and Research www.ijmter.com e-issn No.:2349-9745, Date: 2-4 July, 2015 Analysis of Speech Signal Using Graphic User Interface Solly Joy 1, Savitha
More information2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal.
1 2.1 BASIC CONCEPTS 2.1.1 Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal. 2 Time Scaling. Figure 2.4 Time scaling of a signal. 2.1.2 Classification of Signals
More informationSpeech Signal Analysis
Speech Signal Analysis Hiroshi Shimodaira and Steve Renals Automatic Speech Recognition ASR Lectures 2&3 14,18 January 216 ASR Lectures 2&3 Speech Signal Analysis 1 Overview Speech Signal Analysis for
More informationECE 5650/4650 Exam II November 20, 2018 Name:
ECE 5650/4650 Exam II November 0, 08 Name: Take-Home Exam Honor Code This being a take-home exam a strict honor code is assumed. Each person is to do his/her own work. Bring any questions you have about
More informationDigital Signal Processing
Digital Signal Processing Fourth Edition John G. Proakis Department of Electrical and Computer Engineering Northeastern University Boston, Massachusetts Dimitris G. Manolakis MIT Lincoln Laboratory Lexington,
More informationProject 0: Part 2 A second hands-on lab on Speech Processing Frequency-domain processing
Project : Part 2 A second hands-on lab on Speech Processing Frequency-domain processing February 24, 217 During this lab, you will have a first contact on frequency domain analysis of speech signals. You
More informationDigital Signal Processing PW1 Signals, Correlation functions and Spectra
Digital Signal Processing PW1 Signals, Correlation functions and Spectra Nathalie Thomas Master SATCOM 018 019 1 Introduction The objectives of this rst practical work are the following ones : 1. to be
More informationNoise Measurements Using a Teledyne LeCroy Oscilloscope
Noise Measurements Using a Teledyne LeCroy Oscilloscope TECHNICAL BRIEF January 9, 2013 Summary Random noise arises from every electronic component comprising your circuits. The analysis of random electrical
More informationSpectral Estimation & Examples of Signal Analysis
Spectral Estimation & Examples of Signal Analysis Examples from research of Kyoung Hoon Lee, Aaron Hastings, Don Gallant, Shashikant More, Weonchan Sung Herrick Graduate Students Estimation: Bias, Variance
More informationSignal Processing Toolbox
Signal Processing Toolbox Perform signal processing, analysis, and algorithm development Signal Processing Toolbox provides industry-standard algorithms for analog and digital signal processing (DSP).
More informationspeech signal S(n). This involves a transformation of S(n) into another signal or a set of signals
16 3. SPEECH ANALYSIS 3.1 INTRODUCTION TO SPEECH ANALYSIS Many speech processing [22] applications exploits speech production and perception to accomplish speech analysis. By speech analysis we extract
More informationFourier and Wavelets
Fourier and Wavelets Why do we need a Transform? Fourier Transform and the short term Fourier (STFT) Heisenberg Uncertainty Principle The continues Wavelet Transform Discrete Wavelet Transform Wavelets
More informationChapter 3 Data Transmission COSC 3213 Summer 2003
Chapter 3 Data Transmission COSC 3213 Summer 2003 Courtesy of Prof. Amir Asif Definitions 1. Recall that the lowest layer in OSI is the physical layer. The physical layer deals with the transfer of raw
More informationSAMPLING THEORY. Representing continuous signals with discrete numbers
SAMPLING THEORY Representing continuous signals with discrete numbers Roger B. Dannenberg Professor of Computer Science, Art, and Music Carnegie Mellon University ICM Week 3 Copyright 2002-2013 by Roger
More informationThe Fundamentals of FFT-Based Signal Analysis and Measurement Michael Cerna and Audrey F. Harvey
Application ote 041 The Fundamentals of FFT-Based Signal Analysis and Measurement Michael Cerna and Audrey F. Harvey Introduction The Fast Fourier Transform (FFT) and the power spectrum are powerful tools
More informationDynamic Signal Analysis Basics
Dynamic Signal Analysis Basics James Zhuge, Ph.D., President Crystal Instruments Corporation 4633 Old Ironsides Drive, Suite 304 Santa Clara, CA 95054, USA www.go-ci.com (Part of CoCo-80 User s Manual)
More informationDigital Signal Processing Lecture 1 - Introduction
Digital Signal Processing - Electrical Engineering and Computer Science University of Tennessee, Knoxville August 20, 2015 Overview 1 2 3 4 Basic building blocks in DSP Frequency analysis Sampling Filtering
More informationSignals A Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture 2
Signals A Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture 2 The Fourier transform of single pulse is the sinc function. EE 442 Signal Preliminaries 1 Communication Systems and
More informationReading: Johnson Ch , Ch.5.5 (today); Liljencrants & Lindblom; Stevens (Tues) reminder: no class on Thursday.
