This paper presents a justification for the use of MFCC parameters in automatic pathology detection on speech. While such an application has produced good results up to now, only partial explanations to this good performance had been given before. The herein exposed explanation consists of an interpretation of the mathematical transformations involved in MFCC calculation and a statistical analysis that confirms the conclusions drawn from the theoretical reasoning

Fraile Muñoz, Rubén

Godino Llorente, Juan Ignacio

Gómez Vilda, Pedro

Osma Ruiz, Víctor

Sáenz Lechón, Nicolas

English

Archivo Digital UPM

Automatic detection of voice impairments by means of short-term cepstral parameters and neural network based detectors,”

D e l l e r ,J .G .P r o a k i s ,a n dJ .H .L .H a n s e n , Discrete-time processing of speech signals.N e w

D u d a ,P .E .H a r t ,a n dD .G .S t o r k ,Pattern classiﬁcation.

Dimensionality reduction of a pathological voice quality assessment system based on gaussian mixture models and short-term cepstral parameters,”

Discriminationof pathologicalvoicesusing a time-frequency approach,”

Disordered voice database.”

Fundamentalsof speech recognition.

Identiﬁcation of pathological voices using glottal noise measures,”

Quantiﬁcation of glottal and voiced speech harmonics-to-noise ratios using cepstral-based estimation,”

Use of cepstrum-based parameters for automatic pathology detection on speech. Analysis of performance and theoreticaljustiﬁcation,” in

Use of Mel Frequency Cepstral Coefficients for Automatic Pathology Detection on Sustained Vowel Phonations: Mathematical and Statistical Justification

