Solutions to Exam in Speech Signal Processing EN2300


 Sophia McLaughlin
 3 years ago
 Views:
Transcription
1 Solutions to Exam in Speech Signal Processing EN23 Date: Thursday, Dec 2, 8: 3: Place: Allowed: Grades: Language: Solutions: Q34, Q36 Beta Math Handbook (or corresponding), calculator with empty memory. A: 43 p; B: 37p; C: 3; D: 25p: E: 9 p, out of 48p total. Optional: Swedish or English. To be published on the course web page. Results: Tuesday, Jan 5, 28. Review: At KTHEE/ STEX, Osquldas v.. Good Luck! Please do the Course Evaluation! See the course web page.
2 Please start by browsing through the exam problems. If you do not have enough time to complete tedious numerical computations, just show that you understood the principles. Use shortcuts and reasonable approximations where appropriate. Problems Determine for each of the following statements whether it is true or false (giving +p for each correct answer, p for incorrect). Please note that you can get a negative total result. (a) In a speech transmission system using closedloop adaptive linear prediction with high data rate, the quantization noise is approximately white at the output of the decoder in the receiver. Solution: True (b) For a source signal with uncorrelated sampels, i.e. a white signal spectrum, an optimal scalar quantizer (SQ) achieves exactly the same performance as an optimal vector quantizer (VQ), at any total bitrate, equal for both quantizers. Solution: False (c) A system using the standardized Alaw nonuniform quantization gives a signal to quantization noise ratio which becomes approximately independent of the input signal amplitude in the limit when the signal amplitude decreases towards very low values. Solution: False (d) At time t a hidden Markov model (HMM) has an internal hidden state denoted as S t and observable realvalued output x t. Using a complete observed sequence x,..., x T, the Viterbi algorithm can be used to calculate an exact value of the conditional state probability at time t, P [S t x,..., x T ]. Solution: False. The Viterbi algorithm finds the single most probable state sequence, but other states can still have nonzero probabilities, at any time t. (e) The frequency of the first formant is, on average, about twice as large in female speech, compared to male speech. Solution: False. The vocal tract is on average only slightly shorter for women than for men. (f) A spectrogram, analyzed with a Hamming window of 256 samples, at a sampling rate of 8 samples/s, can show the pitch harmonics in voiced male speech. Solution: True. 2 Fig. shows three synthetic vowel spectra and three polezero configurations for LP whitening filters that may or may not correspond to the displayed vowel spectra. All plots were produced using a sampling rate of 8 Hz.
3 Spectrum Level (db) Frequency/Hz Imaginary Part Real Part (a) (b) Spectrum Level (db) Frequency/Hz Imaginary Part Real Part (c) (d) Spectrum Level (db) Frequency/Hz Imaginary Part Real Part (e) (f) Figure : Synthetic vowel spectra and whitening filter zero plots that may or may not correspond to the vowel spectra, i.e. the mapping is not necessarily onetoone. 2
4 (a) Indicate for each vowel spectrum the most likely corresponding zplane zero plot. (3p) Note: the mapping is not necessarily onetoone, i.e. more than one spectrum may or may not correspond to the same polezero plot! Solution: af, cd, eb. (b) Indicate for each vowel spectrum if the speaker was most probably a man or a woman. (3p) Solution: Spectrum (a) female, (c) male, (e) female. (c) Which of the illustrated vowels has the lowest firstformant frequency? What is, approximately, the first formant frequency in your selected vowel? (2p) Solution: Vowel e has the lowest first formant, at about 3 Hz. 3
