Di it l Si l P iDigital Signal Processing for Speech ...berlin.csie.ntnu.edu.tw/Courses/Speech Recognition/Lectures2009... · Di it l Si l P iDigital Signal Processing for Speech

Di it l Si l P iDigital Signal Processing for Speech Recognition

Berlin ChenDepartment of Computer Science & Information Engineering

R f

g gNational Taiwan Normal University

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 4-63. 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

Digital Signal Processingg g g

• Digital Signalg g– Discrete-time signal with discrete amplitude

[ ]nx

• Digital Signal Processing– Manipulate digital signals in a digital computer

SP- Berlin Chen 2

Two Main Approaches to Digital Signal ProcessingProcessing

• Filteringg

FilterSignal in Signal out

[ ]nx [ ]ny[ ] nx [ ]ny

Amplify or attenuate some frequencycomponents of [ ] nx

• Parameter Extraction

P tSignal in Parameter outParameterExtraction

Signal in Parameter out

[ ] nx11⎥⎤

⎢⎡c 21

⎥⎤

⎢⎡c 1

⎥⎤

⎢⎡ Lc

12

⎥⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢⎢c

22

⎥⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢⎢c

2

⎥⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢⎢

Lce.g.:1. Spectrum Estimation2. Parameters for Recognition

SP- Berlin Chen 3

1⎥⎥

⎦⎢⎢

⎣ mc 2⎥⎦⎢⎣ mc ⎥⎦⎢⎣ Lmc

Sinusoid Signals

: amplitude (振幅)[ ] ( )φω += nAnx cos

A10

frequency normalized:≤≤ f

f

– : amplitude (振幅)– : angular frequency (角頻率), – : phase (相角)

ωφ

A

Tf ππω 22 ==

: phase (相角)φ Period, represented by number of samples

[ ] ⎟⎠⎞

⎜⎝⎛ −=

2cos πωnAnx

⎠⎝

samples 25=T[ ] [ ] )(cos ⎟

⎞⎜⎛ −+=+= NnANnxnx πω

Right-shifted

[ ] [ ]

( ),...2,122

2)(cos

==⇒=⇒

⎟⎠

⎜⎝

+=+=

kN

kkN

NnANnxnx

πωπω

ω

SP- Berlin Chen 4

• E.g., speech signals can be decomposed as sums of sinusoids

Sinusoid Signals (cont.)g ( )

• is periodic with a period of N (samples)[ ]nx is periodic with a period of N (samples)[ ]nx[ ] [ ]nxNnx =+

( ) ( )φφ ANA )(( ) ( )φωφω +=++ nANnA cos)(cos)21( 2 ,...,kkN == πω

π2

• Examples (sinusoid signals)Nπω 2

=

• Examples (sinusoid signals)

– is periodic with period N=8[ ] ( )4/cos1 nnx π=– is periodic with period N=16– is not periodic

[ ] ( )8/3cos2 nnx π=

[ ] ( )nnx cos3 =

SP- Berlin Chen 5


[ ] ( )4/cos1

⎞⎛⎞⎛⎞⎛

= nnx π

( )integers positive are and 824

44cos)(

4cos

4cos

111

11

⋅=⇒⋅=⇒

⎟⎠

⎞⎜⎝

⎛ +=⎟⎠

⎞⎜⎝

⎛ +=⎟⎠

⎞⎜⎝

⎛=

kNkNkN

NnNnn

ππ

ππππ

84

1 =∴ N

[ ] ( )8/3cos2

⎞⎛⎞⎛⎞⎛

= nnx π

( )

( )numberspositive areand 3

1628

3

83

83cos

83cos

83cos

222

22

=⇒⋅=⋅⇒

⎟⎠⎞

⎜⎝⎛ ⋅+⋅=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ ⋅=

kNkNkN

NnNnn

ππ

ππππ

( )

16

p38

2

222

=∴ N

[ ] ( )cos3 nnx =[ ] ( )( ) ( )( ) ( )

ii id2

cos1cos1cos cos

3

33

3

kNkN

NnNnnnnx

⋅=⇒+=+⋅=⋅=

π

SP- Berlin Chen 6

!exist t doesn' integerspositive are and

3

3

NkN

∴Q


• Complex Exponential Signalp p g– Use Euler’s relation to express complex numbers

j

( )φφφ sincos

jAAez

jyxzj +==⇒

+=

( )numberrealais A( )

22 yx + 22 yx

x

+22 yx

y

+

ImImaginary part

φφ

sin cos

AyAx

==

SP- Berlin Chen 7Re

φsinAy =

real part


• A Sinusoid Signalg

[ ] ( )φω nAnx cos +=[ ] ( )( ){ }φω

φωnjAe

nAnxRe

cos=

+=+{ }

{ }φω jnj eAeRe =real part

SP- Berlin Chen 8


• Sum of two complex exponential signals with p p gsame frequency

( ) ( )φωφω ++ + njnj eAeA 10( ) ( )

( )φφω

φφ

+=

+jjnj

jj

eAeAe

eAeA10

10

10

10

( )φω

φω

+

=nj

jnj

Ae

Aee

– When only the real part is considered

( )φ= jAenumbers real are and , 10 AAA

( ) ( ) ( )φωφωφω +=+++ nAnAnA coscoscos 1100

SP- Berlin Chen 9

– The sum of N sinusoids of the same frequency is another sinusoid of the same frequency


