Linear Predictive Analysis 主講人：虞台文. Contents Introduction Basic Principles of Linear...

Linear Predictive Analysis

主講人：虞台文

Contents Introduction Basic Principles of Linear Predictive Analysis The Autocorrelation Method The Covariance Method More on the above Methods Solution of the LPC Equations Lattice Formulations

Introduction

A powerful speech analysis technique. Powerful for estimating speech parameters.

– Pitch– Formants– Spectra– Vocal tract area functions

Especially useful for compression. Linear predictive analysis techniques is often

referred as linear predictive coding or LPC.

Basic Idea A speech sample can be approximated as a

linear combination of past speech samples.

This prediction model corresponds to an all-zero model, whose inverse matches the vocal tract model we developed.

LPC vs. System Identification

The linear prediction have been in use in the areas of control, and information theory under the name of system estimation and system identification.

Using LPC methods, the result system will be modeled as an all-pole linear system.

LPC to Speech Processing

Modeling the speech waveform. There are different formulations. The differences among them are often those of

philosophy or way of viewing the problem.

They almost lead to the same result.

Formulations The covariance method The autocorrelation formulation The lattice method The inverse filter formulation The spectral estimation formulation The maximum likelihood estimation The inner product formulation

Basic Principles of Linear Predictive Analysis

Speech Production Model

Impulse Train Generator

Random NoiseGenerator

Time-VaryingDigital Filter

Vocal TractParameters

Speech Production Model

Impulse Train Generator

Random NoiseGenerator

Time-VaryingDigital Filter

Vocal TractParameters

)()()(1

nGuknsansp

)()()(1

nGuknsansp

Linear Prediction Model

)()()(1

nGuknsansp

)()()(1

nGuknsansp

kk knsαns

)()(ˆLinear Prediction:

)()(ˆ)( nensns Error compensation:

)()()(1

neknsαnsp

)()()(1

neknsαnsp

Speech Production vs. Linear Prediction

)()()(1

nGuknsansp

)()()(1

nGuknsansp

)()()(1

neknsαnsp

)()()(1

neknsαnsp

Speech production:

Linear Prediction:

Vocal Tract Excitation

Linear Predictor Error

ak = k

Prediction Error Filter

)()()(1

neknsαnsp

)()()(1

neknsαnsp

Linear Prediction:

kk knsαnsne

)()()(

)()()(1)(1

zSzAzSzαzEp

1 ,)( 00

αknsαp

kk knsαnsne

)()()(

)()()(1)(1

zSzAzSzαzEp

)(zA )(zAs(n) e(n)

kk zαzA

1 ,)( 00

αknsαp

kk knsαne

)(zA )(zAs(n) e(n)

kk zαzA

Goal: Minimize

ne )(2

kk knsαne

Goal: Minimize

ne )(2

kk knsαne

knsαne2

2 )()(

ii jnsαinsα

ni αjnsinsα

)()( jnsinscn

jjiji αcα

Suppose that cij’s can be estimated from the speech sample.

Our goal now is to find ak’s to minimize the sum of squared errors.

Goal: Minimize

ne )(2

kk knsαne

jjiji αcα

)()( jnsinscn

Fact:cij = cji

,2,1 ,0

Let and solve the equations.

iikijkj

cααcα

i = k j = k

pkαcp

iiki ,,2,1 ,02

iikijkj

cααcα

i = k j = k

pkαcp

iiki ,,2,1 ,02

01212111010 ppαcαcαcαc k = 1:02222121020 ppαcαcαcαc k = 2:

0221100 pppppp αcαcαcαc k = p:

)()( jnsinscn

Fact:cij = cji

)()( jnsinscn

Fact:cij = cji

2222120

1121110

pppppp

011221111 cαcαcαc pp

pppppp cαcαcαc 02211

)()( jnsinscn

Fact:cij = cji

pppppp cαcαcαc 02211

pppppp

)()( jnsinscn

Fact:cij = cji

Remember this equation

)()( jnsinscn

Fact:cij = cji

ΨΦα

ΨΦα 1Such a formulation in fact is unrealistic.

Such a formulation in fact is unrealistic.