5 3 A stationary random signal X(n) has a Gaussian probability density function with zero mean and known variance σ 2. We want to quantize this signal using a rate of bit/sample. (a) We first design an optimal scalar quantizer for this signal, to achieve minimum total distortion power, including both granular and overload distortion. Determine the optimal quantization intervals and reconstruction levels. (3p) Solution: The probability density function for each signal sample is f X (x) = σ (x ) 2 2π e 2σ 2 Because of the symmetry of this distribution around zero, we must design a midrise quantizer with x =, and symmetric reconstruction levels ˆx = a and ˆx 2 = a. To determine the optimal reconstruction levels, we choose a = xf X (x)dx 2 f X (x)dx = 2 xf X (x)dx = σ π (b) Calculate the resulting signaltonoise ratio in db for this scalar quantizer. (3p) Solution: The quantization distortion power is q 2 =2 =2 (x a) 2 f X (x)dx = x 2 f X (x)dx 2a =σ 2 2a 2 + a 2 = σ 2 a 2 = ( =σ 2 2 ) π 2xf X (x)dx + 2a 2 f X (x)dx = using the optimal choice for a. Thus the signaltoquantization noise ratio is ( SNR = lg 2 ) 4.4 (db) π (c) We also consider designing an optimal vector quantizer (VQ) with the same rate of bit per sample, by encoding each pair of consecutive samples using 2 bits. If the input signal is exactly white, does the VQ reduce the quantization noise, compared to the scalar quantizer? Qualitative motivation is required, but no formal calculations. (2p) Solution: With the 2bit VQ we have 4 reconstruction points in the 2dimensional x, x 2 plane, placed symmetrically with one point in each quadrant. If the signal is white, i.e. the signal samples are uncorrelated, then the distortion contribution is exactly the same in both dimensions, and the total distortion is exactly the same as with the scalar quantizer. This is an exception for this special case with only two reconstruction levels in each dimension. Generally, the VQ is better, even for a white signal. 4 Consider a 2dimensional random vector X = (X, X 2 ) T with a distribution as indicated in Fig. 2. 4
6 x 2 a x 2a Figure 2: Probability density for a twodimensional random vector. The probability density is uniform in the shaded rectangle, and zero otherwise. (a) Discuss briefly why direct scalar quantization of the vector elements X and X 2 would not be suitable in this case. (p) Solution: The vector elements X and X 2 are correlated, and therefore a VQ will give much better performance. (b) Sketch an optimally rotated orthonormal coordinate system in which the transformed vectors Y = (Y, Y 2 ) T have a distribution that is maximally suitable for elementwise scalar quantization.(p) Solution: Rotating the coordinate axes 45 degrees in either direction, e.g. as in fig. 3, will make the transformed coordinates (Y, Y 2 ) statistically independent. Then two uniform scalar quantizers will have a good chance to perform well (although a VQ would still be slightly better, because of its more flexible spacefilling properties). (c) Design two scalar quantizers, one for element Y and the other for Y 2, without any overload distortion. You can use a total data rate of R bits per vector, so you are free to allocate w bits for element Y and R w bits for element Y 2. What is the optimal choice of w for a general value of R? Calculate specifically the optimal w for R = 3. (3p) Solution: The total quantization noise variance is σ 2 q = 2 / /2 = (2a) 2 2 2w + a 2 2 2(R w) 5
7 y 2 x 2 a y x 2a Figure 3: Rotated coordinate system corresponding to fig. 2. If w were a continuousvalued variable we would obtain the minimum noise variance when with solution dσ 2 q dw (2a)2 2 2w + a 2 2 2(R w) =; 4 2 2w =2 2(R w) ; 2 2w = 2R + 2w; w =(R + )/2; R w = (R )/2 For simple symmetry reasons, we might have seen directly that we should spend bit more for the dimension with range 2a. The derived result gives exact integervalued solutions for any odd R. For R = 3 we obtain w = 2; R w =. For even values of R we must choose the best of the two closest alternatives: If we choose w = R w = R/2, σ 2 q (2a) 2 2 R + a 2 2 R = 5a 2 2 R If we choose w = (R + 2)/2; R w = (R 2)/2, Thus, both solutions are equally good. σ 2 q (2a) 2 2 R 2 + a 2 2 R+2 = 5a 2 2 R 6
8 (d) How large are the quantization stepsizes and 2 for the two scalar quantizers in the special case of R = 3 bits per vector? (2p) Solution: = 2 = a/2 (e) Instead of the coordinate transformation and scalar quantization, we could have achieved roughly the same performance with a Vector Quantizer. Argue briefly why the transformation and scalar coding may still be preferred (especially at high data rates). (p) Solution: The uniform scalarquantizer is computationally much simpler than the general VQ solution. The VQ encoder must search through the entire VQ codebook for every input vector. 