• Trigonometric IdentitiesTrigonometric Identities

1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )101010

21

20

2

21100

21100

2

sinsincoscos2

sinsincoscos

φφφφ

φφφφ

+++=

+++=

AAAAA

AAAAA

SP- Berlin Chen 10

( )( )1010

21

20

10101010

cos2 φφ

φφφφ

−++= AAAA

Some Digital Signalsg g

SP- Berlin Chen 11

Some Digital Signalsg g

• Any signal sequence can be represented hif l d

[ ]nxas a sum of shift and scaled unit impulse sequences (signals)

∞[ ] [ ] [ ]knkxnxk

−∑=∞

−∞=δ

scale/weightedTime-shifted unit impulse sequence,...3,2,1,0,1,2..., −−=n scale/weighted impulse sequence

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

3

2

−=−= ∑∑−=

∞

−∞=

knkxknkxnxkk

δδ

SP- Berlin Chen 12

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122 −+−−+−+++−++=

−+−+−+++−++−=nnnnnn

nxnxnxnxnxnxδδδδδδ

δδδδδδ

Digital Systemsg y

• A digital system T is a system that, given an input signal x[n], generates an output signal y[n]

[ ] [ ]{ }nxTny =

[ ]nx { }•T [ ]ny[ ]nx { } [ ]ny

SP- Berlin Chen 13

Properties of Digital Systemsg y

• Linear– Linear combination of inputs maps to linear

combination of outputs

[ ] [ ]{ } [ ]{ } [ ]{ }nxbTnxaTnbxnaxT 2121 +=+

• Time-invariant (Time-shift)– A time shift in the input by m samples gives a shift inA time shift in the input by m samples gives a shift in

the output by m samples[ ] [ ]{ } mmnxTmny ∀±=± ,[ ] [ ]{ }y

time sift [ ][ ] lhifl f)(if

samples shift right )0 (if mmmnx ⇒>−

SP- Berlin Chen 14

[ ] samplesshift left )0 (if mmmnx ⇒>+

Properties of Digital Systems (cont.)g y ( )

• Linear time-invariant (LTI)– The system output can be expressed as a

convolution (迴旋積分) of the input x[n] and the i l h[ ]impulse response h[n]

[ ]{ } [ ] [ ] [ ] [ ] [ ] [ ]khknxknhkxnhnxnxT ∑ −=−∑=∗=∞∞

– The system can be characterized by the system’s impulse response h[n], which also is a signal

kk −∞=−∞=

impulse response h[n], which also is a signal sequence

• If the input x[n] is impulse , the output is h[n][ ]nδp [ ] p p [ ]

Digital[ ]nδ [ ]nh

SP- Berlin Chen 15

System


• Linear time-invariant (LTI)Linear time invariant (LTI)– Explanation:

[ ] [ ] [ ]knkxnx −= ∑∞

δ Time-shifted unit impulse sequence[ ] [ ] [ ]

k −∞=

[ ]{ } [ ] [ ]{ }knkxTnxT −=⇒ ∑∞

δ

scaleimpulse sequence

[ ]{ } [ ] [ ]{ }[ ] [ ]{ }knTkx

k

k

−= ∑∞

−∞=

−∞=

δlinear

[ ] [ ][ ] [ ]nhnx

knhkxk

∗=

−= ∑∞

−∞=

Di i l[ ]nδ [ ]nh Time-invariantImpulse response

[ ] [ ]nhnx ∗=

Time invariant

DigitalSystem

[ ]nδ [ ]nh

[ ] [ ]convolution

SP- Berlin Chen 16

[ ] [ ][ ] [ ]knhkn

nhnT

T

−⎯→⎯−

⎯→⎯

δ

δ


• Linear time-invariant (LTI)– Convolution

• Example1

3

Length=M=3

[ ]nδ 0 12

1

-2[ ]nh

[ ]3LTILength=L=3

LTI[ ]nx ?

0 1 2

23

1

3 23

9

Length=L+M-1

[ ]nh⋅3

0

3

0 12

-6

26

[ ]nδ⋅3

3

11

1

Sum up

[ ]ny

[ ]nh3

[ ]12⋅ nh

1

2

1 23

2

-41 1 3

[ ]12 −⋅ nδ 0 1 2

13

-1

4

-2

[ ]12 −⋅ nh

[ ]2nh

SP- Berlin Chen 17

2

1

2 34

1 3

-2

[ ]21 −⋅ nδ[ ]2−nh


• Linear time-invariant (LTI)Linear time invariant (LTI)– Convolution: Generalization

• Reflect h[k] about the origin (→ h[-k])Reflect h[k] about the origin (→ h[ k]) • Slide (h[-k] → h[-k+n] or h[-(k-n)] ), multiply it with

x[k] [ ] [ ] [ ]knhkxny −= ∑∞[ ]

• Sum up [ ]kx

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

−−=

=

∑

∑∞

−∞=

−∞=

[ ]kh Reflect Multiply

[ ]kh −slide

Sum up

SP- Berlin Chen 18

n

21

3

[ ]kh

Convolution0 1

-22

31

[ ]kh

[ ]

