1. Speech Recognition Chapter 3

2. Speech Front-Ends Linear Prediction Analysis
Linear-Prediction Based Processing
Cepstral Analysis
Auditory signal Processing

3. Linear Prediction Analysis
Introduction
Linear Prediction Model
Linear Prediction Coefficients Computation
Linear Prediction for Automatic Speech Recognition
Linear Prediction in Speech Processing
How good is the LP Model.

4. Signal Processing Front End
Convert the speech waveform
in some type of parametric representation. sk
Filterbank
Signal Processing
Front End
Linear Prediction Front End
Linear Prediction Coefficients
O=o(1)o(2)..o(T)

5. Introduction
In short intervals, it provides a good model of the speech.
Mathematical precise and simple.
Easy to implement in software or hardware.
Works fine for recognition applications.
It also has applications in formant and pitch estimation, speech coding and synthesis.

6. Linear Prediction Model
Basic idea:
are called LP(Linear Prediction) coefficients.
By including the excitation signal, we obtain:
where is the normalised excitation and is the gain of the excitation.

7. In the z-domain (secc. 1.1.4, pp. 15, Deller)
leading to the transfer function (Fig. 3.27)

8. LP model retains the spectral magnitude, but it has a minimum phase (Sec. 1.1.7, Deller) feature.
However, in practice, phase is not very important for speech perception.
Observation:
H(z) models the glottal filter(G(z)) and the lips radiation(R(z).

9. Linear Prediction Coefficients Computation
Introduction
Methogologies

10. Linear Prediction Coefficients Computation
LP coefficients can be obtained by solving the next equation system (Secc , Prove ):

11. Methodologies Autocorrelation Method Covariance Method
Not commonly used in Speech Recognition

12. Autocorrelation Method
Assumptions: Each frame is independent (Fig ).
Solution (Juang, secc pp ):
where
(2)
M es el número de parametros LPC.
These equations are know as Yule-Walker equations.

13. Using matrix notation:or

14. Features
Symetric.
Diagonal elements
are the same.
Toeplitz Matriz

15. This matrix is known as Toeplitz
This matrix is known as Toeplitz. A linear system with this matrix can be solved very efficient.
Examples (Fig and )
Example (Fig )
Example (Fig )
Example (Fig )

16. Linear Prediction for Automatic Speech Recogition
To minimise
signal
discontinuity
Flats the
spectrum
equation (2)
usually M=8
Incorporate
signal dynamics
to minimise
noise sensitivity
To Cepstral
Coefficients
Durbin
Algorithm

17. Preemphasis
The transfer function of the glottis can be modelled as follows:
The radiation effect can be modelled as follows:

18. Hence, to obtain the transfer function of the vocal tract
the other pole must be cancelled as follows:.

19. Preemphasis sould be done only for sonorant sounds.
This process can be automated as follows.
where is the autocorrelation function.

20. N samples size frame, M samples frame shift

21. Minimize signal discontinuities at the edges of the frames.
A typical window is the Hamming window.

23. LPC Analysis
Converts the autocorrelations coefficients into LPC “parameter set”.
LPC Parameter set
LPC coefficients
Reflection (PARCOR) coefficients
log area ratio coefficients
The formal method to obtain the LPC parameter set is know as Durbin’s method.

24. Durbin’s method

26. LPC (Typical values)

27. LPC Parameter Conversion
Conversion to Cepstral Coeficients.
Robust feature set for speech recognition.
Algorithm:

28. Parameter weighting
low-order cepstral coefficents are highly sensibles to noise

29. Temporal Cepstral Derivative
First or second order derivatives is enough.
It can be aproximated as follows:

32. Given

33. Hamming Windowed
Large prediction errors
since speech is predicted
form previous samples
arbitray set to zero.

34. Large prediction errors
since speech is predicted
form previous samples
arbitray set to zero.

35. Unvoiced signals
are not position sensitive.
It does not show special
effect at the edges.

36. Observe the
“whitening” phenomena
at the error spectrum.

37. Observe the “whitening
phenomena at the error
specturm

38. Observe the error
wave periodicity
behaviour taken
as bases for the
Pitch Estimators.

39. Observe that a sharp decrease
in the prediction error is obtain
for small M value (M=1...4).
Observe that unvoiced signal
has higher RMS error.

40. Observe the all-pole model
ability to match the spectrum.

41. Linear Prediction in Speech Processing
LPC for Vocal Tract Shape Estimation
LPC for Pitch Detection
LPC for Formant prediction

42. LPC for Vocal Tract Shape Estimation
To minimise
signal
discontinuity
Free of glottis
and radiation
effects
Vocal Tract
Shape
Estimation
Parameter
Calculation
to minimise
noise sensitivity
To Cepstral
Coefficients

43. Parameter Calculation
Durbin’s Method (As in Speech Recognition)
In case, this method is used, first the autocorrelation analysis should be performed.
Lattice Filter

44. Lattice Filter
The reflection coefficients are obtain directly form the signal, avoiding the autocorrelation analysis.
Methods:
Itakura-Saito (Parcor)
Burg
New forms
Advantage:
Easier to implement in Hardware
Disadvantage:
needs around 5 times more calculation.

