Lecture 12_winter_2012.pdf

Digital Speech Processing
Processing—
Lecture 12
Homomorphic
p
Speech Processing
1

General Discrete Discrete-Time Time Model of
S Speech h Production
P d i
pL[ n] = p[ n] ∗hV[ n]
−− Voiced Speech
hV[ n] = AV ⋅g[ n] ∗v[ n] ∗r[
n]
p L [ n ] = u [ n ] ∗ h U [ n ] −− Unvoiced Speech
h [ n] = A ⋅v[ n] ∗r[
n]
U N
2

Basic Speech Model
• short segment of speech can be modeled as
having been generated by exciting an LTI system
either by a quasi-periodic impulse train, or a
random noise signal
• speech analysis => estimate parameters of the
speech model, measure their variations (and
perhaps even their statistical variabilites variabilites-for for
quantization) with time
• speech = excitation * system response
=> want to deconvolve speech into excitation and
system response => > do this using homomorphic
filtering methods
3

Superposition Superposition Principle
Principle
+ +
L{
}
x[ n]
y[ n]
= L{
x[
n]}
x n]
+ x [ n]
L { x n]
} + L { x [ n]
}
1[
2
xn [ ] = ax[ n] + bx[ n]
1 2
1[
2
yn [ ] = L { xn [ ]} = a L { x [ n ]} + b L { x [ n ]}
1 2
4

•
Generalized Superposition for Convolution
*
{ }
xn [ ]
H
y [ n ] = H { x [ n ] }
x n]
∗ x [ n]
H { x n]
} ∗H
{ x [ n]
}
1[
2
for LTI systems we have the result
y [ n ] = x [ n ] ∗ h [ n ] = ∑ x [ k ] h [ n−k ]
∞
k =−∞
*
1[
2
• "generalized" superposition => addition replaced by convolution
xn [ ] = x x1 [ n ] ∗ x x2 [ n ]
yn [ ] = H { x[ n]} = H { x1[ n]} ∗H
{ x2[
n]}
• homomorphic system ffor
or convolution
5

Homomorphic Homomorphic Filter
Filter
• homomorphic filter => homomorphic system ⎡
⎣
⎤
⎦ that H
homomorphic filter homomorphic system ⎡
⎣
⎤
⎦ that
passes the desired signal unaltered, while removing the
undesired signal
H
g
x( n) = x [ n] ∗x
[ n] - with x [ n]
the "undesired" signal
•
1 2 1
xn = x 1 n ∗ 2
H { [ ]} H { [ ]}
H
H
H { x [ n]}
{ x [ n]} → δ ( n) - removal of x [ n]
1 1
{ x [ n]} → x [ n]
2 2
H { xn [ ]} = δ [ n] ∗ x2[ n] = x2[ n]
for linear systems this is analogous to additive noise removal
6

Canonic Form for Homomorphic
Deconvolution
* + + + +
D { } L { }
−1
D { }
∗
x[ n] xn ˆ[ ]
yn ˆ[ ]
yn [ ]
x n x n
ˆ ˆ
ˆ ˆ
1 [ ] ∗ 2 [ ] x 1 [ n ] + x 2 [ n ] y 1 [ n ] + y 2 [ n ]
1 ∗ 2
•
any homomorphic system can be represented as a cascade
of three systems, systems e.g., e g for convolution
∗
*
y [ n ] y [ n ]
1. system takes inputs combined by convolution and transforms
them into additive outputs p
2. system is a conventional linear system
3. inverse of first system--takes additive inputs and transforms
th them iinto t convolutional l ti l outputs
t t
7

Canonic Form for Homomorphic Convolution
*
+ + + +
D { } { }
−1{
}
x[ n]
∗
L
D
[ ] xn ˆ[ [ ] y ˆ[ [ n ] ∗ y [ n ]
x n x n
ˆ ˆ
ˆ ˆ
1[ ] ∗ 2[
] x1[ n] + x2[ n]
y1[ n] + y2[ n]
1 ∗ 2
xn [ ] = x1[ n] ∗x2[
n]
- convolutional relation
xn ˆ[ ] = D { [ ]} ˆ ˆ
∗ xn = x1[ n] + x2[ n]
- additive relation
yn ˆ[ ] = L{
xˆ [ n] + xˆ [ n]} = yˆ [ n] + yˆ [ n]
- conventional linear system
1 2 1 2
−1
∗ ˆ1 ˆ2
1 2
*
y[ n] y [ n]
y[ n] = D { y [ n] + y [ n]} = y [ n] ∗y
[ 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)
8

Properties of Characteristic
Systems
xn ˆ[ ] = D { xn [ ]} = D { x[ n] ∗x[
n]}
∗ ∗
1 2
= D ∗ { x 1 [ n ]} + D ∗ { x 2 [ n ]}
= xˆ [ n] + xˆ [ n]
1 2
D { yn ˆ[ ]} = D { yˆ [ n] + yˆ [ n]}
−1 −1
∗ ∗
1 2
= D { yˆ [ n]} ∗D
{ yˆ [ n]}
−1 −1
∗ 1 ∗ 2
= y [ n ] ∗ y [ n ] = y
[ n ]
1 2
9

