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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249

1

TRNG I HC BCH KHOA H CH MINH

CHNG TRNH K S CHT LNG CAO VIT PHP PFIEV

BO CO THC HNH MATLAB

CC TNH CHT PH

GVHD: PGS.TS DNG HOI NGHA

SINH VIN: TRN NHT HOI BO

MSSV: 41100249

STT: 03

Tp.H Ch Minh, thng 4 nm 2015


2

A. L thuyt

Periodogram

nh ngha

Periodogram l mt qu trnh c lng khng c tham s ca hm mt ph cng sut ca

mt qu trnh dng khi phn tch ra chui Fourier. Periodogram c nh ngha bng cng thc:

2

12

ER

0

1( ).

Nj fnT

P

n

P f T x n eNT

Cc ca s s dng trong vic c lng Periodogram

Ca s ch nht

1 0 1w ( )

0R

n Nn

n

Ca s Hamming

20.54 0.46cos 0

w ( )

0

H

kk N

k N

Ca s Black-man

2 40.42 0.5cos 0.08cos 0

w ( )

0

B

k kk N

k N N

Ca s Hanning

20.5 0.5cos 0

w ( )

0

v

kk N

k N

Correlogram

nh ngha


3

() = ()2()

=

Theo cng thc trn, nu c T trong s m e, tn s f l tn s thc; ngc li, nu khng c T

trong s m e, tn s f l tn s chun ha.

Thng chn =

10

c lng hm t tng quan

Ta c:

() = {()( )} = lim

1

2 ()( )

1

()( )

1

=0

N l s mu d liu, ta c:

Nu () thc th ()

() =1

()( ) , 0

1

=

trong biu thc trn ta c th b T cho gn.

D liu x(0), x(1), ., x(N-1).

B c lng ca hm tng quan

Vi b c lng khng lch ( Unbiased)

V b c lng lch ( Biased)

1*

1*

1( ) ( ) ( ) 0

( ) ( ) 0

1( ) ( ) ( ) 0

( ) ( ) 0

N

k m

N

k m

r m x k x k m mN m

r m r m m

r m x k x k m mN m

r m r m m

1 1*1 1{ ( )} { ( ) ( )} ( ) ( )

N N

x x

k m k m

E r m E x k x k m r m r mN m N m


4

c lng mt ph cng sut

M hnh AR

M hnh AR l m hnh c mi lin h gia u vo e(k) v tn hiu ra x(k) nh sau:

() = 1( 1) 2( 2) ( ) + 0()

c lng cc h s a1, a2,.., an, b0 t cc mu thu thp c, ta c th s dng 3 phng

php ch yu sau.

a. c lng bng phng php bnh phng ti thiu

c lng 1 2 3 0, , ,..., ,na a a a b t d liu (0),..., ( 1)x x N

1

2

0

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

( 1) ( ) ( 1) (1) ( 1)

( 1) ( 2) ( 3) ( 1) ( 1)n

ax n x n x n x e n

ax n x n x n x e nb

ax N x N x N x N n e N

~ ~ ~ ~

0 0x R b e b e x R

2~ ~ ~

0 0 0

~ ~ ~ ~ ~ ~

~ ~ ~

( ) ( )

( ) ( )

2

arg min

T

T

T T T T T

T T

T T

J b e b e b e

J x R x R x x x R R x R R

J R R x R x x

J

~ ~

~ ~1

2 2 0

( )

T T

T T T T

T T T T

LS

dJR R x R R R x R

d

R R R x R R R x

1 1*1 1{ ( )} { ( ) ( )} ( ) ( )

N N

x x

k m k m

N mE r m E x k x k m r m r m

N N N

2

2

( ) ( )

( ) ( )

Mj fmT

cor x

m M

Mj fmT

cor x

m M

P f T r m e

P f T r m e


5

1

2

n

=> 1 2 n

dJ dJ dJ dJ

d d d d

0 1 1 2 21 2 0 1 1 1 2 0 21 0 2

2T Tr r

R R r r rr r

0 1 1 2 1 1 0 2( ) 2 2 ,2 2T Td R R r r r r

d

0 11 2 0 1 1 2 1 1 0 21 0

2 2 2 ,T Tr r

R R r r r rr r

Phng php Least square

~1( )T TLS R R R x

2~

2 1~ ~ ~2 2

0 0 0

1 12 2 2 2 2

0 0 0

0

( )

{ ( )} ( 1)