5 You will now design a Wiener filter for speech enhancement. The filter input is a noisy speech signal y(n) = x(n) + v(n) where x(n) represents the clean speech and v(n) is the additive noise. The speech and noise are assumed to be mutually independent and stationary random processes. The noise is white with power spectral density Φ vv (ω) =, and the speech signal has a power spectral density Φ xx (ω) = + cos(ω) (a) Determine the autocorrelation sequence φ xx (k) for the speech signal. (2p) Solution: π jkω dω φ xx (k) = Φ xx (ω)e 2π = = π π π ( + e+jω + e jω jkω dω )e 2 2π =, k = = /2, k = ±, otherwise (b) Design a causal FIR linearphase Wiener filter with transfer function H(z) = w + w z + w z 2 to produce an optimal estimate of the clean speech signal, in the sense that it minimises a distortion measure Q = E [ (x(n ) ˆx(n)) 2] where ˆx(n) is the output signal from the Wiener filter. (6p) (Note that the delay in this distortion measure is relevant, because the linearphase filter has a group delay of sample, and we do not regard this filter delay as distortion.) Solution: We will need the correlation functions φ xy (k) =E [x(n + k)y(n)] = E [x(n + k)(x(n) + v(n)] = φ xx (k); φ yy (k) =E [y(n + k)y(n)] = E [(x(n + k) + v(n + k))(x(n) + v(n)] = φ xx (k) + φ vv (k); φ vv (k) =δ(k) 7
9 The Wiener filter output signal is ˆx(n) =w y(n) + w y(n ) + w 2 y(n 2) = w (y(n) + y(n 2)) + w y(n ) The necessary conditions for minimal distortion are then = E [ (x(n ) ˆx(n)) 2] [ = E 2(x(n ) ˆx(n)) ˆx(n) ] = w w =E [ 2(x(n ) w (y(n) + y(n 2)) w y(n ))(y(n) + y(n 2))] = = 2(φ xy ( ) + φ xy ()) + 4w (φ yy () + φ yy (2)) + 2w (φ yy ( ) + φ yy ()) = = 4φ xx () + 4w (φ yy () + φ yy (2)) + 4w φ yy (); [ =E 2(x(n ) ˆx(n)) ˆx(n) ] = w =E [ 2(x(n ) w (y(n) + y(n 2)) w y(n ))y(n )] = = 2φ xx () + 4w φ yy () + 2w φ yy () Thus, we obtain the optimal filter coefficient from the linear equations ( ) ( ) ( ) φyy () + φ yy (2) φ yy () w φxx () = ; 2φ yy () φ yy () w φ xx () ( ) ( ) ( ) 2 /2 w /2 = ; 2 w with solution ( w w ) ( ) /7 = ; 3/7 8
10 6 The probability density function of a twodimensional random vector X = (X, X 2 ) T is defined by an Mcomponent Gaussian mixture model (GMM), with a known probability density function as illustrated in Fig. 4. p X (x) = M m= w m (2π) 2/2 det C m e 2 (x µm)t C m (x µm) Figure 4: GMM probability density function plotted with darker gray shading indicating greater probability density. The thin black curves connect points where the probability density equals.9, / e.6, and.2 times the maximal probability density. (a) Estimate approximate values for all the GMM parameters, i.e. weights w m, means µ m, and covariance matrices C m, that define the probability density function shown in Fig. 4. Crude approximations are sufficient, if your theoretical reasoning is correct. (4p) Solution: The symmetry shows that there are M = 2 Gaussian components, with equal weights w = w 2 =.5. The two mean points are located approximately at the two maxima of the pdf, 9
11 i.e. at µ = ( ).5 ; µ 2 = ( ).5. The isodensity curves indicate that X and X 2 are uncorrelated, with equal and diagonal covariance matrices C = C 2. Any oftheisodensity curves can be used, but it is slightly easier to use the second curve. For any scalar Gaussian distribution, the probability density is reduced from the maximum value by the factor e (x µ) 2 2σ 2 This factor equals / e when x µ = σ. Therefore, the standard deviations can be easily estimated from the second isodensity curve in the graph. Thus, σ = and σ 2 2 = 2, and the covariance matrices are ( ) ( ) σ 2 C = C 2 = σ2 2 = 4 (b) Determine the overall mean vector µ = E [X] and covariance matrix C = cov [X] for the random vector X. Express the result in terms of the general parameters w m, µ m, C m of the given GMM, so that the result is valid regardless of your numerical estimates of these parameters. (2p) Solution: We can regard X to be generated in a twostep procedure: First, a discrete random variable U is generated, with probability mass P (U = m) = w m ; m {, 2} This variable indicates which of the two Gaussian components is to be used to generate a vector with the conditional probability density of the selected component. Thus, Similarly, µ =E [X] = E [X U = ] P (U = ) + E [X U = 2] P (U = 2) = =µ w + µ 2 w 2 ; C = cov [X] = E [ (X µ)(x µ) T ] = 2 = E [ (X µ)(x µ) T U = m ] P (U = m) = = = = m= 2 E [ (X µ m + µ m µ)(x µ m + µ m µ) T U = m ] P (U = m) = m= 2 E [ (X µ m )(X µ m ) T + (µ m µ)(µ m µ) T U = m ] P (U = m)+ m= 2 + E [ (X µ m )(µ m µ) T + (µ m µ)(x µ m ) T U = m ] P (U = m) = } {{ } m= = 2 C m w m + (µ m µ)(µ m µ) T w m m=