Discrete Discrete-Time Time Fourier
Transform Representations
10

Canonic Form for Deconvolution Using DTFTs
• need dtto find fida system t that th tconverts t convolution l ti tto addition dditi
xn [ ] = x[ n] ∗x[
n]
1 2
jω = 1
jω ⋅ 2
jω
Xe (
• since
) X( e ) X( e )
D ˆ ˆ ˆ
∗{
xn [ ]} = x1[ n] + x2[ n] = xn [ ]
⎡ jω ( ) ⎤ ˆ jω ˆ jω ˆ jω
D∗
Xe = X1( e ) + X2( e ) = Xe ( )
⎣ ⎦
=> use log function wh ich converts products to sums
ˆ jω ( ) log ⎡ jω ( ) ⎤ log ⎡ jω jω
Xe = Xe = X1( e ) ⋅X2(
e ) ⎤
⎣ ⎦ ⎣ ⎦
ω ω
log ⎡ j ˆ ω ˆ ω
= 1( ) ⎤+ log ⎡ j
2( ) ⎤ j j
X e X e = X1( e ) + X2( e )
⎣ ⎦ ⎣ ⎦
ˆ jω ( ) ⎡ ˆ jω ˆ jω ˆ ˆ
1( ) 2( ) ⎤ jω jω
Ye = L X e + X e = Y1( e ) + Y2( e )
⎣ ⎦
jω ( ) exp ˆ jω = ˆ j
Ye ⎡ ω
Y1( e ) + Y2( e ) ⎤ jω jω
= Y1( e ) ⋅Y2(
e
)
⎣ ⎦

Characteristic System for
Deconvolution Using DTFTs
∞
jω − jωn ∑
Xe ( ) = xne [ ]
n=−∞
ˆ jω jω jω jω
X ( e ) = log g ⎡
⎣ ⎣X ( e ) ⎦
⎤ = log g X( ( e ) + j arg g ⎣
⎡Xe ( )
⎦
⎤
π
1
ˆ[ ] ˆ jω jωn xn = ( ) ω
2 ∫ Xe e d
π −ππ
12

Inverse Characteristic System for Deconvolution Using
DTFTs
∞
jω − jωn Ye ˆ( ) = yne ˆ[
]
∑
n=−∞
j jωω ( ) ˆ j jωω
Y Y( e ) = exp ⎡ ⎡Y( ( )
⎤
⎣
Y e
⎦
π
1
ω ω
[ ] = ∫
j j n
yn [ ] Ye ( ) e d dωω
2
∫ π −π
13

Issues with Logarithms
Logarithms
• it is essential that the logarithm obey the equation
⎡ j j ⎤ ⎡ j ⎤ ⎡ j
log ⎡ jωjω ⎤
1( ) 2( ) ⎤ log ⎡ jω 1( ) ⎤ log ⎡ jω
X e ⋅ X e = X e + X2( e ) ⎤
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
jω jω
• this is trivial if X ( e ) and X ( e ) are real -- however usually
1 2
1( jω ) and 2(
jω
)
X e X e
•
are complex
on the unit circle the complex log can be written in the form:
jω jω
⎡ jω
jarg ⎡ jω
j X ( e ) ⎤
⎣ ⎦
Xe ( ) = | Xe ( )| e
log ⎡ jω ( ) ⎤ 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) z transform) needs
to be a continuous odd function of ω
14

Problems with arg Function
{ (
j jω ) } ≠ { 1(
j jω
) }
jω
+ ARG{ X2( e ) }
ARG X e ARG X e
jω jω
{ Xe } = { X1e }
jω
+ arg { X X2 ( e
) }
arg ( ) arg ( )
15

Complex Cepstrum Properties
i Given a complex logarithm that satisfies the phase continuity
condition, , we have:
π
1
jω jω jωn xn ˆ[ ] = (log | X( e ) | + jarg{ X( e )}) e dω
2π
∫
−π
j jω j jω
i If xn [ ] real, real then log|X ( e ) | is an even function of ωω
and arg{ X ( e )}
is an odd function of ω.
This means that the real and imaginary parts of
the complex log have the appropriate symmetry for xn ˆ[ ] to be a real
sequence, and d xn ˆ[ ˆ[ ] can bbe represented t d as:
xn ˆ[ ] = cn [ ] + dn [ ]
jω
where cn [ ] is the inverse DTFT of log |X ( e )| and the even part of xn ˆ[
], and
jω
dn [ ] is the inverse DTFT of arg{ Xe ( )} and the odd part of xn ˆ[
]:
xn ˆ[ ] + xˆ[ −n] xn ˆ[ ] −xˆ[ −n]
cn [ ] = ; dn [ ] =
2 2
16

Complex Complex and and Real Real Cepstrum
define the inverse Fourier transform of ˆ jω
•
Xe ( ) as
ππ
1
ˆ[ ˆ ω ω
] = ∫
j j n
xn Xe ( ) e dω
2
π
−π
• where xn ˆ[
[ ] called the "complex complex cepstrum" cepstrum since a complex
logarithm is involved in the computation
• can also define a "real
cepstrum" using just the real part of
the logarithm, giving
π
1
[ ] = ∫ Re ⎡ ˆ j
( ) ⎤ j n
cn ∫ Xe e d
2 2π
⎣ ⎦
=
1
2 2π
−π
π
∫
π
−π
ω ω ω
jω jωn log | Xe ( ) | e dω
• can show that cn [ ] is the even part of xn
ˆ[
]
17

Terminology gy
• Spectrum – Fourier transform of signal autocorrelation
• C Cepstrum t – iinverse FFourier i ttransform f of f log l spectrum t
• Analysis – determining the spectrum of a signal
• Alanysis – determining the cepstr cepstrumm of a signal
• Filtering – linear operation on time signal
• Liftering – linear operation on cepstrum
• Frequency – independent variable of spectrum
• Quefrency – independent variable of cepstrum
• Harmonic – integer multiple of fundamental frequency
• Rahmonic – integer multiple of fundamental frequency
Rahmonic integer multiple of fundamental frequency
18

z-Transform z Transform Representation
Representation
i The z − transform of the signal:
xn [ ] = x x1[ [ n n] ]* x x2[ [ n n]
]
is of the form:
X( z) = X1( z) ⋅X2(
z)
i With an appropriate i t ddefinition fi iti of f th the complex l llog, we get: t
Xˆ ( z) = log{ X( z)} = log{ X1( z) ⋅X2(
z)}
= log{ X ( z)} + log{ X ( z)}
= Xˆ ( z) + Xˆ ( z)
1
1 2
2
19

