digital signal processing on speech recognition
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Digital Signal Processing
Related to Speech Recognition
References:
1. A. V. Oppenheim and R. W. Schafer,Discrete-time Signal Processing, 1999
2. X. Huang et. al., Spoken Language Processing, Chapters 5, 6
3. J. R. Deller et. al., Discrete-Time Processing of Speech Signals, Chapters 4-6
4. J. W. Picone, Signal modeling techniques in speech recognition, proceedings of the IEEE,September 1993, pp. 1215-1247
Berlin Chen 2005
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Digital Signal Processing
Digital Signal
Discrete-time signal with discrete amplitude
Digital Signal Processing
Manipulate digital signals in a digital computer
[ ]nx
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Two Main Approaches to Digital Signal
Processing Filtering
Parameter Extraction
Filter
Signal in Signal out
[ ]nx [ ]ny
Parameter
Extraction
Signal in Parameter out
[ ]nx
1
12
11
mc
cc
2
22
21
mc
cc
2
1
Lm
L
L
c
cc
e.g.:
1. Spectrum Estimation
2. Parameters for Recognition
Amplify or attenuate some frequency
components of [ ]nx
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Sinusoid Signals
: amplitude () : angular frequency (
),
: phase ()
E.g., speech signals can be decomposed assums of sinusoids
[ ] ( ) += nAnx cos
Tf
2
2==
[ ]
=
2cos
nAnx
samples25=T
10
frequencynormalized:
f
f
Period, represented
by number of samples
[ ] [ ]
( ),...2,12
2
2)(cos
===
+=+=
kN
kkN
NnANnxnx
Right-shifted
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Sinusoid Signals (cont.)
is periodic with a period ofN(samples)
Examples (sinusoid signals)
is periodic with period N=8 is periodic with period N=16
is not periodic
[ ]nx
[ ] [ ]nxNnx=+
( ) ( ) +=++ nANnA cos)(cos)21(2 ,...,kkN ==
N
2
=
[ ] ( )4/cos1 nnx =[ ] ( )8/3cos2 nnx =
[ ] ( )nnx cos3 =
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Sinusoid Signals (cont.)
[ ] ( )
( )
8
integerspositiveareand824
44cos)(
4cos
4cos
4/cos
1
111
11
1
=
==
+=
+=
=
=
N
kNkNkN
NnNnn
nnx
[ ] ( )
( )
( )
16
numberspositiveareand3
162
8
3
8
3
8
3cos8
3cos8
3cos
8/3cos
2
222
22
2
=
==
+=
+=
=
=
N
kNkNkN
NnNnn
nnx
[ ] ( )( ) ( )( ) ( )
!existtdoesn'
integerspositiveareand
2
cos1cos1cos
cos
3
3
3
33
3
N
kN
kN
NnNnn
nnx
=
+=+==
=
Q
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Sinusoid Signals (cont.)
Complex Exponential Signal
Use Eulers relation to express complex numbers
( ) sincos
jAAez
jyxz
j +==
+=
( )numberrealaisA
Re
Im
sin
cos
Ay
Ax
=
=
22 yx + 22 yxx
+22 yx
y
+
Imaginary part
real part
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Sinusoid Signals (cont.)
A Sinusoid Signal
[ ] ( )( ){ }
{ }
jnj
nj
eAe
Ae
nAnx
Re
Re
cos
=
=
+=+
real part
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Sinusoid Signals (cont.)
Sum of two complex exponential signals with
same frequency
When only the real part is considered
The sum ofNsinusoids of the same frequency isanother sinusoid of the same frequency
( ) ( )
( )
( )
+
++
=
=
+=
+
nj
jnj
jjnj
njnj
Ae
Aee
eAeAe
eAeA
10
10
10
10
( ) ( ) ( ) +=+++ nAnAnA coscoscos 1100
numbersrealareand, 10 AAA
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Sinusoid Signals (cont.)
