1. PATTERN COMPARISON TECHNIQUES
Test Pattern:
Reference Pattern:

2. 4.2 SPEECH (ENDPIONT) DETECTION

3. 4.3 DISTORTION MEASURES-MATHEMATICAL CONSIDERATIONS
x and y: two feature vectors defined on a vector space X
The properties of metric or distance function d:
A distance function is called invariant if

4. PERCEPTUAL CONSIDERATIONS
Spectral changes that do not fundamentally
change the perceived sound include:

5. PERCEPTUAL CONSIDERATIONS
Spectral changes that lead to
phonetically different sounds include:

6. PERCEPTUAL CONSIDERATIONS
Just-discriminable change: known as JND (just-noticeable difference), DL (difference limen), or differential threshold

7. 4.4 DISTORTION MEASURES-PERCEPTUAL CONSIDERATIONS

8. 4.4 DISTORTION MEASURES-PERCEPTUAL CONSIDERATIONS

9. Spectral Distortion Measures
Spectral Density
Fourier Coefficients of
Spectral Density
Autocorrelation Function

10. Spectral Distortion Measures
Short-term autocorrelation
Then is an energy spectral density

11. Spectral Distortion Measures
Autocorrelation matrices

12. Spectral Distortion Measures
If σ/A(z) is the all-pole model for the speech spectrum,
The residual energy resulting from “inverse filtering”
the input signal with an all-zero filter A(z) is:

13. Spectral Distortion Measures
Important properties of all-pole modeling:
The recursive minimization relationship:

14. LOG SPECTRAL DISTANCE

15. LOG SPECTRAL DISTANCE

16. CEPSTRAL DISTANCES The complex cepstrum of a signal is defined as
The Fourier transform of log of the signal spectrum.

17. Truncated cepstral distance
CEPSTRAL DISTANCES
Truncated cepstral distance

18. CEPSTRAL DISTANCES

19. CEPSTRAL DISTANCES

20. Weighted Cepstral Distances and Liftering
It can be shown that under certain regular conditions, the cepstral coefficients, except c0, have:
Zero means
Variances essentially inversed proportional to the square of the coefficient
index:
If we normalize the cepstral
distance by the variance inverse:

21. Weighted Cepstral Distances and Liftering
Differentiating both sides of the Fourier series equation of spectrum:
This is an L2 distance based upon the differences
between the spectral slopes

22. Cepstral Weighting or Liftering Procedure
h is usually
chosen as L/2
and L is typically
10 to 16

23. weighted cepstral distance:
A useful form of
weighted cepstral distance:

24. Likelihood Distortions
Previously defined:
Itakura-Saito
distortion measure
Where and are one-step prediction errors
of and as defined:

26. Likelihood Distortions
The residual energy can be easily evaluated by:

27. Likelihood Distortions
By replacing by its optimal p-th order LPC model spectrum:
If we set σ2 to match the residual energy α :
Which is often referred to as
Itakura distortion measure

28. Likelihood Distortions
Another way to write the Itakura distortion measure is:
Another gain-independent distortion measure is called the
Likelihood Ratio distortion:

29. 4.5.4 Likelihood Distortions

30. 4.5.4 Likelihood Distortions
That is, when the distortion is small, the Itakura distortion measure
is not very different from the LR distortion measure

31. 4.5.4 Likelihood Distortions

32. 4.5.4 Likelihood Distortions
Consider the Itakura-Saito distortion between
the input and output of a linear system H(z)

33. 4.5.4 Likelihood Distortions

34. 4.5.4 Likelihood Distortions

35. 4.5.5 Variations of Likelihood Distortions
Symmetric distortion measures:

36. 4.5.5 Variations of Likelihood Distortions
COSH distortion

37. 4.5.5 Variations of Likelihood Distortions

38. 4.5.6 Spectral Distortion Using a Warped Frequency Scale
Psychophysical studies have shown that human perception of the
frequency Content of sounds does not follow a linear scale.
This research has led to the idea of defining subjective pitch
of pure tones.
For each tone with an actual frequency, f, measured in Hz,
a subjective pitch is measured on a scale called the “mel” scale.
As a reference point, the pitch of a 1 kHz tone, 40 dB above the
perceptual hearing threshold, is defined as 1000 mels.

40. 4.5.6 Spectral Distortion Using a Warped Frequency Scale

41. 4.5.6 Spectral Distortion Using a Warped Frequency Scale

42. 4.5.6 Spectral Distortion Using a Warped Frequency Scale

43. Examples of
Critical bandwidth

44. Warped cepstral distance
b is the frequency in Barks, S(θ(b)) is the spectrum on a
Bark scale, and B is the Nyquist frequency in Barks.

45. 4.5.6 Spectral Distortion Using a Warped Frequency Scale
Where the warping function is defined by

46. 4.5.6 Spectral Distortion Using a Warped Frequency Scale

47. 4.5.6 Spectral Distortion Using a Warped Frequency Scale

48. 4.5.6 Spectral Distortion Using a Warped Frequency Scale

49. 4.5.6 Spectral Distortion Using a Warped Frequency Scale
Mel-frequency cepstrum:
is the output power of the triangular filters
Mel-frequency cepstral distance

50. 4.5.7 Alternative Spectral Representations and Distortion Measures

51. 4.5.7 Alternative Spectral Representations and Distortion Measures

52. 4.5.7 Alternative Spectral Representations and Distortion Measures

53. Summary of Spectral Distortion Measures
Notation
Expression
Computation

54. Summary of Spectral Distortion Measures
Computation
Expression
Notation
Distortion Measure

55. Summary of Spectral Distortion Measures
Computation
Expression
Notation
Distortion Measure

56. 4.6 INCORPORATION OF SPECTRAL DYNAMIC FEATURES INTO THE DISTORTION MEASURE

57. 4.6 INCORPORATION OF SPECTRAL DYNAMIC FEATURES INTO THE DISTORTION MEASURE
Fitting the cepstral trajectory
by a second order polynomial,
Choose h1, h2, h3 such that
E is minimized.
Differentiating E with respect
to h1, h2, and h3 and setting
to zero results in 3 equations:

58. The solutions to these equations are:
4.6 INCORPORATION OF SPECTRAL DYNAMIC FEATURES INTO THE DISTORTION MEASURE
The solutions to these equations are:

59. 4.6 INCORPORATION OF SPECTRAL DYNAMIC FEATURES INTO THE DISTORTION MEASURE

60. 4.6 INCORPORATION OF SPECTRAL DYNAMIC FEATURES INTO THE DISTORTION MEASURE

61. 4.6 INCORPORATION OF SPECTRAL DYNAMIC FEATURES INTO THE DISTORTION MEASURE
A differential spectral distance:
A second differential spectral distance:

62. Cepstral weighting or liftering by differentiating
4.6 INCORPORATION OF SPECTRAL DYNAMIC FEATURES INTO THE DISTORTION MEASURE
Cepstral weighting or liftering by differentiating

63. A weighted differential cepstral distance:
4.6 INCORPORATION OF SPECTRAL DYNAMIC FEATURES INTO THE DISTORTION MEASURE
A weighted differential cepstral distance:

64. Taking the L2 distance as an example:
4.6 INCORPORATION OF SPECTRAL DYNAMIC FEATURES INTO THE DISTORTION MEASURE
Taking the L2 distance as an example: