More On Linear Predictive Analysis 主講人：虞台文

Contents Linear Prediction Error Computation of the Gain Frequency Domain Interpretation of LPC Representations of LPC Coefficients Direct Representation Roots of Predictor Polynomials PARCO Coefficients Log Area Ratio Coefficients Line Spectrum Pair

More On Linear Predictive Analysis Linear Prediction Error

LPC Error

Examples Could be used for Pitch Detection Premphasized Speech Signals

Normalized Mean-Squared Error General Form Autocorrelation Method Covariance Method

Normalized Mean-Squared Error Autocorrelation Method Covariance

Experimental Evaluation on LPC Parameters Frame Width N Filter Order p Conditions: 1. Covariance method and autocorrelation method 2. Synthetic vowel and Nature speech 3. Pitch synchronous and pitch asynchronous analysis

Pitch Synchronous Analysis Covariance method is more suitable for pitch synchronous analysis. Pitch Synchronous Analysis /i/ The frame was beginning at the beginning of a pitch period. Why the error increases? zero: the same order as the synthesizer.

Pitch Asynchronous Analysis Both covariance and autocorrelation methods exhibit similar performance. Pitch Asynchronous Analysis /i/ Monotonically decreasing

The errors resulted by covariance and autocorrelation methods are compatible when N > 2P. Frame Width Variation /i/ Why the errors jump high when the frame size nears the multiples of pitch period?

Pitch Synchronous Analysis Both for synthetic and nature speeches, covariance method is more suitable for pitch synchronous analysis. Pitch Synchronous Analysis

Pitch Asynchronous Analysis Both for synthetic and nature speeches, two methods are compatible. Pitch Asynchronous Analysis

Both for synthetic and nature speeches, the errors resulted by covariance and autocorrelation methods are compatible when N > 2P. Frame Width Variation

More On Linear Predictive Analysis Computation of the Gain

Speech Production Model (Review) Impulse Train Generator Random Noise Time-Varying Digital Filter Vocal Tract Parameters G u(n) s(n)

Speech Production Model (Review) Impulse Train Generator Random Noise Time-Varying Digital Filter Vocal Tract Parameters G u(n) s(n) H(z)

Linear Prediction Model (Review) Error compensation:

Speech Production vs. Linear Prediction Vocal Tract Excitation ak = k Linear Predictor Error Linear Prediction:

Speech Production vs. Linear Prediction

The Gain Generally, it is not possible to solve for G in a reliable way directly from the error signal itself. Instead, we assume Energy of Error Energy of Excitation

Assumptions about u(n) Voiced Speech This requires that both glottal pulse shape and lip radiation are lumped into the vocal tract model. G(z) V(z) R(z) G u(n)=(n) h(n) 1/A(z) Unvoiced Speech

Gain Estimation for Voiced Speech This requires that both glottal pulse shape and lip radiation are lumped into the vocal tract model. G(z) V(z) R(z) G u(n)=(n) h(n) 1/A(z) This requires that p is sufficiently large.

Gain Estimation for Voiced Speech h(n) G(z) V(z) R(z) G u(n)=(n) h(n) 1/A(z)

Correlation Matching Define (n) h(n) Assumed causal. Autocorrelation function of the impulse response.

Correlation Matching Autocorrelation function of the speech signal h(n) Autocorrelation function of the speech signal If H(z) correctly model the speech production system, we should have

Correlation Matching (n) h(n)

Correlation Matching Assumed causal.

Correlation Matching The same formulation as autocorrelation method.

The Gain for voice speech En

More on Autocorrelation H(z) x(n) y(n) Assumed Stationary Define The stationary assumption implies

Properties of LTI Systems H(z) x(n) y(n) Define

Properties of LTI Systems Define

Properties of LTI Systems Independent on n y(n) is also stationary.

Properties of LTI Systems ll+k

Properties of LTI Systems Define Properties of LTI Systems Estimated from input Estimated from output Filter Design

The Gain for Unvoiced Speech s(n)

The Gain for Unvoiced Speech =?

The Gain for Unvoiced Speech Why? The Gain for Unvoiced Speech =?

The Gain for Unvoiced Speech Estimated using rm

The Gain for Unvoiced Speech Once again, we have the same formulation as autocorrelation method. Furthermore,

More On Linear Predictive Analysis Frequency Domain Interpretation of LPC

Spectral Representation of Vocal Tract

Spectra

Frequency Domain Interpretation of Mean-Squared Prediction Error Parseval’s Theorem

Frequency Domain Interpretation of Mean-Squared Prediction Error

Frequency Domain Interpretation of Mean-Squared Prediction Error |Sn(ej)| > |H(ej)| contributes more to the total error than |Sn(ej)| < |H(ej)|. Hence, the LPC spectral error criterion favors a good fit near the spectral peak.

Spectra

More On Linear Predictive Analysis Representations of LPC Coefficients --- Direct Representation

Direct Representation Coding ai’s directly. z1 a1 a2 ap uG[n] uL[n] G G/A(z)

Disadvantages The dynamic ranges of ai’s is relatively large. Quantatization possibly causes instability problems.