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

=

∗=

∑∞

-21

30 1 2

3

[ ]kx

[ ] 0, =nny

[ ] [ ]knhkxk

−= ∑−∞=

0-1-2

-20

13 11

[ ]kh −

[ ] 1, =nny

10-1

1

-21

13 1

11Sum up

[ ]1+−kh[ ],y

[ ] 2=nny [ ]ny

210

1

-22

13 0

3

1 2

13

-1

4

-2

[ ]2+−kh[ ] 2, =nny

[ ] 3=nny

[ ]ny

321

1

-2 -1

3

3

2

[ ]3+−kh[ ] 3, =nny

[ ] 4, =nny

SP- Berlin Chen 19

432

13

-2 -2

4[ ]4+−kh

[ ] 4, nny


• Linear time-invariant (LTI)Linear time invariant (LTI)– Convolution is commutative and distributive

[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx 21

****

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]h

[ ] [ ] [ ]nhnhnx 12 **=

[ ] [ ] [ ]( )nhnhnx 21* +

Commutation

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +[ ] [ ] [ ] [ ]nhnxnhnx 21 ** +=

Distribution

[ ] [ ] [ ][ ] [ ]nxnh

nhnxny==

* *

– An impulse response has finite duration» Finite Impulse Response (FIR)

[ ] [ ]

[ ] [ ]knxkh

knhkxk

−=

−=

∑

∑

∞

∞

−∞=

» Finite-Impulse Response (FIR)

– An impulse response has infinite duration» Infinite-Impulse Response (IIR)

SP- Berlin Chen 20

[ ] [ ]k −∞=

p p ( )


• Prove convolution is commutativeProve convolution is commutative

[ ] [ ] [ ]nhnxny *=

[ ] [ ]knhkxk

−∑=∞

−∞=

[ ] [ ] ( )knmhmnxm

mlet −=∑ −=∞

−∞=

[ ] [ ][ ] [ ]

knxkhk

∑ −=∞

−∞=

[ ] [ ]nxnh * =

SP- Berlin Chen 21


• Linear time-varying SystemLinear time varying System– E.g., is an amplitude modulator [ ] [ ] nnxny 0cos ω=

[ ] [ ] [ ] [ ]0?

101 that suppose nnynynnxnx −=⇒−=[ ] [ ] [ ] [ ]

[ ] [ ] [ ] 00011

0101

coscos

pp

nnnxnnxny

yy

−== ωω[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

cosBut nnωnnxnny

y

−−=−[ ] [ ] ( )0000

SP- Berlin Chen 22


• Bounded Input and Bounded Output (BIBO): stablep p ( )

[ ][ ]

nBnx x ∀∞<≤ and

– A LTI system is BIBO if only if h[n] is absolutely summable

[ ] nBny y ∀∞<≤

[ ] ∞≤∑∞

−∞=kkh

SP- Berlin Chen 23


• Causalityy– 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 ≤ nsequence value for n ≤ n0

[ ] [ ] [ ]∑ ∑= =

−+−=K

k

M

mkkk mnxknyny

1000 βα

k mk1

1

0β [ ]ny[ ]nx-1

1

z-1

z-1

β

2β

1β

z-1

z-1

z-1

1α

2α

z-1Mβ z 1

Nα

SP- Berlin Chen 24

– Any noncausal FIR can be made causal by adding sufficient long delay

Discrete-Time Fourier Transform (DTFT)( )

• Frequency Response ( )ωjeHq y p– Defined as the discrete-time Fourier Transform of

– is continuous and is periodic with period=

[ ]nh( )

π2( )ωjeH ( )

proportional to twoi f h litimes of the sampling

frequency

j

– is a complex function of ω( )ωjeH

njne nj ωωω sincos +=

Imis a complex function of ω( ) ( ) ( )

( ) ( )ω

ωωω

jH

ji

jr

j ejHeHeH∠

+=

( )

SP- Berlin Chen 25

( ) ( )ωω jeHjj eeH ∠=

magnitudephase Re

Discrete-Time Fourier Transform (cont.)( )

• Representation of Sequences by Fourier Transformp q y

( ) [ ]∑∞

−∞=

−= ωω enheHn

njj DTFT

A ffi i t diti f th i t f F i t f

[ ] ( )∫ −= ππ

ωω ωπ

deeHnh njj 2

1 Inverse DTFT

– A sufficient condition for the existence of Fourier transform

[ ] ∞<∑∞

−∞=nnh de mnj

−−

∫ ωππ

ω

21 )(absolutely summable

∞=n

[ ]( )

emnj

mnj

−= −

−

π

πππ

ω

i)(2

12

)(

( ) [ ]enheH njj = ∑∞ − ωω

:invertible is ansformFourier tr

( )( )

nmmn

mn

⎨⎧ =

−−

=ππ

,1

sin( ) [ ]

