<s>
In	O
Fourier	O
analysis	O
,	O
the	O
cepstrum	B-Algorithm
(	O
;	O
plural	O
cepstra	O
,	O
adjective	O
cepstral	B-Algorithm
)	O
is	O
the	O
result	O
of	O
computing	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
IFT	O
)	O
of	O
the	O
logarithm	O
of	O
the	O
estimated	O
signal	O
spectrum	O
.	O
</s>
<s>
The	O
power	O
cepstrum	B-Algorithm
has	O
applications	O
in	O
the	O
analysis	O
of	O
human	O
speech	O
.	O
</s>
<s>
The	O
term	O
cepstrum	B-Algorithm
was	O
derived	O
by	O
reversing	O
the	O
first	O
four	O
letters	O
of	O
spectrum	O
.	O
</s>
<s>
P	O
.	O
Bogert	O
,	O
M	O
.	O
J	O
.	O
R	O
.	O
Healy	O
,	O
and	O
J	O
.	O
W	O
.	O
Tukey	O
,	O
The	O
Quefrency	O
of	O
Time	O
Series	O
for	O
Echoes	O
:	O
Cepstrum	B-Algorithm
,	O
Pseudo	O
Autocovariance	O
,	O
Cross-Cepstrum	O
and	O
Saphe	O
Cracking	O
,	O
Proceedings	O
of	O
the	O
Symposium	O
on	O
Time	O
Series	O
Analysis	O
(	O
M	O
.	O
Rosenblatt	O
,	O
Ed	O
)	O
Chapter	O
15	O
,	O
209-243	O
.	O
</s>
<s>
)	O
,	O
liftering	O
,	O
or	O
cepstral	B-Algorithm
analysis	O
.	O
</s>
<s>
The	O
concept	O
of	O
the	O
cepstrum	B-Algorithm
was	O
introduced	O
in	O
1963	O
by	O
B	O
.	O
P	O
.	O
Bogert	O
,	O
M	O
.	O
J	O
.	O
Healy	O
,	O
and	O
J	O
.	O
W	O
.	O
Tukey	O
.	O
</s>
<s>
Mathematically	O
it	O
deals	O
with	O
the	O
problem	O
of	O
deconvolution	B-Algorithm
of	O
signals	O
in	O
the	O
frequency	O
space	O
.	O
</s>
<s>
The	O
terms	O
"	O
quefrency	O
"	O
,	O
"	O
alanysis	O
"	O
,	O
"	O
cepstrum	B-Algorithm
"	O
and	O
"	O
saphe	O
"	O
were	O
invented	O
by	O
the	O
authors	O
by	O
rearranging	O
the	O
letters	O
in	O
frequency	O
,	O
analysis	O
,	O
spectrum	O
,	O
and	O
phase	O
.	O
</s>
<s>
The	O
cepstrum	B-Algorithm
is	O
the	O
result	O
of	O
following	O
sequence	O
of	O
mathematical	O
operations	O
:	O
</s>
<s>
The	O
cepstrum	B-Algorithm
is	O
used	O
in	O
many	O
variants	O
.	O
</s>
<s>
The	O
following	O
abbreviations	O
are	O
used	O
in	O
the	O
formulas	O
to	O
explain	O
the	O
cepstrum	B-Algorithm
:	O
</s>
<s>
for	O
a	O
continuous	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
a	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
DFT	O
)	O
or	O
even	O
a	O
z-transform	B-Algorithm
,	O
as	O
the	O
z-transform	B-Algorithm
is	O
a	O
generalization	O
of	O
the	O
DFT.Inverse	O
of	O
the	O
fourier	O
transformLogarithm	O
of	O
x	O
.	O
</s>
<s>
The	O
"	O
cepstrum	B-Algorithm
"	O
was	O
originally	O
defined	O
as	O
power	O
cepstrum	B-Algorithm
by	O
the	O
following	O
relationship	O
:	O
</s>
<s>
The	O
power	O
cepstrum	B-Algorithm
has	O
main	O
applications	O
in	O
analysis	O
of	O
sound	O
and	O
vibration	O
signals	O
.	O
</s>
<s>
Due	O
to	O
this	O
formula	O
,	O
the	O
cepstrum	B-Algorithm
is	O
also	O
sometimes	O
called	O
the	O
spectrum	O
of	O
a	O
spectrum	O
.	O
</s>
<s>
which	O
provides	O
a	O
relationship	O
to	O
the	O
real	O
cepstrum	B-Algorithm
(	O
see	O
below	O
)	O
.	O
</s>
<s>
The	O
real	O
cepstrum	B-Algorithm
is	O
directly	O
