Homomorphic Homomorphic Signal Processing [ ] [ ] [ ]

Digital Signal Processing Design
— Lecture 7
Homomorphic Signal
Processing
Superposition Principle
+ +
x[ n]
x 1[
n]
+ x2[
n]
L{
}
y[ n]
= L{
x[
n]}
L { x1[ n]
} + L{
x2[
n]
}
xn ( ) = ax1( n) + bx2( n)
yn ( ) = L xn ( ) = aL x( n) + bL x( n)
[ ] [ ] [ ]
1 2
Homomorphic Filter
• homomorphic filter => homomorphic system ⎡
⎣H⎤ ⎦ that
passes the desired signal unaltered, while removing the
undesired signal
x( n) = x1( n) ∗x2(
n) - with x1( n)
the "undesired" signal
H ⎡⎣xn ( ) ⎤⎦ = H ⎡⎣x1(
n) ⎦⎤∗H⎣⎡x2( n)
⎦⎤
H ⎡⎣x1( n) ⎤⎦
→ δ ( n) - removal of x1( n)
H ⎡⎣x2( n) ⎤⎦
→ x2( n)
H ⎡⎣xn ( ) ⎤⎦
= δ(
n) ∗ x2( n) = x2( n)
• for linear systems this is analogous to additive noise removal
1
3
5
General Discrete Discrete-Time Time Model of
Speech Production
pL( n) = p( n) ∗hV( n)
−− Voiced Speech
hV( n) = AV ⋅g( n) ∗v( n) ∗r(
n)
pL( n) = u( n) ∗hU( n)
−− Unvoiced Speech
hU( n) = AN ⋅v( n) ∗r(
n)
2
Generalized Superposition for Convolution
x[ n]
*
H { }
*
y[ n]
= H { x[
n]
}
x1[ n]
∗ x2[
n]
H { x1[ n]
} ∗H
{ x2[
n]
}
• for LTI systems we have the result
yn ( ) = xn ( ) ∗ hn ( ) = xkhn ( ) ( −k)
∞
∑
k =−∞
• "generalized" superposition => addition replaced by convolution
xn ( ) = x1( n) ∗x2(
n)
yn ( ) = H ⎡⎣xn ( ) ⎤⎦ = H ⎡⎣x1( n) ⎤⎦∗H ⎡⎣x2( n)
⎤⎦
• homomorphic system
for convolution
Canonic Form for Homomorphic Convolution
x[ n]
x n x n
* + + + + *
D ∗{
} L{
}
−1
D { }
xˆ[ n]
yn ˆ[ ] ∗ yn [ ]
ˆ[ ] ˆ [ ]
ˆ [ ] ˆ [ ]
1[ ] ∗ 2[
] x1 n + x2 n y1 n + y2 n
1 ∗ 2
4
y [ n] y [ n]
• any homomorphic system can be represented as a cascade
of three systems, e.g., for convolution
1. system takes inputs combined by convolution and transforms
them into additive outputs
2. system is a conventional linear system
3. inverse of first system--takes additive inputs and transforms
them into convolutional outputs
6
1