[ ] ( ) [ ] deemhdeeHnh nj

m

mjnjj

n

== ∫ ∑∫

∑

−

∞

−∞=

−−

−∞=

ωπ

ωπ

ππ

ωωππ

ωω

21

21

SP- Berlin Chen 26

[ ]mnnm

−=⎩⎨ ≠

=

δ ,0

[ ] [ ] [ ] [ ]nhmnmhdemhm

mnj

m=−== ∑∫∑

∞

−∞=−

−∞

−∞=δω

πππ

ω 21 )(

Discrete-Time Fourier Transform (cont.)( )• Convolution Property

( )∞

( ) [ ]

[ ] [ ] [ ]

ωω

n

njj enheH =

∞

∞

−∞=

−

∑

∑

[ ] [ ] [ ]k

knhkxnhnxny −=∗=−∞=∑][][

knn −='

( ) [ ] [ ] ωω nj

n k

j eknhkxeY −= −∞

−∞=

∞

−∞=∑ ∑ knn

knn−−=−⇒

+=⇒'

'

[ ] [ ]

( ) ( )

ωω nj

nk

kj enhekx ⎟⎠

⎞⎜⎝

⎛= −

∞

−∞=

∞

−∞=

− ∑∑ ' '

'

( ) ( )

( ) ( )

ωω jj eHeX= ( ) ( ) ( )

( ) ( ) ( )ωωω

ωωω

jjj

jjj

eHeXeY

eHeXeY

=⇒

=

SP- Berlin Chen 27

[ ] ( ) ( )ωω jj eHeXnhnx ⇔∗∴ ][( ) ( ) ( )( ) ( ) ( )ωωω jjj eHeXeY

eHeXeY

∠+∠=∠⇒

⇒


• Parseval’s Theorem 222

*

conjugatecomplex :

zyxzz

jyxz jyxz*

*

=+=⋅⇒

−=⇒+=

[ ] ( )∫∑ −

∞

−∞== π

πω ω

πdeXnx j

n

22 21

power spectrumThe total energy of a signalcan be given in either the time or frequency domain

zyxzz =+=⇒

– Define the autocorrelation of signal −∞= πn 2

[ ]nx[ ] ][][ *R

∞

time or frequency domain.

l +[ ]

( ) ][][][][

][][

* nxnxlnxlx

mxnmxnRm

xx

−∗=−−=

+=

∗∞

−∞=

∑

∑)(

lnnlm

nml−−=−=⇒

+=

( ) ][][][][l −∞=∑

( ) ( ) ( ) ( ) 2* ωωωω XXXS xx ==

⇔

( ) ( ) ( ) ( )xx

[ ] ( ) ( )∫∫ −−==

π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21

SP- Berlin Chen 28

[ ] [ ] [ ] [ ] ( )∑ ∫∑∞

−∞=−

∞

−∞=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22*

210

0Set

Discrete-Time Fourier Transform (DTFT)( )

• A LTI system with impulse responseWh i h f

[ ]nh[ ] [ ] ( )φA– What is the output for [ ]ny [ ] ( )φω += nAnx cos

[ ] ( )[ ] [ ] [ ] ( )φω

φωnjA

njAnx

+

+=+

sin1

[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )φω

φω

φω

knj

nj

nj

Aekh

nhAeny

Aenxnxnx

∑=

=

=+=

+−∞

+

+

*0

10

( )[ ]

( ) [ ]

( )

ωφω kj

k

nj

k

ekhAe

Aekh

∑=

∑=

−∞

−∞=

+

−∞=

( )( ) attenuate 1

amplify 1

<

>

ω

ω

j

j

eH

eH

System’s frequency response( ) ( )

( ) ( ) ( )

( ) ( )

ωφω

ωφω

ωeHjjnj

jnj

eeHAe

eHAe

j

j=

=∠+

+

response

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( )( )

ωωωω

φωω

φωφω

ω

jjjj

eHjnjj

eHneHjAeHneHA

eeHAj

∠+++∠++=

= ∠++

sincos

[ ]ny [ ]ny1

SP- Berlin Chen 29

[ ] ( ) ( ) ( )( )ωω φω jj eHneHAny ∠++=⇒ cos


SP- Berlin Chen 30

Z-Transform

• z-transform is a generalization of (Discrete-Time) Fourier g ( )transform

[ ] ( )ωjeHnh

[ ] ( )zHnh

[ ]– z-transform of is defined as

( ) [ ]∑∞

∞=

−=n

nznhzH[ ]nh

• Where , a complex-variable• For Fourier transform

−∞=nωjrez = complex planeIm

ωj

( ) ( ) ωω

jezj zHeH

==

Re

ω

ωjez =

SP- Berlin Chen 31

– z-transform evaluated on the unit circle

e

)1( == zez jω unit circle

Re

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 l lIstability complex plan

R2

Im

• ROC (Region of Converge) R1Re

– Is the set of z for which z-transform exists (converges) ?

[ ] ∞<∑∞

− nznh absolutely summable

– In general, ROC is a ring-shaped region and the Fourier transform exists if ROC includes the unit circle ( )

[ ] ∞<∑−∞=n

znh absolutely summable

1=z

SP- Berlin Chen 32

transform exists if ROC includes the unit circle ( )1=z

Z-Transform (cont.)[ ] [ ] [ ]

[ ] [ ][ ] [ ]knhkx

nxnhnhnxny

k−=

==

∑∞

−∞=

* *

( )

• An LTI system is defined to be causal, if its impulse

[ ] [ ]knxkhk

−= ∑∞

−∞=

y , presponse is a causal signal, i.e.

