sp2005f lecture11 digital signal...
TRANSCRIPT
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Digital Signal Processing Related to Speech Recognition
References:1. A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing, 19992. X. Huang et. al., Spoken Language Processing, Chapters 5, 63. J. R. Deller et. al., Discrete-Time Processing of Speech Signals, Chapters 4-64. 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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SP- Berlin Chen 2
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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SP- Berlin Chen 3
Two Main Approaches to Digital Signal Processing
• Filtering
• Parameter Extraction
FilterSignal in Signal out
[ ] nx [ ] ny
ParameterExtraction
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 Estimation2. Parameters for Recognition
Amplify or attenuate some frequencycomponents of [ ] nx
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Sinusoid Signals
– : amplitude (振幅)– : angular frequency (角頻率), – : phase (相角)
• E.g., speech signals can be decomposed as sums of sinusoids
[ ] ( )φω += nAnx cos
ωφ
A
Tf ππω 22 ==
[ ] ⎟⎠⎞
⎜⎝⎛ −=
2cos πωnAnx
samples 25=T
10frequency normalized:
≤≤ ff
Period, represented by number of samples
[ ] [ ]
( ),...2,122
2)(cos
==⇒=⇒
⎟⎠⎞
⎜⎝⎛ −+=+=
kN
kkN
NnANnxnx
πωπω
πω
Right-shifted
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SP- Berlin Chen 5
Sinusoid Signals (cont.)
• is periodic with a period of N (samples)
• Examples (sinusoid signals)
– is periodic with period N=8– is periodic with period N=16– is not periodic
[ ]nx[ ] [ ]nxNnx =+
( ) ( )φωφω +=++ nANnA cos)(cos)21( 2 ,...,kkN == πω
Nπω 2=
[ ] ( )4/cos1 nnx π=[ ] ( )8/3cos2 nnx π=[ ] ( )nnx cos3 =
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SP- Berlin Chen 6
Sinusoid Signals (cont.)
[ ] ( )
( )8
integers positive are and 824
44cos)(
4cos
4cos
4/cos
1
111
11
1
=∴
⋅=⇒⋅=⇒
⎟⎠
⎞⎜⎝
⎛ +=⎟⎠
⎞⎜⎝
⎛ +=⎟⎠
⎞⎜⎝
⎛=
=
N
kNkNkN
NnNnn
nnx
ππ
ππππ
π
[ ] ( )
( )
( )
16
numbers positive are and 3
1628
3
83
83cos
83cos
83cos
8/3cos
2
222
22
2
=∴
=⇒⋅=⋅⇒
⎟⎠⎞
⎜⎝⎛ ⋅+⋅=⎟
⎠⎞
⎜⎝⎛ +=⎟
⎠⎞
⎜⎝⎛ ⋅=
=
N
kNkNkN
NnNnn
nnx
ππ
πππππ
[ ] ( )( ) ( )( ) ( )
!exist t doesn' integers positive are and
2 cos1cos1cos
cos
3
3
3
33
3
NkN
kNNnNnn
nnx
∴
⋅=⇒+=+⋅=⋅=
=
Q
π
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SP- Berlin Chen 7
Sinusoid Signals (cont.)
• Complex Exponential Signal– Use Euler’s relation to express complex numbers
( )φφφ sincos
jAAez
jyxzj +==⇒
+=
( )number real a is A
Re
Im
φφ
sin cos
AyAx
==
22 yx + 22 yxx
+22 yx
y
+
Imaginary part
real part
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SP- Berlin Chen 8
Sinusoid Signals (cont.)
• A Sinusoid Signal
[ ] ( )( ){ }
{ }φωφω
φω
jnj
nj
eAeAe
nAnx
Re Re
cos
=
=
+=+
real part
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SP- Berlin Chen 9
Sinusoid Signals (cont.)
• Sum of two complex exponential signals with same frequency
– When only the real part is considered
– The sum of N sinusoids of the same frequency is another sinusoid of the same frequency
( ) ( )
( )
( )φω
φω
φφω
φωφω
+
++
=
=
+=
+
nj
jnj
jjnj
njnj
Ae
Aee
eAeAe
eAeA10
10
10
10
( ) ( ) ( )φωφωφω +=+++ nAnAnA coscoscos 1100
numbers real are and , 10 AAA
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SP- Berlin Chen 10
Sinusoid Signals (cont.)
• Trigonometric Identities
1100
1100coscossinsintan
φφφφ
φAAAA
++
=
( ) ( )( )
( )10102120101010
21
20
2
21100
21100
2
cos2
sinsincoscos2
sinsincoscos
φφ
φφφφ
φφφφ
−++=
+++=
+++=
AAAA
AAAAA
AAAAA
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SP- Berlin Chen 11
Some Digital Signals
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Some Digital Signals
• Any signal sequence can be represented as a sum of shift and scaled unit impulse sequences (signals)
[ ]nx
[ ] [ ] [ ]knkxnxk
−∑=∞
−∞=δ
scale/weightedTime-shifted unit impulse sequence
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221
33221101122
3
2
−+−−+−+++−++=−+−+−+++−++−=
−=−= ∑∑−=
∞
−∞=
nnnnnnnxnxnxnxnxnx
knkxknkxnxkk
δδδδδδδδδδδδ
δδ
,...3,2,1,0,1,2..., −−=n
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SP- Berlin Chen 13
Digital Systems
• A digital system T is a system that, given an input signal x[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 [ ][ ] samples shift left )0 (if
samples shift right )0 (if mmmnx
mmmnx⇒>+⇒>−
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SP- Berlin Chen 15
Properties of Digital Systems (cont.)
• Linear time-invariant (LTI)– The system output can be expressed as a
convolution (迴旋積分) of the input x[n] and the impulse response h[n]
– The system can be characterized by the system’s impulse response h[n], which also is a signal sequence
• If the input x[n] is impulse , the output is h[n][ ]nδ
DigitalSystem
[ ]nδ [ ]nh
[ ]{ } [ ] [ ] [ ] [ ] [ ] [ ]khknxknhkxnhnxnxTkk∑ −=−∑=∗=∞
−∞=
∞
−∞=
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SP- Berlin Chen 16
Properties of Digital Systems (cont.)
• Linear time-invariant (LTI)– Explanation:
[ ] [ ] [ ]knkxnxk
−= ∑∞
−∞=δ
[ ]{ } [ ] [ ]{ }[ ] [ ]{ }
[ ] [ ][ ] [ ]nhnx
knhkx
knTkx
knkxTnxT
k
k
k
∗=
−=
−=
−=⇒
∑
∑
∑
∞
−∞=
∞
−∞=
∞
−∞=
δ
δ
scale
Time-shifted unit impulse sequence
Time invariant
DigitalSystem
[ ]nδ [ ]nh
linear
Time-invariant
[ ] [ ][ ] [ ]knhkn
nhnT
T
−⎯→⎯−
⎯→⎯
δ
δ
Impulse response
convolution
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SP- Berlin Chen 17
Properties of Digital Systems (cont.)
• Linear time-invariant (LTI)– Convolution
• Example
[ ]nδ 0 12
13
-2[ ]nh
LTI[ ]nx ?
0 1 2
23
1
0
3
0 12
39
-6
1
2
1 23
26
-4
LTI
2
1
2 34
1 3
-2
Length=M=3
Length=L=3
Length=L+M-1
[ ]nδ⋅3
[ ]12 −⋅ nδ
[ ]21 −⋅ nδ
0
3
1
11
2
13
-1
4
-2
Sum up
[ ]ny
[ ]nh⋅3
[ ]12 −⋅ nh
[ ]2−nh
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SP- Berlin Chen 18
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] or h[-(k-n)] ), multiply it with
x[k]• Sum up [ ]kx
[ ]kh
[ ]kh −
Reflect Multiply
slide
Sum up
n
[ ] [ ] [ ]
[ ] ( )[ ]nkhkx
knhkxny
k
k
−−=
−=
∑
∑∞
−∞=
∞
−∞=
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SP- Berlin Chen 19
0 12
13
-2
0-1-2
13
-2
0 1 2
23
1
0
3
10-1
13
-21
11
210
13
-2
1
2
321
13
-2 -1
3
432
13
-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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SP- Berlin Chen 20
Properties of Digital Systems (cont.)
• Linear time-invariant (LTI)– Convolution is commutative and distributive
[ ] [ ] [ ][ ] [ ]
[ ] [ ]
[ ] [ ]knxkh
knhkx
nxnhnhnxny
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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SP- Berlin Chen 21
Properties of Digital Systems (cont.)