12 (c) Now assume that two random vectors X A and X B are independent of each other and have identical density functions equal to the previously defined GMM density p X (x) in Fig. 4. A new random vector is defined as Y = X A + X B Show that the probability density function for Y, denoted p Y (y), can be expressed exactly by a GMM, and determine all parameters for this GMM, expressed in terms of the given parameters w m, µ m, C m for p X (x). Numerical values are required only for the mixture weights in p Y (y). (4p) Hint, may be used without proof: If any two random vectors both have Gaussian distributions, not necessarily identical, then the sum of the two vectors also has a Gaussian distribution. Furthermore, if the vectors are independent, the mean of the sum is always equal to the sum of the two means, and the covariance of the sum is equal to the sum of the two covariance matrices. Solution: Each of X A and X B may be generated from either component or 2 in the original GMM. We use two discrete random variables U A and U B, each with outcomes or 2, to denote these possibilities. Thus there are three distinct ways that Y can be generated:. Both X A and X B are generated from component, i.e. (U A = U B = ). This happens with probability w Y, =.25, and then µ Y, =E [X A + X B U A = U B = ] = 2µ ; C Y, = cov [X A + X B U A = U B = ] = 2C ; 2. Both X A and X B are generated from component 2, i.e. (U A = 2 U B = 2). This happens with probability w Y,2 =.25, and then µ Y,2 =E [X A + X B U A = U B = 2] = 2µ 2 ; C Y,2 = cov [X A + X B U A = U B = 2] = 2C 2 ; 3. X A and X B are generated from different components, i.e. (U A = U B = 2) (U A = 2 U B = ). This happens with probability w Y,3 =.5, and then µ Y,3 =E [X A + X B U A U B ] = µ + µ 2 ; C Y,3 = cov [X A + X B U A U B ] = C + C 2 ; Thus, the probability density for Y is exactly a 3component GMM, with weight factors w Y, =.25, w Y,2 =.25, and w Y,3 =.5.
TTT4110 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 informationAudio Engineering Society. Convention Paper. Presented at the 129th Convention 2010 November 4 7 San Francisco, CA, USA
Audio Engineering Society Convention Paper Presented at the 129th Convention 2010 November 4 7 San Francisco, CA, USA The papers at this Convention have been selected on the basis of a submitted abstract
More informationLecture 110: Spectrograms
Lecture 110: Spectrograms Overview 1. Spectra of dynamic signals: like many real world signals, speech changes in quality with time. But so far the only spectral analysis we have performed has assumed
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 informationNRZ Bandwidth  HF Cutoff vs. SNR
Application Note: HFAN09.0. Rev.2; 04/08 NRZ Bandwidth  HF Cutoff vs. SNR Functional Diagrams Pin Configurations appear at end of data sheet. Functional Diagrams continued at end of data sheet. UCSP
More informationTTT4120 Digital Signal Processing Suggested Solution to Exam Fall 2008
Norwegian University of Science and Technology Department of Electronics and Telecommunications TTT40 Digital Signal Processing Suggested Solution to Exam Fall 008 Problem (a) The input and the inputoutput
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationComponent Ordering in Independent Component Analysis Based on Data Power
Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals
More informationAutomatic Detection of Emergency Vehicles for Hearing Impaired Drivers
Automatic Detection of Emergency Vehicles for Hearing Impaired Drivers Sungwon ark and Jose Trevino Texas A&M UniversityKingsville, EE/CS Department, MSC 92, Kingsville, TX 78363 TEL (36) 5932638, FAX
More informationAdvanced Signal Processing and Digital Noise Reduction
Advanced Signal Processing and Digital Noise Reduction Saeed V. Vaseghi Queen's University of Belfast UK WILEY HTEUBNER A Partnership between John Wiley & Sons and B. G. Teubner Publishers Chichester New
More informationSome probability and statistics
Appendix A Some probability and statistics A Probabilities, random variables and their distribution We summarize a few of the basic concepts of random variables, usually denoted by capital letters, X,Y,
More informationBroadband Networks. Prof. Dr. Abhay Karandikar. Electrical Engineering Department. Indian Institute of Technology, Bombay. Lecture  29.