[ ] 0f0 <h Ri ht id d

– Similarly, anti-causal can be defined as[ ] 0for 0 <= nnh Right-sided sequence

A LTI t i d fi d t b t bl if f

[ ] 0for 0 >= nnh Left-sided sequence

• An LTI system is defined to be stable, if for every bounded input it produces a bounded output– Necessary condition:Necessary condition:

Th t i F i t f i t d th f t f

[ ] ∞<∑∞

−∞=nnh

SP- Berlin Chen 33

• That is Fourier transform exists, and therefore z-transform includes the unit circle in its region of converge


• Right-Sided Sequence– E.g., the exponential signal

[ ] [ ] [ ] ⎨⎧ ≥

==0for 1

erewh1 1n

nunuanh n

0

[ ] [ ] [ ]⎩⎨ < 0for 0

ere wh, .1 1 nnunuanh

( ) ( ) 11

11∞ −−∞

=== ∑∑ azzazHnnn

have a pole at az =( ) ( ) 11 1 −−∞=−∞= −aznn

If 11 <−az

p(Pole: z-transform goes to infinity)

azROC >∴ is 1Imth it l Im

[ ] 1ifexistsofansformFourier tr<anh×

the unit cycle

SP- Berlin Chen 34

Rea[ ] 1 ifexists1 <anh×


• Left-Sided Sequenceq– E.g.

[ ] [ ] ).,-0,-1,-2,..(n 1 .2 2 ∞=−−−= nuanh n

0

[ ] [ ] )(2

( ) [ ] 12 1 −−

−∞=

−∞

−∞=−=−−−= ∑∑ zaznuazH n

n

nn

n

n If 11 <− za

( ) 11

1

10

1

1 11

11111 −−

−

−

∞

=

−∞

=

−

−=

−−

−=−

−=−=−= ∑∑azza

zaza

zazan

n

n

nn

azROC <∴ is 2

Imthe unit cycle

Im

[ ] [ ]<

nhnha

gowillbecauseexisttdoesn'of ansformFourier tr the,1when

SP- Berlin Chen 35

Rea[ ] [ ]

−∞→nnhnh

aslly exponentia gowill becauseexist,t doesn 22×


• Two-Sided Sequenceq– E.g.

[ ] [ ] [ ]121

31 .3 3 −−⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−= nununh

nn0

23 ⎠⎝⎠⎝[ ]

31 ,

311

1 31

1>

+⎯→←⎟

⎠⎞

⎜⎝⎛ −

−z

znu z

n

( )⎟⎞

⎜⎛⎟⎞

⎜⎛

⎟⎠⎞

⎜⎝⎛ −

=+=11

1212

11

11

113

zzzH

[ ]21 ,

211

1 121

1<

−⎯→←−−⎟

⎠⎞

⎜⎝⎛−

−z

znu z

n ⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ +−+ −−

21

31

211

311 11 zzzz

Imthe unit cycle 3

1 and 21 is 3 ><∴ zzROC

Im

31

− [ ] existtdoesn'ofansformFourier tr nh

SP- Berlin Chen 36

Re××

21

3 [ ]circleunit theincludet doesn' because

exist,t doesnofansformFourier tr

3

3

ROCnh


• Finite-length Sequence– E.g.

[ ]⎩⎨⎧ −≤≤

=th0

10 , .3 4

Nnanh

n

13221 ..... −−−− ++++ NNNN azaazz

0 N-1

[ ]⎩ others ,0

( ) ( ) ( ) azazazzazHNN

N

NN nN nn −

=−

===−− −− − ∑∑ 11

11 114

11( ) ( )azzaz Nnn −− −−==

∑∑ 11004 1

0except plane- entire is4 =∴ zzROC pp4

Imthe unit cycle ( ) 11 ,

2−== ,..,N kaez N

kjk

π

×31

− 4π If N=8

Re

SP- Berlin Chen 37

7 poles at zero A pole and zero at is cancelled az =


• Properties of z-transformp1. 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

[ ]nh [ ] 1 ,0 nnnh ≤=0rz >z

will be in ROC

• If ,ROC will include 01 ≥n ∞=z

A causal sequence is right-sided with ROC is the exterior of circle including

01 ≥n∞=z∴

2 If is left sided sequence i e the ROC is[ ]nh [ ] 0 nnnh ≥=2. If is left-sided sequence, i.e. , the ROC is the interior of some circle,

• If ,ROC will include

[ ]nh [ ] 2 ,0 nnnh ≥=

02 <n 0=z

SP- Berlin Chen 38

3. If is two-sided sequence, the ROC is a ring4. The ROC can’t contain any poles

[ ]nh

Summary of the Fourier and z-transforms

SP- Berlin Chen 39

LTI Systems in the Frequency Domainy y

• Example 1: A complex exponential sequence– System impulse response [ ]nh

[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]knhkx

nxnhnhnxny

k−=

==

∑∞

−∞=

* *

[ ] [ ] [ ] [ ] knjekhnhnxny ω=∗= −∞

∞=∑ )(

-k( ) ofansformFo rier trthe:ωjH

[ ] [ ]knxkhk

−= ∑∞

−∞=

[ ]