Why?Why?

pppppp

Error Energy

ne )(2

kk knsαne

kk knsα

j nji insinsαα

jjiji αcα

10 11 12 11

20 21 22 22

p p p ppp

c c c cα

jjjαc

=0, i 0

kkkcαc

Short-Time Analysis

Original Goal: Minimize

neE )(2

Vocal tract is a slowly time-varying system.

Minimizing the error energy for whole speech signal is unreasonable.

Short-Time Analysis

Original Goal: Minimize

neE )(2

New Goal: Minimize m

nn meE )(2

Short-Time Analysis

New Goal: Minimize m

nn meE )(2

nnn msmsE 2)(ˆ)(

knk kmsα

The Autocorrelation Method

)()()( mwmnsmsn ]1,0[for ,0)( Nmmw

Usually, we use a Hamming window.Usually, we use a Hamming window.

)()()( mwmnsmsn ]1,0[for ,0)( Nmmw

2 )(pN

mnn meEError energy

n me )(2

So, the original formulation can be directly applied to find the prediction coefficients.

)()( jmsimsc nm

nnn ΨαΦ

nnn ΨΦα 1

s?' estimate toHow nijc s?' estimate toHow n

ijc What properties they have?What properties they have?

For convenience, I’ll drop the sup/subscripts n in the following discussion.

Properties of cij’s

)()( jmsimscm

Property 1: jiij cc

Property 2:

)]([)( jimsmscim

)]([)( jimsmsm

)(,0 jic

Its value depends on the difference |ij|.

|| jiij rc

The Equations for the Autocorrelation Methods

A Toeplitz Matrix

nnn ΨαΦ nnn ΨΦα 1

The Error Energy

2 )(pN

mnn meE

nkn kmsαms

0 1 11

2 )()()()(2)(pN

nknn jmsαimsαkmsαmsms

2 )()()()(2)(pN

nkn jmsimsααkmsmsαms

nn rααrαrE

The Error Energy

nn rααrαrE

n rαrαr11

nn rαrαrE

010 10

The Error Energy

nn rαrαrE

010 10

nnnn E

The Covariance

Method

The Covariance Method

)()( mnsmsn

Goal: Minimize

mnn meE

)()( mnsmsn

Goal: Minimize

mnn meE

The range for evaluating error

energy is different from the

autocorrelation method.The range for evaluating error

energy is different from the

autocorrelation method.

Goal: Minimize

mnn meE

nkn kmsαE

jmsimsαα n

( ) ( )p pN

n ni n j n

α s m i α s m j

jmsimsc n

jimsms n

ijmsms n

Property:nji

nij cc

jmsimsc n

jimsms n

ijmsms n

0 N1i j

)(msn)(msn i

)( jimsn )( jimsn

jmsimsc n

jimsms n

ijmsms n

0 N1i j

)(msn)(msn i

)( jimsn )( jimsn

cij is, in fact, a cross-correlation function.cij is, in fact, a cross-correlation function.

The samples involved in computation of cij’s are values of sn(m) in the interval pmN1.

The value of cij depends on both i and j.The value of cij depends on both i and j.

The Equations for the Covariance Methods

3333231

2232221

1131211

Symmetric but not Toeplitz

nnn ΨαΦ nnn ΨΦα 1

The Error Energy

mnn meE

nkn kmsαms

0 1 11

2 )()()()(2)(N

nknn jmsαimsαkmsαmsms

2 )()()()(2)(N

nkn jmsimsααkmsmsαms

nn cααcαcE

1 11000 2

The Error Energy

nn cααcαcE

1 11000 2

3333231

2232221

1131211

nn cαcαcE

1000 10

n cαcαc1

More on the above Methods

The Equations to be Solved

nnn ΨαΦ nnn ΨΦα 1The Autocorrelation

Method

The CovarianceMethod

3333231

2232221

1131211

n and n for the Autocorrelation Method

Define

)1()2()()1()0(0000

0)1()1()()1()0(000

00)2()3()2()3()0(00

00)1()2()1()2()1()0(0

000)1()()1()2()1()0(

NsNspNspNss

NspNspNsss

NsNspspss

NsNspspsss

Nspspssss

nnnnnn

Ψ TnnSS

n is positive definite.