L105/205 Phonetics Scarborough Handout 7 10/18/05 Reading: Johnson Ch.2.3.3-2.3.6, Ch.5.5 (today); Liljencrants & Lindblom; Stevens (Tues) reminder: no class on Thursday Spectral Analysis 1. There are
More informationStructure of Speech. Physical acoustics Time-domain representation Frequency domain representation Sound shaping
Structure of Speech Physical acoustics Time-domain representation Frequency domain representation Sound shaping Speech acoustics Source-Filter Theory Speech Source characteristics Speech Filter characteristics
More informationOutline. EECS 3213 Fall Sebastian Magierowski York University. Review Passband Modulation. Constellations ASK, FSK, PSK.
EECS 3213 Fall 2014 L12: Modulation Sebastian Magierowski York University 1 Outline Review Passband Modulation ASK, FSK, PSK Constellations 2 1 Underlying Idea Attempting to send a sequence of digits through
More informationURBANA-CHAMPAIGN. CS 498PS Audio Computing Lab. Audio DSP basics. Paris Smaragdis. paris.cs.illinois.
UNIVERSITY ILLINOIS @ URBANA-CHAMPAIGN OF CS 498PS Audio Computing Lab Audio DSP basics Paris Smaragdis paris@illinois.edu paris.cs.illinois.edu Overview Basics of digital audio Signal representations
More informationMichael F. Toner, et. al.. "Distortion Measurement." Copyright 2000 CRC Press LLC. <
Michael F. Toner, et. al.. "Distortion Measurement." Copyright CRC Press LLC. . Distortion Measurement Michael F. Toner Nortel Networks Gordon W. Roberts McGill University 53.1
More informationTE 302 DISCRETE SIGNALS AND SYSTEMS. Chapter 1: INTRODUCTION
TE 302 DISCRETE SIGNALS AND SYSTEMS Study on the behavior and processing of information bearing functions as they are currently used in human communication and the systems involved. Chapter 1: INTRODUCTION
More informationTopic 6. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)
Topic 6 The Digital Fourier Transform (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith) 10 20 30 40 50 60 70 80 90 100 0-1 -0.8-0.6-0.4-0.2 0 0.2 0.4
More information6 Sampling. Sampling. The principles of sampling, especially the benefits of coherent sampling
Note: Printed Manuals 6 are not in Color Objectives This chapter explains the following: The principles of sampling, especially the benefits of coherent sampling How to apply sampling principles in a test
More informationSignals. Continuous valued or discrete valued Can the signal take any value or only discrete values?
Signals Continuous time or discrete time Is the signal continuous or sampled in time? Continuous valued or discrete valued Can the signal take any value or only discrete values? Deterministic versus random
More informationPROBLEM SET 6. Note: This version is preliminary in that it does not yet have instructions for uploading the MATLAB problems.