• Prove convolution is commutative
[ ] [ ] [ ][ ] [ ]
[ ] [ ] ( )
[ ] [ ][ ] [ ]nxnh
knxkh
knmhmnx
knhkx
nhnxny
k
m
k
*
mlet
*
=
∑ −=
−=∑ −=
−∑=
=
∞
−∞=
∞
−∞=
∞
−∞=
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SP- Berlin Chen 22
Properties of Digital Systems (cont.)
• Linear time-varying System– E.g., is an amplitude modulator [ ] [ ] nnxny 0cos ω=
[ ] [ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] ( )0000
00011
0?
101
cosBut
coscos
that suppose
nnωnnxnny
nnnxnnxny
nnynynnxnx
−−=−
−==
−=⇒−=
ωω
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SP- Berlin Chen 23
Properties of Digital Systems (cont.)
• Bounded Input and Bounded Output (BIBO): stable
– A LTI system is BIBO if only if h[n] is absolutely summable
[ ] ∞≤∑∞−∞=k
kh
[ ][ ] nBny
nBnx
y
x
∀∞
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SP- Berlin Chen 24
Properties of Digital Systems (cont.)
• Causality– A system is “casual” if for every choice of n0, the output
sequence value at indexing n=n0 depends on only the input sequence value for n ≤ n0
– Any noncausal FIR can be made causal by adding sufficient long delay
[ ] [ ] [ ]∑ ∑= =
−+−=K
k
M
mkkk mnxknyny
1000 βα
z-1
z-1
z-1
Mβ
2β
0β [ ]ny[ ]nx1β
z-1
z-1
z-1
1α
2α
Nα
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SP- Berlin Chen 25
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( )ωjeH
π2( )ωjeH
ω( ) ( ) ( )
( ) ( )ωωωωω
jeHjj
ji
jr
j
eeH
ejHeHeH∠=
+=
magnitudephase
( )ωjeH
proportional to twotimes of the samplingfrequency
njne nj ωωω sincos +=
Im
Re
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SP- Berlin Chen 26
Discrete-Time Fourier Transform (cont.)
• Representation of Sequences by Fourier Transform
– A sufficient condition for the existence of Fourier transform
[ ] ∞
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SP- Berlin Chen 27
Discrete-Time Fourier Transform (cont.)• Convolution Property
( ) [ ]
[ ] [ ] [ ]
( ) [ ] [ ]
[ ] [ ]
( ) ( )
[ ] ( ) ( )ωω
ωω
ωω
ωω
ωω
jj
jj
nj
nk
kj
nj
n k
j
k
n
njj
eHeXnhnx
eHeX
enhekx
eknhkxeY
knhkxnhnxny
enheH
⇔∗∴
=
⎟⎠
⎞⎜⎝
⎛=
−=
−=∗=
=
−∞
−∞=
∞
−∞=
−
−∞
−∞=
∞
−∞=
∞
−∞=
∞
−∞=
−
∑∑
∑ ∑
∑
∑
][
'
][][
'
'
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω
ωωω
ωωω
jjj
jjj
jjj
eHeXeY
eHeXeY
eHeXeY
∠+∠=∠⇒
=⇒
=
knnknnknn
−−=−⇒+=⇒−=
''
'
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SP- Berlin Chen 28
Discrete-Time Fourier Transform (cont.)
• Parseval’s Theorem
– Define the autocorrelation of signal
[ ] ( )∫∑ −∞−∞=
= ππω ω
πdeXnx j
n
22 21
power spectrum
[ ]nx[ ]
( ) ][][][][
][][
*
*
nxnxlnxlx
mxnmxnR
l
mxx
−∗=−−=
+=
∗∞
−∞=
∞
−∞=
∑
∑
( ) ( ) ( ) ( ) 2* ωωωω XXXS xx ==⇔
[ ] ( ) ( )∫∫ −− ==π
π
ωπ
π
ω ωωπ
ωωπ
deXdeSnR njnjxxxx2
21
21
[ ] [ ] [ ] [ ] ( )∑ ∫∑∞
−∞=−
∞
−∞=
===
=
mmxx dXmxmxmxR
nπ
πωω
π22*
210
0Set
The total energy of a signalcan be given in either the time or frequency domain.