Trigonometric Identities
1100
1100
coscos
sinsintan
AA
AA
+
+=
( ) ( )
( )
( )10102120
1010102
120
2
21100
21100
2
cos2
sinsincoscos2
sinsincoscos
++=
+++=
+++=
AAAA
AAAAA
AAAAA
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Some Digital Signals
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Some Digital Signals
Any signal sequence can be represented
as a sum ofshift and scaled unit impulse
sequences (signals)
[ ]nx
[ ] [ ] [ ]knkxnxk
=
=
scale/weighted
Time-shifted unit
impulse sequence
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221
33221101122
3
2
+++++++=
+++++++=
== =
=
nnnnnn
nxnxnxnxnxnx
knkxknkxnxkk
,...3,2,1,0,1,2..., =n
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Digital Systems
A digital system Tis a system that, given an
input signalx[n], generates an output signal y[n]
[ ] [ ]{ }nxTny =
[ ]nx { }T [ ]ny
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Properties of Digital Systems
Linear
Linear combination of inputs maps to linear
combination of outputs
Time-invariant (Time-shift) A time shift in the input by m samples gives a shift in
the output by m samples
[ ] [ ]{ } [ ]{ } [ ]{ }nxbTnxaTnbxnaxT 2121 +=+
[ ] [ ]{ } mmnxTmny = ,
time sift [ ][ ] samplesshiftleft)0(if
samplesshiftright)0(if
mmmnx
mmmnx
>+
>
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Properties of Digital Systems (cont.)
Linear time-invariant (LTI)
The system output can be expressed as a
convolution () of the inputx[n] and theimpulse response h[n]
The system can be characterized by the systems
impulse response h[n], which also is a signal
sequence
If the inputx[n] is impulse , the output is h[n][ ]n
Digital
System
[ ]n [ ]nh
[ ]{ } [ ] [ ] [ ] [ ] [ ] [ ]khknxknhkxnhnxnxTkk
===
=
=
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Properties of Digital Systems (cont.)
Linear time-invariant (LTI)
Explanation:
[ ] [ ] [ ]knkxnxk
=
=
[ ]{ } [ ] [ ]{ }[ ] [ ]{ }
[ ] [ ]
[ ] [ ]nhnx
knhkx
knTkx
knkxTnxT
k
k
k
=
=
=
=
=
=
=
scale
Time-shifted unitimpulse sequence
Time invariant
Digital
System
[ ]n [ ]nh
linear
Time-invariant
[ ] [ ]
[ ] [ ]knhkn
nhn
T
T
Impulse response
convolution
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Properties of Digital Systems (cont.)
Linear time-invariant (LTI) Convolution
Example[ ]n
0 1
21
3
-2[ ]nh
LTI[ ]nx
?0 1 2
2
3
1
0
3
0 1
23
9
-6
1
2
1 2
32
6
-4
LTI
2
1
2 3
41 3
-2
Length=M=3
Length=L=3
Length=L+M-1
[ ]n3
[ ]12 n
[ ]21 n
0
3
1
11
2
13
-1
4
-2
Sum up
[ ]ny
[ ]nh3
[ ]12 nh
[ ]2nh
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Properties of Digital Systems (cont.)
Linear time-invariant (LTI)
Convolution: Generalization
Reflect h[k] about the origin ( h[-k]) Slide (h[-k] h[-k+n] orh[-(k-n)] ), multiply it withx[k]
Sum up[ ]kx
[ ]kh
[ ]kh
Reflect Multiply
slide
Sum up
n
[ ] [ ] [ ]
[ ] ( )[ ]nkhkx
knhkxny
k
k
=
=
=
=
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0 1
21
3
-2
0-1
-21
3
-2
0 1 2
2
3
1
0
3
10
-11
3
-2
1
11
21
01
3
-2
1
2
32
11
3
-2 -1
3
432
1
3
-2 -2
4
0
3
1
11
2
13
-1
4
-2
Sum up
[ ]kh
[ ]kx
[ ]kh [ ] 0, =nny
[ ]1+ kh
[ ]2+ kh
[ ]3+ kh
[ ]4+ kh
[ ] 1, =nny
[ ] 2, =nny
[ ] 3, =nny
[ ] 4, =nny
[ ]ny
Reflect [ ] [ ] [ ]
[ ] [ ]knhkx
nhnxny
k=
=
=
Convolution
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Properties of Digital Systems (cont.)