Why?Why?

n and n for the Covariance Method

Define

)1()2()0()1()2()1()(

)()1()1()0()3()2()1(

)3()4()2()3()0()1()2(

)2()3()1()2()1()0()1(

)1()2()()1()2()1()0(

pNspNssspspsps

NsNspspssss

nnnnnnn

n is positive definite.

Why?Why?

,11,13,12,11,1

,21,2222120

,11,1121110

,01,0020100

Solution of the LPC Equations

Covariance Method---Cholesky Decomposition Method

Also called the square root method.

ppppppp

3333231

2232221

1131211

Symmetric and

positive definite.

Φα ΨΦα Ψ

TVDVΦ

ppp vvv

A lower triangularmatrix

A diagonalmatrix

ΨαVDV T

Φα ΨΦα Ψ

TVDVΦ ΨαVDV T

ppppp ψ

11 ψy 11 ψy

12122 yvψy

23213133 yvyvψy

jjijii yvψy

Y can be recursively solved.

TVDVΦ ΨαVDV T

YαDV T YDαV 1T

ppp dyα / ppp dyα /

pppppp αvdyα 1,111 /

ppppppppp αvαvdyα 2,12,1222 /

jipjpipipip αvdyα

0,/ jp

jipjpipipip αvdyα

TVDVΦ How?

3333231

2232221

1131211

ppppppppp

vdvdvdd

223222

113112111

kikij vdvc

kjkkikjjjij vdvvdv

TVDVΦ jkk

kikij vdvc

kjkkikjjjij vdvvdv

kjkkikijjij vdvcdv

Considerdiagonalelements

kikkikiiiii vdvcdv

kkikiii dvcd 111 cd

3333231

2232221

1131211

ppppppppp

3333231

2232221

1131211

ppppppppp

kjkkikijjij vdvcdv

TVDVΦ jkk

kikij vdvc

kjkkikjjjij vdvvdv

kkikiii dvcd 111 cd

3333231

2232221

1131211

ppppppppp

3333231

2232221

1131211

ppppppppp

111 ii cdv 111 / dcv ii

kjkkikijjij vdvcdv

TVDVΦ jkk

kikij vdvc

kjkkikjjjij vdvvdv

kkikiii dvcd

kkk dvcd

3333231

2232221

1131211

ppppppppp

3333231

2232221

1131211

ppppppppp

2 2 2 21

i i ik k kk

v d c v d v

2 2 2 21

i i ik k kk

v c v d v d

The story is, then, continued.The story is, then, continued.

Error Energy

kkkn cαcE

ppppppp

3333231

2232221

1131211

ppppppp

3333231

2232221

1131211

ΨαTn cE 00

ΨVDY 1100

YDY 100

kkk dYc

T VDV α ΨT VDV α Ψ VY ΨVY ΨT DV α Y

1( )T α DV Y1 1( )T α V D Y

1 1T T α Y D V

Autocorrelation Method---Durbin’s Recursive Solution

The recursive solution proceeds in steps.

In each step, we already have a solution for a lower order predictor, and we use that solution to compute the coefficients for the higher order predictor.

Notations:Coefficients forthe nthorderpredictor:

Error energy forthe nth order predictor:

The Toeplitz matrix forthe nth order predictor:

( )nE (0) ?E (0) ?E

( ) ( )2

( ) ( )2( )1

(0) (0) 1 α α (0) (0) 1 α α

(0)0rR (0)

The equation for the autocorrelation method:

)()()( How the

procedure proceeds recursively?

How the procedure proceeds recursively?

(0)0E r

(0) 1α (0) 1α (0)0rR (0)

Permutation Matrix

44434241

34333231

24232221

14131211

44434241

34333231

24232221

14131211

24232221

34333231

44434241

34333231

24232221

14131211

41424344

31323334

21222324

11121314

Row inversing

Column inversing

Property of a Toeplitz Matrix

n PRRP

A Toeplitz Matrix

)()()(

0PαRP

0PαPR

)()(nnnn

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

rR )()(

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

)1()1()1(

)1()1(

)1()1(0 0

rR )()(

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

)1()1()1(

)1()1(

)1()1(0 0

)()()(

This is what we want.This is what we want.