Canonic Form for Homomorphic Convolution
x[ n]
x n x n
* + + + + *
D ∗{
} L{
}
−1
D { }
xˆ[ n]
yn ˆ[ ] ∗ yn [ ]
ˆ ˆ
ˆ ˆ
1[ ] ∗ 2[
] x1[ n] + x2[ n]
y1[ n] + y2[ n]
1 ∗ 2
y [ n] y [ n]
xn ( ) = x1( n) ∗x2(
n)
- convolutional relation
xn ˆ( ) = D ˆ ˆ
∗ ⎡⎣xn ( ) ⎤⎦
= x1( n) + x2( n)
- additive relation
yn ˆ( ) = L ⎡ ⎣xˆ ˆ ˆ ˆ
1( n) + x2( n) ⎤ ⎦ = y1( n) + y2( n)
- conventional linear system
−1
yn ( ) = D ˆ ˆ
∗ ⎡ ⎣y1( n) + y2( n) ⎤ ⎦ = y1( n) ∗y2(
n)
- inverse of convolutional relation
=> design converted back to linear system, L
D∗
⎡⎣ ⎤⎦
- fixed (called the characteristic system for homomorphic deconvolution)
−1
D∗
⎡⎣ ⎤⎦
- fixed (characteristic system
for inverse homomorphic deconvolution)
x[ n]
*
Characteristic Systems for Homomorphic Deconvolution
D∗{
}
• • + +
Z { } log{
} -1
X( z)
Xˆ Z { }
( z)
+
7
xˆ[ n]
x1[ n]
∗ x2[
n]
X ( z) X ( z)
Xˆ ( z) + Xˆ ( z)
x ˆ1[ n]
+ xˆ
2[
n]
∞
∑
Xz ( ) = xnz ( )
n=−∞
1
ˆ( ) ˆ ( )
1 ⋅ 2
1 2
−n
[ ] [ ]
Xˆ ( z) = log X( z) = log X( z) + jarg X( z)
∫
n−1
xn = Xzz dz
2π
j C
Issues with Logarithms
• it is essential that the logarithm obey the equation
log ⎡⎣X1( z) ⋅ X2( z) ⎤⎦ = log ⎡⎣X1( z) ⎤⎦+ log ⎡⎣X2( z)
⎤⎦
• this is trivial if X1( z) and X2( z)
are real -- however usually
X1( z) and X2( z)
are complex
ω
• consider
evaluating ( ) only on the unit circle, e.g., = j
Xz z e
• then the complex log can be written in the form:
arg ⎡ jω
( ) ⎤
jω jω
j X e
Xe ( ) = | Xe ( )| e ⎣ ⎦
ω
log ⎡ j
( ) ⎤ ˆ jω ω ω
= ( ) = log ⎡ j
| ( )| ⎤+ arg ⎡ j
Xe Xe Xe j Xe ( ) ⎤
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
• no problems
with log magnitude term; uniqueness problems
arise in defining the imaginary part of the log; can show that
the imaginary part (the phase angle of the z-transform) needs
to be a continuous odd function of ω
9
11
X( z)
Frequency Domain Canonic Form
X z X z
• need to find a system that converts convolution to addition
xn ( ) = x1( n) ∗x2(
n)
Xz ( ) = X1( z) ⋅ X2( z)
• since
D ( ) ˆ ˆ ˆ
∗ ⎡⎣xn⎤⎦ = x1( n) + x2( n) = xn ( )
D ( ) ˆ ˆ ˆ
∗ ⎡⎣Xz⎤⎦ = X1( z) + X2( z) = Xz ( )
=> use log function which converts products to sums
Xz ˆ ( ) = log ⎡⎣⎡⎣ Xz ( ) ⎤⎦⎤⎦ = log ⎡⎣⎡⎣ X 1 ( z ) ⋅ X 2 ( z ) ⎤⎦⎤⎦
= log ⎡ ˆ ˆ
⎣X1( z) ⎤⎦+ log ⎡⎣X2( z) ⎤⎦
= X1( z) + X2( z)
Yz ˆ( ) = L ⎡Xˆ ˆ ˆ ˆ
1( z) + X2( z) ⎤ = Y1( z) + Y2( z)
⎣ ⎦
Yz ( ) = exp ⎡Yˆ ˆ
1( z) + Y2( z) ⎤ = Y1( z) ⋅Y2(
z)
⎣ ⎦
• + + + + •
log { } L
Xˆ z{
} exp{
}
( z)
Yˆ ( z)
Y( z)
ˆ ˆ
ˆ ˆ
1( ) ⋅ 2(
) X1( z) + X2( z)
Y1( z) + Y2( z)
1 ⋅ 2
Y( z) Y ( z)
Characteristic System for Homomorphic Deconvolution
−1
D∗
{ }
+ + + • • *
Z { } exp{
} -1
yn ˆ[ ]
Yˆ Z { }
( z)
Y( z)
yn [ ]
y ˆ [ n] + y ˆ [ n]
Y ˆ ˆ( z) + Yˆ ˆ ( z)
Y y1 [ n ] ∗ y2
[ n ]
1( z) ⋅Y2(
z)
1 2
1 2
∞
∑
Yz ˆ( ) = ynz ˆ(
)
n=−∞
−n
Yz ( ) = exp ⎡ ˆ(
) ⎤
⎣
Yz
⎦
( )
1
( )
∫
n−1
y n = Y z z dz
2π
j C
Problems with arg Function
jω jω jω
{ ( ) } ≠ { 1( ) } +
{ 2(
) }
ARG X e ARG X e ARG X e
10
12
2