)(
lnnlmnml
−−=−=⇒+=
222
*
conjugatecomplex :
zyxzz
jyxz jyxz*
*
=+=⋅⇒
−=⇒+=
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SP- Berlin Chen 29
Discrete-Time Fourier Transform (DTFT)
• A LTI system with impulse response– What is the output for
[ ]nh[ ]ny [ ] ( )φω += nAnx cos
[ ] ( )[ ] [ ] [ ] ( )
[ ] ( ) [ ]
[ ] ( )( )
( ) [ ]( ) ( )( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )
[ ] ( ) ( ) ( )( )ωωωωωω
φωω
ωφω
ωφω
ωφω
φω
φω
φω
φω
φωφω
φω
ω
ω
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
( )( ) attenuate 1
amplify 1
<
>
ω
ω
j
j
eH
eH
System’s frequency response
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SP- Berlin Chen 30
Discrete-Time Fourier Transform (cont.)
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SP- Berlin Chen 31
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 on the unit circle
( ) [ ]∑∞−∞=
−=n
nznhzH
[ ] ( )zHnh [ ] ( )ωjeHnh
[ ]nh
ωjrez =
( ) ( ) ωω jezj zHeH ==)1( == zez jω
complex plane
unit circle
Im
Re
ω
ωjez =
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SP- Berlin Chen 32
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 of z for 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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SP- Berlin Chen 33
Z-Transform (cont.)
• An LTI system is defined to be causal, if its impulse response is a causal signal, i.e.
– Similarly, anti-causal can 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
[ ] 0for 0 = nnh
Right-sided sequence
Left-sided sequence
[ ] ∞
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SP- Berlin Chen 34
Z-Transform (cont.)
• Right-Sided Sequence– E.g., the exponential signal
[ ] [ ] [ ]⎩⎨⎧
<≥
==0for 00for 1
ere wh, .1 1 nn
nunuanh n
( ) ( ) 111 11
−
∞
−∞=
−−∞
−∞= −=== ∑∑
azazzazH
n
nn
n
n
If 11 ∴ is 1
Re
Im
a[ ] 1 if exists
ofansformFourier tr
1
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SP- Berlin Chen 35
Z-Transform (cont.)
• Left-Sided Sequence– E.g.
[ ] [ ] ).,-0,-1,-2,..(n 1 .2 2 ∞=−−−= nuanh n
( ) [ ]
( ) 111
10
1
1
12
11
11111
1
−−
−
−
∞
=
−∞
=
−
−−
−∞=
−∞
−∞=
−=
−−
−=−
−=−=−=
−=−−−=
∑∑
∑∑
azzaza
zazaza
zaznuazH
n
n
n
nn
n
n
nn
n
n If 11
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SP- Berlin Chen 36
Z-Transform (cont.)
• Two-Sided Sequence– E.g.
[ ] [ ] [ ]121
31 .3 3 −−⎟
⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−= nununh
nn
[ ]
[ ]21 ,
211
1 121
31 ,
311
1 31
1
1
<−
⎯→←−−⎟⎠⎞
⎜⎝⎛−
>+
⎯→←⎟⎠⎞
⎜⎝⎛ −
−
−
zz
nu
zz
nu
zn
zn
Re
Im
×
the unit cycle
×21
31
− [ ]circleunit theincludet doesn' because
exist,t doesn' of ansformFourier tr
3
3
ROCnh
31 and
21 is 3 >
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SP- Berlin Chen 37
Z-Transform (cont.)
• Finite-length Sequence– E.g.
[ ]⎩⎨⎧ −≤≤
=others ,0
10 , .3 4
Nnanh
n
( ) ( ) ( )azaz
zazazazzazH
NN
N
NN
n
nN
n
nn
−−
=−
−=== −−
−−
=
−−
=
− ∑∑ 11
11
0
11
04
11
1
0except plane- entire is 4 =∴ zzROC
13221 ..... −−−− ++++ NNNN azaazz
Im
×
the unit cycle
31
−
7 poles at zero A pole and zero at is cancelled az =
4π
( ) 11 ,2
−== ,..,N kaez Nkj
k
π
If N=8
0 N-1
Re
-
SP- Berlin Chen 38
Z-Transform (cont.)