Characteristic System for Deconvolution
∞
−
( ) = ∑
n
Xz xnz [ ] = Xz ( ) e
n=−∞
{ }
jarg X( z)
Xˆ ( z) = log X( z) = log X( z) + jarg X( z)
[ ] [ ]
1
ˆ[ ] ∫
ˆ n
xn [ ] = ∫ X ( z ) zdz
2π
j
20

Inverse Characteristic System for
Deconvolution
∞
Yz ˆ( ) = ∑ ynz ˆ[
]
n=−∞
−n
Yz ( ) = exp ⎡ ˆ(
) ⎤
⎣
Yz
⎦
= log Yz ( ) + jarg Yz ( )
1
[ ] ∫
n
[ ] = ∫
n
yn Y ( zzdz ) d
2π
j
[ ]
21

z-Transform
z Transform Cepstrum Alanysis
•
i
consider digital systems with rational z-transforms of the general type
Xz ( ) =
M
M M0
−1 −1 −1
i
∏( 1− k ) ∏(
1−
)
k
A a z b z
k= 1 k=
1
N
i
− 1
∏ ( 1−
cz k )
k = 1
we can express the above equation as:
Xz ( ) =
−M0 M M0 ∏ −
M Mi
−1 k ∏ 1− k
M M0
−1
∏ 1−
k
k= 1 k = 1
k=
1
Ni
− 1
( ) ( ) ( )
z A b a z b z
∏ ( 1 1−
cz k )
k = 1
• with all coefficients ak, bk, ck < 1 => all ck
poles and
a ak zeros are inside the unit circle; all b bk
zeros
are outside the unit circle;
22

z-Transform
z Transform Cepstrum Alanysis
i
express X( z)
as product of minimum-phase and
maximum maximum-phase phase signals signals, ii.e., e
i
i
i
X z X z z X z
where
−M
0
( ) = ( ) ⋅ ( )
X ( z ) =
min
min max
Mi
( −1
∏ 1−
) k
A a z
k = 1
N
i
( 1
1 cz ) k
−
∏ −
k = 1
all ll poles l and d zeros iinside id unit it circle i l
max
M
i
∏
( 1 )
X ( z) b 1
−
= − − bz
k
Mi
∏
k = 1
k = 1
all zeros outside unit circle
( )
k
23

z-Transform
z Transform Cepstrum Alanysis
i
i
i
can express xn [ ] as the convolution:
xn [ ] = xmin[ n] ∗xmax[ n−M0] minimum-phase p component p is causal
xmin [ n] = 0, n<
0
maximum-phase maximum phase component is anti-causal anti causal
x [ n] = 0, n>
0
max
M0
factor z is the shift in time ori
− 0 i factor z is the shift in time ori gin by M 0
samples required so that the overall sequence,
xn
[ ] be causal
24

z-Transform z Transform Cepstrum Alanysis
• the complex logarithm of Xz ( ) is
0
ˆ
−1
− 0
( ) = log ⎡⎣ ( ) ⎤⎦
= log | | + ∑log | | + log[ ] +
M
M
Xz Xz A bk z
k = 1
M M
N
i 0
i
∑ ( −1) ( ) ( −1
log 1− az + ∑log 1− −∑log 1−
)
k bz k cz k
k = 1 k = 1 k = 1
• evaluating Xz ˆ ( ) on the unit circle we can ignore the term
⎡ jωM ⎤
e
⎣ ⎦
part and is a linear phase shift)
0
related to log (as this contributes only to the imaginary
25

z-Transform
z 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:
ππ
1
ˆ
ˆ ω ω
( ) = ∫
j j n
xn Xe ( ) e dω
2
π
−π
= log | A | +
M
0
∑
k = 1
Ni n
c k
Mi
n
a k
∑ ∑
k= 1 k=
1
M0 −n
bk
−1
k
log | b | n =
= − n >
n n
∑ k
= n < 0
n
k = 1
0
0
∞ n
Z
log(1 − Z) =− ∑ , | Z | < 1
n
n=
1
26

Cepstrum Properties
1. complex cepstrum is non-zero and of infinite extent for
both positive and negative n, even though x[ n]
may be
causal, or even of finite duration ( X ( z ) has only zeros).
2. complex cepstrum is a decaying sequence that is bounded by:
| n|
α
| xn ˆ[
]| < β , for | n|
→∞
| n |
3. zero-quefrency value of complex cepstrum (and the cepstrum)
depends on the gain constant and the zeros outside the unit circle.
Setting x ˆ[0] [0] = 0 (and therefore c [0] = 0 0)
) is equivalent to normalizing
the log magnitude spectrum to a gain constant of:
M
0
∏
k = 1
∏
1
( − b k ) = 1
A b −
4. If X( z) has no zeros outside the unit circle (all bk=
0), then:
xn ˆ[ ] = 0, n<
0 (minimum-phase
signals)
5. If X( z) has no poles or zeros inside the unit circle (all ak, ck
= 0), then:
xn ˆ[ ] = 0, n>
0 (maximum-phase signals)
27

