1. More On Linear Predictive Analysis
主講人：虞台文

2. 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

3. More On Linear Predictive Analysis
Linear Prediction Error

4. LPC Error

5. Examples
Could be used for
Pitch Detection
Premphasized
Speech Signals

6. Normalized Mean-Squared Error
General Form
Autocorrelation
Method
Covariance
Method

7. Normalized Mean-Squared Error
Autocorrelation
Method
Covariance

8. 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

9. 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.

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

11. 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?

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

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

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

15. More On Linear Predictive Analysis
Computation of the Gain

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

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

18. Linear Prediction Model (Review)
Error compensation:

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

20. Speech Production vs. Linear Prediction

21. 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

22. 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

23. 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.

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

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

26. 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

27. Correlation Matching
(n)
h(n)

28. Correlation Matching
Assumed causal.

29. Correlation Matching
The same formulation as
autocorrelation method.

30. The Gain for voice speechEn

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

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

33. Properties of LTI Systems
Define

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

35. Properties of LTI Systems
ll+k

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

37. The Gain for Unvoiced Speech
s(n)

38. The Gain for Unvoiced Speech=?

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

40. The Gain for Unvoiced Speech
Estimated using rm

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

42. More On Linear Predictive Analysis
Frequency Domain Interpretation of LPC

43. Spectral Representation of Vocal Tract

44. Spectra

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

46. Frequency Domain Interpretation of Mean-Squared Prediction Error

47. 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.

48. Spectra

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

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

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

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

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

54. The Application
Formant Analysis Application.

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

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

57. PARCO Coefficients
Step-Up Procedure:

58. 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:

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

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

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

62. 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.

63. 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.

64. 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

65. LSP Polynomials

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

67. Proof

68. Proof

69. Proof

70. Proof
>0

71. 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.

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

73. 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

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

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

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

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

78. 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

79. 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).

80. 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.

81. 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.

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

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

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

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

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

87. 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

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

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

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

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

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

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

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

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

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

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

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