Linear time-invariant (LTI)
Convolution is commutative and distributive
[ ] [ ] [ ]
[ ] [ ][ ] [ ]
[ ] [ ]knxkh
knhkx
nxnh
nhnxny
k
k
=
==
=
=
=
*
*
[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1
[ ]nh 2
[ ]nh 1
[ ] [ ]nhnh 21 +
An impulse response has finite duration
Finite-Impulse Response (FIR)
An impulse response has infinite duration
Infinite-Impulse Response (IIR)
[ ] [ ] [ ][ ] [ ] [ ]nhnhnx
nhnhnx
12
21
****
=
[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx
nhnhnx
21
21
**
*
+=
+
Commutation
Distribution
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Properties of Digital Systems (cont.)
Prove convolution is commutative
[ ] [ ] [ ][ ] [ ]
[ ] [ ] ( )
[ ] [ ]
[ ] [ ]nxnh
knxkh
knmhmnx
knhkx
nhnxny
k
m
k
*
mlet
*
=
=
= =
=
=
=
=
=
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Properties of Digital Systems (cont.)
Linear time-varying System
E.g., is an amplitude modulator[ ] [ ] nnxny 0cos =
[ ] [ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] ( )0000
00011
0
?
101
cosBut
coscos
thatsuppose
nnnnxnny
nnnxnnxny
nnynynnxnx
=
==
==
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Properties of Digital Systems (cont.)
Bounded Input and Bounded Output (BIBO): stable
A LTI system is BIBO if only ifh[n] is absolutely summable
[ ] =k
kh
[ ][ ] nBnynBnx
y
x
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Properties of Digital Systems (cont.)
Causality
A system is casual if for every choice ofn0, the output
sequence value at indexing n=n0 depends on only the inputsequence value forn n0
Any noncausal FIR can be made causal by adding sufficient long
delay
[ ] [ ] [ ] = =
+=K
k
M
mkkk mnxknyny
1000
z-1
z-1
z-1
M
2
0[ ]ny[ ]nx
1
z-1
z-1
z-1
1
2
N
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Discrete-Time Fourier Transform (DTFT)
Frequency Response
Defined as the discrete-time Fourier Transform of
is continuous and is periodic with period=
is a complex function of
[ ]nh( )jeH
2( )jeH
( ) ( )
jeHjj
j
i
j
r
j
eeH
ejHeHeH
=
+=
magnitudephase
( )j
eH
proportional to twotimes of the sampling
frequency
njne nj sincos +=
Im
Re
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Discrete-Time Fourier Transform (cont.)
Representation of Sequences by Fourier Transform
A sufficient condition for the existence of Fourier transform
[ ]
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Discrete-Time Fourier Transform (cont.)
Convolution Property
( ) [ ]
[ ] [ ] [ ]
( ) [ ] [ ]
[ ] [ ]
( ) ( )
[ ] ( ) ( )
jj
jj
nj
nk
kj
nj
n k
j
k
n
njj
eHeXnhnx
eHeX
enhekx
eknhkxeY
knhkxnhnxny
enheH
=
=
=
==
=
=
=
=
=
=
=
][
'
][][
'
'
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
jjj
jjj
jjj
eHeXeY
eHeXeY
eHeXeY
+=
=
=
knnknn
knn
=+=
=
''
'
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Discrete-Time Fourier Transform (cont.)