Broadband Networks Prof. Dr. Abhay Karandikar Electrical Engineering Department Indian Institute of Technology, Bombay Lecture  29 Voice over IP So, today we will discuss about voice over IP and internet
More informationVoiceis analog in character and moves in the form of waves. 3important wavecharacteristics:
Voice Transmission Basic Concepts Voiceis analog in character and moves in the form of waves. 3important wavecharacteristics: Amplitude Frequency Phase Voice Digitization in the POTS Traditional
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationEnhancing the SNR of the Fiber Optic Rotation Sensor using the LMS Algorithm
1 Enhancing the SNR of the Fiber Optic Rotation Sensor using the LMS Algorithm Hani Mehrpouyan, Student Member, IEEE, Department of Electrical and Computer Engineering Queen s University, Kingston, Ontario,
More informationBlind Deconvolution of Barcodes via Dictionary Analysis and Wiener Filter of Barcode Subsections
Blind Deconvolution of Barcodes via Dictionary Analysis and Wiener Filter of Barcode Subsections Maximilian Hung, Bohyun B. Kim, Xiling Zhang August 17, 2013 Abstract While current systems already provide
More informationPHASE ESTIMATION ALGORITHM FOR FREQUENCY HOPPED BINARY PSK AND DPSK WAVEFORMS WITH SMALL NUMBER OF REFERENCE SYMBOLS
PHASE ESTIMATION ALGORITHM FOR FREQUENCY HOPPED BINARY PSK AND DPSK WAVEFORMS WITH SMALL NUM OF REFERENCE SYMBOLS Benjamin R. Wiederholt The MITRE Corporation Bedford, MA and Mario A. Blanco The MITRE
More informationLezione 6 Communications Blockset
Corso di Tecniche CAD per le Telecomunicazioni A.A. 20072008 Lezione 6 Communications Blockset Ing. Marco GALEAZZI 1 What Is Communications Blockset? Communications Blockset extends Simulink with a comprehensive
More informationEricsson T18s Voice Dialing Simulator
Ericsson T18s Voice Dialing Simulator Mauricio Aracena Kovacevic, Anna Dehlbom, Jakob Ekeberg, Guillaume Gariazzo, Eric Lästh and Vanessa Troncoso Dept. of Signals Sensors and Systems Royal Institute of
More informationHardware Implementation of Probabilistic State Machine for Word Recognition
IJECT Vo l. 4, Is s u e Sp l  5, Ju l y  Se p t 2013 ISSN : 22307109 (Online) ISSN : 22309543 (Print) Hardware Implementation of Probabilistic State Machine for Word Recognition 1 Soorya Asokan, 2
More informationUnderstanding and Applying Kalman Filtering
Understanding and Applying Kalman Filtering Lindsay Kleeman Department of Electrical and Computer Systems Engineering Monash University, Clayton 1 Introduction Objectives: 1. Provide a basic understanding
More informationLinear Codes. Chapter 3. 3.1 Basics
Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length
More informationAdaptive Equalization of binary encoded signals Using LMS Algorithm
SSRG International Journal of Electronics and Communication Engineering (SSRGIJECE) volume issue7 Sep Adaptive Equalization of binary encoded signals Using LMS Algorithm Dr.K.Nagi Reddy Professor of ECE,NBKR
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 informationElectronic Communications Committee (ECC) within the European Conference of Postal and Telecommunications Administrations (CEPT)
Page 1 Electronic Communications Committee (ECC) within the European Conference of Postal and Telecommunications Administrations (CEPT) ECC RECOMMENDATION (06)01 Bandwidth measurements using FFT techniques
More informationCoding and decoding with convolutional codes. The Viterbi Algor
Coding and decoding with convolutional codes. The Viterbi Algorithm. 8 Block codes: main ideas Principles st point of view: infinite length block code nd point of view: convolutions Some examples Repetition
More informationPCM Encoding and Decoding:
PCM Encoding and Decoding: Aim: Introduction to PCM encoding and decoding. Introduction: PCM Encoding: The input to the PCM ENCODER module is an analog message. This must be constrained to a defined bandwidth
More informationAutocovariance and Autocorrelation
Chapter 3 Autocovariance and Autocorrelation If the {X n } process is weakly stationary, the covariance of X n and X n+k depends only on the lag k. This leads to the following definition of the autocovariance
More informationPrinciples of Digital Communication
Principles of Digital Communication Robert G. Gallager January 5, 2008 ii Preface: introduction and objectives The digital communication industry is an enormous and rapidly growing industry, roughly comparable
More informationTCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS
TCOM 370 NOTES 994 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 informationSchool Class Monitoring System Based on Audio Signal Processing
C. R. Rashmi 1,,C.P.Shantala 2 andt.r.yashavanth 3 1 Department of CSE, PG Student, CIT, Gubbi, Tumkur, Karnataka, India. 2 Department of CSE, Vice Principal & HOD, CIT, Gubbi, Tumkur, Karnataka, India.
More informationAnswer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade
Statistics Quiz Correlation and Regression  ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements
More informationLecture 16: Noise and Filters
Lecture 16: Noise and Filters Overview 1. Periodic and Aperiodic Signals Review: by periodic signals, we mean signals that have a waveform shape that repeats. The time taken for the waveform to repeat
More informationExample/ an analog signal f ( t) ) is sample by f s = 5000 Hz draw the sampling signal spectrum. Calculate min. sampling frequency.
