Sinusoidal Modeling. Yannis Stylianou SPCC University of Crete, Computer Science Dept., Greece,

Size: px
Start display at page:

Download "Sinusoidal Modeling. Yannis Stylianou SPCC University of Crete, Computer Science Dept., Greece,"

Transcription

1 Sinusoidal Modeling Yannis Stylianou University of Crete, Computer Science Dept., Greece, SPCC 2016

2 1 Speech Production 2 Modulators 3 Sinusoidal Modeling Sinusoidal Models Voiced Speech Unvoiced Speech Synthesis Amplitude and Phase Interpolation 4 Harmonic Modeling HNM Speech Models 5 References yannis@csd.uoc.gr Sinusoidal Modeling 2/44

3 A Simple view Used after permission: [1] Sinusoidal Modeling 3/44

4 Downward-looking into the larynx: Vocal folds Left: Voicing, Right: Breathing Used after permission: [1] Sinusoidal Modeling 4/44

5 Glottal airflow velocity Used after permission: [1] Sinusoidal Modeling 5/44

6 Vocat tract shapes Used after permission: [1] Sinusoidal Modeling 6/44

7 Example Let the excitation of vocal tract, h[n], be: u[n] = g[n] p[n] then, the output speech, x[n, τ], is given by: x[n, τ] = w[n, τ]{h[n] (g[n] p[n])} and X (ω, τ) = 1 P H(ω k )G(ω k )W (ω ω k, τ) k= yannis@csd.uoc.gr Sinusoidal Modeling 7/44

8 Harmonics and Formants Used after permission: [1] Sinusoidal Modeling 8/44

9 Sinusoidal Models Sinusoidal Model: K x(t) = γ k e j2πf kt k= K Special case, Harmonic Model: K x(t) = γ k e j2πkf 0t k= K Estimation of parameters (Linear approach): γ = F fk s Model Evaluation, Mean-Squared Error (MSE): ɛ = s(t) x(t) 2 dt w yannis@csd.uoc.gr Sinusoidal Modeling 9/44

10 Sinusoidal Models Sinusoidal Model: K x(t) = γ k e j2πf kt k= K Special case, Harmonic Model: K x(t) = γ k e j2πkf 0t k= K Estimation of parameters (Linear approach): γ = F fk s Model Evaluation, Mean-Squared Error (MSE): ɛ = s(t) x(t) 2 dt w yannis@csd.uoc.gr Sinusoidal Modeling 9/44

11 Sinusoidal Models Sinusoidal Model: K x(t) = γ k e j2πf kt k= K Special case, Harmonic Model: K x(t) = γ k e j2πkf 0t k= K Estimation of parameters (Linear approach): γ = F fk s Model Evaluation, Mean-Squared Error (MSE): ɛ = s(t) x(t) 2 dt w yannis@csd.uoc.gr Sinusoidal Modeling 9/44

12 Sinusoidal Models Sinusoidal Model: K x(t) = γ k e j2πf kt k= K Special case, Harmonic Model: K x(t) = γ k e j2πkf 0t k= K Estimation of parameters (Linear approach): γ = F fk s Model Evaluation, Mean-Squared Error (MSE): ɛ = s(t) x(t) 2 dt w yannis@csd.uoc.gr Sinusoidal Modeling 9/44

13 Source-Filter with Sinusoids Source: where: u(t) = φ k (t) = K(t) k= K(t) t 0 α k (t)e jφ k(t) Ω k (σ)dσ + φ k Filter: h(t, τ) with Fourier Transform (FT): H(t, Ω) = M(t, Ω)e jφ(t,ω) yannis@csd.uoc.gr Sinusoidal Modeling 10/44

14 Source-Filter with Sinusoids Source: where: u(t) = φ k (t) = K(t) k= K(t) t 0 α k (t)e jφ k(t) Ω k (σ)dσ + φ k Filter: h(t, τ) with Fourier Transform (FT): H(t, Ω) = M(t, Ω)e jφ(t,ω) yannis@csd.uoc.gr Sinusoidal Modeling 10/44

15 Source-Filter with Sinusoids where: s(t) = K(t) k=1 K(t) A k (t)e jθ k(t) A k (t) = α k (t)m [t, Ω k (t)] θ k (t) = φ k (t) + Φ [t, Ω k (t)] = t 0 Ω k (σ)dσ + Φ [t, Ω k (t)] + φ k yannis@csd.uoc.gr Sinusoidal Modeling 11/44

16 Sinusoidal Analysis: Stationarity Assumption We assume stationarity inside the analysis window: which leads to: and to: s(t) = In discrete time: s[n] = K k= K K k= K A k (t) = A k Ω k (t) = Ω k K(t) = K θ k (t) = Ω k (t t l ) + θ k A k e jθ k e jω k(t t l ) γ k e jω kn N w 1 2 t l T 2 t t l + T 2 n N w 1 2 yannis@csd.uoc.gr Sinusoidal Modeling 12/44

17 Sinusoidal Analysis: Stationarity Assumption We assume stationarity inside the analysis window: which leads to: and to: s(t) = In discrete time: s[n] = K k= K K k= K A k (t) = A k Ω k (t) = Ω k K(t) = K θ k (t) = Ω k (t t l ) + θ k A k e jθ k e jω k(t t l ) γ k e jω kn N w 1 2 t l T 2 t t l + T 2 n N w 1 2 yannis@csd.uoc.gr Sinusoidal Modeling 12/44

18 Sinusoidal Analysis: Stationarity Assumption We assume stationarity inside the analysis window: which leads to: and to: s(t) = In discrete time: s[n] = K k= K K k= K A k (t) = A k Ω k (t) = Ω k K(t) = K θ k (t) = Ω k (t t l ) + θ k A k e jθ k e jω k(t t l ) γ k e jω kn N w 1 2 t l T 2 t t l + T 2 n N w 1 2 yannis@csd.uoc.gr Sinusoidal Modeling 12/44

19 Sinusoidal Analysis: Stationarity Assumption We assume stationarity inside the analysis window: which leads to: and to: s(t) = In discrete time: s[n] = K k= K K k= K A k (t) = A k Ω k (t) = Ω k K(t) = K θ k (t) = Ω k (t t l ) + θ k A k e jθ k e jω k(t t l ) γ k e jω kn N w 1 2 t l T 2 t t l + T 2 n N w 1 2 yannis@csd.uoc.gr Sinusoidal Modeling 12/44