Terminology
• spectrum – Fourier transform of signal autocorrelation
• cepstrum – Inverse Fourier transform of logarithm of
spectrum
• analysis – determining the spectrum of a signal
• alanysis y – determining g the cepstrum p of a signal g
• filtering – linear operations on time signal
• liftering – linear operations on log spectrum
• frequency – independent variable of spectrum
• quefrency – independent variable of cepstrum
• harmonic – integer multiple of fundamental frequency
• rahmonic – integer multiple of fundamental quefrency
Fourier Transform Representation: The
Complex Cepstrum
D∗{
}
* • • + + +
F {} log{
}
-1
F {}
( )
ω j
x [n]
xˆ
[ n]
X e
ˆ jω
( ) X ( e )
j
X e
( )
j
X e
∞
∑
jω − jωn jω
Xe ( ) = xne ( ) = Xe ( ) e
n=−∞
1 ˆ jω jωn xn ˆ( ) = ( ) ω
2π
∫ Xe e d
jω
{ }
jarg X( e )
ˆ jω jω jω jω
Xe ( ) = log ⎡
⎣
Xe ( ) ⎤
⎦
= log Xe ( ) + jarg ⎡
⎣
Xe ( ) ⎤
⎦
π
−π
The Complex Cepstrum Cepstrum-DFT DFT Implementation
2π j k
N
∞
∑
n n=−∞∞
2π
− j kn
N
X ( k) = X( e ) = x( n) e k = 01 , ,..., N −1,
p
jω
i Xp( k) is the N point DFT corresponding to X( e )
Xˆ ( k) = log X ( k) = log X ( k) + jarg X ( k)
{ } { }
N−1
∑ ˆ
2π
j kn
N
∞
∑
p p p p
1
xˆ ( n) = X ( k) e = xˆ( n+ rN) n = 01 , ,..., N −1
p p
N k= 0
r=−∞
• xˆp( n) is an aliased version of xn ˆ(
)
⇒use
as large a value of N as possible to minimize aliasing
13
15
17
Complex and Real Cepstrum
define the inverse Fourier transform of ˆ jω
•
Xe ( ) as
π
1
ˆ( ) ˆ jω jωn xn = Xe ( ) e dω
2π
∫
−π
• where xn ˆ(
) called the "complex cepstrum" since a complex
logarithm g is involved in the computation
• can also define a "real
cepstrum" using just the real part of
the logarithm, giving
π
1 ˆ jω jωn cn ( ) = Re ⎡Xe ( ) ⎤ e dω
2π
∫ ⎣ ⎦
−π
π
1
jω jωn = log | Xe ( ) | e dω
2π
∫
−π
• can show that cn ( ) is the even part of xn ˆ(
)
Fourier Transform Representation:
The Complex Cepstrum
−1
D∗
{ }
+ + + • • *
F { } exp{
}
-1
F { }
yn [ ]
ˆ jω
Y ( e ) ( )
ω j
yn ˆ[ ]
( ) Y e
j
Y e
( )
j
Y e
jω ∞
= ∑
n=−∞
− jωn jω jω jω jω
Ye ˆ( ) yne ˆ[
]
Ye ( ) = exp ⎡ ˆ ⎤ = + ⎡
⎣
⎤
⎣
Ye ( )
⎦
log Ye ( ) jarg Ye ( )
⎦
π
1
jω jωn yn [ ] =
ω
2π
∫ Ye ( ) e d
−π
The Cepstrum-DFT Cepstrum DFT Implementation
2π 2π
j k
∞
− j kn
N N
Xp( k) = X( e ) = ∑ x( n) e n = 01 , ,..., N −1
n=−∞
jarg { Xp( k)
}
Xp( k) = Xp( k) e
jω jω
Ce ( ) = log Xe ( )
2π
j k
N
Cp( k) = log Xp( e )
π
1
jω jωn cn ( ) = ( ) ω ˆ( ) ˆ(
) / 2
2π
∫ Ce e d = ⎡⎣xn + x−n⎤⎦ −π
1
c ( n) = C ( k) e = c( n+ rN) n = 01 , ,..., N −1
N−12π
j kn
∞
N
p ∑ p ∑
N k= 0
r=−∞
• cp( n) is an aliased version of c( n) ⇒use
as large a value of N
as possible to minimize aliasing
14
16
18
3