( )

kjnj ekhe ωω= −∞

∞=∑

-k

( )

responsefrequencysystem theas toreferredoften isIt

response. impulse system the ofansformFourier trthe:ωjeH

– Therefore, a complex exponential input to an LTI systemlt i th l ti l t th t t b t

( ) njj eeH ωω= response.frequency system

results in the same complex exponential at the output, but modified by

• The complex exponential is an eigenfunction of an LTI i l( )

( )jωeH

SP- Berlin Chen 40

system, and is the associated eigenvalue[ ]{ } ( ) [ ]nxeHnxT jω=

( )jωeHscalar

LTI Systems in the Frequency Domain (cont.)y y ( )

• Example 2: A sinusoidal sequence [ ] ( )φ+= nwAnx 0cosp q ( )[ ] ( )

njjnjj AAnAnx cos 0

ωφωφ

φω

−−

+=

( )

θ

θ

θθ

θθj

j

ieie

− −=

+=

1sincos

sincos

– System impulse response

njjnjj eeee 00

22 ωφωφ −−+=

[ ]nh

( )θθθ jj ee −+=⇒21cos

( )ωω

ω

ω

i*

sincos * jj

jejyxzj

j +=⇒+=

[ ] ( ) ( ) 0000 njjjωnjjjω eeAeHeeAeHny += −−− ωφωφ

( )( ) ( )

ωωωω

ωωω

ω

sincos sincos

sincos *

jje

jejyxzj

j

−=−+−=

−=⇒−=−

[ ] ( ) ( )

( ) ( )[ ]0000 )()(

22njjω*njjω eeHeeHA

eeeHeeeHny

+=

+=

+−+ φωφω

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω*

jω*jω

eeHeH

eHeH∠−

−

=

=

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]00

000

0 )()(

2

2

njeHjjωnjeHjjω eeeHeeeHA

eeHeeH

jωjω

+=

+

+−∠−+∠ φωφω

SP- Berlin Chen 41

( ) ( )[ ]( ) ( )[ ]00

0cos 2

jωjω eHneHA ∠++= φωmagnitude response phase response


• Example 3: A sum of sinusoidal sequencesp q

[ ] ( )∑ +=K

kkk nAnx cos φω=k 1

[ ] ( ) ( )[ ]∑ ∠++=K

jkk

jk

kk eHneHAny cos ωω φωh

– A similar expression is obtained for an input consisting of a sum f l ti l

=k 1 magnitude response phase response

of complex exponentials

SP- Berlin Chen 42


• Example 4: Convolution Theorem [ ] [ ] ( ) ( )ωω jj eHeXnhnx ⇔∗

[ ] [ ]∑∞

−∞=−=

kkPnnx δ ( ) ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −= ∑

∞

kPP

eXk

j πωδπω 22DTFT

[ ] [ ] ( ) ( )

[ ] [ ] 1 , <= ∑∞

−∞=

anuanhk

n⎦⎣ ⎠⎝−∞= PPk

( ) ωω

jj

aeeH −−

=1

1DTFT

( ) ( ) ( )ωωω

πωδπ

jjj

k

eXeHeY∞

⎥⎤

⎢⎡

⎟⎞

⎜⎛

∑

=

221

ω

ω

π

ωδ

j

kj

aeP

kPPae

−

−∞=−

=

⎥⎦

⎢⎣

⎟⎠

⎜⎝

−∑−

=

112

1

has a nonzero jaeP −1 value when

πω2

Pk =

SP- Berlin Chen 43


• Example 5: Windowing Theorem [ ] [ ] ( ) ( )ωω jj eXeWnwnx ∗⇔21 ( ) ( )π2

[ ] [ ]∑∞

−∞=−=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

−=⎟⎠⎞

⎜⎝⎛

−−=

otherwise 0

1,......,1,0 ,1

2cos46.054.0 NnN

nnw

πHamming window

⎩( ) ( ) ( )

( ) ∑∞

⎟⎠⎞

⎜⎝⎛ −∗=

∗=

j

jjj

keW

eXeWeY

ω

ωωω

πωδππ

22121

( )

( )∑

∑∞

−∞=

−∞=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ −∗=

⎟⎠

⎜⎝

k

j

k

kP

eWP

kPP

eW

ω πωδ

ωδπ

21

2

( )

∑

∑ ∑∞ ⎟

⎠⎞

⎜⎝⎛ −

∞

−∞=

∞=

−∞=

⎟⎞

⎜⎛

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ −−=

kj

k

m

m

jm mkP

eWP

πω

πωδ

21

2 1

has a nonzero

SP- Berlin Chen 44

∑−∞=

⎟⎠

⎜⎝

⎟⎟⎠

⎞⎜⎜⎝

⎛=

k

Pj

eWP1 value when

kP

m πω 2−=

Difference Equation Realization for a Digital Filterfor a Digital Filter

• The relation between the output and input of a digital p p gfilter can be expressed by