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

)1()1()1(

)1()1(

)1()1(0 0

)()()(

0 )1(1

)1(1)1(

0 )1(1

)1(1)1(

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

)1()1()1(

)1()1(

)1()1(0 0

)()()(

0 )1(1

)1(1)1(

0 )1(1

)1(1)1(

)1()( 0

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

)1()1()1(

)1()1(

)1()1(0 0

)()()(

)1()( 0

)1( nTnq αr )1( nT

nq αr

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

)()()(

)1()( 0

)1( nTnq αr )1( nT

nq αr

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

)1( nTnq αr )1( nT

nq αr

)1( nn E

qk )1()1(

kk Er 1)(

qkEE nnn )1()( )( )1()1( n

nnn EkkE

)1(2 )1( nn Ek

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

)1( nn E

qk )1()1(

kk Er 1)(

qkEE nnn )1()( )( )1()1( n

nnn EkkE

)1(2 )1( nn Ek

)1()( 0

1)(0 n

nik ninn

ni ,,2,1,)1()1()(

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

)1( nn E

qk )1()1(

kk Er 1)(

qkEE nnn )1()( )( )1()1( n

nnn EkkE

)1(2 )1( nn Ek

)1()( 0

αα What can you say about kn?What can you say about kn?

11 nk 11 nk1)(0 n

nik ninn

ni ,,2,1,)1()1()(

(0)0E r

(0)0E r (0) (0) 1 α α

(0) (0) 1 α α

(0)0rR (0)

Summary: Construct a pth order linear predictor.

Step1. Compute the values of r0, r1, , rp.

Step2. Set E(0) = r0.

Step3. Recursively compute the following terms from n=1 to p.

)1()1(

kkn Erk )1(2)( )1( n

nn EkE

1)(0 n

1,,2,1 ,)1()1()(

nik ninn

Lattice Formulations

The Steps for Finding LPC Coefficients

Both the covariance and the autocorrelation methods consist of two steps:– Computation of a matrix of correlation values.– Solution of a set of linear equations.

Lattice method:– Combine them into one.

The Clue from Autocorrelation Method

Consider the system function of an nth order the linear predictor.

( ) ( 1) ( 1) , 1, 2, , 1n n ni i n n ik i n

( ) ( )

( ) 1n

n n ii

A z a z

The recursive relation from autocorrelation method:

( ) ( 1) ( 1)

( ) 1n

n n n k ni n n i n

A z k z k z

( ) ( 1) ( 1)

( ) 1n n

n n i n i ni n n i

A z z k z z

1 1( ) ( 1) ( 1)

( ) 1n n

n n i n i ni n n i

A z z k z z

1( ) ( 1) ( 1)

( ) ( )n i n

n n n n i nn i

A z A z k z z

Change indexi n iA(n 1) (z)

1( 1) ( 1)

n n n i nn i

A z k z z

1( 1) ( 1)

n n n in i

A z k z z

A(n 1) (z 1 )

)()()( 1)1()1()( zAzkzAzA nnn

nn )()()( 1)1()1()( zAzkzAzA nnn

1( ) ( 1) ( 1)

( ) ( )n i n

n n n n i nn i

A z A z k z z

Interpretation

)()()()()()( 1)1()1()( zSzAzkzSzAzSzA nnn

)()()()()( 1)1()1()( zSzAzkzSzAzE nnn

nn 1 1

( ) ( 1) ( 1)

( ) ( ) ( )n n

n n ni n i

e m a s m i k a s m n i

e(n 1) (m) b(n 1) (m 1)

)()()( 1)1()1()( zAzkzAzA nnn

1( 1) ( 1)

( ) ( ) ( )n

e m s m a s m i

( 1) ( 1)

( ) ( ) ( )n

e m s m a s m i

1( 1) ( 1)

( 1) ( ) ( )n

b m s m n a s m n i

( 1) ( 1)

( 1) ( ) ( )n

b m s m n a s m n i

order n 1

Interpretation

Forward PredictionError Filter What is this?