z-Transform Transform Cepstrum Alanysis
i The main z-transform formula for cepstrum alanysis is based on
the power series expansion:
∞ ( −1)
∑
n+
1
n
log( 1+ x) = x x < 1
n=
1 n
i EExample l 1
--Apply A l thi this fformula l tto th the exponential ti l sequence
n
1
x1( n)
= aun ( ) ⇔ X1( z)
= −1
1−
az
n+
1
∞
ˆ −1( −1)
n −n
X1( z) = log[ X1( z)] = −log( 1−
az ) = −∑( −a)
z
n=
1 n
n ∞ n
a 1
ˆ 1( ) ( 1) ˆ
− ⎛a ⎞ −n
x n = u n− ⇔ X1( z) = −log( 1−
az ) = ∑⎜
⎟z
n n=
1 ⎝ n ⎠
z-Transform Transform Cepstrum Alanysis
• the complex logarithm of Xz ( ) is
⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
⎡ r ⎤
⎣ ⎦
Mi
∑
k = 1
M N
N
−1
( 1 k )
Xz ˆ ( ) = log Xz ( ) = log A + log z + log −az
0 i
0
− 1
∑ ∑log ( 1 bz k ) ∑ ∑log ( 1 cz k ) ∑ ∑log
( 1 dz k )
+ − − − − −
k= 1 k= 1 k=
1
• evaluating Xz ˆ ( ) on the unit circle we can ignore the term
related to log ⎡ jωr e ⎤ (as this contributes only to the imaginary
⎣ ⎦
part and is a linear phase shift)
z-Transform Transform Cepstrum Alanysis for 2
Pulses
• Example 2--an
input sequence of two pulses of the form
x3( n) = δ( n) + αδ( n− Np)
( 0 < α < 1)
−Np
X3( z) = 1+
αz
ˆ
−Np
X3( z) = log ⎡⎣X3( z) ⎤⎦
= log(
1+
αz
)
∞ n + 1
( −1)
n −nNp
= ∑ α z
n
n=
1
∞
k
+ 1 α
ˆ k
x3( n) = ∑(
−1) δ ( n−kNp) k
k = 1
• the cepstrum is an impulse train with impulses spaced at Np
samples
N p
2N 2Np 3N 3Np 19
21
23
z-Transform Transform Cepstrum Alanysis
• consider digital systems with rational z-transforms of the general type
Xz ( ) =
M
M
Az a z b z
i
r
∏( 1− k
0
−1
) ∏(
1−
k )
k= 1 k=
1
Ni
N0
− 1
∏ ( 1 1−cz k ) ∏ ( 1 1−
d dkz )
k= 1 k=
1
• with all coefficients ak, bk, ck, dk < 1 => all ck
poles and
ak zeros
are inside the unit circle; all dk poles and bk
zeros
r
are outside the unit circle; the factor z is just a shift of the time
origin and has no effect
z-Transform Transform Cepstrum Alanysis
• we can then evaluate the remaining terms, use power series
expansion for logarithmic terms (and take the inverse
transform to give the complex cepstrum) giving:
π
1
ˆ( ) ˆ jω jωn xn ( ) = ∫ Xe ( ) e dωω
2π
∫ Xe e d
−π
= log⎡⎣A⎤⎦
n = 0
Ni n Mi
n
ck ak
= ∑ −
n ∑ n
k= 1 k=
1
n > 0
M0 −n N0
−n
bk dk
= ∑ −
n ∑ n
n < 0
k= 1 k=
1
Cepstrum for Train of Impulses
• an important special case is a train of impulses
of the form:
∑αδ = 0
M
r p
r
M
N
α
0
−rNp
∑ r
r =
xn ( ) = ( n−rN) Xz ( ) = z
−Np −1
• clearly Xz ( ) is a polynomial in z rather than z ;
thus Xz ( ) can be expressed as a product of factors
−Np
Np
of the form ( 1−az ) and ( 1−bz
), giving a complex
cepstrum, xn ˆ(
), that is non-zero only at integer multiples of N
20
22
p
24
4