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

USE OF MEL FREQUENCY CEPSTRAL COEFFICIENTS FOR
AUTOMATIC PATHOLOGY DETECTION ON SUSTAINED VOWEL
PHONATIONS: MATHEMATICAL AND STATISTICAL JUSTIFICATION
R. Fraile, N. Sa´enz-Lecho´n, J.I. Godino-Llorente, V. Osma-Ruiz, P. Go´mez-Vilda
EUIT Telecomunicacio´n. Circuits & Systems Engineering Department
Carretera de Valencia Km.7, 28031 Madrid (SPAIN)
ABSTRACT
This paper presents a justification for the use of MFCC
parameters in automatic pathology detection on speech.
While such an application has produced good results up
to now, only partial explanations to this good performance
had been given before. The herein exposed explanation
consists of an interpretation of the mathematical transfor-
mations involved in MFCC calculation and a statistical
analysis that confirms the conclusions drawn from the the-
oretical reasoning.
1. INTRODUCTION
Automatic detection of pathologies on speech has been tra-
ditionally realised through the analysis of distortion and
noise measurements taken from records of sustained vow-
els [1]. However, recently alternative approaches have also
been taken in two directions: using running text [1] and
different parameterization schemes, such as those based on
Mel-frequency Cepstral Coefficients (MFCC) [2]. While
the use of this Mel-cepstral analysis produces low error
rates, to authors’ knowledge a complete justification for
its use other than the empirical results has not been given
yet, though the same authors have approached the problem
in [3] and [4] and, previously, other authors have proven
the presence of noise information in the speech cepstrum
[5].
In this paper, a justification for the use of MFCC for
pathology detection on speech records is provided. This
consists in both a theoretical interpretation of the math-
ematical transformations involved in MFCC computation
(section 2) and a statistical analysis of the parameters ex-
tracted from records belonging to a commercial database
(section 3). In the statistical analysis, both parametric and
non-parametric approaches have been taken. While the
parametric analysis permits the use of analytic statistical
tools, the non-parametric analysis allows confirming the
This research was carried out within projects funded by the Ministry
of Science and Technology of Spain (TEC2006-12887-C02) and the Uni-
versidad Polite´cnica de Madrid (AL06-EX-PID-033). It has also been
realised within the framework of European COST action 2103.
results of the parametric analysis avoiding the model as-
sumptions.
2. MATHEMATICAL FORMULATION OF
CEPSTRAL COEFFICIENTS
The presence of pathologies in speech is closely related
to signal variability, hence the traditional use of distor-
tion measurements. The need for detecting such variabil-
ity leads to the convenience of employing short-time tech-
niques for speech processing. A mathematical framework
for short-time processing of speech is provided in [6]. Ac-
cording to it, if a discrete-time speech signal x [n] is seg-
mented in speech frames gp [n] of length L, being p the
frame index, then the short-time Discrete Fourier Trans-
form (stDFT) of each frame may be written as:
Sp (k) =
L−1∑
n=0
gp [n] · e−j·
2πn
NDFT
·k (1)
where NDFT is the number of points of the DFT and k is
the index of the DFT elements (k = 0 . . .NDFT − 1).
2.1. Short-time Cepstrum
The short-time cepstrum can be derived from the stDFT as
follows [6]:
cp [q] =
1
NDFT
NDFT−1∑
k=0
log |Sp (k)| · ej·
2πk
NDFT
·q (2)
The use of cepstrum in the assessment of pathological
voices is supported by two arguments: on the one hand,
cepstrum analysis is appropriate for estimating the noise
level of the voice signal [5] and, on the other hand, for
the case of sustained vowels, the variability of the glot-
tal waveform can also be easily detected from cepstral pa-
rameters [4]. A cepstral set of parameters calculated from
a smoothed spectrum is proposed in [4]. This smooth-
ing consists in the convolution in frequency domain of the
stDFT with a set of triangular filters, much like in the case
of MFCC [6]:
S′p (i) =
∑
fk∈Ii
(
1 − |fk − i · Δf/2|
Δf/2
)
· |Sp (k)| (3)
where Ii = [Δf · (i − 1) /2,Δf · (i + 1) /2] and fk is the
frequency associated to Sp(k). The number of filters M
depends on the choice of Δf . Consequently: i = 1 . . .M ,
and (2) becomes:
c′p [q] =
1
M + 1
M∑
k=1
log
∣∣S′p (k)∣∣ · cos
(
πk
M + 1
· q
)
(4)
If the value of Δf is chosen to be higher than the fun-
damental frequency of the voice, then the pitch informa-
tion is lost in the modified cepstrum (4). However, this
fact does not significantly affect its pathology detection
capability [4]. Nevertheless, up to now, no work on the op-
timisation of Δf (or M , equivalently) has been reported.
2.2. Short-time MFCC
Another approach for speech parametrisation in the cep-
stral domain is MFCC calculation. This approach is simi-
lar to the previous one in that it includes spectrum smooth-
ing prior to the transformation into cepstral domain. How-
ever, it differs in that the spectrum smoothing is done in
the Mel-frequency domain, hence resulting in a set of nar-
rower filters for low frequencies and wider for high fre-
quencies. Namely, the following frequency transformation
is applied [7]:
fmk = 2595 · log10
(
1 +
fk
700
)
(5)
and the smoothing is done in the transformed domain:
S˜p (i) =
∑
fm
k
∈Im
i
⎛
⎝1 −
∣∣∣fmk − Fm · iM+1 ∣∣∣
Δfm/2
⎞
⎠ · |Sp (k)|
(6)
where Imi =
[
Fm · i−1M+1 , Fm · i+1M+1
]
, M is the number
of filters, Δfm = 2M+1 · Fm corresponds to Δf and Fm
is the maximum frequency in Mel domain.
The number of Mel-band filters is commonly chosen
to be: M = 3 · log fs, but, again, the impact of the
selection of M on the performance of the whole pathol-
ogy detector has not been assessed up to now. After spec-
trum smoothing, the return to the linear frequency scale
can be done considering that the central frequencies of the
smoothing filters are fci = 700 · 10
Fm·i
2595·(M+1) . Then (4)
becomes:
c′′p [q] =
1
M + 1
M∑
k=1
log
∣∣∣S˜p (k)∣∣∣ · cos
(
2πfck
fs
· q
)
(7)
A graphical representation of both kinds of cepstral co-
efficients obtained after spectrum smoothing, namely mod-
ified cepstrum (4) and Mel-band cepstrum (7), is depicted
in figure 1. It can be noticed that the use of Mel-bandfilters
instead of the lineally spaced filters allows a reduction on
the length of cepstrum, hence a dimensionality reduction,
while keeping most of the information of the first cepstral
coefficients. This is due to the Mel-band filters providing
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Quefrency (ms)
N
or
m
al
is
ed
 c
ep
st
ra
l a
m
pl
itu
de
 