Parsevals Theorem
Define the autocorrelation of signal
[ ] ( )
= =
deXnxj
n
22
2
1
power spectrum
[ ]nx[ ]
( ) ][][][][
][][
*
*
nxnxlnxlx
mxnmxnR
l
mxx
==
+=
=
=
( ) ( ) ( ) ( )2*
XXXSxx ==
[ ] ( ) ( ) ==
deXdeSnR njnjxxxx2
2
1
2
1
[ ] [ ] [ ] [ ] ( )
=
=
===
=
mm
xx dXmxmxmxR
n
22*
2
10
0Set
The total energy of a signal
can be given in either thetime or frequency domain.
)(
lnnlm
nml
==
+=
222
*
conjugatecomplex:
zyxzz
jyxzjyxz
*
*
=+=
=+=
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Discrete-Time Fourier Transform (DTFT)
A LTI system with impulse response
What is the output for
[ ]nh[ ]ny [ ] ( ) += nAnx cos
[ ] ( )
[ ] [ ] [ ] ( )
[ ] ( ) [ ]
[ ] ( )( )
( ) [ ]
( ) ( )( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( ) ( )( )
[ ] ( ) ( ) ( )( )
jj
jjjj
eHjnjj
eHjjnj
jnj
kj
k
nj
knj
k
nj
nj
eHneHAny
eHneHjAeHneHA
eeHA
eeHAe
eHAe
ekhAe
Aekh
nhAeny
Aenxnxnx
njAnx
j
j
++=
+++++=
=
=
=
=
=
=
=+=
+=
++
+
+
=
+
+
=
+
+
cos
sincos
*
sin
0
10
1
[ ]ny [ ]ny1
( )
( ) attenuate1
amplify1
j
j
eH
eH
Systems frequency
response
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Discrete-Time Fourier Transform (cont.)
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Z-Transform
z-transform is a generalization of (Discrete-Time) Fourier
transform
z-transform of is defined as
Where , a complex-variable
For Fourier transform
z-transform evaluated onthe unit circle
( ) [ ]
=
=n
nznhzH
[ ] ( )zHnh [ ] ( )
j
eHnh
[ ]nh
jrez =
( ) ( ) jez
jzHeH
==
)1( == zez j
complex plane
unit circle
Im
Re
jez=
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Z-Transform (cont.)
Fourier transform vs. z-transform
Fourier transform used to plot the frequency response of a filter
z-transform used to analyze more general filter characteristics, e.g.
stability
ROC (Region of Converge) Is the set ofzfor which z-transform exists (converges)
In general, ROC is a ring-shaped region and the Fourier
transform exists if ROC includes the unit circle ( )
[ ]
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Z-Transform (cont.)
An LTI system is defined to be causal, if its impulse
response is a causal signal, i.e.
Similarly, anti-causalcan be defined as
An LTI system is defined to be stable, if for every
bounded input it produces a bounded output
Necessary condition:
That is Fourier transform exists, and therefore z-transform
includes the unit circle in its region of converge
[ ] 0for0 = nnh
Right-sided sequence
Left-sided sequence
[ ]
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Z-Transform (cont.)
Right-Sided Sequence
E.g., the exponential signal
[ ] [ ] [ ]
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Z-Transform (cont.)
Finite-length Sequence
E.g.
[ ]
= others,0
10,
.3 4Nna
nh
n
( ) ( ) ( )az
az
zaz
azazzazH
NN
N
NN
n
nN
n
nn
=
===
=
=
11
11
0
11
04
1
1
1
0exceptplane-entireis4 = zzROC
13221 ..... ++++ NNNN azaazz
Im
the unit cycle
3
1
7 poles at zero A pole and zero atis cancelledaz=
4
( )11,
2
== ,..,Nkaez Nkj
k
IfN=8
0 N-1
Re
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Z-Transform (cont.)