. . .s(m)

s(m2)s(m3)

s(mn+3)s(mn+2)

s(mn+1)

s(mn)1)1(

)1(2na

)1(3na

)()()( 1)1()1()( zAzkzAzA nnn

1( 1) ( 1)

( ) ( ) ( )n

e m s m a s m i

( 1) ( 1)

( ) ( ) ( )n

e m s m a s m i

1( 1) ( 1)

( 1) ( ) ( )n

b m s m n a s m n i

( 1) ( 1)

( 1) ( ) ( )n

b m s m n a s m n i

order n 1

. . .s(m)

s(m2)s(m3)

s(mn+3)s(mn+2)

s(mn+1)

Interpretation

Backward PredictionError Filter

)1(1na

)1(2na

)1(3na

)()()( 1)1()1()( zAzkzAzA nnn

1( 1) ( 1)

( ) ( ) ( )n

e m s m a s m i

( 1) ( 1)

( ) ( ) ( )n

e m s m a s m i

1( 1) ( 1)

( 1) ( ) ( )n

b m s m n a s m n i

( 1) ( 1)

( 1) ( ) ( )n

b m s m n a s m n i

Backward Prediction Defined

Define )()( 1)(1)( zAzzB nnn

)()()( )1()1()( zBkzAzA nn

nn )()()( )1()1()( zBkzAzA nn

( ) ( )

( ) 1n

n n ii

A z a z

1)()0( zA

)()()( 1)1()1()( zAzkzAzA nnn

Backward Prediction Defined

( ) 1 ( )

( ) 1n

n n n ii

B z z a z

1 ( ) 1

nn n n i

)()()( )1()1()( zBkzAzA nn

nn )()()( )1()1()( zBkzAzA nn

1)0( )( zzB

( ) ( )

( ) 1n

n n ii

A z a z

1)()0( zA

Forward Prediction vs. Backward Prediction

1)()0( zA 1)()0( zA

1)0( )( zzB1)0( )( zzB

)()()( )1()1()( zBkzAzA nn

nn )()()( )1()1()( zBkzAzA nn

)]()([)( 1)1(1)1(1)( zBkzAzzB nn

)()( )(11)( zAzzB nnn

)()( )1(11)1(1 zAzkzAz nn

)()( )1(1)1(1 zAzkzBz nn

)]()([)( )1()1(1)( zBzAkzzB nnn

n )]()([)( )1()1(1)( zBzAkzzB nnn

The Prediction Errors

1)()0( zA 1)()0( zA

1)0( )( zzB1)0( )( zzB

)()()( )1()1()( zBkzAzA nn

nn )()()( )1()1()( zBkzAzA nn

)]()([)( )1()1(1)( zBzAkzzB nnn

n )]()([)( )1()1(1)( zBzAkzzB nnn

( ) 1 ( ) 1

n n n n ii

B z z a z

( ) ( )

( ) 1n

n n ii

A z a z

)()()( )( zSzAzE nn

( ) ( ) ( )n

e m s m a s m i

)()()( )( zSzBzE n

( ) ( 1) ( 1)n

e m s m n a s m n i

)()()( 11 mekmeme nnnn

)1()1()( 11

memekme nnnn)1()1()( 11

memekme nnnn

)()(0 msme )()(0 msme

)1()(0 msme )1()(0 msme

The forwardprediction error

The backwardprediction error

The Lattice Structure

)1()1()( 11

memekme nnnn

)()(0 msme )()(0 msme

)1()(0 msme )1()(0 msme

s(m))(0 me

)(0 me

)(1 me

)(2 me

)(mep)(1 mep

)(1 mep

The Lattice Structure

ki=?ki=?Throughout the discussion, we have assumed that ki’s are the same as that developed for the autocorrelation method.

So, ki’s can be found using the autocorrelation method.

s(m))(0 me

)(0 me

)(1 me

)(2 me

)(mep)(1 mep

)(1 mep

Another Approach to Find ki’s

mn meFor the nth order predictor, our goal is to minimize

So, we want to minimize

)( )]()([N

n mekmeE

s(m))(0 me

)(0 me

)(1 me

)(2 me

)(mep)(1 mep

)(1 mep

So, we want to minimize

)( )]()([N

n mekmeE

)()]()([2 1

memekmek

2111 )]([)()(2

mnnnn mekmeme 0

21 )]([)]([

mn meme

)]([)]([

PARCOR

kpCORRCORRk1CORRCORR

s(m) )(0 me

)(0 mez1

k2CORRCORR

)(1 mep

)(1 me

)(1 mez1

)(2 me

CORRCORR

)()( meme p )()( meme p

)]([)]([

)()( 0 mems )()( 0 mems

( )( )( ) 1 1

( ) ( )

p pp p i i

i ii i

E zE zA z α z α z

S z E z

( )( )( ) 1 1

( ) ( )

p pp p i i

i ii i

E zE zA z α z α z

S z E z

Given kn’s, can you find i’s?