More On Linear Predictive Analysis Representations of LPC Coefficients --- Roots of Predictor Polynomials

Roots of the Predictor Polynomial Coding p/2 zk’s. Dynamic range of rk’s? Dynamic range of k’s?

The Application Formant Analysis Application.

Implementation G/A(z) Each Stage represents one formant frequency and its corresponding bandwidth.

More On Linear Predictive Analysis Representations of LPC Coefficients --- PARCO Coefficients

PARCO Coefficients Step-Up Procedure:

PARCO Coefficients Dynamic range of ki’s? Step-Down Procedure: where n goes from p to p1, down to 1 and initially we set:

More On Linear Predictive Analysis Representations of LPC Coefficients --- Log Area Ratio Coefficients

Log Area Ratio Coefficients ki’s: Reflection Coefficients gi’s: Log Area Ratios

More On Linear Predictive Analysis Representations of LPC Coefficients --- Line Spectrum Pair

LPC Coefficients where m is the order of the inverse filter. If the system is stable, all zeros of the inverse filter are inside the unit circle. Line Spectrum Pair (LSP) is an alternative LPC spectral representation.

Line Spectrum Pair LSP contains two polynomials. The zeros of the the two polynomials have the following properties: Lie on unit circle Interlaced Through quantization, the minimum phase property of the filter is kept. Useful for vocoder application.

Recursive Relation of the inverse filter where km+1 is the reflection coefficient of the m+1th tube. Special cases: Recall that ki =1: Ai+1 ki =1: Ai+1=0

LSP Polynomials

Properties of LSP Polynomials Show that The zeros of P(z) and Q(z) are on the unit circle and interlaced.

Proof

Proof

Proof

Proof >0

Proof P(z)=0 iff H(z) = 1. Q(z)=0 This concludes that the zeros of P(z) and Q(z) are on the unit circle. P(z)=0 Q(z)=0 iff H(z) = 1.

Proof (interlaced zeros) Fact: H(z) is an all-pass filter. One can verify that (0) = 0 and (2) = 2(m+1)  Phase

Proof (interlaced zeros) zeros of Q(z) zeros of P(z) Therefore, z=1 is a zero of Q(z). One can verify that (0) = 0 and (2) = 2(m+1)  Phase

Proof (interlaced zeros) (0) = 0 (2) = 2(m+1)  Proof (interlaced zeros) 2 () 2(m+1)   Is this possible?

Proof (interlaced zeros) (0) = 0 (2) = 2(m+1)  Proof (interlaced zeros) 2 () 2(m+1)   Is this possible?

Proof (interlaced zeros) (0) = 0 (2) = 2(m+1)  Proof (interlaced zeros) Group Delay > 0 () is monotonically decreasing.

Proof (interlaced zeros) (0) = 0 (2) = 2(m+1)  Proof (interlaced zeros) 2 () 2(m+1)    2 3 4 5 Typical shape of () .

Proof (interlaced zeros) 2 () 2(m+1)    2 3 4 5 . Typical shape of () Q(ej)=0 P(ej)=0 Q(ej)=0 P(ej)=0 Q(ej)=0 P(ej)=0

Proof (interlaced zeros) 2 () 2(m+1)    2 3 4 5 . Typical shape of () Q(ej)=0 P(ej)=0 There are 2(m+1) cross points from 0    , these constitute the 2(m+1) interlaced zeros of P(z) and Q(z).

Quantization of LSP Zeros Is such a quantization detrimental? Quantization of LSP Zeros For effective transmission, we quantize i’s into several levels, e.g., using 5 bits.

Minimum Phase Preserving Property Show that in quantizing the LSP frequencies, the reconstructed all-pole filter preserves its minimum phase property as long as the zeros has the properties shown in the left figure.

Find the Roots of P(z) and Q(z) Symmetric Anti-symmetric

Find the Roots of P(z) and Q(z) . . . .

Find the Roots of P(z) and Q(z) . We only need compute the values on 1  i  m/2.

Find the Roots of P(z) and Q(z) . . . .

Find the Roots of P(z) and Q(z) . We only need compute the values on 1  i  m/2.

Find the Roots of P(z) and Q(z) Both P’(z) and Q’(z) are symmetric. P’(z) Find the Roots of P(z) and Q(z) zero on 1 Q’(z) zero on +1

Find the Roots of P(z) and Q(z) To find its zeros.

Find the Roots of P(z) and Q(z)

Find the Roots of P(z) and Q(z) Define

Find the Roots of P(z) and Q(z) .

Find the Roots of P(z) and Q(z) Consider m=10.

Find the Roots of P(z) and Q(z)

Find the Roots of P(z) and Q(z)

Find the Roots of P(z) and Q(z)

Find the Roots of P(z) and Q(z) We want to find i’s such that

Find the Roots of P(z) and Q(z) Algorithm:  We only need to find zeros for this half.

Find the Roots of P(z) and Q(z)   2 3