Properties ofz-transform
1. If is right-sided sequence, i.e. and ifROC
is the exterior of some circle, the all finite for which
will be in ROC
If ,ROCwill include
2. If is left-sided sequence, i.e. , the ROCisthe interior of some circle,
If ,ROCwill include
3. If is two-sided sequence, the ROCis a ring
4. The ROC cant contain any poles
[ ]nh [ ] 1,0 nnnh =
01 n
0
rz >z
=z
[ ]nh [ ] 2,0 nnnh =
02
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Summary of the Fourier and z-transforms
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LTI Systems in the Frequency Domain
Example 1: A complex exponentialsequence System impulse response
Therefore, a complex exponential input to an LTI systemresults in the same complex exponential at the output, butmodified by
The complex exponential is an eigenfunction of an LTI
system, and is the associated eigenvalue
[ ]nh
[ ] [ ] [ ] [ ]
[ ]
( ) njjkjnj
knj
eeH
ekhe
ekhnhnxny
=
=
==
=
=
-k
)(
-k
( )
response.frequencysystem
theastoreferredoftenisItresponse.impulsesystemthe
ofansformFourier trthe:jeH
[ ] njenx =[ ] [ ] [ ]
[ ] [ ]
[ ] [ ][ ] [ ]knxkh
knhkx
nxnh
nhnxny
k
k
=
=
=
=
=
=
*
*
[ ]{ } [ ]nxeHnxT j=
jeH
jeH
scalar
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[ ] ( ) ( )
( ) ( )[ ]( ) ( ) ( ) ( )[ ]
( ) ( )[ ]00
00
000
0
0000
0000
0
)()(
)()(
cos
2
2
22
jj
njeHjjnjeHjj
njj*njj
njjjnjjj
eHneHA
eeeHeeeHA
eeHeeHA
eeA
eHeeA
eHny
jj
++=
+=
+=
+=
++
++
LTI Systems in the Frequency Domain (cont.)
Example 2: A sinusoidal sequence
System impulse response
[ ] ( )+= nwAnx 0cos
[ ] ( )
njjnjj eeAeeA
nAnx
00
22
cos 0
+=
+=
[ ]nh
( )
jj
j
j
ee
ie
ie
+=
=
+=
2
1cos
sincos
sincos
( ) ( )
( ) ( )( )000
00
jeHjjj*
j*j
eeHeH
eHeH
=
=
( )
( ) ( )
sincossincos
sincos*
sincos
*
jje
jejyxz
jejyxz
j
j
j
=+=
==
+=+=
magnitude response phase response
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LTI Systems in the Frequency Domain (cont.)
Example 3: A sum of sinusoidal sequences
A similar expression is obtained for an input consisting of a sum
of complex exponentials
[ ] ( )= +=
K
kkkk nAnx 1 cos
[ ] ( ) ( )[ ]=
++=K
k
j
kk
j
kkk eHneHAny
1
cos
magnitude response phase response
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LTI Systems in the Frequency Domain (cont.)
Example 4: Convolution Theorem
[ ] [ ]
=
=k
kPnnx
[ ] [ ] 1,
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LTI Systems in the Frequency Domain (cont.)
Example 5: Windowing Theorem [ ] [ ] ( ) ( )
jj eXeWnwnx 2
1
[ ] [ ]
=
=k
kPnnx
[ ]
=
=
otherwise0
1,......,1,0,1
2cos46.054.0 Nn
N
n
nw
Hamming window
( ) ( ) ( )( )
( )
( )
=
=
=
=
=
=
=
=
=
=
=
k
kP
j
k
m
m
jm
k
j
k
j
jjj
eWP
mkP
eWP
kP
eWP
kPP
eW
eXeWeY
21
2
1
21
22
2
1
2
1
has a nonzero
value when
k
P
m
2
=
Difference Equation Realization
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Difference Equation Realization
for a Digital Filter
The relation between the output and input of a digital
filter can be expressed by
[ ] [ ] [ ] = = +=N
k
M
kkk knxknyny
1 0
( ) ( ) ( ) kN
k
M
kk
k
k zzXzzYzY
= =
+=
1 0
delay property
( )( )( )
=
=
==
N
k
k
k
M
k
k
k
z
z
zX
zYzH
1
0
1
[ ] ( )
[ ] ( )0
0
n
zzXnnx
zXnx
linearity and delay properties
A rational transfer function
Causal:
Rightsided, the ROC outside the
outmost pole
Stable:
The ROC includes the unit circle
Causal and Stable:
all poles must fall inside the unitcircle (not including zeros)
z-1
z-1
z-1
M
2
0[ ]ny[ ]nx
1
z-1
z-1
z-1
1
2
N
Difference Equation Realization
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Difference Equation Realization
for a Digital Filter (cont.)