 
Modified cepstrum
Mel−band cepstrum
MFCC
Figure 1. Modified cepstrum, Mel-band cepstrum and
MFCC; all of them correspond to a speech record contain-
ing a sustained vowel /ah/.
a more detailed description of the low frequency bands of
speech and, on the opposite, averaging the less significant
high frequency bands. Nevertheless, (7) is not the com-
mon way to express cepstral coefficients obtained from
Mel-band spectrum smoothing. Instead, once the spec-
trum has been smoothed, the frequency transformation is
not reversed, usually. This gives as a result what are com-
monly known as MFCC:
c′′′p [q] =
1
M + 1
M∑
k=1
log
∣∣∣S˜p (k)∣∣∣ · cos
(
πk
M + 1
· q
)
(8)
though a frequency-shiftedversion is more frequently used
[7]:
cIVp [q] =
1
M + 1
M∑
k=1
log
∣∣∣S˜p (k)∣∣∣ · cos
(
π (k − 0.5)
M + 1
· q
)
(9)
In fact, both (8) and (9) give similar results. A repre-
sentation of the MFCC obtained using (8) is also plot in
figure 1. It can be observed that the Mel-frequency trans-
formation produces in cepstral domain somewhat of a que-
frency compression of the slower component of cepstrum,
thus reducing the quefrency interval containing significant
values of cepstrum.
As a conclussion to this section, it can be said that the
capability of cepstrum to carry information of both noise
level of sustained vowels [5] and their associated glottal
waveform [4] is kept when the Mel-band filters are used
for spectrum smoothing. Moreover, the Mel-frequency
transformation produces in cepstral domain a compression
of the quefrency axis that, in principle, should be useful in
providing some reduction in the size of speech feature vec-
tors.
3. PERFORMANCE ANALYSIS
Within this section, the performance in pathology detec-
tion of each one of the three above-described approaches
for speech parametrisation in cepstral domain is analysed.
A three-step analysis is presented. The first step consists
in assessing the relevance of each individual parameter for
pathology detection using the Fisher linear discriminant
0 200 400 600 800 1000 1200
0
0.5
1
Modified cepstrum
Quefrency (ms)
F
is
he
r 
cr
ite
rio
n 
F
0 200 400 600 800 1000 1200
0
1
2
Mel−band cepstrum
Quefrency (ms)
F
is
he
r 
cr
ite
rio
n 
F
0 200 400 600 800 1000 1200
0
1
2
MFCC
Quefrency (ms)
F
is
he
r 
cr
ite
rio
n 
F
Figure 2. Value of the Fisher criterion for each cepstral pa-
rameter: modified cepstrum (4) -up-, Mel-band cepstrum
(7)-middle- and MFCC (8) -down-.
criterion. The second step is based on the typical “Gaus-
sian” assumption for the distribution of the speech seg-
ments in the feature vector space. The last part consists in
studying the performance of non-parametric classifiers so
as to confirm the results obtained in the second part.
For all three parts, the well-known Kay speech record
database is used [8]. More specifically, a subset containing
53 normal and 173 pathological records, each consisting of
a sustained phonation (1-3 s long) of the vowel /ah/ [9].
This subset covers a wide variety of voice disorders and
the distribution of speakers is balanced in age and gender.
The sampling rate of speech records has been made equal
to 25 kHz, while coding has a resolution of 16 bits. For
short-time processing, each speech record has been split
in 20 ms frames and cepstral analysis has been performed
for each individual frame.
3.1. Relevance of individual parameters
The first step in the analysis has consisted in assessing the
relevance of each individual cepstral parameter for pathol-
ogy detection. More specifically, the parameters given by
(4) for Δf = 200 Hz and (7) and (8) for M = 31 have
been analysed by means of the Fisher linear discriminant
[10]. The greater values of the discriminant correspond to
the greater relevances of the parameters for detection.
Figure 2 shows the calculated values of the Fisher dis-
criminant for all the cepstral coefficients of each scheme.
The obtained results confirm the observations made in the
previous section. Firstly, the spectrum smoothing using
Mel-band filters allows both a reduction in the length of
cepstrum and an increase in the relevance of computed pa-
rameters for pathology detection. This can be concluded
from comparison of the top and middle graphs in figure
2. Secondly, observation of both the middle and bottom
graphs indicates that keeping the Mel-frequency transfor-
mation when passing from the spectral to the cepstral do-
main further increases the relevance of the first cepstral
parameters while reducing that of the rest.
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
Length of the feature vector (num. of parameters)
M
ah
al
an
ob
is
 d
is
ta
nc
e
 