i
i
z-Transform z Transform Cepstrum Alanysis
The main z-transform formula for cepstrum alanysis is based on
the h power series i expansion: i
( 1)
∞ −
∑
n+
1
n
log( g( 1+ x ) = x x < 1
n
n=
1
Example 1--Apply
this formula to the exponential sequence
x1( n)
n
1
= aun ( ) ⇔ X( z)
=
1−
az
1 −1
( −1
)
ˆ
1 ( ) = l log[ [ 1 ( )] = −l log( ( 1 1−
∞
− 1
) = − ∑
n=
1
n+
1
( −
n
n
)
− n
n
a
ˆ 1 ( ) =
n
( − 1 ) ⇔ ˆ
1 ( ) = − log( 1 1−
∞ n
−1⎛a ⎞
) = ∑ ⎜ ⎟
n=
1 n
−n
X z X z az a z
x n u n X z az z
⎝ ⎠
28

z-Transform z Transform Cepstrum Alanysis
• Example 2 2--consider
consider the case of a digital system with a
single zero outside the unit circle ( b < 1)
x x2 ( n ) = δδ ( n ) + b bδδ ( n + 1 )
X2( z) = 1+ bz (zero at z = −1/
b)
X Xˆ ( z ) = l log X ( z ) = llog
1 1+
b z
xˆ
2
2 2
[ ] ( )
∞ + 1
( −1)
= ∑ ( )
n
∑
n n
( b) z
n=
1 n
n+ 1 n
( −1)
b
( n ) = u ( −n−1 )
n
29

•
z-Transform Transform Cepstrum Alanysis for 2 Pulses
Example 3--an
input sequence of two pulses of the form
x 3 ( n ) = δδ ( n ) + αδ αδ ( n n− N Np
) ( 0 < αα
<
1 )
X ( z) = 1+
αz
3
−N
( −N
1 αα
)
X Xˆ 3 ( z ) = log ⎡⎣⎡⎣ X X3 ( z ) ⎤⎦⎤⎦
= log + z
∞ n+
1
( −1)
= ∑ n
p
α
n
z
−nN
n=
1
∞
k
ˆ + 1 α
3(
) = ∑
k
( −1) k
k = 1
δ ( − p )
x n n kN
p
• the cepstrum is an impulse train with impulses spaced at p samples
N
N p
2N 2Np p
3N 3Np 30

•
•
Cepstrum Cepstrum for Train of Impulses
an important special case is a train of impulses
of the form:
∑ M
xn ( ) = αδ(
n−rN) r = 0
M
Xz ( ) = ∑αr
z
r = 0
r p
−rN
p
−N −1
clearly Xz ( ) is a polynomial in z p rather than z ;
thus Xz ( ) can be expressed
as a product of factors
−N
N
of the form ( 1−az p) and ( 1−bz
p),
giving a complex
cepstrum, xn ˆ ( ), that is non non-zero zero only at integer multiples of N
p
31

z-Transform Transform Cepstrum Alanysis for
C Convolution l ti of f 2 2 Sequences
S
i Example 4 4--consider
consider the convolution of sequences 1and3 1 and 3, ie i.e.,
n
x4( n) = x1( n) ∗ x3( n) = ⎡
⎣
aun ( ) ⎤
⎦
∗ ⎡
⎣δ( n) + αδ(
n−N) ⎤ p ⎦
n
n n− N Np
= aun ( ) + α a un ( − N )
i The complex cepstrum is therefore the sum of the complex
cepstra
of the two sequences (since convolution inthetimedomainis
in the time domain is
converted to addition in the cepstral domain)
xˆ ( n) = xˆ ( n) + xˆ ( n)
4 1 3
n ∞ k+ 1 k
a ( −1)
α
un 1 ∑ δ n kNp
n k = 1 k
= ( − ) + ( − )
p
32

i
z-Transform Transform Cepstrum Alanysis for
C Convolution l ti of f 3 3 Sequences
S
Example 5--consider
the convolution of sequences 1, 2 and 3, i.e.,
x ( n) = x ( n) ∗x ( n) ∗x
( n)
5 1 2 3
[ 1 ]
n
= ⎡
⎣
⎡
⎣
aun ( ) ⎤
⎦
⎤
⎦
∗ δδ ( n ) + b bδδ ( n + ) ∗ ⎡
⎣
⎡
⎣δδ ( n ) + αδ δ ( n n− N ) ⎤
⎦
⎤ p ⎦
n n−Np = aun ( ) + αaun ( − N
n
n− Np+
1
) + baun ( + 1) + αbaun
( − N + 1)
p p
i The complex cepstrum is therefore the sum of the complex cepstra
of the three sequences
xˆ ( n) = xˆ ( n) + xˆ ( n) + xˆ ( n)
5 1 2 3
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
33

Example: a=.9, b=.8, a=.7, Np=15
( 1)
+
−
n
b
n 1 n
a n
a
n
( 1+
bz)
p
Xz ( ) = + z
( 1 1−
az )
( 1)
k
∞ −
∑
k = 1
k+ 1 k
( 1 α )
1
−
−
N
( )
az
α
δ [ n−kN ]
p
34

Homomorphic Analysis of
Speech Model
35

Homomorphic Analysis of Speech Model
•
i
•
i
the transfer function for voiced speech is of the form
H ( z) = A ⋅G(
z) V( z) R( z)
V V
with ith effective ff ti impulse i l response ffor voiced i d speech h
h [ n] = A ⋅g[ n] ∗v[ n] ∗r[
n]
V V
similarly for unvoiced speech we have
H ( z) = A ⋅V(
z) R( z)
U
U
with effective impulse response for unvoiced speech
h [ n] = A ⋅v[ n] ∗r[
n]
U U
36