related	O
to	O
the	O
power	O
cepstrum	B-Algorithm
:	O
</s>
<s>
It	O
is	O
derived	O
from	O
the	O
complex	O
cepstrum	B-Algorithm
(	O
defined	O
below	O
)	O
by	O
discarding	O
the	O
phase	O
information	O
(	O
contained	O
in	O
the	O
imaginary	O
part	O
of	O
the	O
complex	O
logarithm	O
)	O
.	O
</s>
<s>
The	O
complex	O
cepstrum	B-Algorithm
was	O
defined	O
by	O
Oppenheim	O
in	O
his	O
development	O
of	O
homomorphic	O
system	O
theory	O
.	O
</s>
<s>
Therefore	O
:	O
The	O
complex	O
cepstrum	B-Algorithm
can	O
be	O
also	O
written	O
as	O
:	O
</s>
<s>
The	O
complex	O
cepstrum	B-Algorithm
retains	O
the	O
information	O
about	O
the	O
phase	O
.	O
</s>
<s>
phase	O
spectrum	O
=	O
(	O
complex	O
cepstrum	B-Algorithm
−	O
time	O
reversal	O
of	O
complex	O
cepstrum	B-Algorithm
)	O
2	O
.	O
</s>
<s>
The	O
independent	O
variable	O
of	O
a	O
cepstral	B-Algorithm
graph	O
is	O
called	O
the	O
quefrency	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
the	O
sampling	O
rate	O
of	O
an	O
audio	O
signal	O
is	O
44100Hz	O
and	O
there	O
is	O
a	O
large	O
peak	O
in	O
the	O
cepstrum	B-Algorithm
whose	O
quefrency	O
is	O
100	O
samples	O
,	O
the	O
peak	O
indicates	O
the	O
presence	O
of	O
a	O
fundamental	O
frequency	O
that	O
is	O
44100/100	O
=	O
441Hz	O
.	O
</s>
<s>
This	O
peak	O
occurs	O
in	O
the	O
cepstrum	B-Algorithm
because	O
the	O
harmonics	O
in	O
the	O
spectrum	O
are	O
periodic	O
and	O
the	O
period	O
corresponds	O
to	O
the	O
fundamental	O
frequency	O
,	O
since	O
harmonics	O
are	O
integer	O
multiples	O
of	O
the	O
fundamental	O
frequency	O
.	O
</s>
<s>
The	O
kepstrum	O
,	O
which	O
stands	O
for	O
"	O
Kolmogorov-equation	O
power-series	O
time	O
response	O
"	O
,	O
is	O
similar	O
to	O
the	O
cepstrum	B-Algorithm
and	O
has	O
the	O
same	O
relation	O
to	O
it	O
as	O
expected	O
value	O
has	O
to	O
statistical	O
average	O
,	O
i.e.	O
</s>
<s>
cepstrum	B-Algorithm
is	O
the	O
empirically	O
measured	O
quantity	O
,	O
while	O
kepstrum	O
is	O
the	O
theoretical	O
quantity	O
.	O
</s>
<s>
It	O
was	O
in	O
use	O
before	O
the	O
cepstrum	B-Algorithm
.	O
</s>
<s>
The	O
autocepstrum	O
is	O
defined	O
as	O
the	O
cepstrum	B-Algorithm
of	O
the	O
autocorrelation	O
.	O
</s>
<s>
The	O
autocepstrum	O
is	O
more	O
accurate	O
than	O
the	O
cepstrum	B-Algorithm
in	O
the	O
analysis	O
of	O
data	O
with	O
echoes	O
.	O
</s>
<s>
Playing	O
further	O
on	O
the	O
anagram	O
theme	O
,	O
a	O
filter	O
that	O
operates	O
on	O
a	O
cepstrum	B-Algorithm
might	O
be	O
called	O
a	O
lifter	O
.	O
</s>
<s>
The	O
cepstrum	B-Algorithm
can	O
be	O
seen	O
as	O
information	O
about	O
the	O
rate	O
of	O
change	O
in	O
the	O
different	O
spectrum	O
bands	O
.	O
</s>
<s>
It	O
was	O
originally	O
invented	O
for	O
characterizing	O
the	O
seismic	O
echoes	O
resulting	O
from	O
earthquakes	O
and	O
bomb	B-Device
explosions	O
.	O
</s>
<s>
It	O
has	O
also	O
been	O
used	O
to	O
determine	O
the	O
fundamental	O
frequency	O
of	O
human	O