45. Itakura-Saito (PARCOR)
where
Accumulates over time (n).
It can be shown that the PARCOR
coefficients, obtain for the Itakura-Saito
method are exactly the same as the
reflection coefficients obtained by the
Levison Durbin algorithm.
Example

46. Burg
where
Example

47. Example
Itakura-Saito
Burg

48. New Forms
Stroback, New forms of Levinson and Schur algorithms, IEEE Signal Processing Magazine, pp , 1991.

49. Vocal Tract Shape Estimation
From:
We obtain
Therefore, by setting the the lips area to an arbitrary value
we can obtain the vocal tract configuration relative to the initial condition.
This technique as been succesfully used to train deaf persons.

50. LPC for Pitch Detection
Speech
Sampled
at 10KHz
Inverse
Filering
A(z)
LPF
800Hz
DownSampler
5:1
Peak
finding
Autocorrelation
LPC Analysis
V/U decision or
Pitch

51. LPC for Formant Detection
Sampled
Speech
Formants
LPC
Spectrum
Emphasis Peaks
(second derivative)
Peak
finding
LPC Analysis

52. LPC Spectrum
LP assumes that the vocal tract system can be modelled with an all-pole system:
The spectrum can be obtain by
In order to emphasis formant peaks we can set

53. In order to increase the spectral resolution we pad with zeros:
Therefore
Spectrum (DTFT)
Spectrum (DFT)
In order to increase the spectral resolution we pad with zeros:
In order to use an FFT algorithm

54. Caclulate the Spectral magnitude(DFT)
Invert the Spectral magnitude(DFT)
This spectrum is called the LPC Spectrum.

55. How good is the LP Model
As shown by the physiological analysis of the vocal tract the speech model is as follows:
However, it can be shown ( ), that LP Model is good for estimating the magnitude of pole-zero system.

56. Prove
According to lema 1 ( ) and lema 2 ( ) , can be written as follows:
The estimates are calculated such that it correspond to the of this model.
All pass
component

57. Since hence
therefore, if the estimators, are exacts, then at least we obtain a model with a correct magnitude.

58. Lema 1 Lema 1(System Decomposition): Any causal ration system
can be descomponed as (prove ):
Minimal phase
component

59. Prove For two poles and two zeros: Lets define:
Re-arranging this equation:

60. With the knowledge that:
Hence:

61. Therefore:
End of prove.

62. Lema 2
Lema 2: Minimum phase component can be expresed as an all-pole system:
in theory goes to infinity, in practice is limited.

63. Linear Prediction Based Procesing
Critics to the Linear Prediction Model
Perceptual Linear Prediction (PLP)
LP Cepstra

64. Critics to the Linear Prediction Model
The LP spectrum approximate the speech spectrum equally well at all frequencies of the analysis band.
This property is inconsistent with the human hearing.

65. Precepual Linear Prediction (PLP)
Critical Band
Spectral Analysis
Equal Loudness
Pre-emphasis
Intensity
Loudness
IDFT
Yule-Walker
Equations Solutions

66. Critical Band Analysis
Speech
Signal
Frame
Critical Band
Spectral
Resolution
Short-Term
Spectra
Windowing
DFT
(20 ms)
(200 samples
56 zeros for
padding for Ts=10KHz)c
DFT
(20 ms Hamming Window

67. Critical-Band Spectral Resolution
Frequency
Warping
(Hertz -> Barks)
Convolution
and
Downsampling
filter-bank masking curve
approximation

68. Equal Loudness Pre-emphasis
Approximate the non-equal sensitivity
of the human hearing at different frequencies.

69. Intensitive Loudnes Power Law
Approximate the non-linear relation
between the intensity of sound and
its perceived loudness.

70. Cepstral Analysis Introduction Homomorphic Processing
Cepstral Spectrum
Cepstrum
Mel-Cepstrum
Cepstrum in Speech Processing

71. Introduction When speech is pre-emphasised
The excitation is not necessary for estimate the vocal tract function.
Therefore, it is desirable to separate the excitation information form
the vocal tract information.

72. We can think the speech spectrum as a signal,
we can observer that is composed for the multiplication
of a slow signal, and a fast signal,
Therefore, we can try to obtain the best of this knowledge.
The formal technique which exploit this feature is called
“Homomorphic Processing”.

73. Homomorphic Processing
It is a technique to filter no-lineal systems.
In Homomorphic Processing the non-linear related signals are transform the signal to a linear domain.
H[ ]
F(z)
H-1[ ]

74. log[ ] S+(z) exp[ ] In order to obtain a linear system a complex
log transformation is applied to the speech spectrum.
log[ ]
S+(z)
exp[ ]

75. Cepstral Spectrum
Definition.
where
is the STFT

76. Cepstrum
Definition.

77. Cepstrum In Speech Processing
Pitch Estimation
Format Estimation
Pitch and Formant Estimation

78. Pitch Estimation Sampled Speech High-Pass Liftering Emphasis Peaks
(second derivative)
Peak
finding
Cepstrum
Pitch

79. Formant Estimation Sampled Speech Low-Pass Liftering Emphasis Peaks
(second derivative)
Peak
finding
Cepstrum
Formants

80. Pitch and Formant Estimation
Sampled
Speech
High-Pass
Liftering
Emphasis Peaks
(second derivative)
Peak
finding
Cepstrum
Pitch
Low-Pass
Liftering
Emphasis Peaks
(second derivative)
Peak
finding
Formants