• Properties of z-transform1. If is right-sided sequence, i.e. and if ROC
is the exterior of some circle, the all finite for which will be in ROC
• If ,ROC will include
2. If is left-sided sequence, i.e. , the ROC is the interior of some circle,
• If ,ROC will include 3. If is two-sided sequence, the ROC is a ring4. The ROC can’t contain any poles
[ ]nh [ ] 1 ,0 nnnh ≤=
01 ≥n
0rz >z
∞=z
[ ]nh [ ] 2 ,0 nnnh ≥=
02
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SP- Berlin Chen 39
Summary of the Fourier and z-transforms
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SP- Berlin Chen 40
LTI Systems in the Frequency Domain
• Example 1: A complex exponential sequence– System impulse response
– Therefore, a complex exponential input to an LTI systemresults in the same complex exponential at the output, but modified by
• The complex exponential is an eigenfunction of an LTI system, and is the associated eigenvalue
[ ]nh
[ ] [ ] [ ] [ ]
[ ]
( ) njjkjnj
knj
eeH
ekhe
ekhnhnxny
ωω
ωω
ω
=
=
=∗=
−∞
∞=
−∞
∞=
∑
∑
-k
)(
-k( )
response.frequency system theas toreferredoften isIt
response. impulse system the of ansformFourier tr the:ωjeH
[ ] njenx ω= [ ] [ ] [ ][ ] [ ][ ] [ ]
[ ] [ ]knxkh
knhkx
nxnhnhnxny
k
k
−=
−=
==
∑
∑
∞
−∞=
∞
−∞=
* *
[ ]{ } ( ) [ ]nxeHnxT jω=( )jωeH
( )jωeH
scalar
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SP- Berlin Chen 41
[ ] ( ) ( )
( ) ( )[ ]
( ) ( ) ( ) ( )[ ]( ) ( )[ ]00
00
000
0
0000
0000
0
)()(
)()(
cos 2
2
22
jωjω
njeHjjωnjeHjjω
njjω*njjω
njjjωnjjjω
eHneHA
eeeHeeeHA
eeHeeHA
eeAeHeeAeHny
jωjω
∠++=
+=
+=
+=
+−∠−+∠
+−+
−−−
φω
φωφω
φωφω
ωφωφ
LTI Systems in the Frequency Domain (cont.)
• Example 2: A sinusoidal sequence
– System impulse response
[ ] ( )φ+= nwAnx 0cos[ ] ( )
njjnjj eeAeeAnAnx
00
22
cos 0ωφωφ
φω
−−+=
+=
[ ]nh( )θθ
θ
θ
θ
θθ
θθ
jj
j
j
ee
ieie
−
−
+=⇒
−=
+=
21cos
sincossincos
( ) ( )( ) ( ) ( )000
00
jωeHjjωjω*
jω*jω
eeHeH
eHeH∠−
−
=
=
( )( ) ( )
ωωωω
ωω
ωω
ω
ω
ω
sincos sincos
sincos *
sincos *
jje
jejyxz
jejyxz
j
j
j
−=−+−=
−=⇒−=
+=⇒+=
−
magnitude response phase response
-
SP- Berlin Chen 42
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
1cos φω
[ ] ( ) ( )[ ]∑=
∠++=K
k
jkk
jk
kk eHneHAny1
cos ωω φωmagnitude response phase response
-
SP- Berlin Chen 43
LTI Systems in the Frequency Domain (cont.)
• Example 4: Convolution Theorem
[ ] [ ]∑∞−∞=
−=k
kPnnx δ
[ ] [ ] 1 ,
-
SP- Berlin Chen 44
LTI Systems in the Frequency Domain (cont.)