20 Mean-Squared Error Given the original measured waveform, y[n] and the synthetic speech waveform, s[n], estimate the unknown parameters A l k, ωl k, and θk l by minimizing the MSE criterion: n= (N w 1)/2 ɛ l = n=(n w 1)/2 n= (N w 1)/2 y[n] s[n] 2 which can be written as: n=(n w 1)/2 K l ( Y ) ɛ l = y[n] 2 + N w (ω l k ) γl k 2 Y (ωk l ) 2 which can be reduced further to: ɛ l = n=(n w 1)/2 n= (N w 1)/2 k=1 y[n] 2 N w Y (ωk l ) 2 K l k=1 yannis@csd.uoc.gr Sinusoidal Modeling 13/44

21 Mean-Squared Error Given the original measured waveform, y[n] and the synthetic speech waveform, s[n], estimate the unknown parameters A l k, ωl k, and θk l by minimizing the MSE criterion: n= (N w 1)/2 ɛ l = n=(n w 1)/2 n= (N w 1)/2 y[n] s[n] 2 which can be written as: n=(n w 1)/2 K l ( Y ) ɛ l = y[n] 2 + N w (ω l k ) γl k 2 Y (ωk l ) 2 which can be reduced further to: ɛ l = n=(n w 1)/2 n= (N w 1)/2 k=1 y[n] 2 N w Y (ωk l ) 2 K l k=1 yannis@csd.uoc.gr Sinusoidal Modeling 13/44

22 Mean-Squared Error Given the original measured waveform, y[n] and the synthetic speech waveform, s[n], estimate the unknown parameters A l k, ωl k, and θk l by minimizing the MSE criterion: n= (N w 1)/2 ɛ l = n=(n w 1)/2 n= (N w 1)/2 y[n] s[n] 2 which can be written as: n=(n w 1)/2 K l ( Y ) ɛ l = y[n] 2 + N w (ω l k ) γl k 2 Y (ωk l ) 2 which can be reduced further to: ɛ l = n=(n w 1)/2 n= (N w 1)/2 k=1 y[n] 2 N w Y (ωk l ) 2 K l k=1 yannis@csd.uoc.gr Sinusoidal Modeling 13/44

23 Karhunen-Loève Expansion Karhunen-Loève expansion allows constructing a random process from harmonic sinusoids with uncorrelated complex amplitudes. Estimated power spectrum should not vary too much over consecutive frequencies. Following the above necessary constraints, for unvoiced speech, and for a window width to be at least 20ms, an 100 Hz harmonic structure provides good results. yannis@csd.uoc.gr Sinusoidal Modeling 14/44

24 Karhunen-Loève Expansion Karhunen-Loève expansion allows constructing a random process from harmonic sinusoids with uncorrelated complex amplitudes. Estimated power spectrum should not vary too much over consecutive frequencies. Following the above necessary constraints, for unvoiced speech, and for a window width to be at least 20ms, an 100 Hz harmonic structure provides good results. yannis@csd.uoc.gr Sinusoidal Modeling 14/44

25 Karhunen-Loève Expansion Karhunen-Loève expansion allows constructing a random process from harmonic sinusoids with uncorrelated complex amplitudes. Estimated power spectrum should not vary too much over consecutive frequencies. Following the above necessary constraints, for unvoiced speech, and for a window width to be at least 20ms, an 100 Hz harmonic structure provides good results. yannis@csd.uoc.gr Sinusoidal Modeling 14/44

26 Karhunen-Loève Expansion Karhunen-Loève expansion allows constructing a random process from harmonic sinusoids with uncorrelated complex amplitudes. Estimated power spectrum should not vary too much over consecutive frequencies. Following the above necessary constraints, for unvoiced speech, and for a window width to be at least 20ms, an 100 Hz harmonic structure provides good results. yannis@csd.uoc.gr Sinusoidal Modeling 14/44

27 Sinusoidal Analysis: Peak picking Used after permission: [1] Sinusoidal Modeling 15/44

28 Sinusoidal Synthesis: OLA, a simple solution yannis@csd.uoc.gr Sinusoidal Modeling 16/44

29 Problem of frequency matching Sinusoidal Modeling 17/44

30 Frame-to-Frame Peak Matching Sinusoidal Modeling 18/44

31 The birth/death process Sinusoidal Modeling 19/44

32 A birth/death process in speech Sinusoidal Modeling 20/44

33 Amplitude and Phase Interpolation Linear amplitude interpolation model: ( ( n ) A l k [n] = Al k + A l+1 k Ak) l n = 0, 1, 2,, L 1 L Cubic Phase interpolation model θ(t) = ζ + γt + αt 2 + βt 3 yannis@csd.uoc.gr Sinusoidal Modeling 21/44

34 Block diagram of the Synthesis System Sinusoidal Modeling 22/44

35 Synthesis: Reconstruction Example Used after permission: [1] Sinusoidal Modeling 23/44

36 Harmonic model Sinusoidal Model: K x(t) = γ k e j2πf kt k= K Special case, Harmonic Model: K x(t) = γ k e j2πkf 0t k= K Estimation of parameters (Linear approach): γ = F fk s Model Evaluation, Mean-Squared Error (MSE): ɛ = s(t) x(t) 2 dt w yannis@csd.uoc.gr Sinusoidal Modeling 24/44

37 Brief overview of HNM HNM is a pitch-synchronous harmonic plus noise representation of the speech signal. Speech spectrum is divided into a low and a high band delimited by the so-called maximum voiced frequency. The low band of the spectrum (below the maximum voiced frequency) is represented solely by harmonically related sine waves. The upper band is modeled as a noise component modulated by a time-domain amplitude envelope. HNM allows high-quality copy synthesis and prosodic modifications. yannis@csd.uoc.gr Sinusoidal Modeling 25/44

38 HNM in equations Harmonic part: h(t) = L(t) k= L(t) A k (t)e j kω 0(t) t Noise part: Speech: n(t) = e(t) [v(τ, t) b(t)] s(t) = h(t) + n(t) yannis@csd.uoc.gr Sinusoidal Modeling 26/44