[ ] [ ] [ ]∑ ∑ −+−=N M

kk knxknyny βα 0β [ ]ny[ ]nx[ ] [ ] [ ]∑ ∑= =

+k k

kk knxknyny1 0

βα

delay property[ ] ( )zXnx →

linearity and delay properties z-1

z-1

2β

1β

z-1

z-11α

2α

( ) ( ) ( ) kN Mk

kk zzXzzYzY −−∑ ∑+= βα

[ ] ( )[ ] ( ) 0

0nzzXnnx −→−

z-1Mβ z-1

Nα

( ) ( ) ( )k k

kk zzXzzYzY= =∑ ∑+

1 0βα

A rational transfer functionCausal:Rightsided, the ROC outside the

( ) ( )( )

∑=

−

== N

M

k

kk z

XzYzH 0

βoutmost poleStable:The ROC includes the unit circleC l d St bl

SP- Berlin Chen 45

( ) ( ) ∑=

−−N

k

kk zzX

11 α Causal and Stable:

all poles must fall inside the unit circle (not including zeros)

Difference Equation Realization for a Digital Filter (cont )for a Digital Filter (cont.)

SP- Berlin Chen 46

Magnitude-Phase Relationshipg

• Minimum phase system:p y– The z-transform of a system impulse response sequence ( a

rational transfer function) has all zeros as well as poles inside the unit cycleunit cycle

– Poles and zeros called “minimum phase components”– Maximum phase: all zeros (or poles) outside the unit cyclep ( p ) y

• All-pass system:– Consist a cascade of factor of the form

1

111

±

− ⎥⎦

⎤⎢⎣

⎡

− azz-a*

– Characterized by a frequency responsewith unit (or flat) magnitude for all frequencies 1

11

1 =− −az

z-a*

SP- Berlin Chen 47

Poles and zeros occur at conjugate reciprocal locations

Magnitude-Phase Relationship (cont.)g ( )

• Any digital filter can be represented by the cascade of a y g p yminimum-phase system and an all-pass system

( ) ( ) ( )zHzHzH apmin=( ) ( ) ( )apmin

( )( )

1)( 1 zero oneonly has that Suppose <aa*zH

( )( ) ( )( ) ( )

( )filter) phase minimum a is ( 1

:asexpressed becan circle.unit theoutside

*1

*1 −= zHzazHzH

zH

( )( ) ( )( )

:where

111 1

*1

1 −−

−−

−=az

zaazzH

( )( )( )1

filter. phase minimum a also is 1

:where

*

11

−−

za

azzH

SP- Berlin Chen 48

( )( ) filter. pass-all a is 11 1−−−

azza

FIR Filters

• FIR (Finite Impulse Response)( p p )– The impulse response of an FIR filter has finite duration– Have no denominator in the rational function

• No feedback in the difference equation ( )zH

[ ] [ ]M [ ]ny[ ]nx[ ] [ ]∑

=−=

M

rr rnxny

0β

z-1

z-1

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ ≤≤

=h i0

0 , Mnnh nβ

z-1Mβ

[ ]⎩⎨ otherwise ,0

( ) ( )( ) ∑ −==

M

k

kk z

XzYzH

0β

– Can be implemented with simple a train of delay, multiple, and add operations

( ) =kzX 0

SP- Berlin Chen 49

First-Order FIR Filters

• A special case of FIR filters p

[ ] [ ] [ ]1−+= nxnxny α ( ) 11 −+= zzH α

( ) jj( ) ωω α jj eeH −+= 1( ) ( )

( ) ( ) ++=++=

−+=

ωααωαωα

ωωαω

cos21sincos1

sincos1

222

22j jeHRe Im

( ) ⎟⎠

⎞⎜⎝

⎛+

−=ωα

ωαθ ωcos1

sinarctanje: pre-emphasis filter0<α

Re

( ) 2( ) 2log10 ωjeH

SP- Berlin Chen 50

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 sequenceable to recover the sequence– For a sequence of finite length N, sampling yields the new

transform referred to as discrete Fourier transform (DFT)

( ) [ ] 10 , 21

0−≤≤∑=

−−

=NnenxkX

knN

jN

n

π

DFT, Analysis

[ ] ( ) 10 , 1 21

0

−≤≤∑=−

=

NnekXN

nxkn

NjN

nπ

Inverse DFT, Synthesis0=N k

SP- Berlin Chen 51

Discrete Fourier Transform (cont.)( )

10 k

[ ] [ ] ≤≤

−≤≤∀

−−

∑ 10

1021

NnenxkX

Nkkn

NjN π

[ ] [ ]

[ ] [ ] ⎤⎡⎤⎡⎤⎡

−≤≤==∑

00111

10 ,0

Xx

NnenxkX N

n

L( )

[ ][ ]

[ ][ ]

⎥⎥⎥⎤

⎢⎢⎢⎡

=⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

−⋅−⋅−10

10

1

111112112

XX

xx

eeN

Nj

Nj

MMMMM

Lππ

( ) ( ) ( ) [ ] [ ]⎥⎥

⎦⎢⎢

⎣ −⎥⎥

⎦⎢⎢

⎣ −⎥⎥⎥

⎦⎢⎢⎢

⎣−⋅−−⋅−− 111112112

NXNxeeNN

NjN

Nj

MM

L

MLMMππ

Orthogonal

⎦⎣

SP- Berlin Chen 52

Discrete Fourier Transform (cont.)( )