M it d Ph R l ti hi
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Magnitude-Phase Relationship
Minimum phase system:
The z-transform of a system impulse response sequence ( a
rational transfer function) has all zeros as well as poles inside the
unit cycle
Poles and zeros called minimum phase components
Maximum phase: all zeros (or poles) outside the unit cycle
All-pass system: Consist a cascade of factor of the form
Characterized by a frequency response
with unit (or flat) magnitude for all frequencies
1
11
1
az
z-a*
Poles and zeros occur at conjugate
reciprocal locations
11
11
= az
z-a*
M it d Ph R l ti hi ( t )
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Magnitude-Phase Relationship (cont.)
Any digital filter can be represented by the cascade of a
minimum-phase system and an all-pass system
( ) ( ) ( )zHzHzH apmin=( )
( )
( ) ( )( ) ( )
( )( ) ( )( )
( )( )( )
( )filter.pass-allais
1
1
filter.phaseminimumaalsois1
:where
1
11
filter)phaseminimumais(1
:asexpressedbecancircle.unittheoutside
1)(1zerooneonlyhasthatSuppose
1
*
1
1
1
*1
1
1
*
1
=
=
sampling rate=8kHz
Digital Signal:Discrete-time
signal with discrete
amplitude
Analog Signal to Digital Signal (cont )
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Analog Signal to Digital Signal (cont.)
Impulse Train
To
Sequence [ ] ( ))( nTxnxa
=
Sampling
( )
( ) ( ) ( ) ( )
( ) ( ) [ ] ( )
=
=
=
==
==
nn
a
n
aa
s
nTtnxnTtnTx
nTttxtstx
tx
( )tx a
Continuous-Time
Signal
Discrete-TimeSignal
Digital Signal
Discrete-time
signal with discrete
amplitude
[ ]nx
Continuous-Time to Discrete-Time Conversion
( ) ( )
==
nnTtts
Periodic Impulse Train ( ) [ ]nxtxs
byspecifieduniquelybecan
switch
( )tx a
( ) ( )
==
nnTtts
( )
( )
=
=
1
0,0
dtt
tt
1
T0 2T 3T 4T 5T 6T 7T 8T-T-2T
Analog Signal to Digital Signal (cont )
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Analog Signal to Digital Signal (cont.)
A continuous signal sampled at different periods
T
1
( )tx a ( )tx a
( )
( ) ( ) ( ) ( )
( ) ( ) [ ] ( )
=
=
=
==
==
nn
a
n
aa
s
nTtnxnTtnTx
nTttxtstx
tx
Analog Signal to Digital Signal (cont )
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Analog Signal to Digital Signal (cont.)
( )jXa
frequency)(sampling22
ss FT
==
=
TsN
2
1
aliasing distortion
( ) ( )
==
ksk
TjS
2
( ) ( ) ( )
( ) ( )( )
=
=
=
k
sas
as
kjXT
jX
jSjXjX
1
21
N
( )
>
>
2 Ns
NNsQ
( )
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Analog Signal to Digital Signal (cont.)
To avoid aliasing (overlapping, fold over) The sampling frequency should be greater than two times of
frequency of the signal to be sampled (Nyquist) sampling theorem
To reconstruct the original continuous signal
Filtered with a low pass filter with band limit
Convolved in time domain
s
( ) tth s= sinc
( ) ( ) ( )
( ) ( )
=
=
=
=
n
sa
n
aa
nTtnTx
nTthnTxtx
sinc
Ns > 2
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