39 Periodic part HNM 1 : Sum of exponential functions without slope h 1 [n] = L(n i a) k= L(n i a) a k (n i a)e j2πkf 0(n i a)(n n i a) HNM 2 : Sum of exponential function with complex slope L(n a i ) h 2 [n] = R A k (n) exp j2πkf 0(na i )(n ni a ) where k=1 A k (n) = a k (n i a) + (n n i a)b k (n i a) with a k (n i a), b k (n i a) to be complex numbers (amplitude and slope respectively). R denotes taking the real part. yannis@csd.uoc.gr Sinusoidal Modeling 27/44

40 Periodic part HNM 1 : Sum of exponential functions without slope h 1 [n] = L(n i a) k= L(n i a) a k (n i a)e j2πkf 0(n i a)(n n i a) HNM 2 : Sum of exponential function with complex slope L(n a i ) h 2 [n] = R A k (n) exp j2πkf 0(na i )(n ni a ) where k=1 A k (n) = a k (n i a) + (n n i a)b k (n i a) with a k (n i a), b k (n i a) to be complex numbers (amplitude and slope respectively). R denotes taking the real part. yannis@csd.uoc.gr Sinusoidal Modeling 27/44

41 Periodic part continuing HNM 3 : Sum of sinusoids with time-varying real amplitudes where h 3 [n] = L(n i a) k=0 a k (n)cos(ϕ k (n)) a k (n) = c k0 + c k1 (n na) i c kp (n na) i p(n) ϕ k (n) = ɛ k + 2π kζ (n na) i where p(n) is the order of the amplitude polynomial, which is, in general, a time-varying parameter. yannis@csd.uoc.gr Sinusoidal Modeling 28/44

42 Residual (Noise) part The non-periodic part is just the residual signal obtained by subtracting the periodic-part (harmonic part) from the original speech signal in the time-domain r[n] = s[n] h[n] where h[n] is either h 1 [n], h 2 [n], or h 3 [n]. yannis@csd.uoc.gr Sinusoidal Modeling 29/44

43 Amplitudes and phases estimation Having f 0 estimated for voiced frames, amplitudes and phases are estimated by minimizing the criterion: ɛ = n i a+n n=n i a N w 2 [n](s[n] ĥ[n]) 2 where na i = na i 1 + P(na i 1 ), and P(na i 1 ) denotes the pitch period at na i 1. for HNM 1 and HNM 2, this criterion has a quadratic form and is solved by inverting an over-determined system of linear equations. For HNM 3, however,a non-linear system of equations has to be solved. The residual signal r[n] is estimated by ˆr[n] = s[n] ĥ[n] yannis@csd.uoc.gr Sinusoidal Modeling 30/44

44 Amplitudes and phases estimation Having f 0 estimated for voiced frames, amplitudes and phases are estimated by minimizing the criterion: ɛ = n i a+n n=n i a N w 2 [n](s[n] ĥ[n]) 2 where na i = na i 1 + P(na i 1 ), and P(na i 1 ) denotes the pitch period at na i 1. for HNM 1 and HNM 2, this criterion has a quadratic form and is solved by inverting an over-determined system of linear equations. For HNM 3, however,a non-linear system of equations has to be solved. The residual signal r[n] is estimated by ˆr[n] = s[n] ĥ[n] yannis@csd.uoc.gr Sinusoidal Modeling 30/44

45 Time domain characteristics of ˆr[n] 1.5 x Original speech signal (a) Number of samples Error signal using the HNM (c) Number of samples Error signal using HNM (a) Number of samples Error signal using HNM (d) Number of samples yannis@csd.uoc.gr Sinusoidal Modeling 31/44

46 Variance of the residual signal The variance of the residual signal is given as: E(rr h ) = I WP(P h W h WP) 1 P h W h 1 Variance using the three harmonic models Variance Time in samples yannis@csd.uoc.gr Sinusoidal Modeling 32/44

47 Modeling error 3 x 104 (a) Original signal Error in db Time in sec Time in sec yannis@csd.uoc.gr Sinusoidal Modeling 33/44

48 Modeling the residual signal Full bandwidth representation using a low-order (10th) AR filter Time-domain characteristics of the residual signal can be modeled using various functions yannis@csd.uoc.gr Sinusoidal Modeling 34/44

49 Time domain energy modulation Time in samples Time in samples yannis@csd.uoc.gr Sinusoidal Modeling 35/44

50 Triangular envelope Sinusoidal Modeling 36/44

51 Signal envelope There are many ways to obtain the envelope of a signal, as: Hilbert Transform (analytic signal) Low-pass local energy (energy envelope): e[n] = 1 2N + 1 N k= N where r[n] denotes the residual signal. r[n k] yannis@csd.uoc.gr Sinusoidal Modeling 37/44

52 Signal envelope There are many ways to obtain the envelope of a signal, as: Hilbert Transform (analytic signal) Low-pass local energy (energy envelope): e[n] = 1 2N + 1 N k= N where r[n] denotes the residual signal. r[n k] yannis@csd.uoc.gr Sinusoidal Modeling 37/44

53 Example of energy envelope Example of Energy Envelope, with N = Time in samples The energy envelope can be efficiently parameterized with a few Fourier coefficients: ê[n] = where L e is set to be 3 to 4 L e k= L e A k e j2πk(f0/fs)n yannis@csd.uoc.gr Sinusoidal Modeling 38/44

54 Comparing time-domain modulations Noise Part Triangular Envelope Hilbert Envelope Energy Envelope yannis@csd.uoc.gr Sinusoidal Modeling 39/44

55 Sound Examples using HNM 1 Examples with HNM 1 and Maximum Voiced frequency fixed at 4000 Hz Male Male Female Female Original Triangular Hilbert Energy yannis@csd.uoc.gr Sinusoidal Modeling 40/44

56 Sound Examples using HNM 2 Examples with HNM 2 and Maximum Voiced frequency fixed at 4000 Hz Male Male Female Female Original Triangular Hilbert Energy yannis@csd.uoc.gr Sinusoidal Modeling 41/44

57 Sensitivity on f Maginute (db) (a) Frequency in Hz 100 Maginute (db) (b) Frequency in Hz yannis@csd.uoc.gr Sinusoidal Modeling 42/44