z-Transform Transform Cepstrum Alanysis for
Convolution of 3 Sequences
i Example 3 -- consider the convolution of three sequences, i.e.,
xn ( ) = x1( n) ∗x2( n) ∗x3(
n)
n
= ⎡
⎣
aun ( ) ⎤
⎦
∗ [ δ( n) + bδ( n+ 1)
] ∗ ⎡
⎣δ( n) + αδ(
n−N) ⎤ p ⎦
n n−N p n
n− N p + 1
α p α
p
= a u ( n ) + a u ( n n− N ) + ba u ( n n+ 1 ) + ba u ( n n− N + 1 )
i The complex cepstrum
is therefore the sum of the complex cepstra
of the three sequences
xn ˆ( ) = xˆ1( n) + xˆ ˆ
2( n) + x3( n)
n ∞ k+ 1 k n+ 1 n
a ( −1) α ( −1)
b
= un ( − 1) + ∑ δ ( n− kNp) + u( −n−1) n k n
k = 1
Properties of the Complex Cepstrum
• in general, the complex cepstrum is non-zero and of
infinite extent for both positive and negative n,
even when
xn ( ) is causal, stable, and even of finite duration
• the complex cepstrum is a decaying sequence which is
bounded byy
| n|
α
xn ( ) < β for | n|
→∞
| n |
• where α is the maximum absolute value of the quantities
ak, bk, ck, dk
and β is a constant multiplier
• if Xz ( ) has no poles or zeros outside the unit circle ( bk = dk
= 0)
then xn ˆ(
) = 0 for n<
0 => such signals are called "minimum phase"
signals
27
The Phase Issue in Computation
i In order for the complex cepstrum to be computable, we need
jω jω jω
arg { Xe ( ) } = arg { X1( e ) } + arg { X2( e ) }
i when
Xe ( ) = X( e ) + X( e )
jω jω jω
1 2
jω jω jω
{ ( ) } ≠ { 1( ) } + { 2(
) }
ARG X e ARG X e ARG X e
25
29
Example: a=.9, b=.8, α=.7, N p=15
( 1)
+
− b
n
n 1 n
a
n
n
∞
∑
k = 1
−1
( )
( 1 α )
−Np
( 1+
bz)
Xz ( ) = + z
1−
az
k+ 1 k
( −1)
α
δ [ n−kNp] k
Relation of Cepstrum to Complex Cepstrum
i for the sequence xn ( ) we have:
∞
jω
jω − jωn jω
jarg { X( e ) }
Xe ( ) = ∑ xne ( ) = Xe ( ) e
n=−∞
ˆ jω jω jω jω
Xe ( ) = log Xe ( ) = log Xe ( ) + jarg Xe ( )
i giving the complex cepstrum
{ } { }
π
1 ˆ jω jωn xˆ( n) =
2 ∫ ( ) ω
π ∫ X e e d
−π
i with cepstrum
π
1
jω jωn cn ( ) = log ( ) ω
2π
∫ Xe e d
−π
jω jω
i because of the symmetries of log Xe ( ) (even), and jarg { Xe ( ) } (odd),
along with the symmetries of cos( ωn) (even) and sin( ωn)
(odd), it can easily
be shown
that
xn ˆ( ) + xˆ( −n)
cn ( ) = Ev{ xn ˆ(
) } =
2
The “Phase Unwrapping” Problem
26
28
30
5

Cepstrum for Minimum Phase Signals
• for minimum phase signals (no poles or zeros outside unit circle) the complex cepstrum
can be completely represented by the real part of the Fourier transforms
• this means we can represent the complex
cepstrum of minimum phase signals by the log
of the magnitude of the FT alone
• since the real part of the FT is the FT of the even part of the sequence
ˆ
ˆ( ) ˆ(
)
Re ⎡ jω ⎡ + − ⎤
( ) ⎤
xn x n
Xe = FT
⎣ ⎦ ⎢
2
⎥
⎣ 2 ⎦
jω
FT ⎡⎣c( n)
⎤⎦
= log Xe ( )
xn ˆ( ) + xˆ( −n)
cn ( ) =
2
• giving
xn ˆ(
) = 0 n<
0
= cn ( ) n=
0
= 2cn ( ) n>
0
• thus the complex cepstrum (for minimum phase signals) can be computed by computing
the cepstrum and using the equation above
31
Complex Cepstrum for Speech
• the models for the speech components are as follows:
Mi
M0
−M−1 ∏ 1− k ∏ 1−
k
k= 1 k=
1
Ni
−1
∏(
1−
cz k )
k = 1
a k = b k = 0
Az ( a z ) ( b z)
1. vocal tract: Vz ( ) =
--for o voiced o ced speec speech, , oonly y po poles es => a b , aall
k
--unvoiced speech and nasals,
need pole-zero model but all poles are
inside the unit circle => ck
< 1
--all speech has complex poles and zeros that occur in complex conjugate
pairs
−1
2. radiation model: Rz ( ) ≈1−z(high frequency emphasis)
3. glottal pulse model: finite duration pulse with transform
Li
L0
−1
Gz ( ) = B∏( 1−αkz) ∏(
1−βkz)
k= 1 k=
1
with zeros both inside and outside the unit circle
Complex Cepstrum for Voiced Speech
33
35
Characteristic System for
Homomorphic Convolution
• still need to define (and design) the L
operator part (the linear system
component) p ) of the system y to completely p y
define the characteristic system for
homomorphic convolution for speech
– to do this properly and correctly, need to look
at the properties of the complex cepstrum for
speech signals
Complex Cepstrum for Voiced Speech
Complex Cepstrum for Voiced Speech
32
34
36
6