T N

k n

N N

e

k n k n

J x R

J b e b e e b e k

E J b E e k b b N n

Jb

N n

ngha hnh hc ca pp LS:


6

Nu

~

~

1

~

( 1)

( 1)

( )

min ( 1) min (

arg min 0

T T

x N R

x N R

R R R R

J x N R R

J

Tng qut :

~ ~ ~

~

~ ~

~ ~

~

( 1)

: [0]

_

T

x N x x

x R

x R x

x R x x R

x R x

b. c lng bng phng php bnh phng ti thiu quy

Theo m hnh AR : () = 1( 1) 2( 2) ( ) + 0()

Ti thi im k

Vi cch t cc k hiu nh sau:

1

2

0

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

( 1) ( ) ( 1) . . . (1) ( 1)

.. . . . .

.. . . . .

.. . . . .

( ) ( 1) ( 2) . . . ( ) ( )n

ax n x n x n x e n

ax n x n x n x e n

b

ax n k x k x k x k n e k


7

V

Ti thi im k+1

[

( + 1)] = [

] + [

( + 1)] 0

Vi = [() ( + 1) ( + 1)]

= ()

1

V +1 = (+1 +1)

1+1 +1

Sau khi tnh ton ta c

+1 = +(

)1(+1

)

1 + (

)1

t = ()

1 :

+1 = +(+1

)

1 +

t tip +1 =

1+

+1 = + +1(+1 )

T ta tm ra cch c lng cc h s bng phng php bnh phng ti thiu quy.

( )

( 1)

.( )

.

.

( )

x n

x n

x k

x n k

( )

( 1)

.

.

.

( )

k

e n

e n

e

e k

1

2

.

.

.

n

a

a

a

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

( ) ( 1) . . . (1)

. . .

. . .

. . .

( 1) ( 2) . . . ( )

k

x n x n x

x n x n x

R

x k x k x k n


8

c. c lng bng phng php bnh phng Jule Walker

Phng trnh Jule-Walker

e(k): n trng , 20, 1e em

0

1 2

1 2

1 2 0

( )1 ...

( ) ( 1) ( 2) ... ( ) ( )

n

n

n

bH z

a z a z a z

x k a x k a x k a x k n b e k

2

* 0

ex

, 0( ) ( ) ( )

0 , 0

eb mr m E e k x k mm

* * * *

ex 1 2

*

0

: 0

(0) ( ) ( ) ( )[ ( 1) ( 2) ... ( )

( ) ( )

n

TH m

r E e k x k E e k a x k a x k a x k n

b E e k e k

Vi * *1( )[ ( 1) ... ( )] 0nE e k a x k a x k n

* * *

ex 1

*

0

: 0

( ) ( ) ( ) ( )[ ( 1 ) ... ( )]

( ) ( ) 0

n

TH m

r m E e k x k m E e k a x k m a x k n m

b E e k e k m

V e(k) l tn trng nn ch tng quan vi chnh n

Vy

2

* 0

ex

, 0( ) ( ) ( )

0 , 0

eb mr m E e k x k mm

*

*

1 2 0

1 2 0 ex

( ) ( ) ( )

[ ( 1) ( 2) ... ( ) ( )]x ( )

( 1) ( 2) ... ( ) ( )

x

n

x x n x

r m E x k x k m

E a x k a x k a x k n b e k k m

a r m a r m a r m n b r m

H(Z) e(k)

x(k)


9

2

1 2 0

1 2

1 2

1 2

* 0 (0) ( 1) ( 2) ... ( )

* 1 (1) (0) ( 1) ... (1 )

1 * 2 (2) (1) (0) ... (2 )

...

* ( ) ( 1) ( 2) ... (0)

x x x n x e

x x x n x

x x x n x

x x x n x

m r a r a r a r n b

m r a r a r a r n

n pt m r a r a r a r n

m n r n a r n a r n a r

Phng trnh Jule-Walker

1

2

3

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

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

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

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

x x x x x

x x x x x

x x x x x

x x x x n x

r r r r n a r

r r r r n a r

r r r r n a r

r n r n r n r a r n

1 2

2

0 1 2

, ,...,

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

n

x x n x x

a a a

b a r a r a r n r


10

B. Bi tp Matlab

Xt hm truyn

e(k)

() =0

4

[( 1)2 22][( 3)2 4

2]

Vi cc thng s

1 = 0.5 4 = 0.62 = 0.4 0 = 0.4

3 = 0.5

e(k) l n trng vi me=0 . To 1000 d liu.