1 2 3 4 Example/ an analog signal f ( t) = 1+ cos(4000πt ) is sample by f s = 5000 Hz draw the sampling signal spectrum. Calculate min. sampling frequency. Sol/ H(f) 7KHz 5KHz 3KHz 2KHz 0 2KHz 3KHz
More informationBy choosing to view this document, you agree to all provisions of the copyright laws protecting it.
This material is posted here with permission of the IEEE Such permission of the IEEE does not in any way imply IEEE endorsement of any of Helsinki University of Technology's products or services Internal
More informationANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 21
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 informationCommunication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 2, FEBRUARY 2002 359 Communication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel Lizhong Zheng, Student
More informationSignal Detection C H A P T E R 14 14.1 SIGNAL DETECTION AS HYPOTHESIS TESTING
C H A P T E R 4 Signal Detection 4. SIGNAL DETECTION AS HYPOTHESIS TESTING In Chapter 3 we considered hypothesis testing in the context of random variables. The detector resulting in the minimum probability
More informationAvailable from Deakin Research Online:
This is the authors final peered reviewed (post print) version of the item published as: Adibi,S 2014, A low overhead scaled equalized harmonicbased voice authentication system, Telematics and informatics,
More informationIN current film media, the increase in areal density has
IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 1, JANUARY 2008 193 A New Read Channel Model for Patterned Media Storage Seyhan Karakulak, Paul H. Siegel, Fellow, IEEE, Jack K. Wolf, Life Fellow, IEEE, and
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationExample: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.
Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation:  Feature vector X,  qualitative response Y, taking values in C
More informationBackground 2. Lecture 2 1. The Least Mean Square (LMS) algorithm 4. The Least Mean Square (LMS) algorithm 3. br(n) = u(n)u H (n) bp(n) = u(n)d (n)
Lecture 2 1 During this lecture you will learn about The Least Mean Squares algorithm (LMS) Convergence analysis of the LMS Equalizer (Kanalutjämnare) Background 2 The method of the Steepest descent that
More informationModule 3: Correlation and Covariance
Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis
More informationApplications to Data Smoothing and Image Processing I
Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is
More informationSTUDY OF MUTUAL INFORMATION IN PERCEPTUAL CODING WITH APPLICATION FOR LOW BITRATE COMPRESSION
STUDY OF MUTUAL INFORMATION IN PERCEPTUAL CODING WITH APPLICATION FOR LOW BITRATE COMPRESSION Adiel BenShalom, Michael Werman School of Computer Science Hebrew University Jerusalem, Israel. {chopin,werman}@cs.huji.ac.il
More informationRANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA
RANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA ABSTRACT Random vibration is becoming increasingly recognized as the most realistic method of simulating the dynamic environment of military
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationMIMO CHANNEL CAPACITY
MIMO CHANNEL CAPACITY Ochi Laboratory Nguyen Dang Khoa (D1) 1 Contents Introduction Review of information theory Fixed MIMO channel Fading MIMO channel Summary and Conclusions 2 1. Introduction The use
More informationDepartment of Electrical and Computer Engineering BenGurion University of the Negev. LAB 1  Introduction to USRP
Department of Electrical and Computer Engineering BenGurion University of the Negev LAB 1  Introduction to USRP  11 Introduction In this lab you will use software reconfigurable RF hardware from National
More informationVISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
More informationMiniproject in TSRT04: Cell Phone Coverage
Miniproject in TSRT04: Cell hone Coverage 19 August 2015 1 roblem Formulation According to the study Swedes and Internet 2013 (Stiftelsen för Internetinfrastruktur), 99% of all Swedes in the age 1245
More information(2) (3) (4) (5) 3 J. M. Whittaker, Interpolatory Function Theory, Cambridge Tracts
Communication in the Presence of Noise CLAUDE E. SHANNON, MEMBER, IRE Classic Paper A method is developed for representing any communication system geometrically. Messages and the corresponding signals
More informationTutorial about the VQR (Voice Quality Restoration) technology
Tutorial about the VQR (Voice Quality Restoration) technology Ing Oscar Bonello, Solidyne Fellow Audio Engineering Society, USA INTRODUCTION Telephone communications are the most widespread form of transport
More informationRECOMMENDATION ITUR BO.786 *
Rec. ITUR BO.786 RECOMMENDATION ITUR BO.786 * MUSE ** system for HDTV broadcastingsatellite services (Question ITUR /) (992) The ITU Radiocommunication Assembly, considering a) that the MUSE system
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationLinear Predictive Coding