58 T. F. Quatieri, Discrete-Time Speech Signal Processing. Engewood Cliffs, NJ: Prentice Hall, Sinusoidal Modeling 43/44

59 THANK YOU for your attention on this part Sinusoidal Modeling 43/44

Timbral, Scale, Pitch modifications

Timbral, Scale, Pitch modifications Introduction Timbral, Scale, Pitch modifications M2 Mathématiques / Vision / Apprentissage Audio signal analysis, indexing and transformation Page 1 / 40 Page 2 / 40 Modification of playback speed Modifications

More information

Feature extraction 2

Feature extraction 2 Centre for Vision Speech & Signal Processing University of Surrey, Guildford GU2 7XH. Feature extraction 2 Dr Philip Jackson Linear prediction Perceptual linear prediction Comparison of feature methods

More information

CS578- Speech Signal Processing

CS578- Speech Signal Processing CS578- Speech Signal Processing Lecture 7: Speech Coding Yannis Stylianou University of Crete, Computer Science Dept., Multimedia Informatics Lab yannis@csd.uoc.gr Univ. of Crete Outline 1 Introduction

More information

SPEECH ANALYSIS AND SYNTHESIS

SPEECH ANALYSIS AND SYNTHESIS 16 Chapter 2 SPEECH ANALYSIS AND SYNTHESIS 2.1 INTRODUCTION: Speech signal analysis is used to characterize the spectral information of an input speech signal. Speech signal analysis [52-53] techniques

More information

representation of speech

representation of speech Digital Speech Processing Lectures 7-8 Time Domain Methods in Speech Processing 1 General Synthesis Model voiced sound amplitude Log Areas, Reflection Coefficients, Formants, Vocal Tract Polynomial, l

More information

GATE EE Topic wise Questions SIGNALS & SYSTEMS

GATE EE Topic wise Questions SIGNALS & SYSTEMS www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)

More information

Signal representations: Cepstrum

Signal representations: Cepstrum Signal representations: Cepstrum Source-filter separation for sound production For speech, source corresponds to excitation by a pulse train for voiced phonemes and to turbulence (noise) for unvoiced phonemes,

More information

Voiced Speech. Unvoiced Speech

Voiced Speech. Unvoiced Speech Digital Speech Processing Lecture 2 Homomorphic Speech Processing General Discrete-Time Model of Speech Production p [ n] = p[ n] h [ n] Voiced Speech L h [ n] = A g[ n] v[ n] r[ n] V V V p [ n ] = u [

More information

Lab 9a. Linear Predictive Coding for Speech Processing

Lab 9a. Linear Predictive Coding for Speech Processing EE275Lab October 27, 2007 Lab 9a. Linear Predictive Coding for Speech Processing Pitch Period Impulse Train Generator Voiced/Unvoiced Speech Switch Vocal Tract Parameters Time-Varying Digital Filter H(z)

More information

SPEECH COMMUNICATION 6.541J J-HST710J Spring 2004

SPEECH COMMUNICATION 6.541J J-HST710J Spring 2004 6.541J PS3 02/19/04 1 SPEECH COMMUNICATION 6.541J-24.968J-HST710J Spring 2004 Problem Set 3 Assigned: 02/19/04 Due: 02/26/04 Read Chapter 6. Problem 1 In this problem we examine the acoustic and perceptual

More information

L6: Short-time Fourier analysis and synthesis

L6: Short-time Fourier analysis and synthesis L6: Short-time Fourier analysis and synthesis Overview Analysis: Fourier-transform view Analysis: filtering view Synthesis: filter bank summation (FBS) method Synthesis: overlap-add (OLA) method STFT magnitude

More information

L7: Linear prediction of speech

L7: Linear prediction of speech L7: Linear prediction of speech Introduction Linear prediction Finding the linear prediction coefficients Alternative representations This lecture is based on [Dutoit and Marques, 2009, ch1; Taylor, 2009,

More information

Mel-Generalized Cepstral Representation of Speech A Unified Approach to Speech Spectral Estimation. Keiichi Tokuda

Mel-Generalized Cepstral Representation of Speech A Unified Approach to Speech Spectral Estimation. Keiichi Tokuda Mel-Generalized Cepstral Representation of Speech A Unified Approach to Speech Spectral Estimation Keiichi Tokuda Nagoya Institute of Technology Carnegie Mellon University Tamkang University March 13,

More information

Chapter 5 Frequency Domain Analysis of Systems

Chapter 5 Frequency Domain Analysis of Systems Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this

More information

Chapter 5 Frequency Domain Analysis of Systems

Chapter 5 Frequency Domain Analysis of Systems Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this

More information

L8: Source estimation

L8: Source estimation L8: Source estimation Glottal and lip radiation models Closed-phase residual analysis Voicing/unvoicing detection Pitch detection Epoch detection This lecture is based on [Taylor, 2009, ch. 11-12] Introduction

More information

Digital Signal Processing

Digital Signal Processing Digital Signal Processing Introduction Moslem Amiri, Václav Přenosil Embedded Systems Laboratory Faculty of Informatics, Masaryk University Brno, Czech Republic amiri@mail.muni.cz prenosil@fi.muni.cz February

More information

Improved Method for Epoch Extraction in High Pass Filtered Speech

Improved Method for Epoch Extraction in High Pass Filtered Speech Improved Method for Epoch Extraction in High Pass Filtered Speech D. Govind Center for Computational Engineering & Networking Amrita Vishwa Vidyapeetham (University) Coimbatore, Tamilnadu 642 Email: d

More information

Chapter 9. Linear Predictive Analysis of Speech Signals 语音信号的线性预测分析

Chapter 9. Linear Predictive Analysis of Speech Signals 语音信号的线性预测分析 Chapter 9 Linear Predictive Analysis of Speech Signals 语音信号的线性预测分析 1 LPC Methods LPC methods are the most widely used in speech coding, speech synthesis, speech recognition, speaker recognition and verification

More information

Review of Fundamentals of Digital Signal Processing

Review of Fundamentals of Digital Signal Processing Chapter 2 Review of Fundamentals of Digital Signal Processing 2.1 (a) This system is not linear (the constant term makes it non linear) but is shift-invariant (b) This system is linear but not shift-invariant

More information

Speech Signal Representations

Speech Signal Representations Speech Signal Representations Berlin Chen 2003 References: 1. X. Huang et. al., Spoken Language Processing, Chapters 5, 6 2. J. R. Deller et. al., Discrete-Time Processing of Speech Signals, Chapters 4-6