 
Cepstrum from smoothed spectrum
Mel−band cepstrum
MFCC
Figure 3. Mahalanobis distance as a function of the length
of the feature vector for the three types of cepstral analysis
under study.
3.2. Parametric analysis of feature vectors
In general, the analysis of individual parameters, such as
the one reported before, is not fully significant when such
individual parameters belong to multidimensional feature
vectors. This is due to the joint relevance of parameters
not depending only on their individual relevances, but also
on the correlation between their values. It is hard to find an
statistical test that considers both individual relevance and
correlation for arbitrarily distributed feature vectors. For
this reason, such distribution has been assumed Gaussian
and, based on it, the Mahalanobis distance [10] has been
taken as a performance indicator for each combination of
parameters. Its value is directly related to the performance
of the feature vector, that is, the greater the Mahalanobis
distance associated to a feature vector, the less detection
error rate should be expected from that vector.
In the herein reported analysis, the values of the Ma-
halanobis distance have been calculated over the above-
mentioned database. The selection of parameters in order
to compose the feature vectors has been realised as fol-
lows: first, the parameter yielding the highest Fisher dis-
criminant value has been chosen and, afterwards, an iter-
ative process has been run in order to, at each step, add
to the feature vector the parameter that gave the greatest
increase in the Mahalanobis distance. Results are plot in
figure 3.
The relation of the calculated Mahalanobis distance
with the length of the feature vector indicates that simi-
lar performances are to be expected from the three sets
of cepstral parameters, but the use of Mel-band spectrum
smoothing allows achieving that performance with shorter
feature vectors. To give a figure, while the asymptotic be-
haviour of the MFCC and Mel-band cepstral vectors is
reached with vector lengths between 15 and 20, lengths
well above 26 are necessary for modified cepstral vec-
tors. Looking at another aspect of the graph in figure
3, the performance of Mel-band cepstrum and MFCC is
very similar, though MFCC provide better performance
for very short feature vectors. This is coherent with pre-
vious results, since the first element of the MFCC vector
is more significant than the first element of the Mel-band
cepstrum.
The same approach including the evaluation of the Ma-
0 5 10 15 20
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
Smoothed Spectrum
F
ra
m
e 
E
qu
al
 E
rr
or
 R
at
e
Num. of parameters
0 5 10 15 20
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
Mel−band Cepstrum
F
ra
m
e 
E
qu
al
 E
rr
or
 R
at
e
Num. of parameters
0 5 10 15 20
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
MFCC
F
ra
m
e 
E
qu
al
 E
rr
or
 R
at
e
Num. of parameters
0 5 10 15 20
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
Num. of parameters
F
ra
m
e 
E
qu
al
 E
rr
or
 R
at
e
Comparison or regressions
 