• Example 5: Windowing Theorem [ ] [ ] ( ) ( )ωωπ
jj eXeWnwnx ∗⇔21
[ ] [ ]∑∞−∞=
−=k
kPnnx δ
[ ]⎪⎩
⎪⎨⎧
−=⎟⎠⎞
⎜⎝⎛
−−=
otherwise 0
1,......,1,0 ,1
2cos46.054.0 NnN
nnw
πHamming window
( ) ( ) ( )
( )
( )
( )
∑
∑ ∑
∑
∑
∞
−∞=
⎟⎠⎞
⎜⎝⎛ −
∞
−∞=
∞=
−∞=
∞
−∞=
∞
−∞=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ −−=
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ −∗=
⎟⎠⎞
⎜⎝⎛ −∗=
∗=
k
kP
j
k
m
m
jm
k
j
k
j
jjj
eWP
mkP
eWP
kP
eWP
kPP
eW
eXeWeY
πω
ω
ω
ωωω
πωδ
πωδ
πωδππ
π
21
2 1
21
2221
21
has a nonzero value when
kP
m πω 2−=
-
SP- Berlin Chen 45
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 0βα
( ) ( ) ( ) kNk
M
kk
kk zzXzzYzY
−
= =
−∑ ∑+=1 0
βα
delay property
( ) ( )( ) ∑
∑
=
−
=
−
−== N
k
kk
M
k
kk
z
z
zXzYzH
1
0
1 α
β
[ ] ( )[ ] ( ) 00 nzzXnnx
zXnx−→−
→linearity and delay properties
A rational transfer functionCausal:Rightsided, the ROC outside the outmost poleStable:The ROC includes the unit circleCausal and Stable:all poles must fall inside the unit circle (not including zeros)
z-1
z-1
z-1
Mβ
2β
0β [ ]ny[ ]nx1β
z-1
z-1
z-1
1α
2α
Nα
-
SP- Berlin Chen 46
Difference Equation Realization for a Digital Filter (cont.)
-
SP- Berlin Chen 47
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 responsewith unit (or flat) magnitude for all frequencies
1
111
±
− ⎥⎦
⎤⎢⎣
⎡
− azz-a*
Poles and zeros occur at conjugate reciprocal locations
111
1 =− −azz-a*
-
SP- Berlin Chen 48
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-all a is 11
filter. phase minimum a also is 1
:where
111
filter) phase minimum a is ( 1
:as expressed becan circle.unit theoutside
1)( 1 zero oneonly has that Suppose
1
*
11
1
*1
1
1*
1
−
−
−−
−−
−
−−
−=
−=
<
azza
azzH
azzaazzH
zHzazHzH
zH
aa*zH
-
SP- Berlin Chen 49
FIR Filters
• FIR (Finite Impulse Response)– The impulse response of an FIR filter has finite duration– Have no denominator in the rational function
• No feedback in the difference equation
– Can be implemented with simple a train of delay, multiple, and add operations
( )zH
[ ] [ ]∑=
−=M
rr rnxny
0β
z-1
z-1
z-1
Mβ
1β0β
[ ]ny[ ]nx
[ ]⎩⎨⎧ ≤≤
=otherwise ,00 , Mn
nh nβ
( ) ( )( ) ∑=−==
M
k
kk zzX
zYzH0β
-
SP- Berlin Chen 50
First-Order FIR Filters
• A special case of FIR filters
[ ] [ ] [ ]1−+= nxnxny α ( ) 11 −+= zzH α( ) ωω α jj eeH −+= 1( ) ( )
( ) ( )
( ) ⎟⎠⎞
⎜⎝⎛
+−=
+=++=
−+=
ωαωαθ
ωαωαωα
ωωα
ω
ω
cos1sinarctan
cos21sincos1
sincos122
22
j
j
e
jeH
( ) 2log10 ωjeH
: pre-emphasis filter0
-
SP- Berlin Chen 51
Discrete Fourier Transform (DFT)
• The Fourier transform of a discrete-time sequence is a continuous function of frequency– We need to sample the Fourier transform finely enough to be
able to recover the sequence– For a sequence of finite length N, sampling yields the new
transform referred to as discrete Fourier transform (DFT)
( ) [ ]
[ ] ( ) 10 , 1
10 ,
21
0
21
0
−≤≤∑=
−≤≤∑=
−
=
−−
=
NnekXN
nx
NnenxkX
knN
jN
k
knN
jN
nπ
π
Inverse DFT, Synthesis
DFT, Analysis
-
SP- Berlin Chen 52
Discrete Fourier Transform (cont.)
[ ] [ ]
( )
( ) ( ) ( )
[ ][ ]
[ ]
[ ][ ]
[ ]⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−≤≤=
−≤≤∀
−⋅−−⋅−−
−⋅−⋅−
−−
=∑
1
10
1
10
1
1
111
10 ,
10
112112
112112
21
0
NX
XX
Nx
xx
ee
ee
NnenxkX
Nk
NNN
jNN
j
NN
jN
j
knN
jN
n
MM
L
MLMM
L
L
ππ
ππ
π
Orthogonal
-
SP- Berlin Chen 53
Discrete Fourier Transform (cont.)