More information

ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization

ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals 1. Sampling and Reconstruction 2. Quantization 1 1. Sampling & Reconstruction DSP must interact with an analog world: A to D D to A x(t)

More information

Vocoding approaches for statistical parametric speech synthesis

Vocoding approaches for statistical parametric speech synthesis Vocoding approaches for statistical parametric speech synthesis Ranniery Maia Toshiba Research Europe Limited Cambridge Research Laboratory Speech Synthesis Seminar Series CUED, University of Cambridge,

More information

convenient means to determine response to a sum of clear evidence of signal properties that are obscured in the original signal

convenient means to determine response to a sum of clear evidence of signal properties that are obscured in the original signal Digital Speech Processing Lecture 9 Short-Time Fourier Analysis Methods- Introduction 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)

More information

EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, Cover Sheet

EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, Cover Sheet EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, 2012 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 5,7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet Calculators

More information

2A1H Time-Frequency Analysis II

2A1H Time-Frequency Analysis II 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 209 For any corrections see the course page DW Murray at www.robots.ox.ac.uk/ dwm/courses/2tf. (a) A signal g(t) with period

More information

Linear Prediction Coding. Nimrod Peleg Update: Aug. 2007

Linear Prediction Coding. Nimrod Peleg Update: Aug. 2007 Linear Prediction Coding Nimrod Peleg Update: Aug. 2007 1 Linear Prediction and Speech Coding The earliest papers on applying LPC to speech: Atal 1968, 1970, 1971 Markel 1971, 1972 Makhoul 1975 This is

More information

Feature extraction 1

Feature extraction 1 Centre for Vision Speech & Signal Processing University of Surrey, Guildford GU2 7XH. Feature extraction 1 Dr Philip Jackson Cepstral analysis - Real & complex cepstra - Homomorphic decomposition Filter

More information

Question Paper Code : AEC11T02

Question Paper Code : AEC11T02 Hall Ticket No Question Paper Code : AEC11T02 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B. Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)

More information

Digital Signal Processing Lecture 9 - Design of Digital Filters - FIR

Digital Signal Processing Lecture 9 - Design of Digital Filters - FIR Digital Signal Processing - Design of Digital Filters - FIR Electrical Engineering and Computer Science University of Tennessee, Knoxville November 3, 2015 Overview 1 2 3 4 Roadmap Introduction Discrete-time

More information

Es e j4φ +4N n. 16 KE s /N 0. σ 2ˆφ4 1 γ s. p(φ e )= exp 1 ( 2πσ φ b cos N 2 φ e 0

Es e j4φ +4N n. 16 KE s /N 0. σ 2ˆφ4 1 γ s. p(φ e )= exp 1 ( 2πσ φ b cos N 2 φ e 0 Problem 6.15 : he received signal-plus-noise vector at the output of the matched filter may be represented as (see (5-2-63) for example) : r n = E s e j(θn φ) + N n where θ n =0,π/2,π,3π/2 for QPSK, and

More information

Design Criteria for the Quadratically Interpolated FFT Method (I): Bias due to Interpolation

Design Criteria for the Quadratically Interpolated FFT Method (I): Bias due to Interpolation CENTER FOR COMPUTER RESEARCH IN MUSIC AND ACOUSTICS DEPARTMENT OF MUSIC, STANFORD UNIVERSITY REPORT NO. STAN-M-4 Design Criteria for the Quadratically Interpolated FFT Method (I): Bias due to Interpolation

More information

A.1 THE SAMPLED TIME DOMAIN AND THE Z TRANSFORM. 0 δ(t)dt = 1, (A.1) δ(t)dt =

A.1 THE SAMPLED TIME DOMAIN AND THE Z TRANSFORM. 0 δ(t)dt = 1, (A.1) δ(t)dt = APPENDIX A THE Z TRANSFORM One of the most useful techniques in engineering or scientific analysis is transforming a problem from the time domain to the frequency domain ( 3). Using a Fourier or Laplace

More information

Homework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt

Homework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t

More information

ω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the

ω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the he ime-frequency Concept []. Review of Fourier Series Consider the following set of time functions {3A sin t, A sin t}. We can represent these functions in different ways by plotting the amplitude versus

More information

Topic 3: Fourier Series (FS)

Topic 3: Fourier Series (FS) ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties

More information

CEPSTRAL ANALYSIS SYNTHESIS ON THE MEL FREQUENCY SCALE, AND AN ADAPTATIVE ALGORITHM FOR IT.

CEPSTRAL ANALYSIS SYNTHESIS ON THE MEL FREQUENCY SCALE, AND AN ADAPTATIVE ALGORITHM FOR IT. CEPSTRAL ANALYSIS SYNTHESIS ON THE EL FREQUENCY SCALE, AND AN ADAPTATIVE ALGORITH FOR IT. Summarized overview of the IEEE-publicated papers Cepstral analysis synthesis on the mel frequency scale by Satochi

More information

Chapter 4 The Fourier Series and Fourier Transform

Chapter 4 The Fourier Series and Fourier Transform Chapter 4 The Fourier Series and Fourier Transform Representation of Signals in Terms of Frequency Components Consider the CT signal defined by N xt () = Acos( ω t+ θ ), t k = 1 k k k The frequencies `present

More information

Lecture 16: Zeros and Poles

Lecture 16: Zeros and Poles Lecture 16: Zeros and Poles ECE 401: Signal and Image Analysis University of Illinois 4/6/2017 1 Poles and Zeros at DC 2 Poles and Zeros with Nonzero Bandwidth 3 Poles and Zeros with Nonzero Center Frequency

More information

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz Discrete Time Signals and Systems Time-frequency Analysis Gloria Menegaz Time-frequency Analysis Fourier transform (1D and 2D) Reference textbook: Discrete time signal processing, A.W. Oppenheim and R.W.

