Topics covered in this chapter

1

• Topics covered in this chapter
• Three basic problems in pattern comparison
• How to detect the speech signal in a recording
  interval (i.e. separate speech from background)
• How to locally compare spectra from two speech
  utterances (local spectral distortion measure),
  and
• How to globally align and normalize the distance
  between two speech patterns (sequences of
  spectral vectors) which may or may not represent
  the same linguistic sequence of sounds (word,
  phrase, sentence, etc.)

2
Distortion Measures

• Mathematical considerations to find out the
  dissimilarity between two feature vectors.
• Let x and y are two vectors defined on a vector
  space X.
• A metric or distance function d on the vector
  space X as a real valued function on the
  Cartesian product X?X is defined as

3
Distortion Measures
4
Distortion Measures

• If a measure of a distance d, satisfies only the
  positive definiteness property then it is called
  as distortion measure if vectors are
  representation of the speech spectra.
• Distance in speech recognition means measure of
  dissimilarity.
• For speech processing, an important consideration
  in choosing a measure of distance is its
  subjective meaningfulness
• The mathematical measure of distance to be useful
  in speech processing should consider the
  lingustic characteristics.

5
Distortion Measures

• For example a large difference in the waveform
  error does not always imply large subjective
  differences.

6
Distortion Measures

• Perceptual considerations the choice of an
  appropriate measure of spectral dissimilarity is
  the concept of subjective judgment of sound
  difference or phonetic relevance.
• Spectral changes that keep the sound the same
  perceptually should be associated with small
  distances.
• And spectral changes that keep the sound the
  different perceptually should be associated with
  large distances

7
Distortion Measures

• Consider comparing two spectral representations,
  S(w) and S(w) using a distance measure d(S,S)
• If the spectral content of two signal are
  phonetically same (same sound) then the distance
  measure d is ideally very small

8
Distortion Measures

• Spectral changes due to large phonetic distance
  include
• Significant differences in formant locations. i.e
  the spectral resonance of S(w) and S(w) occure
  at very different frequencies.
• Significant differences in formant bandwidths.
  i.e the frequency widths of spectral resonance of
  S(w) and S(w) are very different.
• For each of these cases sounds are different so
  the spectral distance measure d(S,S) is ideally
  very large

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Distortion Measures

• To relate a physical measure of difference to
  subjective perceived measure of difference it is
  important to understand auditory sensitivity to
  changes in frequencies, bandwidths of the speech
  spectrum, signal sensitivity and fundamental
  frequency.

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Distortion Measures

• This sensitivity is presented in the form of just
  discriminable change the change in a physical
  parameter such that the auditory system can
  reliably detect the change as measured in
  standard listening test.

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Spectral-distortion measures

• Measuring the difference between two speech
  patterns in terms of average spectral distortion
  is reasonable way both in terms of its
  mathematically tractability and its computational
  efficiency
• Perceived sound differences can be interpreted in
  terms of differences of spectral features

12
Log spectral distance

• Consider two spectra S(w) and S(w). The
  difference between two spectra on a log magnitude
  versus frequency scale is defined by
• A distance or distortion measure between S and S
  can be defined by

13
This is related to how humans perceive sound
differences
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Log spectral distance

• For P1 the above equation defines the mean
  absolute log spectral distortion
• For P2, equation defines the rms log spectral
  distortion that has application in many speech
  processing systems
• For P tends to infinity, equation reduces to the
  peak log spectral distrotion

15
Log spectral distance

• Since perceived loudness of a signal is
  approximately logarithmic, the log spectral
  distance family appears to be closely tied to the
  subjective assessment of sound differences
  hence, it is perceptually relevant distortion
  measure
• We can calculate the distortion using short time
  FFT power spectra and by LPC model spectra (all
  pole smooth model spectra)
• The smooth spectral difference allows a closer
  examination of the properties of the distortion
  measure.

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Cepstral distances

• For the Cepstral coefficients we use the rms log
  spectral distance.

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Cepstral distances
18
Cepstral distances

• Since the cepstrum is a decaying sequence, the
  summation in equation 3 does not require an
  infinite number of terms
• The number of terms must be no less than p
  (cepstral coefficients)
• The truncated cepstral distance is defined as
• The truncated cepstral distance is a very
  efficient method for estimation the rms log
  spectral distance.

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Weighted cepstral distances and liftering

• Several other properties of the cepstrum when
  properly utilized are beneficial for speech
  recognition applications
• It can be shown that under certain regular
  conditions, the cepstral coefficients except c0
  have
• Zero means
• Variance essentially inversely proportional to
  the square of the coefficient index, s.t

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Weighted cepstral distances and liftering

• Liftering makes the system more robust to noise,
• Liftering is done to obtain the equal variance
• Liftering is significant for the improvement for
  the recognition performance
• If we incorporate n2 factor into the cepstral
  distance to normalize the contribution from each
  cepstarl term, the distance

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Weighted cepstral distances and liftering