Complex Complex Cepstrum for Speech
•
the models for the speech components are as follows:
1. vocal tract: Vz ( ) =
Mi
−M ∏ 1− M0
− 1
∏ 1−
= 1 = 1
−1
∏( 1−
)
i
M
k k
k k
N
∏ cz k
k = 1
Az ( a z ) ( b z)
--for voiced speech, only poles => ak = bk = 0,
all 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
2. radiation model: Rz ( ) ≈1−z(high frequency emphasis)
3. glottal pulse model: finite duration pulse with transform
L
i
0
− 1
∏ ∏
−1
G ( z ) = B ( 1−αz ) ( 1−βz
)
k k
k= 1 k=
1
with zeros both inside and outside the unit circle
L
37

Complex Cepstrum for Voiced
S Speech h
• combination of vocal tract, glottal pulse and
radiation will be non-minimum phase p =>
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
38

•
•
Simplified Speech Speech Model
Model
short-time speech model
[ ∗ ∗ ∗ ]
xn [ ] = wn [ ] ⋅ pn [ ] ∗ gn [ ] ∗ vn [ ] ∗ rn [ ]
≈ p [ n] ∗h
[ n]
w v
short-time complex cepstrum
xn ˆ [ ] = p pˆ [ n ] + gn ˆ [ ] + vn ˆ [ ] + rn ˆ [ ]
w
39

Analysis y of Model for Voiced Speech p
i Assume sustained /AE/ vowel with fundamental frequency of 125 Hz
i Use glottal pulse model of the form:
⎧0.5
[1− cos( π ( n+ 1) / N1)] ⎪
gn [ ] = ⎨cos(0.5
π ( n+ 1 −N1)/ N2) ⎪
⎩ 0
0 ≤n≤ N1−1
N1 ≤n≤ N1+ N2
−2
otherwise
N = 25, N = 10
⇒ 34 sample impulse response, with transform
i
1 2
33
−33 ∏
33
−1
k ∏ k
k= 1 k=
1
Gz ( ) = z ( −b) (1 −bz) ⇒ all roots outside unit circle ⇒ maximum phase
Vocal tract system specified by 5 formants (frequencies and bandwidths)
V( z)
=
1
5
−2πσ kT −1 −4πσ
kT
−2
∏ (1 (1− 2 e cos(2 ππ
FT k ) z + e z )
k = 1
{ Fk,
σ k}
= [(660,60),(1720,100),(2410,120),(3500,175),(4500,250)]
i Radiation load is simple first difference
−1
R( z) = 1 − γz, γ =
0.96
40

Time Domain Analysis
Analysis
41

Pole Pole-Zero Zero Analysis of Model
Components
42

Spectral Analysis Analysis of of Model
Model
43

Speech Speech Model Output
44

Complex Cepstrum of Model
i
The voiced speech signal is modeled as:
x [ n ] = A AV ⋅gn [ ] ∗vn [ ] ∗rn [ ] ∗ pn [ ]
i with complex cepstrum:
sn ˆ [ ] = log| A | δ [ n ] + gn ˆ [ ] + vn ˆ [ ] + rn ˆ [ ] + pn ˆ
V δ n + gn + vn + rn + pn [ ]
i glottal pulse is maximum phase ⇒ gn ˆ[ ] = 0, n>
0
i vocal tract
and radiation systems y are minimum phase p
⇒ vn ˆ[ ] = 0, n< 0, rn ˆ[
] = 0, n<
0
∞ k
ˆ β
Pz ( ) = − log(1 − ββ
z ) = ∑ z
k
−N −kN
= =∑
k = 1
∞ k
β
p pˆ[
[ n ] = ∑
δ [ n−kN p ]
k
k = 1
p p
45

Cepstral Analysis Analysis of of Model
Model
46

Resulting Complex and Real
Cepstra
47

Frequency Frequency Domain Representations
48

Frequency
Frequency-Domain Domain Representation of
C Complex l Cepstrum C t
49

The Complex Cepstrum-DFT Cepstrum DFT Implementation
i
•
2π∞2π j k − j kn
N N
∑
Xk [ ] = Xe ( ) = xne [ ] k= 01 , ,..., N−1,
n=−∞
jω
X [ k ] is the N point p DFT corresponding p g to X( ( e )
p
{ } { }
Xk ˆ = Xe ˆ = Xk = Xk + j Xk
j2πk/ N
[ ] ( ) log [ ] log [ ] arg [ ]
N−12π∞
1
j kn
ˆ ∑ ˆ N
xn [ ] = ∑Xke [ ] = ∑ xn ˆ ˆ[
+ rN] n= 01 , ,..., N−1
N k= 0
r=−∞
x ˆ [ n ] is an aliased version of xˆ [ n ]
⇒use as large a value of N
as possible to minimize aliasing
50

Inverse System- System y DFT Implementation
p
51

The Cepstrum Cepstrum-DFT DFT Implementation
π
1
jω jωn cn [ ] = ∫ log| Xe ( )| e dωω −∞< ∞< n

Cepstral Computation Aliasing
N=256, N Np=75, =75,
α=0 α=0.8 =0 =0.8 8
Circle dots are
cepstrum
values in
correct
locations; all
other dots are
results of
aliasing due to
finite range
computations
53

Summary
1. Homomorphic System for Convolution:
x[n] X(z) X(z) ^ X(z) Y(z) ^ Y(z) Y(z)
y[n]
z( ) log( ) L( ) exp( ) z-1 ( )
X(z)
^
z 1 ( ) -1 ( )
2. Practical Case:
z( ) → DFT
−1
z () → IDFT
x[n] ^ [ ] Linear
System
y[n] ^ y[ y[n] ]
jω jω
Xe ( ) = Xe ( ) e
jω
{ }
jarg X( e )
z( ( )
Y(z)
^
{ }
jω jω jω
log ⎡
⎣
Xe ( ) ⎤
⎦
= log Xe ( ) + jarg Xe ( )
54