Linear Predictive Coding Jeremy Bradbury December 5, 2000 0 Outline I. Proposal II. Introduction A. Speech Coding B. Voice Coders C. LPC Overview III. Historical Perspective of Linear Predictive Coding
More informationSpeech Compression. 2.1 Introduction
Speech Compression 2 This chapter presents an introduction to speech compression techniques, together with a detailed description of speech/audio compression standards including narrowband, wideband and
More informationManual Analysis Software AFD 1201
AFD 1200  AcoustiTube Manual Analysis Software AFD 1201 Measurement of Transmission loss acc. to Song and Bolton 1 Table of Contents Introduction  Analysis Software AFD 1201... 3 AFD 1200  AcoustiTube
More informationCCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York
BME I5100: Biomedical Signal Processing Linear Discrimination Lucas C. Parra Biomedical Engineering Department CCNY 1 Schedule Week 1: Introduction Linear, stationary, normal  the stuff biology is not
More informationFEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL
FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint
More informationPERCENTAGE ARTICULATION LOSS OF CONSONANTS IN THE ELEMENTARY SCHOOL CLASSROOMS
The 21 st International Congress on Sound and Vibration 1317 July, 2014, Beijing/China PERCENTAGE ARTICULATION LOSS OF CONSONANTS IN THE ELEMENTARY SCHOOL CLASSROOMS Dan Wang, Nanjie Yan and Jianxin Peng*
More information521493S Computer Graphics. Exercise 2 & course schedule change
521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 1618 in TS128 Question 2.1 Given two nonparallel,
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 informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationLogLikelihood Ratiobased Relay Selection Algorithm in Wireless Network
Recent Advances in Electrical Engineering and Electronic Devices LogLikelihood Ratiobased Relay Selection Algorithm in Wireless Network Ahmed ElMahdy and Ahmed Walid Faculty of Information Engineering
More informationMODULATION Systems (part 1)
Technologies and Services on Digital Broadcasting (8) MODULATION Systems (part ) "Technologies and Services of Digital Broadcasting" (in Japanese, ISBN4339622) is published by CORONA publishing co.,
More informationSPEAKER IDENTIFICATION FROM YOUTUBE OBTAINED DATA
SPEAKER IDENTIFICATION FROM YOUTUBE OBTAINED DATA Nitesh Kumar Chaudhary 1 and Shraddha Srivastav 2 1 Department of Electronics & Communication Engineering, LNMIIT, Jaipur, India 2 Bharti School Of Telecommunication,
More informationExperiment 7: Familiarization with the Network Analyzer
Experiment 7: Familiarization with the Network Analyzer Measurements to characterize networks at high frequencies (RF and microwave frequencies) are usually done in terms of scattering parameters (S parameters).
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 03 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
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 information1 Example of Time Series Analysis by SSA 1
1 Example of Time Series Analysis by SSA 1 Let us illustrate the 'Caterpillar'SSA technique [1] by the example of time series analysis. Consider the time series FORT (monthly volumes of fortied wine sales
More informationTaking the Mystery out of the Infamous Formula, "SNR = 6.02N + 1.76dB," and Why You Should Care. by Walt Kester
ITRODUCTIO Taking the Mystery out of the Infamous Formula, "SR = 6.0 + 1.76dB," and Why You Should Care by Walt Kester MT001 TUTORIAL You don't have to deal with ADCs or DACs for long before running across
More informationPrinciple Component Analysis and Partial Least Squares: Two Dimension Reduction Techniques for Regression
Principle Component Analysis and Partial Least Squares: Two Dimension Reduction Techniques for Regression Saikat Maitra and Jun Yan Abstract: Dimension reduction is one of the major tasks for multivariate
More informationCHAPTER 3: DIGITAL IMAGING IN DIAGNOSTIC RADIOLOGY. 3.1 Basic Concepts of Digital Imaging
Physics of Medical XRay Imaging (1) Chapter 3 CHAPTER 3: DIGITAL IMAGING IN DIAGNOSTIC RADIOLOGY 3.1 Basic Concepts of Digital Imaging Unlike conventional radiography that generates images on film through
More informationmin ǫ = E{e 2 [n]}. (11.2)
C H A P T E R 11 Wiener Filtering INTRODUCTION In this chapter we will consider the use of LTI systems in order to perform minimum meansquareerror (MMSE) estimation of a WSS random process of interest,
More informationAN1200.04. Application Note: FCC Regulations for ISM Band Devices: 902928 MHz. FCC Regulations for ISM Band Devices: 902928 MHz
AN1200.04 Application Note: FCC Regulations for ISM Band Devices: Copyright Semtech 2006 1 of 15 www.semtech.com 1 Table of Contents 1 Table of Contents...2 1.1 Index of Figures...2 1.2 Index of Tables...2
More information4F7 Adaptive Filters (and Spectrum Estimation) Least Mean Square (LMS) Algorithm Sumeetpal Singh Engineering Department Email : sss40@eng.cam.ac.