PROBLEM SET 6 Issued: 2/32/19 Due: 3/1/19 Reading: During the past week we discussed change of discrete-time sampling rate, introducing the techniques of decimation and interpolation, which is covered
More informationCommunications II. Professor Kin K. Leung EEE Departments Imperial College London
Communications II Professor Kin K. Leung EEE Departments Imperial College London Acknowledge Contributions by Darren Ward, Maria Petrou and Cong Ling Lecture 1: Introduction and Review 2 What does communication
More informationBasic Definitions and The Spectral Estimation Problem
Informal Definition of Spectral Estimation Given: A finite record of a signal Basic Definitions and The Spectral Estimation Problem Determine: The distribution of signal power over frequency signal t=1,
More information4. Design of Discrete-Time Filters
4. Design of Discrete-Time Filters 4.1. Introduction (7.0) 4.2. Frame of Design of IIR Filters (7.1) 4.3. Design of IIR Filters by Impulse Invariance (7.1) 4.4. Design of IIR Filters by Bilinear Transformation
More informationDSP Laboratory (EELE 4110) Lab#10 Finite Impulse Response (FIR) Filters
Islamic University of Gaza OBJECTIVES: Faculty of Engineering Electrical Engineering Department Spring-2011 DSP Laboratory (EELE 4110) Lab#10 Finite Impulse Response (FIR) Filters To demonstrate the concept
More informationDYNAMIC SIGNAL ANALYSIS BASICS
CI PRODUCT NOTE No. 001 DYNAMIC SIGNAL ANALYSIS BASICS (Included in the CoCo-80 User s Manual) WWW.CRYSTALINSTRUMENTS.COM TABLE OF CONTENTS Frequency Analysis PAGE 1 Basic Theory of FFT Frequency Analysis
More informationFAULT DETECTION OF ROTATING MACHINERY FROM BICOHERENCE ANALYSIS OF VIBRATION DATA
FAULT DETECTION OF ROTATING MACHINERY FROM BICOHERENCE ANALYSIS OF VIBRATION DATA Enayet B. Halim M. A. A. Shoukat Choudhury Sirish L. Shah, Ming J. Zuo Chemical and Materials Engineering Department, University
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science. OpenCourseWare 2006
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341: Discrete-Time Signal Processing OpenCourseWare 2006 Lecture 6 Quantization and Oversampled Noise Shaping
More informationEE 215 Semester Project SPECTRAL ANALYSIS USING FOURIER TRANSFORM
EE 215 Semester Project SPECTRAL ANALYSIS USING FOURIER TRANSFORM Department of Electrical and Computer Engineering Missouri University of Science and Technology Page 1 Table of Contents Introduction...Page
More informationLinear Frequency Modulation (FM) Chirp Signal. Chirp Signal cont. CMPT 468: Lecture 7 Frequency Modulation (FM) Synthesis
Linear Frequency Modulation (FM) CMPT 468: Lecture 7 Frequency Modulation (FM) Synthesis Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 26, 29 Till now we
More information1B Paper 6: Communications Handout 2: Analogue Modulation
1B Paper 6: Communications Handout : Analogue Modulation Ramji Venkataramanan Signal Processing and Communications Lab Department of Engineering ramji.v@eng.cam.ac.uk Lent Term 16 1 / 3 Modulation Modulation
More informationLinguistic Phonetics. Spectral Analysis
24.963 Linguistic Phonetics Spectral Analysis 4 4 Frequency (Hz) 1 Reading for next week: Liljencrants & Lindblom 1972. Assignment: Lip-rounding assignment, due 1/15. 2 Spectral analysis techniques There
More informationCG401 Advanced Signal Processing. Dr Stuart Lawson Room A330 Tel: January 2003
CG40 Advanced Dr Stuart Lawson Room A330 Tel: 23780 e-mail: ssl@eng.warwick.ac.uk 03 January 2003 Lecture : Overview INTRODUCTION What is a signal? An information-bearing quantity. Examples of -D and 2-D
More informationSignal Processing Summary
Signal Processing Summary Jan Černocký, Valentina Hubeika {cernocky,ihubeika}@fit.vutbr.cz DCGM FIT BUT Brno, ihubeika@fit.vutbr.cz FIT BUT Brno Signal Processing Summary Jan Černocký, Valentina Hubeika,
More informationElectrical & Computer Engineering Technology
Electrical & Computer Engineering Technology EET 419C Digital Signal Processing Laboratory Experiments by Masood Ejaz Experiment # 1 Quantization of Analog Signals and Calculation of Quantized noise Objective:
More informationHow to sniff the G3 and Prime data and detect the interfere attack
How to sniff the G3 and Prime data and detect the interfere attack Abstract This topic will talk about how to get the PLC data stream in a PLC communication system which would use G3 or Prime standard,
More information21/01/2014. Fundamentals of the analysis of neuronal oscillations. Separating sources
21/1/214 Separating sources Fundamentals of the analysis of neuronal oscillations Robert Oostenveld Donders Institute for Brain, Cognition and Behaviour Radboud University Nijmegen, The Netherlands Use
More informationChapter 2 Direct-Sequence Systems
Chapter 2 Direct-Sequence Systems A spread-spectrum signal is one with an extra modulation that expands the signal bandwidth greatly beyond what is required by the underlying coded-data modulation. Spread-spectrum
More informationSignal segmentation and waveform characterization. Biosignal processing, S Autumn 2012
Signal segmentation and waveform characterization Biosignal processing, 5173S Autumn 01 Short-time analysis of signals Signal statistics may vary in time: nonstationary how to compute signal characterizations?