speech	O
and	O
to	O
analyze	O
radar	B-Application
signal	O
returns	O
.	O
</s>
<s>
Cepstrum	B-Algorithm
pitch	O
determination	O
is	O
particularly	O
effective	O
because	O
the	O
effects	O
of	O
the	O
vocal	O
excitation	O
(	O
pitch	O
)	O
and	O
vocal	O
tract	O
(	O
formants	O
)	O
are	O
additive	O
in	O
the	O
logarithm	O
of	O
the	O
power	O
spectrum	O
and	O
thus	O
clearly	O
separate	O
.	O
</s>
<s>
The	O
cepstrum	B-Algorithm
is	O
a	O
representation	O
used	O
in	O
homomorphic	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
,	O
to	O
convert	O
signals	O
combined	O
by	O
convolution	B-Language
(	O
such	O
as	O
a	O
source	O
and	O
filter	O
)	O
into	O
sums	O
of	O
their	O
cepstra	O
,	O
for	O
linear	O
separation	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
power	O
cepstrum	B-Algorithm
is	O
often	O
used	O
as	O
a	O
feature	O
vector	O
for	O
representing	O
the	O
human	O
voice	O
and	O
musical	O
signals	O
.	O
</s>
<s>
For	O
these	O
applications	O
,	O
the	O
spectrum	O
is	O
usually	O
first	O
transformed	O
using	O
the	O
mel	B-Architecture
scale	I-Architecture
.	O
</s>
<s>
The	O
result	O
is	O
called	O
the	O
mel-frequency	B-Algorithm
cepstrum	I-Algorithm
or	O
MFC	O
(	O
its	O
coefficients	O
are	O
called	O
mel-frequency	B-Algorithm
cepstral	I-Algorithm
coefficients	I-Algorithm
,	O
or	O
MFCCs	O
)	O
.	O
</s>
<s>
The	O
cepstrum	B-Algorithm
is	O
useful	O
in	O
these	O
applications	O
because	O
the	O
low-frequency	O
periodic	O
excitation	O
from	O
the	O
vocal	O
cords	O
and	O
the	O
formant	O
filtering	O
of	O
the	O
vocal	O
tract	O
,	O
which	O
convolve	O
in	O
the	O
time	O
domain	O
and	O
multiply	O
in	O
the	O
frequency	O
domain	O
,	O
are	O
additive	O
and	O
in	O
different	O
regions	O
in	O
the	O
quefrency	O
domain	O
.	O
</s>
<s>
Note	O
that	O
a	O
pure	O
sine	O
wave	O
can	O
not	O
be	O
used	O
to	O
test	O
the	O
cepstrum	B-Algorithm
for	O
its	O
pitch	O
determination	O
from	O
quefrency	O
as	O
a	O
pure	O
sine	O
wave	O
does	O
not	O
contain	O
any	O
harmonics	O
and	O
does	O
not	O
lead	O
to	O
quefrency	O
peaks	O
.	O
</s>
<s>
An	O
important	O
property	O
of	O
the	O
cepstral	B-Algorithm
domain	O
is	O
that	O
the	O
convolution	B-Language
of	O
two	O
signals	O
can	O
be	O
expressed	O
as	O
the	O
addition	O
of	O
their	O
complex	O
cepstra	O
:	O
</s>
<s>
The	O
concept	O
of	O
the	O
cepstrum	B-Algorithm
has	O
led	O
to	O
numerous	O
applications	O
:	O
</s>
<s>
Recently	O
,	O
cepstrum-based	O
deconvolution	B-Algorithm
was	O
used	O
on	O
surface	O
electromyography	O
signals	O
,	O
to	O
remove	O
the	O
effect	O
of	O
the	O
stochastic	O
impulse	O
train	O
,	O
which	O
originates	O
an	O
sEMG	O
signal	O
,	O
from	O
the	O
power	O
spectrum	O
of	O
the	O
sEMG	O
signal	O
itself	O
.	O
</s>
<s>
A	O
short-time	O
cepstrum	B-Algorithm
analysis	O
was	O
proposed	O
by	O
Schroeder	O
and	O
Noll	O
in	O
the	O
1960s	O
for	O
application	O
to	O
pitch	O
determination	O
of	O
human	O
speech	O
.	O
</s>
