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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 XX is defined as ……
3
Distortion Measures
y)d(x,z)y z,d(x d)
ifinvariant called isfunction distortion headdition tin
condition) inequality
r (triangula Xyfor x, z)d(y,),(y)d(x, c)
(symmetry) X yfor x, x)d(y,y)d(x, b)
property) sdefinitnes (positive
y xiff 0y)d(x, and Xyfor x, ),(0 a)
zxd
yxd
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
9
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.
10
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.
11
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
-1---------)(logS'-)S(log)V(
22
)()()S'(S, p
d
VddP
p
13
22
)',()()S'(S, p
d
SSVddP
p
This is related to how humans perceive sound differences
14
Log spectral distance
• For P=1 the above equation defines the mean absolute log spectral distortion
• For P=2, 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
16
Cepstral distances• For the Cepstral coefficients we use the
rms log spectral distance.
1|)S(|log
spectrumpower The
)(log)(log)(log
)S( of log Taking
domainfrequency in the )()()S(
excitement and components tract vocalx(t)*h(t)S(t)
2
jn
nnec
XHS
XH
17
Cepstral distances
ly.respective )(S' and )S( of tscoefficien
cepstral are 'c and c where)'(
32
)('log)(log)',(
tscoefficien LPC
thefrom obtained becan tscoefficien Cepstral The
nn2
2
2
nnn
ceps
cc
dSSSSd
20
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
n
nnn
nnw ncncccnd 2)()( 2'2'222
24
25
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
26
Weighted cepstral distances and liftering
• A useful form of weighted cepstral distance is
• Where w(n) is any lifter function.
L
nnncw cnwcnwd
1
2'2 ))()((
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
28
Itakura and Saito
-
2
22
2
2
)(
2
dw S(w) logexp
where
ly.respective (w)S' and
S(w) of errors prediction are ' and where
1'
log2)('
)()',(
21)()',(
dw
wS
wSSSd
dwwVeSSd
IS
wVIS
29
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
energy residual theis where, 2
)(
gain theis where
1loglog2
)(1
,
2
2
222
22
2
dweAwS
dweAwS
eASd
jw
jw
jwIs
30
Itakura
2)(
)(log
1,
1
is measure distortion Itakura then the
consider uslet
2
2
22
2
dw
eA
eA
AAd
jwp
jw
p
I
31
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
2222
1,
11,
1
AAd
AAd
p
LR
p
I
32
Likelihood Distortions
When the distortion is very small the Itakura distortion measure is not very different from the likelihood distortion measure.
33
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
34
COSH distortion
• COSH distortion is given by
• The COSH distortion is almost identical to twice the log spectral distance for small distortions
12)('
)(logcosh
dw
wS
wSdCOSH
35
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
36
Weighted likelihood ratio distortion
lyrespective A'
' and
Afor
sequencesation autocorrel are )('ˆ and )(ˆ and
A'
1log and
A
1log
of tscoefficien cepstral are c' and c where
''
)('ˆ)(ˆ
2
2
2
2
22
nn
22
nrnr
ccnrnr
d nnWLR
37
Comparison of dWLR and d22
tenvironmennoisy in n recognitiospeech assuch
necessary, is peaks spectral of emphasisary extraordin
wherensapplicatio in the required isproperty This
)A'
1log-
A
1(log
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emphasisheavier shows which )A'
1-
A
1(
deviationlinear by replaced is thisdin
)A'
1log-
A
1(log dIn
22
22
WLR
2222
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
39
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
considerd bands critical
of no. total theisK and S' and Sbetween ),(')(
difference slope spectral band critical afor tscoefficien
weighting theis u(i) ,S' and Sbetween '
energy absolutefor constant weighting theis where
)(')()(')',(1
2
ii
EE
u
iiiuEEuSSd
ss
ss
E
K
issssEWSM
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.
42
Summary
• Log spectral: Lp metric requires large amount of calculations because we need 2 FFT’s to obtain S(w) and S’(w), logarithms of all values of S and S’ and an integral
p
P
p
dwSwSd
/1
2)('log)(log(
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
2'
1
2
1
2'2
)()(
)()(
nn
L
nCW
L
nnnc
ccnWd
ccLd
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
45
Summary
12)('
)(logcosh
12
2log
1'
log2)('
)()',(
2
2
2
2
2
2
dw
wS
wSd
dw
A
Ad
dw
A
Ad
dw
wS
wSSSd
COSH
p
LR
p
I
IS
46
Summary
• Weighted likelihood ratio distortion: Requires L operations, similar to that of the cepstral measures
2'
122
)('
)('ˆ)(ˆnn
L
nWLR cc
nrnrd
47
Summary
• Weighted Slope metric (WSM): Requires K operations, where K is the number of frequency bands used in computations (32-64)
K
issssEWSM iiiuEEuSSd
1
2)(')()(')',(
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