All-Pole Lattice

)1()1()( 11

memekme nnnn

)1()1()( 11

memekme nnnn

s(m))(0 me

)(0 me

)(1 me

)(2 me

)(mep)(1 mep

)(1 mep

All-Pole Lattice

s(m))(0 me

)(0 me

)(1 me

)(2 me

)(men)(1 men

)(1 men

)1()1()( 11

memekme nnnn

)()()( 11 zEkzEzE nnnn

)()()( 11 zEzEkzzE nnnn

All-Pole Lattice

s(m))(0 me

)(0 me

)(1 me

)(2 me

)(men)(1 men

)(1 men

)(zzEn

)(1 zEn

)(1 zzEn

All-Pole Lattice

)(zzEn

)(1 zEn

)(1 zzEn

)(1 zE

)(1 zzE

)(2 zE

)(2 zzE

)(1 zEp

)(1 zzEp

)(2 zEp

)(2 zzEp

10 )( zzE

)()( zEzEp )()( zEzEp

)(0 zE

)(0 zzE

All-Pole Lattice)()( zEzE p

)()( zEzE p )()( 0 zEzS )()( 0 zEzS

e(m) s(m)

p zαzαzE

)(1 zE

)(1 zzE

)(2 zE

)(2 zzE

)(1 zEp

)(1 zzEp

)(2 zEp

)(2 zzEp

)(0 zE

)(0 zzE

Comparison

)(1 zE p

)(1 zEp

)(2 zE

)(0 zE )(1 zE

)(1 zE

)(0 zE

e(m)s(m)

PARCOR

)(1 zE

)(1 zzE

)(2 zE

)(2 zzE

)(1 zEp

)(1 zzEp

)(2 zEp

)(2 zzEp

)(0 zE

)(0 zzE

Normalize Lattice

)(zzEn

)(1 zEn

)(1 zzEn

)(zzEn

)(1 zEn

)(1 zzEn

Sectionn

Normalize Lattice

Section1

)(0 zE

)(0 zzE

)(1 zE

)(1 zzE

Section2

)(2 zE

)(2 zzE

Sectionp

)(1 zEp

)(1 zzEp

)(zzE p

10 )( zzE

)()( zEzEp )()( zEzEp

)(zzEn

)(1 zEn

)(1 zzEn

Sectionn

Normalize Lattice

)()]()([)( 11 zEzEkzEkzzE nnnnnn

)()1()( 12 zEkzEk nnnn

)()1()()( 12 zEkzEkzzE nnnnn

Normalize Lattice

Three multiplier form

)()1()()( 12 zEkzEkzzE nnnnn

)(zzEn

)(1 zEn

)(1 zzEn

Normalize Lattice

Three multiplier form

Let nn kθ sin21cos nn kθ

Four multiplier form

)(zzEn

)(1 zEn

)(1 zzEn

)(zzEn

)(1 zEn

)(1 zzEn

sin nθsin nθ

cos nθ

Normalize Lattice

Kelly-Lochbaum form

)(zzEn

)(1 zEn

)(1 zzEn

)(zzEn

)(1 zEn

)(1 zzEn

sin nθsin nθ

cos nθ

)(zzEn

)(1 zEn

)(1 zzEn

Normalize Lattice1

( ) 1p

A z α z

( ) 1p

A z α z

zAzV )(

cos)( 1

)1()( 1

Section1

)(0 zE

)(0 zzE

)(1 zE

)(1 zzE

Section2

)(2 zE

)(2 zzE

Sectionp

)(1 zEp

)(1 zzEp

)(zzE p

)(zE )(zS1

)(zzEn

)(1 zEn

)(1 zzEn

)(zzEn z1

)(1 zEn

)(1 zzEn

sin nθsin nθ

cos nθ

)(zzEn

)(1 zEn

)(1 zzEn

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