4F7 Adaptive Filters (and Spectrum Estimation) Least Mean Square (LMS) Algorithm Sumeetpal Singh Engineering Department Email : sss40@eng.cam.ac.uk 1 1 Outline The LMS algorithm Overview of LMS issues
More information1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number
1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x  x) B. x 3 x C. 3x  x D. x  3x 2) Write the following as an algebraic expression
More informationCluster Algorithms. Adriano Cruz adriano@nce.ufrj.br. 28 de outubro de 2013
Cluster Algorithms Adriano Cruz adriano@nce.ufrj.br 28 de outubro de 2013 Adriano Cruz adriano@nce.ufrj.br () Cluster Algorithms 28 de outubro de 2013 1 / 80 Summary 1 KMeans Adriano Cruz adriano@nce.ufrj.br
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationTechnical Bulletin on the Evaluation of the Kinetics Tuned Absorber/Diffuser Panel by Peter D'Antonio RPG Diffusor Systems, Inc Upper Marlboro, MD
Technical Bulletin on the Evaluation of the Kinetics Tuned Absorber/Diffuser Panel by Peter D'Antonio RPG Diffusor Systems, Inc Upper Marlboro, MD August 25 TABLE OF CONTENTS INTRODUCTION 1 ONEDIMENSIONAL
More informationBasics of FloatingPoint Quantization
Chapter 2 Basics of FloatingPoint Quantization Representation of physical quantities in terms of floatingpoint numbers allows one to cover a very wide dynamic range with a relatively small number of
More informationImplementation of Digital Signal Processing: Some Background on GFSK Modulation
Implementation of Digital Signal Processing: Some Background on GFSK Modulation Sabih H. Gerez University of Twente, Department of Electrical Engineering s.h.gerez@utwente.nl Version 4 (February 7, 2013)
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationCapacity Limits of MIMO Channels
Tutorial and 4G Systems Capacity Limits of MIMO Channels Markku Juntti Contents 1. Introduction. Review of information theory 3. Fixed MIMO channels 4. Fading MIMO channels 5. Summary and Conclusions References
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More information1 Multichannel frequency division multiplex frequency modulation (FDMFM) emissions
Rec. ITUR SM.8531 1 RECOMMENDATION ITUR SM.8531 NECESSARY BANDWIDTH (Question ITUR 77/1) Rec. ITUR SM.8531 (19921997) The ITU Radiocommunication Assembly, considering a) that the concept of necessary
More informationNonData Aided Carrier Offset Compensation for SDR Implementation
NonData Aided Carrier Offset Compensation for SDR Implementation Anders Riis Jensen 1, Niels Terp Kjeldgaard Jørgensen 1 Kim Laugesen 1, Yannick Le Moullec 1,2 1 Department of Electronic Systems, 2 Center
More informationLuigi Piroddi Active Noise Control course notes (January 2015)
Active Noise Control course notes (January 2015) 9. Online secondary path modeling techniques Luigi Piroddi piroddi@elet.polimi.it Introduction In the feedforward ANC scheme the primary noise is canceled
More informationDeveloping an Isolated Word Recognition System in MATLAB
MATLAB Digest Developing an Isolated Word Recognition System in MATLAB By Daryl Ning Speechrecognition technology is embedded in voiceactivated routing systems at customer call centres, voice dialling
More informationJPEG compression of monochrome 2Dbarcode images using DCT coefficient distributions
Edith Cowan University Research Online ECU Publications Pre. JPEG compression of monochrome Dbarcode images using DCT coefficient distributions Keng Teong Tan Hong Kong Baptist University Douglas Chai
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
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 informationThe Effect of Network Cabling on Bit Error Rate Performance. By Paul Kish NORDX/CDT
The Effect of Network Cabling on Bit Error Rate Performance By Paul Kish NORDX/CDT Table of Contents Introduction... 2 Probability of Causing Errors... 3 Noise Sources Contributing to Errors... 4 Bit Error
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationQuarterly Progress and Status Report. Measuring inharmonicity through pitch extraction
Dept. for Speech, Music and Hearing Quarterly Progress and Status Report Measuring inharmonicity through pitch extraction Galembo, A. and Askenfelt, A. journal: STLQPSR volume: 35 number: 1 year: 1994
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationHow to Win the Stock Market Game
How to Win the Stock Market Game 1 Developing ShortTerm Stock Trading Strategies by Vladimir Daragan PART 1 Table of Contents 1. Introduction 2. Comparison of trading strategies 3. Return per trade 4.
More informationA Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution
A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September
More information