More information

Linear Prediction 1 / 41

Linear Prediction 1 / 41 Linear Prediction 1 / 41 A map of speech signal processing Natural signals Models Artificial signals Inference Speech synthesis Hidden Markov Inference Homomorphic processing Dereverberation, Deconvolution

More information

Review of Fundamentals of Digital Signal Processing

Review of Fundamentals of Digital Signal Processing Solution Manual for Theory and Applications of Digital Speech Processing by Lawrence Rabiner and Ronald Schafer Click here to Purchase full Solution Manual at http://solutionmanuals.info Link download

More information

EE 224 Signals and Systems I Review 1/10

EE 224 Signals and Systems I Review 1/10 EE 224 Signals and Systems I Review 1/10 Class Contents Signals and Systems Continuous-Time and Discrete-Time Time-Domain and Frequency Domain (all these dimensions are tightly coupled) SIGNALS SYSTEMS

More information

x[n] = x a (nt ) x a (t)e jωt dt while the discrete time signal x[n] has the discrete-time Fourier transform x[n]e jωn

x[n] = x a (nt ) x a (t)e jωt dt while the discrete time signal x[n] has the discrete-time Fourier transform x[n]e jωn Sampling Let x a (t) be a continuous time signal. The signal is sampled by taking the signal value at intervals of time T to get The signal x(t) has a Fourier transform x[n] = x a (nt ) X a (Ω) = x a (t)e

More information

Applications of Linear Prediction

Applications of Linear Prediction SGN-4006 Audio and Speech Processing Applications of Linear Prediction Slides for this lecture are based on those created by Katariina Mahkonen for TUT course Puheenkäsittelyn menetelmät in Spring 03.

More information

CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation

CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 23, 2006 1 Exponentials The exponential is

More information

Ch.11 The Discrete-Time Fourier Transform (DTFT)

Ch.11 The Discrete-Time Fourier Transform (DTFT) EE2S11 Signals and Systems, part 2 Ch.11 The Discrete-Time Fourier Transform (DTFT Contents definition of the DTFT relation to the -transform, region of convergence, stability frequency plots convolution

More information

EDISP (NWL2) (English) Digital Signal Processing Transform, FT, DFT. March 11, 2015

EDISP (NWL2) (English) Digital Signal Processing Transform, FT, DFT. March 11, 2015 EDISP (NWL2) (English) Digital Signal Processing Transform, FT, DFT March 11, 2015 Transform concept We want to analyze the signal represent it as built of some building blocks (well known signals), possibly

More information

Class of waveform coders can be represented in this manner

Class of waveform coders can be represented in this manner Digital Speech Processing Lecture 15 Speech Coding Methods Based on Speech Waveform Representations ti and Speech Models Uniform and Non- Uniform Coding Methods 1 Analog-to-Digital Conversion (Sampling

More information

LECTURE NOTES IN AUDIO ANALYSIS: PITCH ESTIMATION FOR DUMMIES

LECTURE NOTES IN AUDIO ANALYSIS: PITCH ESTIMATION FOR DUMMIES LECTURE NOTES IN AUDIO ANALYSIS: PITCH ESTIMATION FOR DUMMIES Abstract March, 3 Mads Græsbøll Christensen Audio Analysis Lab, AD:MT Aalborg University This document contains a brief introduction to pitch

More information

Lecture 7: Wireless Channels and Diversity Advanced Digital Communications (EQ2410) 1

Lecture 7: Wireless Channels and Diversity Advanced Digital Communications (EQ2410) 1 Wireless : Wireless Advanced Digital Communications (EQ2410) 1 Thursday, Feb. 11, 2016 10:00-12:00, B24 1 Textbook: U. Madhow, Fundamentals of Digital Communications, 2008 1 / 15 Wireless Lecture 1-6 Equalization

More information

Sound 2: frequency analysis

Sound 2: frequency analysis COMP 546 Lecture 19 Sound 2: frequency analysis Tues. March 27, 2018 1 Speed of Sound Sound travels at about 340 m/s, or 34 cm/ ms. (This depends on temperature and other factors) 2 Wave equation Pressure

More information

Chapter 2: Problem Solutions

Chapter 2: Problem Solutions Chapter 2: Problem Solutions Discrete Time Processing of Continuous Time Signals Sampling à Problem 2.1. Problem: Consider a sinusoidal signal and let us sample it at a frequency F s 2kHz. xt 3cos1000t

More information

I. Signals & Sinusoids

I. Signals & Sinusoids I. Signals & Sinusoids [p. 3] Signal definition Sinusoidal signal Plotting a sinusoid [p. 12] Signal operations Time shifting Time scaling Time reversal Combining time shifting & scaling [p. 17] Trigonometric

More information

2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit dwm/courses/2tf

2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit   dwm/courses/2tf Time-Frequency Analysis II (HT 20) 2AH 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 20 For hints and answers visit www.robots.ox.ac.uk/ dwm/courses/2tf David Murray. A periodic

More information

3.2 Complex Sinusoids and Frequency Response of LTI Systems

3.2 Complex Sinusoids and Frequency Response of LTI Systems 3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n]. LTI system: LTI System Output = A weighted superposition of the system response to each complex

More information

Review of Discrete-Time System

Review of Discrete-Time System Review of Discrete-Time System Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by Profs. K.J. Ray Liu and Min Wu.

More information

Series FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis

Series FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 24 g.s.mcdonald@salford.ac.uk 1. Theory

More information

Source/Filter Model. Markus Flohberger. Acoustic Tube Models Linear Prediction Formant Synthesizer.

Source/Filter Model. Markus Flohberger. Acoustic Tube Models Linear Prediction Formant Synthesizer. Source/Filter Model Acoustic Tube Models Linear Prediction Formant Synthesizer Markus Flohberger maxiko@sbox.tugraz.at Graz, 19.11.2003 2 ACOUSTIC TUBE MODELS 1 Introduction Speech synthesis methods that

More information

5 Kalman filters. 5.1 Scalar Kalman filter. Unit delay Signal model. System model

5 Kalman filters. 5.1 Scalar Kalman filter. Unit delay Signal model. System model 5 Kalman filters 5.1 Scalar Kalman filter 5.1.1 Signal model System model {Y (n)} is an unobservable sequence which is described by the following state or system equation: Y (n) = h(n)y (n 1) + Z(n), n

More information

NAME: 11 December 2013 Digital Signal Processing I Final Exam Fall Cover Sheet

NAME: 11 December 2013 Digital Signal Processing I Final Exam Fall Cover Sheet NAME: December Digital Signal Processing I Final Exam Fall Cover Sheet Test Duration: minutes. Open Book but Closed Notes. Three 8.5 x crib sheets allowed Calculators NOT allowed. This test contains four