Yu cu :

- c lng v v 2( )j fxP e

dng Periodogram v correlogram

- c lng m hnh AR dung phng php:

+ Bnh phng ti thiu

+ Bnh phng ti thiu quy

+ Phng php Jule-Walker

- V 2( )j fxP e

H(z) x(k)


11

S KHI CTHC HIN

Tn hiu vo l nhiu trng vi = 0 v variance 2 = 1


12

Thay th cc gi tr vo trong H(z) ta c hm:

() =0.44

4 23 + 2.022 1.02 + 0.2501

T ta c:

() = 2( 1) 2.02( 2) + 1.02( 3) 0.2501( 4) + 0.4()

1. c lng v v 2( )j fxP e

dng Periodogram v Correlogram

Code Periodogram

clear all

clc

load Project.mat YY;

TT=YY(1,:);

XX=YY(2,:);

length=size(XX,2);

f=0:1/1000:1;

% Xac dinh P[exp(j2*Pi*f)] voi cua so chu nhat

[Pper,f]= periodogram(XX,window,f,1);

subplot(2,2,1);

plot(f,Pper);

title('Periodogram with Rectangular Window');

xlabel('Frequency (Hz)');

ylabel('Power Spectral Density');

% Xac dinh P[exp(j2*Pi*f)] voi cua so Hamming

[Pper,f]= periodogram(XX,Hamming(length),f,1);

subplot(2,2,2);

plot(f,Pper);

title('Periodogram with Hamming window');



% Xac dinh P[exp(j2*Pi*f)] voi cua so Blackman

[Pper,f]= periodogram(XX,Blackman(length),f,1);

subplot(2,2,3);

plot(f,Pper);

title('Periodogram with Blackman window');




13

Kt qu c lng Periodogram vi cc ca s

Kt lun: Vi cc ca s ging nhau, v vy 2( )j fxP e

thu c cng khng ging nhau hon

ton.

% Xac dinh P[exp(j2*Pi*f)] voi cua so Hanning

[Pper,f]= periodogram(XX,Hann(length),f,1);

subplot(2,2,4);

plot(f,Pper);

title('Periodogram with Hanning window');




14

Code Correlogram

Kt qu

clear all

clc


TT=YY(1,:);

XX=YY(2,:);

N=size(XX,2);

f=0:1/1000:1;

T=1;

M=round(N/10);

min=M+1;

max=2*M+1;

%Correlogram with Unbiased estimate

rm=xcorr(XX,M,'unbiased');

PcorwithinT=zeros(1,length(f));

for i=1:length(f)

test=0;

for m=min:max

test=test+rm(m)*2*cos(2*pi*f(i)*(m-101)*T);

end

PcorwithinT(i)=test;

end

Pcor=(PcorwithinT-rm(min))*T;

subplot(1,2,1);

plot(f,Pcor);

grid on;

title('Correlogram with Unbiased estimate');



%Correlogram with Biased estimate

rmb=xcorr(XX,M,'biased');

PcorwithinTb=zeros(1,length(f));

for i=1:length(f)

test=0;

for m=min:max

test=test+rmb(m)*2*cos(2*pi*f(i)*(m-101)*T);

end

PcorwithinTb(i)=test;

end

Pcorb=(PcorwithinTb-rmb(min))*T;

subplot(1,2,2);

plot(f,Pcorb);

grid on;

title('Correlogram with Biased estimate');




15

Kt lun: 2( )j fxP e

thu c khi dng Correlogram c dng ging vi 2( )j fxP e

khi s dng

Periodogram

2. c lng m hnh AR bng 3 phng php

Phng php bnh phng ti thiu

Code Least squares

clear all


TT= YY(1,:);

XX= YY(2,:);

NN = size(YY,2);

nTheta = 4;

%Least squares

X = XX(1,5:NN)';

R = [-XX(1,4:NN-1)' -XX(1,3:NN-2)' -XX(1,2:NN-3)' -

XX(1,1:NN-4)'];

ThetaLS = inv(R'*R)*R'*X;

Xtest= X-R*ThetaLS;

J=norm(Xtest);

B0_LS=J*sqrt(1/(NN-4))

Leastsquares =[ThetaLS]


16

Kt qu

Kt lun: Cc h s c c lng tin gn cc h s c ch ra u bi. Mt s h s c

c lng sai s ng k.