More informationChapter 2. Signals and Spectra
Chapter 2 Signals and Spectra Outline Properties of Signals and Noise Fourier Transform and Spectra Power Spectral Density and Autocorrelation Function Orthogonal Series Representation of Signals and Noise
More informationADSP ADSP ADSP ADSP. Advanced Digital Signal Processing (18-792) Spring Fall Semester, Department of Electrical and Computer Engineering
ADSP ADSP ADSP ADSP Advanced Digital Signal Processing (18-792) Spring Fall Semester, 201 2012 Department of Electrical and Computer Engineering PROBLEM SET 5 Issued: 9/27/18 Due: 10/3/18 Reminder: Quiz
More informationProblem Set 8 #4 Solution
Problem Set 8 #4 Solution Solution to PS8 Extra credit #4 E. Sterl Phinney ACM95b/100b 1 Mar 004 4. (7 3 points extra credit) Bessel Functions and FM radios FM (Frequency Modulated) radio works by encoding
More informationEstimation of Sinusoidally Modulated Signal Parameters Based on the Inverse Radon Transform
Estimation of Sinusoidally Modulated Signal Parameters Based on the Inverse Radon Transform Miloš Daković, Ljubiša Stanković Faculty of Electrical Engineering, University of Montenegro, Podgorica, Montenegro
More information6.02 Practice Problems: Modulation & Demodulation
1 of 12 6.02 Practice Problems: Modulation & Demodulation Problem 1. Here's our "standard" modulation-demodulation system diagram: at the transmitter, signal x[n] is modulated by signal mod[n] and the
More informationThe Periodogram. Use identity sin(θ) = (e iθ e iθ )/(2i) and formulas for geometric sums to compute mean.
The Periodogram Sample covariance between X and sin(2πωt + φ) is 1 T T 1 X t sin(2πωt + φ) X 1 T T 1 sin(2πωt + φ) Use identity sin(θ) = (e iθ e iθ )/(2i) and formulas for geometric sums to compute mean.
More informationShort-Time Fourier Transform and Its Inverse
Short-Time Fourier Transform and Its Inverse Ivan W. Selesnick April 4, 9 Introduction The short-time Fourier transform (STFT) of a signal consists of the Fourier transform of overlapping windowed blocks
More informationCommunication Channels
Communication Channels wires (PCB trace or conductor on IC) optical fiber (attenuation 4dB/km) broadcast TV (50 kw transmit) voice telephone line (under -9 dbm or 110 µw) walkie-talkie: 500 mw, 467 MHz
More informationECE438 - Laboratory 7a: Digital Filter Design (Week 1) By Prof. Charles Bouman and Prof. Mireille Boutin Fall 2015
Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 7a: Digital Filter Design (Week 1) By Prof. Charles Bouman and Prof. Mireille Boutin Fall 2015 1 Introduction
More informationQuantification of glottal and voiced speech harmonicsto-noise ratios using cepstral-based estimation
Quantification of glottal and voiced speech harmonicsto-noise ratios using cepstral-based estimation Peter J. Murphy and Olatunji O. Akande, Department of Electronic and Computer Engineering University
More informationMeasurement of RMS values of non-coherently sampled signals. Martin Novotny 1, Milos Sedlacek 2
Measurement of values of non-coherently sampled signals Martin ovotny, Milos Sedlacek, Czech Technical University in Prague, Faculty of Electrical Engineering, Dept. of Measurement Technicka, CZ-667 Prague,
More informationTheory of Telecommunications Networks
Theory of Telecommunications Networks Anton Čižmár Ján Papaj Department of electronics and multimedia telecommunications CONTENTS Preface... 5 Introduction... 6. Mathematical models for communication channels...