Summary
3. Complex Cepstrum:
π
1
ˆ[
] = ∫
ˆ j j n
xn Xe ( ) e d
2 2π
4. Cepstrum:
π −π
π
ω ω ω
1
[ ] = ∫
j j n
cn log Xe ( ) e d
2π
−π
ω ω ω
xn ˆ[ ] + xˆ[ −n]
cn [ ] = = even part of xn ˆ[
]
2
55

x(n)
Summary
X(k X(k) ^
X(k X(k)
x(n x(n) ^
DFT Complex Log IDFT
~
5. Practical Implementation of Complex Cepstrum:
∞
j N k j N kn
∑
( 2 2ππ / ) − ( 2 2ππ
/ )
Xk [ ] = X Xe ( ) = x [ n ] e )
6. Examples:
n=−∞
{ p } p { p }
Xˆ [ k] = log X ( k) = log X ( k) + jarg X ( k)
N N−
1
∞
1 j 2π
/ N kn
(
xn ˆ[
] = ∑Xke ˆ [ ]
N k= 0
)
= ∑ xn ˆ[
+ rN]
⇒aliasing
r=−∞
xn ˆ[
] = aliased version
of xn ˆ[
]
∞
1
( ) = ⇔ ˆ[
] = ∑ δ [ − ]
1−
r
a
Xz xn n r
−1
az r = 1 r
∞ r
b
Xz ( ) = 1−bz⇔ xn ˆ[
] = − ∑ δ [ n+ r]
r
r = 1
56

Complex Cepstrum Without Phase
Unwrapping
i short-time analysis uses finite-length windowed segments, xn [ ]
i
M
( ) = ∑ [ ]
n=
0
−n
,
thh
-order polynomial
X z x n z M
Find polynomial roots
M M
−1 −1 −1
m m
m= 1 m=
1
i
0
∏ ∏
X( z) = x[0] (1 −a z ) (1 −b
z )
i am
roots are inside unit circle (minimum-phase
part)
i b roots t are outside t id unit it circle i l ( (maximum-phase i h part) t)
i
m
−1 −1
Factor out terms of form − bmz giving:
MiM0 −M 0
−1
X ( z ) = Az A ∏ (1 −a a mz ) ∏ (1 −b
b mz
)
m= 1 m=
1
M0
M0
−1
A= x[0]( −1)
∏bm
mm=
= 1
i Use polynomial root finder
to find the zeros that lie inside
and outside the unit circle and solve directly for xn
ˆ[ ] .
57

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(
)
Re ⎡ ˆ jω ( ) ⎤
⎡ xn + x−n⎤ Xe = FT
⎣ ⎦ ⎢
2
⎥
⎣ ⎦
jω
FT ⎡ ⎣ ⎣c( n) ⎤ ⎦ = log Xe ( )
xn ˆ( ) + xˆ( −n)
cn ( ) =
2
• giving
xˆ ( n) = 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
58

Recursive Relation for Complex
C Cepstrum t f for Mi Minimum i Phase Ph Signals Si l
• th the complex l cepstrum t for f minimum i i phase h signals i l
can be computed recursively from the input signal,
xn ( ) using the relation
xn ˆ( ) = 0 n<
0
= log ⎡⎣x( 0) ⎤⎦
n = 0
n−1
xn ( ) ⎛ k ⎞ xn ( − k )
= − ˆ( )
>
( 0) ⎜ ⎟xk
n
x ⎝n⎠ x(
0)
∑ k
= 0
0
59

Recursive Relation for Complex
C Cepstrum t f for Mi Minimum i Phase Ph Signals Si l
xn ( ) ←⎯→ Xz ( ) 1. basic z-transform
dX ( z )
nx( n) ←⎯→− z =−zX′
( z) dz
2. scale by n rule
xn ˆ( ) ←⎯→ Xz ˆ ( ) = log Xz ( ) 3. definition of complex cepstrum
[ ]
dXˆ ( z) d X′ ( z)
= [ log[ Xz ( )] ] =
dz dz X( z)
4. differentiation of z-transform
dX dXˆ ( z )
−z Xz ( ) =−zX′
( z)
dz
5. multiply both sides of equation
60

Recursive Relation for Complex
C Cepstrum t f for Mi Minimum i Phase Ph Signals Si l
dXˆ ( z)
nxˆ( ( n ) ∗x ( n ) ←⎯→− z X ( z ) =−zX′ ( z ) ←⎯→ nx ( n )
ddz
∞
∑
( )
nx( n) = xˆ ( k) x( n −k)
k
k =−∞
i for minimum phase systems we have xˆ ( n) = 0for n < 0,
xn ( ) = 0for n<
0,
giving:
n
( ) ˆ
⎛k⎞ xn ( ) = ∑ xkxn ( ) ( − k ) ⎜ ⎟
k = 0
⎝n⎠ i separating out the term for k = n we get:
n−1
⎛k ˆ
⎛k ⎞
xn ( ) = ∑ xkxn ( ) ( − k) ⎜ ⎟+
x( 0)
xn ˆ(
)
k = 0
⎝n⎠ n−1
xn ( ) xn ( − k) ˆ( ) ˆ
⎛k⎞ x( n) = − ∑ x ( k ) , > 0
( 0) ( 0)
⎜ ⎟,
n
x( 0) 0 ( 0)
⎜ ⎟
k = x ⎝ ⎝n ⎠
xˆ( 0) = log x( 0) , xˆ( n) = 0, n
< 0
[ ]
61