 
Smoothed Spectrum
Mel−band Cepstrum
MFCC
Figure 4. Comparison between the three schemes in terms
of performance as the length of the feature vector in-
creases. Potential regressions have been plotted so as to
ease interpretation.
halanobis distance has been undertaken so as to assess the
impact of the spectral filter width selection on the per-
formance of each parameter set. For Mel-band cepstrum
the option M = 3 · log fs = 31 has been found to pro-
vide somewhat of an upper limit on the performance. For
MFCC slight improvements might be achieved by increas-
ing this value; however, the impact of increasing M has
shown to be much more significant for M < 31 than for
M > 31; thus M = 31 appears to be a sensible option
too. For the case of modified cepstrum, the same trend oc-
curs: for narrow filters (larger M ) performance is better
than for wider filters (lower M ). Namely, Δf = 100 Hz
and Δf = 200 Hz exhibit similar performace, while for
Δf > 200 Hz the performance degrades.
3.3. Non-parametric analysis
In order to confirm the previous results, a non-parametric
analysis has been realised. This consists in measuring the
equal error rate (EER) achieved with each one of the three
schemes when the classification of the feature vectors is
carried out by a Multilayer Perceptron (MLP), which is
a general purpose non-parametric classifier [10]. Depen-
dence of the achieved EER on the length of the feature
vector is depicted in figure 4. The selection of features for
each case has been done according to the criterion based
on the Mahalanobis distance and explained before. The
graph is fully coherent with that of figure 3: MFCC and
Mel-band cepstrum have very similar performances, with
a slight advantage of MFCC for short vectors, whilst modi-
fied cepstrum needs longer feature vectors to achieve com-
parable performance.
4. CONCLUSION
Within this paper, it has been shown that MFCC are rele-
vant parameters for automatic pathology detection. Math-
ematically, it has been shown that their computation allows
to concentrate the noise information present in speech cep-
strum in a few cepstral coefficients, namely those corre-
sponding to the lowest quefrencies. Such conclusion has
been confirmed by means of a statistical analysis. This
analysis has also demonstrated that MFCC provide certain
advantage over other similar cepstral analyses in terms of
dimensionality reduction.
5. REFERENCES
[1] K. Umapathy, S. Krishnan, V. Parsa, and D. G.
Jamieson, “Discrimination of pathological voices us-
ing a time-frequency approach,” IEEE Trans. on
Biomed. Eng., vol. 52, no. 3, pp. 421–430, Mar 2005.
[2] J. I. Godino-Llorente and P. Go´mez-Vilda, “Auto-
matic detection of voice impairments by means of
short-term cepstral parameters and neural network
based detectors,” IEEE Trans. on Biomed. Eng.,
vol. 51, no. 2, pp. 380–384, Feb 2004.
[3] J. I. Godino-Llorente, P. Go´mez-Vilda, and
M. Blanco-Velasco, “Dimensionality reduction of a
pathological voice quality assessment system based
on gaussian mixture models and short-term cepstral
parameters,” IEEE Trans. on Biomed. Eng., vol. 53,
no. 10, pp. 1943–1953, Oct 2006.
[4] R. Fraile, J. I. Godino-Llorente, N. Sa´enz-Lecho´n,
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cepstrum-based parameters for automatic pathology
detection on speech. Analysis of performance and
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vol. 1, Jan 2008, pp. 85–91.
[5] P. J. Murphy and O. O. Akande, “Quantification of
glottal and voiced speech harmonics-to-noise ratios
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LISP’05), 2005, pp. 224–232.
[6] J. R. Deller, J. G. Proakis, and J. H. L. Hansen,
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[7] L. Rabiner and B. H. Juang, Fundamentals of speech
recognition. Englewood Cliffs (USA): Prentice-
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[8] Kay Elemetrics Corp., “Disordered voice database.”
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[9] V. Parsa and D. G. Jamieson, “Identification of
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[10] R. O. Duda, P. E. Hart, and D. G. Stork, Pattern clas-
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Use of Mel Frequency Cepstral Coefficients for Automatic Pathology Detection on Sustained Vowel Phonations: Mathematical and Statistical Justification

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