More information

Randomly Modulated Periodic Signals

Randomly Modulated Periodic Signals Randomly Modulated Periodic Signals Melvin J. Hinich Applied Research Laboratories University of Texas at Austin hinich@mail.la.utexas.edu www.la.utexas.edu/~hinich Rotating Cylinder Data Fluid Nonlinearity

More information

Final Exam 14 May LAST Name FIRST Name Lab Time

Final Exam 14 May LAST Name FIRST Name Lab Time EECS 20n: Structure and Interpretation of Signals and Systems Department of Electrical Engineering and Computer Sciences UNIVERSITY OF CALIFORNIA BERKELEY Final Exam 14 May 2005 LAST Name FIRST Name Lab

More information

Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes (cont d)

Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes (cont d) Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes (cont d) Electrical & Computer Engineering North Carolina State University Acknowledgment: ECE792-41 slides

More information

Zeros of z-transform(zzt) representation and chirp group delay processing for analysis of source and filter characteristics of speech signals

Zeros of z-transform(zzt) representation and chirp group delay processing for analysis of source and filter characteristics of speech signals Zeros of z-transformzzt representation and chirp group delay processing for analysis of source and filter characteristics of speech signals Baris Bozkurt 1 Collaboration with LIMSI-CNRS, France 07/03/2017

More information

Fourier Analysis of Signals Using the DFT

Fourier Analysis of Signals Using the DFT Fourier Analysis of Signals Using the DFT ECE 535 Lecture April 29, 23 Overview: Motivation Many applications require analyzing the frequency content of signals Speech processing study resonances of vocal

More information

Frequency Domain Speech Analysis

Frequency Domain Speech Analysis Frequency Domain Speech Analysis Short Time Fourier Analysis Cepstral Analysis Windowed (short time) Fourier Transform Spectrogram of speech signals Filter bank implementation* (Real) cepstrum and complex

More information

Automatic Speech Recognition (CS753)

Automatic Speech Recognition (CS753) Automatic Speech Recognition (CS753) Lecture 12: Acoustic Feature Extraction for ASR Instructor: Preethi Jyothi Feb 13, 2017 Speech Signal Analysis Generate discrete samples A frame Need to focus on short

More information

Discrete-Time David Johns and Ken Martin University of Toronto

Discrete-Time David Johns and Ken Martin University of Toronto Discrete-Time David Johns and Ken Martin University of Toronto (johns@eecg.toronto.edu) (martin@eecg.toronto.edu) University of Toronto 1 of 40 Overview of Some Signal Spectra x c () t st () x s () t xn

More information

Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes

Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes Electrical & Computer Engineering North Carolina State University Acknowledgment: ECE792-41 slides were adapted

More information

Example: Bipolar NRZ (non-return-to-zero) signaling

Example: Bipolar NRZ (non-return-to-zero) signaling Baseand Data Transmission Data are sent without using a carrier signal Example: Bipolar NRZ (non-return-to-zero signaling is represented y is represented y T A -A T : it duration is represented y BT. Passand

More information

Final Exam ECE301 Signals and Systems Friday, May 3, Cover Sheet

Final Exam ECE301 Signals and Systems Friday, May 3, Cover Sheet Name: Final Exam ECE3 Signals and Systems Friday, May 3, 3 Cover Sheet Write your name on this page and every page to be safe. Test Duration: minutes. Coverage: Comprehensive Open Book but Closed Notes.

More information

Multimedia Signals and Systems - Audio and Video. Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2

Multimedia Signals and Systems - Audio and Video. Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2 Multimedia Signals and Systems - Audio and Video Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2 Kunio Takaya Electrical and Computer Engineering University of Saskatchewan December

More information

MITIGATING UNCORRELATED PERIODIC DISTURBANCE IN NARROWBAND ACTIVE NOISE CONTROL SYSTEMS

MITIGATING UNCORRELATED PERIODIC DISTURBANCE IN NARROWBAND ACTIVE NOISE CONTROL SYSTEMS 17th European Signal Processing Conference (EUSIPCO 29) Glasgow, Scotland, August 24-28, 29 MITIGATING UNCORRELATED PERIODIC DISTURBANCE IN NARROWBAND ACTIVE NOISE CONTROL SYSTEMS Muhammad Tahir AKHTAR

More information

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency Response-Design Method

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency Response-Design Method .. AERO 422: Active Controls for Aerospace Vehicles Frequency Response- Method Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. ... Response to

More information

Signals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters

Signals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters Signals, Instruments, and Systems W5 Introduction to Signal Processing Sampling, Reconstruction, and Filters Acknowledgments Recapitulation of Key Concepts from the Last Lecture Dirac delta function (

More information

LTI Systems (Continuous & Discrete) - Basics

LTI Systems (Continuous & Discrete) - Basics LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying

More information

Multidimensional digital signal processing

Multidimensional digital signal processing PSfrag replacements Two-dimensional discrete signals N 1 A 2-D discrete signal (also N called a sequence or array) is a function 2 defined over thex(n set 1 of, n 2 ordered ) pairs of integers: y(nx 1,

More information

Discrete Time Fourier Transform

Discrete Time Fourier Transform Discrete Time Fourier Transform Recall that we wrote the sampled signal x s (t) = x(kt)δ(t kt). We calculate its Fourier Transform. We do the following: Ex. Find the Continuous Time Fourier Transform of

More information

Transform Representation of Signals

Transform Representation of Signals C H A P T E R 3 Transform Representation of Signals and LTI Systems As you have seen in your prior studies of signals and systems, and as emphasized in the review in Chapter 2, transforms play a central

More information

SIGNAL PROCESSING. B14 Option 4 lectures. Stephen Roberts

SIGNAL PROCESSING. B14 Option 4 lectures. Stephen Roberts SIGNAL PROCESSING B14 Option 4 lectures Stephen Roberts Recommended texts Lynn. An introduction to the analysis and processing of signals. Macmillan. Oppenhein & Shafer. Digital signal processing. Prentice

More information

Digital Speech Processing Lecture 10. Short-Time Fourier Analysis Methods - Filter Bank Design

Digital Speech Processing Lecture 10. Short-Time Fourier Analysis Methods - Filter Bank Design Digital Speech Processing Lecture Short-Time Fourier Analysis Methods - Filter Bank Design Review of STFT j j ˆ m ˆ. X e x[ mw ] [ nˆ m] e nˆ function of nˆ looks like a time sequence function of ˆ looks

More information

Figure 3.1 Effect on frequency spectrum of increasing period T 0. Consider the amplitude spectrum of a periodic waveform as shown in Figure 3.2.