• Orthogonality of Complex Exponentials ( )
⎩⎨⎧ =
=∑−
=
−
otherwise ,0 if ,11 1
0
2 mNk-re
N
N
n
nrkN
j π
[ ] [ ]
[ ] [ ]( )
[ ]( )
[ ]
[ ] [ ]kn
Nj
nrkN
j
nrkN
jrnN
j
knN
j
enxkX
rX
eN
kX
ekXN
enx
ekXN
nx
N
n
N
n
N
k
N
n
N
k
N
n
N
k
π
π
ππ
π
2
2
22
2
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1
1
−
−
−−
∑=⇒
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡∑∑=
∑ ∑=∑⇒
∑=
−
=
−
=
−
=
−
=
−
=
−
=
−
=
[ ] [ ][ ]rX
mNrXkX=
+=
-
SP- Berlin Chen 54
Discrete Fourier Transform (DFT)
• Parseval’s theorem
[ ] ( )∑∑−
=
−
=
=1
0
21
0
2 1 N
k
N
nkX
Nnx Energy density
-
SP- Berlin Chen 55
Analog Signal to Digital Signal
[ ] ( ) period sampling : , TnTxnx a=
rate sampling 1TFs =
Analog Signal
Discrete-time Signal or Digital Signal
nTt =
sampling period=125μs=>sampling rate=8kHz
Digital Signal:Discrete-time
signal with discreteamplitude
-
SP- Berlin Chen 56
Analog Signal to Digital Signal (cont.)
Impulse TrainTo
Sequence [ ] ( ))( ˆ nTxnx a=
Sampling
( )
( ) ( ) ( ) ( )
( ) ( ) [ ] ( )∑∑
∑∞
−∞=
∞
−∞=
∞
−∞=
−=−=
−==
nna
naa
s
nTtnxnTtnTx
nTttxtstx
tx
δδ
δ
( )tx a
Continuous-TimeSignal
Discrete-TimeSignal
Digital Signal
Discrete-time signal with discrete
amplitude
[ ] nx
Continuous-Time to Discrete-Time Conversion
( ) ( )∑∞−∞=
−=n
nTtts δPeriodic Impulse Train ( ) [ ]nxtxs by specifieduniquely becan
switch
( )tx a
( ) ( )∑∞−∞=
−=n
nTtts δ
( )
( )∫∞
∞−
=
≠∀=
1
0 ,0
dtt
tt
δ
δ1
T0 2T 3T 4T 5T 6T 7T 8T-T-2T
-
SP- Berlin Chen 57
Analog Signal to Digital Signal (cont.)
• A continuous signal sampled at different periods
T
1
( )tx a ( )tx a
( )
( ) ( ) ( ) ( )
( ) ( ) [ ] ( )∑∑
∑∞
−∞=
∞
−∞=
∞
−∞=
−=−=
−==
nna
naa
s
nTtnxnTtnTx
nTttxtstx
tx
δδ
δ
-
SP- Berlin Chen 58
Analog Signal to Digital Signal (cont.)
( )ΩjXa
frequency) (sampling22 ss FTππ ==Ω
⎟⎠⎞
⎜⎝⎛=ΩΩ
TsNπ
21
aliasing distortion
( ) ( )∑∞−∞=
Ω−Ω=Ωk
skTjS δπ2
( ) ( ) ( )
( ) ( )( )∑∞
−∞=
Ω−Ω=Ω
Ω∗Ω=Ω
ksas
as
kjXT
jX
jSjXjX
121π
NΩ
( )⎭⎬⎫
⎩⎨⎧
Ω>Ω⇒Ω>Ω−Ω
2 NsNNsQ
( )⎭⎬⎫
⎩⎨⎧
Ω
-
SP- Berlin Chen 59
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 sΩ
( ) tth sΩ= sinc( ) ( ) ( )
( ) ( )∑
∑∞
−∞=
∞
−∞=
−Ω=
−=
nsa
naa
nTtnTx
nTthnTxtx
sinc
Ns Ω>Ω 2
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