• Orthogonality of Complex Exponentials g y p p( )

⎩⎨⎧ =

=∑− −

otherwise0 if ,11 1 2 mNk-r

eN

N nrkN

j π

⎩∑= otherwise ,00N n

knjNπ2

11[ ] [ ]

[ ] [ ]( )nrk

Njrn

Nj

knN

j

kX

ekXN

nx

N NN

N

kππ 22

1 11

1

0

1

1

−−

∑ ∑∑⇒

∑=

− −−

−

=

[ ] [ ]

[ ]( )nrk

Nj

ekX

ekXN

enx

NN

n knπ2

11

0 00

1 −

⎥⎥⎤

⎢⎢⎡

∑∑=

∑ ∑=∑⇒

−−

= ==

[ ] [ ][ ]rX

mNrXkX=

+=[ ]

[ ]knj

rX

N

N

nk

π21

00

=

⎥⎦⎢⎣∑∑==

[ ]rX

SP- Berlin Chen 53

[ ] [ ]kn

Nj

enxkXN

n

1

0

−

∑=⇒−

=

Discrete Fourier Transform (DFT)( )

• Parseval’s theorem

[ ] ( )∑∑−−

=1

21

2 1 NN

kXN

nx Energy density[ ] ( )∑∑== 00 kn N

SP- Berlin Chen 54

Analog Signal to Digital Signalg g g g

Analog SignalAnalog Signal

Digital Signal:

[ ] ( ) periodsampling:TT

Discrete-time Signal or Digital SignalDigital Signal:

Discrete-time signal with discrete

amplitude[ ] ( ) periodsampling: , TnTxnx a=nTt =

SP- Berlin Chen 55

rate sampling 1TFs =

sampling period=125μs=>sampling rate=8kHz

Analog Signal to Digital Signal (cont.)g g g g ( )

Impulse TrainSamplingContinuous-Time

SignalDiscrete-Time

SignalContinuous-Time to Discrete-Time Conversion

Impulse TrainTo

Sequence [ ] ( ))( ˆ nTxnx a=

p g

( )∞

s tx

( )tx a

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )∑∑

∑∞∞

∞

−∞=

−=−=

−==

a

naa

nTtnxnTtnTx

nTttxtstx

δδ

δ

( ) ( )∑∞

−= nTtts δ

switch

−∞=−∞= nn

Digital Signal

[ ] nx−∞=n

Periodic Impulse Train ( ) [ ]nxtxs by specifieduniquely becan

Discrete-time signal with discrete

amplitude

( )tx a

amplitude

( ) ( )∞

( )∞

≠∀= 0 ,0 ttδ1

SP- Berlin Chen 56

( ) ( )∑∞

−∞=−=

nnTtts δ ( )∫

∞

∞−

= 1dttδ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T


• A continuous signal sampled at different periodsg p p

( )tx a ( )tx a

TT

1

( )tx ( )

( ) ( ) ( ) ( )∑∞∞

∞

−∞=

−==n

aa

s

nTttxtstx

tx

δ

SP- Berlin Chen 57

( ) ( ) [ ] ( )∑∑∞

−∞=

∞

−∞=

−=−=nn

a nTtnxnTtnTx δδ


( )ΩjXa

• Spectra( )ja

2π2NΩ

frequency) (sampling22ss F

Tππ

==Ω( ) ( )∑∞

−∞=Ω−Ω=Ω

ksk

TjS δπ2

⎞⎛ π1

( ) ( ) ( )∞

Ω∗Ω=Ω as jSjXjX

121π ( )

⎩⎨⎧ Ω<Ω

=ΩΩ otherwise 0 2/ sT

jRs

( ) ( ) ( )ΩΩ=Ω Ω jXjRjX pa s

⎟⎠⎞

⎜⎝⎛=Ω<Ω

TsNπ

21( ) ( )( )∑

−∞=

Ω−Ω=Ωk

sas kjXT

jX 1

( )⎬⎫

⎨⎧ Ω>Ω−Ω NNsQ

Low-pass filter

⎟⎠⎞

⎜⎝⎛=Ω>Ω

TsNπ

21

⎭⎬

⎩⎨ Ω>Ω⇒ 2 Ns

high frequency componentsgot superimposed onlow frequency

SP- Berlin Chen 58

⎠⎝ T2

aliasing distortion

( )⎭⎬⎫

⎩⎨⎧

Ω<Ω⇒Ω<Ω−Ω

2 Ns

NNsQ

( ) ( )ΩΩ jXjX pa from recovered bet can'

components


• 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

Ns Ω>Ω 2– (Nyquist) sampling theorem

• To reconstruct the original continuous signalFil d i h l fil i h b d li i– Filtered with a low pass filter with band limit

• Convolved in time domain sΩ

( ) ( ) ( )∑∞

= nTthnTxtx( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )∑

∑∞

−∞=

−∞=

−Ω=

−=

nsa

naa

nTtnTx

nTthnTxtx

sinc

SP- Berlin Chen 59

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