Figure 3.1 Effect on frequency spectrum of increasing period T 0. Consider the amplitude spectrum of a periodic waveform as shown in Figure 3.2. 3. Fourier ransorm From Fourier Series to Fourier ransorm [, 2] In communication systems, we oten deal with non-periodic signals. An extension o the time-requency relationship to a non-periodic signal

More information

Virtual Array Processing for Active Radar and Sonar Sensing

Virtual Array Processing for Active Radar and Sonar Sensing SCHARF AND PEZESHKI: VIRTUAL ARRAY PROCESSING FOR ACTIVE SENSING Virtual Array Processing for Active Radar and Sonar Sensing Louis L. Scharf and Ali Pezeshki Abstract In this paper, we describe how an

More information

Music 270a: Complex Exponentials and Spectrum Representation

Music 270a: Complex Exponentials and Spectrum Representation Music 270a: Complex Exponentials and Spectrum Representation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) October 24, 2016 1 Exponentials The exponential

More information

Design of a CELP coder and analysis of various quantization techniques

Design of a CELP coder and analysis of various quantization techniques EECS 65 Project Report Design of a CELP coder and analysis of various quantization techniques Prof. David L. Neuhoff By: Awais M. Kamboh Krispian C. Lawrence Aditya M. Thomas Philip I. Tsai Winter 005

More information

NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF ELECTRONICS AND TELECOMMUNICATIONS

NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF ELECTRONICS AND TELECOMMUNICATIONS page 1 of 5 (+ appendix) NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF ELECTRONICS AND TELECOMMUNICATIONS Contact during examination: Name: Magne H. Johnsen Tel.: 73 59 26 78/930 25 534

More information

COMP Signals and Systems. Dr Chris Bleakley. UCD School of Computer Science and Informatics.

COMP Signals and Systems. Dr Chris Bleakley. UCD School of Computer Science and Informatics. COMP 40420 2. Signals and Systems Dr Chris Bleakley UCD School of Computer Science and Informatics. Scoil na Ríomheolaíochta agus an Faisnéisíochta UCD. Introduction 1. Signals 2. Systems 3. System response

More information

EE 521: Instrumentation and Measurements

EE 521: Instrumentation and Measurements Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA November 1, 2009 1 / 27 1 The z-transform 2 Linear Time-Invariant System 3 Filter Design IIR Filters FIR Filters

More information

Massachusetts Institute of Technology

Massachusetts Institute of Technology Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing QUIZ, April 1, 010 QUESTION BOOKLET Your

More information

A METHOD FOR NONLINEAR SYSTEM CLASSIFICATION IN THE TIME-FREQUENCY PLANE IN PRESENCE OF FRACTAL NOISE. Lorenzo Galleani, Letizia Lo Presti

A METHOD FOR NONLINEAR SYSTEM CLASSIFICATION IN THE TIME-FREQUENCY PLANE IN PRESENCE OF FRACTAL NOISE. Lorenzo Galleani, Letizia Lo Presti A METHOD FOR NONLINEAR SYSTEM CLASSIFICATION IN THE TIME-FREQUENCY PLANE IN PRESENCE OF FRACTAL NOISE Lorenzo Galleani, Letizia Lo Presti Dipartimento di Elettronica, Politecnico di Torino, Corso Duca

More information

Homework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, Signals & Systems Sampling P1

Homework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, Signals & Systems Sampling P1 Homework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, 7.49 Signals & Systems Sampling P1 Undersampling & Aliasing Undersampling: insufficient sampling frequency ω s < 2ω M Perfect

More information

KLT for transient signal analysis

KLT for transient signal analysis The Labyrinth of the Unepected: unforeseen treasures in impossible regions of phase space Nicolò Antonietti 4/1/17 Kerastari, Greece May 9 th June 3 rd Qualitatively definition of transient signals Signals

More information

Digital Image Processing Lectures 13 & 14

Digital Image Processing Lectures 13 & 14 Lectures 13 & 14, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2013 Properties of KL Transform The KL transform has many desirable properties which makes

More information

Lecture 7 Discrete Systems

Lecture 7 Discrete Systems Lecture 7 Discrete Systems EE 52: Instrumentation and Measurements Lecture Notes Update on November, 29 Aly El-Osery, Electrical Engineering Dept., New Mexico Tech 7. Contents The z-transform 2 Linear

More information

Discrete-Time Signals: Time-Domain Representation

Discrete-Time Signals: Time-Domain Representation Discrete-Time Signals: Time-Domain Representation 1 Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the

More information

Cast of Characters. Some Symbols, Functions, and Variables Used in the Book

Cast of Characters. Some Symbols, Functions, and Variables Used in the Book Page 1 of 6 Cast of Characters Some s, Functions, and Variables Used in the Book Digital Signal Processing and the Microcontroller by Dale Grover and John R. Deller ISBN 0-13-081348-6 Prentice Hall, 1998

More information

Solutions. Number of Problems: 10

Solutions. Number of Problems: 10 Final Exam February 9th, 2 Signals & Systems (5-575-) Prof. R. D Andrea Solutions Exam Duration: 5 minutes Number of Problems: Permitted aids: One double-sided A4 sheet. Questions can be answered in English

More information

Each problem is worth 25 points, and you may solve the problems in any order.

Each problem is worth 25 points, and you may solve the problems in any order. EE 120: Signals & Systems Department of Electrical Engineering and Computer Sciences University of California, Berkeley Midterm Exam #2 April 11, 2016, 2:10-4:00pm Instructions: There are four questions

More information

6.003 (Fall 2011) Quiz #3 November 16, 2011

6.003 (Fall 2011) Quiz #3 November 16, 2011 6.003 (Fall 2011) Quiz #3 November 16, 2011 Name: Kerberos Username: Please circle your section number: Section Time 2 11 am 3 1 pm 4 2 pm Grades will be determined by the correctness of your answers (explanations

More information