More information(Time )Frequency Analysis of EEG Waveforms
(Time )Frequency Analysis of EEG Waveforms Niko Busch Charité University Medicine Berlin; Berlin School of Mind and Brain niko.busch@charite.de niko.busch@charite.de 1 / 23 From ERP waveforms to waves
More informationAdvanced Digital Signal Processing Part 2: Digital Processing of Continuous-Time Signals
Advanced Digital Signal Processing Part 2: Digital Processing of Continuous-Time Signals Gerhard Schmidt Christian-Albrechts-Universität zu Kiel Faculty of Engineering Institute of Electrical Engineering
More informationSignal Processing for Digitizers
Signal Processing for Digitizers Modular digitizers allow accurate, high resolution data acquisition that can be quickly transferred to a host computer. Signal processing functions, applied in the digitizer
More informationFinal Exam Practice Questions for Music 421, with Solutions
Final Exam Practice Questions for Music 4, with Solutions Elementary Fourier Relationships. For the window w = [/,,/ ], what is (a) the dc magnitude of the window transform? + (b) the magnitude at half
More informationPART II Practical problems in the spectral analysis of speech signals
PART II Practical problems in the spectral analysis of speech signals We have now seen how the Fourier analysis recovers the amplitude and phase of an input signal consisting of a superposition of multiple
More informationRemoval of Line Noise Component from EEG Signal
1 Removal of Line Noise Component from EEG Signal Removal of Line Noise Component from EEG Signal When carrying out time-frequency analysis, if one is interested in analysing frequencies above 30Hz (i.e.
More informationTime-Frequency Analysis of Shock and Vibration Measurements Using Wavelet Transforms
Cloud Publications International Journal of Advanced Packaging Technology 2014, Volume 2, Issue 1, pp. 60-69, Article ID Tech-231 ISSN 2349 6665, doi 10.23953/cloud.ijapt.15 Case Study Open Access Time-Frequency
More informationDiscrete Fourier Transform, DFT Input: N time samples
EE445M/EE38L.6 Lecture. Lecture objectives are to: The Discrete Fourier Transform Windowing Use DFT to design a FIR digital filter Discrete Fourier Transform, DFT Input: time samples {a n = {a,a,a 2,,a
More informationLocal Oscillator Phase Noise and its effect on Receiver Performance C. John Grebenkemper
Watkins-Johnson Company Tech-notes Copyright 1981 Watkins-Johnson Company Vol. 8 No. 6 November/December 1981 Local Oscillator Phase Noise and its effect on Receiver Performance C. John Grebenkemper All
More informationJOURNAL OF OBJECT TECHNOLOGY
JOURNAL OF OBJECT TECHNOLOGY Online at http://www.jot.fm. Published by ETH Zurich, Chair of Software Engineering JOT, 2009 Vol. 9, No. 1, January-February 2010 The Discrete Fourier Transform, Part 5: Spectrogram
More informationForced Oscillation Detection Fundamentals Fundamentals of Forced Oscillation Detection
Forced Oscillation Detection Fundamentals Fundamentals of Forced Oscillation Detection John Pierre University of Wyoming pierre@uwyo.edu IEEE PES General Meeting July 17-21, 2016 Boston Outline Fundamental
More informationECE 556 BASICS OF DIGITAL SPEECH PROCESSING. Assıst.Prof.Dr. Selma ÖZAYDIN Spring Term-2017 Lecture 2
ECE 556 BASICS OF DIGITAL SPEECH PROCESSING Assıst.Prof.Dr. Selma ÖZAYDIN Spring Term-2017 Lecture 2 Analog Sound to Digital Sound Characteristics of Sound Amplitude Wavelength (w) Frequency ( ) Timbre
More informationWindows and Leakage Brief Overview
Windows and Leakage Brief Overview When converting a signal from the time domain to the frequency domain, the Fast Fourier Transform (FFT) is used. The Fourier Transform is defined by the Equation: j2πft
More information