Complex Cepstrum for Voiced Speech
Complex Cepstrum for Voiced Speech
Complex Cepstrum for Voiced Speech
37
39
41
Complex Cepstrum for Voiced
Speech
• combination of vocal tract, glottal pulse and
radiation will be non-minimum phase =>
complex cepstrum exists for all values of n
• the complex cepstrum will decay rapidly for
large n (due to polynomial terms in expansion
of complex cepstrum)
• effect of the voiced source is a periodic pulse
train for multiples of the pitch period
Complex/Real Cepstrum for Voiced Speech
Homomorphic Filtering of Voiced
Speech
• goal is to separate out the
excitation impulses from the
remaining components of the
complex cepstrum
• use cepstral window, l(n), to
separate excitation pulses from
combined vocal tract
– l(n)=1 for |n|

Homomorphic Analysis of Voiced Speech
Homomorphic Analysis of Voiced Speech
Homomorphic Analysis of Voiced Speech
43
45
47
Homomorphic Analysis of Voiced Speech
Homomorphic Analysis of Voiced Speech
Homomorphic Analysis of Unvoiced Speech
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8

Homomorphic Analysis of Unvoiced Speech
Homomorphic Analysis
Example 11—single
single pole sequence
(computed using all 3 methods)
[ ] = [ ]
n
xn aun
n
a
xn ˆ[ ] = un [ −1]
n
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53
Homomorphic Analysis
Homomorphic Analysis of a sequence of 15
frames from /SH/ to /EY/ (between the two
demarked windows in the above plot).
Review of Cepstral Calculation
• 3 potential methods for computing cepstral
coefficients, xn ˆ[ ] , of sequence x[n]
– analytical method; assuming X(z) is a rational
function; find poles and zeros and expand using log
power series i
– recursion method; assuming X(z) is either a minimum
phase (all poles and zeros inside unit circle) or
maximum phase (all poles and zeros outside unit
circle) sequence
– DFT implementation; using windows, with phase
unwrapping (for complex cepstra)
Example 22—voiced
voiced speech frame
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9

Example 33—low
low quefrency liftering
Example 4—effects 4 effects of low quefrency lifter
Example 66—phase
phase unwrapping
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57
59
Example 33—high
high quefrency liftering
Example 55—phase
phase unwrapping
Homomorphic Spectrum Smoothing
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10

Homomorphic Spectrum Smoothing
3200 Hz
Delta Cepstrum
200 Hz
Δ cmel [ n, ] = cmel [ n, ] −cmel [ n−1,
]
Δ log ( E [ n, ] ) = log ( E [ n, ] ) −log ( E [ n−1,
]
)
mel mel mel
Differencing removes effect of linear filtering.
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63
x[n]
w[ n − m]
Mel Cepstrum
Xnω [ , k ]
1
mel [ , ] = ∑ ( ω ) [ , ω ]
U
E n Vk X n k
A
Vl ( ωk
)
R filters
2
U
l k= L
k= L
mel[ ,
1 1
] log (
0
mel[
, ] ) cos
2 1 ( 2 )
π
R−
=
R =
+
R
= ∑ ( ω )
A V
⎡ ⎤
c n m ∑ E n ⎢ m⎥
⎣ ⎦
Summary
Emel[ n,
l]
• Introduced the concept of the cepstrum of a signal,
defined as the inverse Fourier transform of the log of the
signal spectrum
1
j
ˆ[ ] log ( )
xn F X e ω
− = ⎡
⎣
⎤
⎦
• Showed cepstrum p reflected pproperties p of both the
excitation (high quefrency) and the vocal tract (low
quefrency)
– short quefrency window filters out excitation; long quefrency
window filters out vocal tract
• Mel-scale cepstral coefficients used as feature set for
speech recognition
• Delta and delta-delta cepstral coefficients used as
indicators of spectral change over time 64
k
2
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11