Recursive Relation for Complex
C Cepstrum t f for Mi Minimum i Phase Ph Signals Si l
i why is xˆ( 0) = log[ x(
0)]?
i assume we have a finite sequence xn ( ) ), n n= 01 , ,..., N N−
1
i we can write xn ( ) as:
xn ( ) = x( 0) δ( n) + x( 1) δ( n− 1) + ... + xN ( −1) δ(
N−1)
⎡ x () 1 x ( N − 1 ) ⎤
= x( 0) ⎢δ( n) + δ( n− 1) + ... + δ(
N −1)
⎣ x( 0) x(
0)
⎥
⎦
i taking z − transforms, we get:
N−1
∑
−n NZ1 ∏ 1 k
NZ 2
−1
∏ 1 k
n= 0 k= 1 k=
1
Xz ( ) = xnz ( ) = G ( −az) ( −bz)
i where the first term is the gain, G = x(
0),
and the two product terms are the
zeros inside and outside the
unit circle.
i for minimum phase systems we have all zeros inside the unit circle so the
second product term is gone, and we have the result that
xˆ ( 0 ) = log[ g[ G] ] = log[ g[ x ( 0)]; )] xˆ ( n ) = 0, n < 0
NZ1 a
xn ˆ( ) = 0
k=1
n ⎛ ⎞ k − ∑⎜
⎟ n >
⎝ n ⎠
62

Cepstrum p for Maximum Phase Signals g
•
for maximum phase signals (no poles or
zeros iinside id unit it circle) i l )
ˆ( ) + ˆ(
− )
( ) =
2
xn x n
cn
2
• giving
xn ˆ( ( ) = 0 n > 0
= cn ( ) n=
0
= 2cn ( ) n<
0
• thus the complex cepstrum (for maximum
phase signals) can be computed
by computing
the cepstrum and using the equation above
63

•
Recursive Relation for Complex
C Cepstrum t f for M Maximum i Phase Ph Signals Si l
the complex cepstrum for maximum phase signals
can be computed recursively from the input signal, signal
xn ( ) using the relation
xn ˆ( ( ) = 0 n > 0
= log ⎡⎣x( 0) ⎤⎦
n = 0
0
xn ( ) ⎛k⎞ xn ( − k)
= − ∑ ˆ( )
<
( 0) ⎜ ⎟xk
n
x ⎝n⎠ x(
0)
k= n+
1
0
64

Computing Short-Time
Short Time
Cepstrums from Speech
Speech
Using Polynomial Roots
65

Cepstrum From Polynomial Roots
66

Cepstrum From Polynomial Polynomial Roots
Roots
67

Computing Short-Time
Short Time
Cepstrums from Speech
Speech
Using Using g the DFT
68

Practical Considerations
• window to define short-time short time analysis
• window duration (should be several pitch
periods long)
• size of FFT (to minimize aliasing)
• elimination of linear phase components
(positioning signals within frames)
• cutoff quefrency of lifter
• type of lifter (low/high quefrency)
69

Computational Considerations
70

Voiced Speech Example
Hamming window
40 msec duration
(section beginning
at sample 13000
in file
test_16k.wav)
71

Voiced Speech Speech Example
Example
wrapped phase
unwrapped
phase
72

Voiced Speech Speech Example
Example
73

Characteristic System for
H Homomorphic hi Convolution
C l i
• still need to define (and design) the L
operator p p part ( (the linear system y
component) of the system to completely
define the characteristic system y for
homomorphic convolution for speech
– to do this properly p p y and correctly, y, need to look
at the properties of the complex cepstrum for
speech signals
74

Complex Cepstrum Cepstrum of of Speech
Speech
• model of speech:
– voiced speech produced by a quasi-periodic
pulse train exciting slowly time time-varying varying linear
system => p[n] convolved with hv[n] – unvoiced speech produced by random noise
exciting slowly time-varying linear system =>
u[n] convolved with hv[n] • time to examine full model and see what
the complex p cepstrum p of speech p
looks like
75

Homomorphic Filtering of Voiced
• 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
– this window removes combined
vocal tract
• the filtered signal is processed by
the inverse characteristic system
to recover the combined vocal
tract component
S Speech h
– l(n)=1 for |n|

Voiced Speech Speech Example
Example
Cepstrally
smoothed log
magnitude, 50
quefrencies
cutoff
Cepstrally
unwrapped
phase, 50
quefrencies
cutoff
77

Voiced Speech Speech Example
Example
Combined
impulse
response of
glottal pulse,
vocal tract
system, and
radiation system
78

Voiced Speech Speech Example
Example
High quefrency
liftering; cutoff
quefrency=50;
log magnitude
and unwrapped
phase
79

Voiced Speech Speech Example
Example
Estimated
excitation
function for
voiced speech
(Hamming
window
weighted) i ht d)
80

Unvoiced Speech Speech Example
Example
Hamming window
40 msec duration
(section beginning
at sample 3200 in
file test test_16k.wav) 16k.wav)
81

Unvoiced Speech Speech Example
Example
wrapped phase
unwrapped
phase
82

Unvoiced Speech Speech Example
Example
83

Unvoiced Speech Speech Example
Example
Cepstrally
smoothed log
magnitude, 50
quefrencies
cutoff
Cepstrally
unwrapped
phase, 50
quefrencies
cutoff
84

Unvoiced Speech Speech Example
Example
Estimated
excitation source
for unvoiced
speech section
(Hamming
window
weighted)
85

Short Short-Time Time Homomorphic Analysis
STFT
86

Review 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
– 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)
87

Example 11—single
single pole sequence
( (computed t d using i all ll 3 3 methods) th d )
[ ] = [ ]
n
xn aun
n
a
xn ˆ[ [ ] = un [ − 1 ]
n
88

