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