• The variability of higher capstral coefficients
  are more influenced by the inherent objects
  (artifacts) of LPC analysis than that of lower
  cepstral coefficients.
• For speech recognition, therefore, suppression of
  higher cepstral coefficients in the calculation
  of a cepstral distance should lead to a more
  reliable measurement of spectral differences than
  otherwise

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Weighted cepstral distances and liftering

• The Lpc spectrum also includes components that
  are strong functions of the speakers glottal
  shape and vocal cord duty cycles.
• These components affects mainly the first few
  cepstral coefficients.
• For speech recognition the phonetic content of
  the sound is important and not these components
  so these components are need to be de-emphasized

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Weighted cepstral distances and liftering

• A cepstral weighting or liftering procedure, w(n)
  can therefore be designed to control the non
  information-bearing cepstral variabilities for
  reliable discrimination of sounds.
• The index weighting as used in equation 2 is the
  example of the simple form of cepstral weighting

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Weighted cepstral distances and liftering

• The original sharp spectral peaks are highly
  sensitive to the LPC analysis condition and the
  resulting peakiness creates unnecessary
  sensitivity in spectral comparison
• The liftering process tends to reduce the
  sensitivity without altering the fundamental
  formant structure.
• i.e the undesirable (noiselike) components of the
  LPC spectrum are reduced or removed, while
  essential characteristics of the formant
  structure are retained

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Weighted cepstral distances and liftering

• A useful form of weighted cepstral distance is
• Where w(n) is any lifter function.

27
Itakura and Saito

• The log spectral difference V(w) is defined by
  V(w) log S(w) log S(w) is the basis of many
  distortion measures
• The distortion measure proposed by Itakura and
  Saito in their formulation of linear prediction
  as an approximate maximum likelihood estimation is

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Itakura and Saito
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Itakura and Saito

• The Itakura Satio distortion measure can be used
  to illustrate the spectral matching properties by
  replacing S(w) with the pth order all pole
  spectrum

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Itakura
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Likelihood Distortions

• The role of the gain terms is not explicit in the
  Itakura distortion because the signal level
  essentially makes no difference in the human
  understanding of speech so long as it is
  unambiguously heard.
• Gain independent distortion measure called
  likelihood ration distortion can be derived
  directly from IS distortion measure

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Likelihood Distortions

• When the distortion is very small the Itakura
  distortion measure is not very different from the
  likelihood distortion measure.

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Variations of likelihood distortions

• Compare to the cepstral distance likelihood
  distortions are asymmetric.
• To symmetries the distortion measure there are
  two methods
• COSH distortion
• Weighted likelihood distortion

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COSH distortion

• COSH distortion is given by
• The COSH distortion is almost identical to twice
  the log spectral distance for small distortions

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Weighted likelihood ratio distortion

• The purpose of weighting is to take the spectral
  shape into account as a weighting function such
  that different spectral components along
  frequency axis can be emphasized or de-emphasized
  to reflect some of the observed perceptual
  effects

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Weighted likelihood ratio distortion
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Comparison of dWLR and d22
38
Weighted slope metric distortion measure

• Based on a series of experiments designed to
  measure the subjective phonetic distance
  between pairs of synthetic vowels and fricatives,
  it is found that by controlled variation of
  several acoustic parameters and spectral
  distortions including formant frequency, formant
  amplitude, spectral tilt, highpass, lowpass, and
  notch filtering only formant frequency deviation
  was phonetically relevant

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Weighted slope metric distortion measure

• WSM attach a weight on the spectral slope
  difference near spectral peaks, rather than the
  spectral amplitude difference, and take the
  overall energy difference explicitly into
  consideration

S
40
Summary

• The spectral distortion measures are designed to
  measure dissimilarity or distance between two
  (power) spectra of speech
• Many of these dissimilarity measures are not
  metrics because they do not satisfy the symmetry
  property
• If an objective speech distortion measure needs
  to reflect the subjective reality of human
  perception of sound differences, or even phonetic
  disparity, the asymmetry seems to be actual
  desirable.

S
41
Summary

• All distortion measures are equally important
  because certain distortion measures may be better
  for an less noisy environment, while others may
  be robust when the background is more noisy.

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Summary

• Log spectral Lp metric requires large amount of
  calculations because we need 2 FFTs to obtain
  S(w) and S(w), logarithms of all values of S and
  S and an integral

43
Summary

• Truncated and weighted cepstral Requires only L
  operations where L is of the order of 12-16 hence
  calculations required are less compared to Lp
  metric

44
Summary

• The likelihood, Itakura-Saito, Itakura and COSH
  measurements all requires on the order of p is
  the LPC order of all pole polynomial (8-12).
  Hence the computations are same for cepstral
  measures

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Summary
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Summary

• Weighted likelihood ratio distortion Requires L
  operations, similar to that of the cepstral
  measures

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Summary

• Weighted Slope metric (WSM) Requires K
  operations, where K is the number of frequency
  bands used in computations (32-64)

48
Summary

• From all these points we can say that all the
  measures are both physically reasonable and
  computationally tractable for speech recognition
  except for the Lp metrics.
• Hence, practically we are going to use all the
  measures to study the speech recognition system