Cepstral Computation Aliasing
i
i
i
i
Effect of quefrency aliasing via a simple example
xn [ ] = δδ [ n ] + αδ αδ [ n n− N ]
with discrete-time Fourier transform
( j jω
) 1
j jω N
α −
X e e ω
= +
p
We can express the complex logarithm as
p
ˆ j
j Np
X( e ) log{1 e }
ω
ω
α −
∞ m+ 1 m ⎛( −1)
α ⎞ − jωmNp = + = ∑⎜
⎟e
m=
1⎝
m ⎠
giving a complex cepstrum in the form
( −1)
α
xn ˆ[ [ ] = ∑
⎜ ⎟ ⎟δδ
[ n n− mN mNp
]
m=
1⎝
m ⎠
∞ m+ 1 m ⎛ ⎞
89

Example Example 22—voiced
2 voiced voiced speech frame
90

Example 33—low
low quefrency liftering
91

Example 33—high
high quefrency liftering
92

Example 4—effects 4 effects of low quefrency lifter
93

Example 55—phase
phase unwrapping
94

Example 66—phase
phase unwrapping
95

Homomorphic Spectrum Spectrum Smoothing
Smoothing
96

Running Cepstrum
97

Running Cepstrum
98

Running Cepstrums
99

Cepstrum Applications
100

Cepstrum Distance Distance Measures
Measures
i The cepstrum forms a natural basis for comparing
patterns in speech recognition or vector quantization
because of its stable mathematical characterization
for speech signals
i A typical "cepstral distance
measure" is of the form:
nco
∑
D= ([ c n] −c[
n])
n=
1
2
where cn [ ] and cn [ ] are cepstral p sequences q corresponding p g
to frames of signal, and D is the cepstral distance between
the pair of sequences.
i Using Parseval's Parseval s theorem, we can express the cepstral
distance in the frequency domain as:
1
π
jω jω
2
D (log | H( e ) | log | H( e ) |) d
2π
π −
= ∫
−
2π
i Thus we see that the cepstral distance is actually a log
magnitude spectral
distance
ω
101

Mel Frequency Cepstral Coefficients
i Basic idea is to compute a frequency analysis based on a filter
bank with approximately critical band spacing of the filters and
bandwidths. For 4 kHz bandwidth, , approximately pp y 20 filters are used.
i First perform a short-time Fourier analysis, giving Xm[ k], k = 0,1,..., NF / 2
where m is the frame number and k is the frequency index (1 to half
thesizeoftheFFT)
the size of the FFT)
i Next the DFT values are grouped together in critical bands and weighted
by triangular weighting functions.
102

Mel Frequency Cepstral Coefficients
th th
i The mel-spectrum of the m frame for the r filter ( r = 1,2,..., R)
is defined as:
MF
U r 1
[] r = ∑ | V [] k X []| k
A
m r m
r k= L
r
2
th
where Vr[ k] is the weighting function for the r filter, ranging from
DFT index L to U , and
U
r r
r
Ar = ∑ | Vr[ k]|
k k= L
r
2
th
is the normalizing factor for the r mel-filter. (Normalization guarantees
that if the input spectrum is flat, the mel-spectrum is flat).
i A discrete cosine transform
of the log magnitude of the filter outputs is
computed to form the function mfcc [ n]
as:
R 1 ⎡2π⎛ 1⎞
⎤
mfcc mfccm [ n ] = ∑ log( MF MFm
[ r ])cos ⎢ ⎜ r r+ ⎟ n n⎥ ,
R
⎢ ⎜ + ⎟
r 1
R 2
⎥
=
⎣ ⎝ ⎠ ⎦
n n= 1,2, 1,2,..., , N Nmfcc
f
i Typically N = 13 and R=
24 for 4 kHz bandwidth speech signals.
mfcc
103

Delta Cepstrum
i The set of mel frequency cepstral coefficients provide perceptually
meaningful and smooth estimates of speech spectra, over time
i Since speech is inherently a dynamic signal, it is reasonable to seek
a representation t ti that th t includes i l d some aspect t of f the th dynamic d i nature t off
the time derivatives (both first and second order derivatives) of the shortterm
cepstrum
i The resulting parameter sets are called th the e delta cepstrum (first derivative)
and the delta-delta cepstrum (second derivative).
i The simplest method of computing delta cepstrum parameters is a first
difference of cepstral vectors, of the form:
Δ mfccm[ n] = mfccm[ n] −mfccm−1[
n]
i The simple difference is a poor approximation to the first derivative and is
not generally used. Instead a least-squares approximation to the local slope
(over a region
around the current sample) is used, and is of the form:
Δ mfcc [ n]
=
M
∑
k=−M m M
k( mfcc [ n])
∑
k=−M k
m+ k
2
where the region is M frames before and after the current frame
104

Homomorphic Homomorphic Vocoder
Vocoder
• time-dependent p complex p cepstrum p retains all the
information of the time-dependent Fourier
transform => exact representation of speech
• time i ddependent d real l cepstrum lloses phase h
information -> not an exact representation of
speech
• quantization of cepstral parameters also loses
information
• cepstrum gives good estimates of pitch, voicing,
formants => can build homomorphic vocoder
105

Homomorphic Homomorphic Vocoder
Vocoder
1. compute p cepstrum p every y 10-20 msec
2. estimate pitch period and
voiced/unvoiced decision
3. quantize and encode low-time cepstral
values
44. at t synthesizer-get th i t approximation i ti to t h hv(n) ( )
or hu(n) from low time quantized cepstral
values
5. convolve hv(n) or hu(n) with excitation
created from pitch, p , voiced/unvoiced, , and
amplitude information
106

Homomorphic Vocoder
• l( l(n) ) iis cepstrum t window i d th that t selects l t llow-time ti
values and is of length 26 samples homomorphic
vocoder
107

Homomorphic Vocoder Impulse Responses
108

Summary
• 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 reflected properties of both the
excitation (high quefrency) and the vocal tract (low
quefrency) f )
– 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-delta delta cepstral coefficients used as
indicators of spectral change over time 109