Phng php bnh phng ti thiu quy

Code Recursive Least Squares

Kt qu

B0_LS =

0.4016

Leastsquares =

-1.9919

2.0560

-1.0797

0.2930

clear all


TT= YY(1,:);

XX= YY(2,:);

NN = size(YY,2);

nTheta = 4;

% Recursive Least Squares

ThetaRLS = zeros(nTheta,1);

GG = 100*eye(nTheta);

for k = 5 : NN

Phi = [-XX(k-1); -XX(k-2); -XX(k-3); -XX(k-4)];

KK = GG*Phi/(1+Phi'*GG*Phi);

ThetaRLS = ThetaRLS + KK*(XX(k) - Phi'*ThetaRLS);

GG = (eye(nTheta)-KK*Phi')*GG;

end

RecursiveLeastSquares=[ ThetaRLS]

Xtestt=X-R*ThetaRLS;

Jt=norm(Xtestt);

B0_RLS=Jt*sqrt(1/(NN-4));


17

Kt lun: Cc h s c c lng kh ging vi phng php bnh phng ti thiu ( Least

Squares)

Phng php bnh phng Jule Walker

Code Jule Walker ( dng hm Matlab v da vo l thuyt)

RecursiveLeastSquares =

-1.9915

2.0550

-1.0788

0.2926

B0_RLS =

0.4016

clear all


TT= YY(1,:);

XX= YY(2,:);

NN = size(YY,2);

nTheta = 4;

%Jule Walker Matlab

[JuleWalker,NoiseVariance] = aryule(XX,4);

B0_JuleWalker=sqrt(NoiseVariance);

[Pxx,f] = pyulear(XX,4,1000,1);

subplot(212);

plot(f,10*log10(Pxx));

%Jult Walker

M=NN/10;

RR=xcorr(XX,M,'biased');

Rx=[-RR(1,101:104)' -RR(1,100:103)' -RR(1,99:102)' -

RR(1,98:101)'];

Rxx=RR(1,102:105)';

ThetaJW=inv(Rx)*Rxx;

R_square=RR(1,102:105);

B0_square=0;

for i=1:4

B0_square=B0_square+ThetaJW(i)*RR(101+i);

end

B0=sqrt(B0_square+RR(101));


18

S dng 2 phng php tnh Jule Walker ra cng 1 kt qu

Kt lun

Phng php Jule Walker cho ra cc h s c c lng chnh xc hn 2 phng php bnh

phng ti thiu v bnh phng ti thiu quy. Cc h s c sai s ng k 2 phng php

trn, trong phng php Jule Walker cho ra gn vi h s cho hn.

V 2( )j fxP e

vi cc h s c lng trn ( lm vi Periodogram dng ca s ch

nht).

JuleWalker =

1.0000 -1.9577 1.9799 -1.0056 0.2617

NoiseVariance =

0.1694

B0_JuleWalker =

0.4116

ThetaJW =

-1.9577

1.9799

-1.0056

0.2617

B0 =

0.4116


19

c lng Periodogram vi phng php bnh phng ti thiu cho ta ph 2( )j fxP e

c lng Periodogram vi phng php bnh phng Jule Walker cho ta ph 2( )j fxP e


20

c lng Periodogram vi phng php bnh phng ti thiu quy cho ta ph 2( )j fxP e

Kt lun: Da vo cc kt qu ph mt cng sut thu c, so snh vi tn hiu ban u, cc

ph cho kt qu kh ging nhau.


21

PH LC

Hnh trn l tn hiu ng ra c v t cc s liu thu nhn c ti Project.mat

Hnh hin th c t Scope, trong pha trn l tn hiu ng ra x(k), hnh di l tn hiu

nhiu e(k)


22

Cc mu d liu trong file Project.mat

Code Matlab v 2( )j fxP e

dng Periodogram vi cc phng php bnh phng ti thiu

clear all

clc

load ProjectLS.mat YY;

TT=YY(1,:);

XX=YY(2,:);

length=size(XX,2);

f=0:1/1000:1;



plot(f,Pper);

title('Periodogram LS with Rectangular Window');




23

Phng php bnh phng ti thiu quy

V phng php bnh phng Jule Walker

Cc file cha d liu 1000 mu c lu trong cc file Project.mat (vi cu 1, 2); ProjectLS.mat,

ProjectRLS.mat, ProjectJW.mat (vi cc cu v 2( )j fxP e

) ca cc tn hiu c c lng

h s.

Cc file Project.mdl ( dng cho cu 1,2) ; cc file ProjectLS.mdl, ProjectRLS.mdl,

ProjectJW.mdl dng ly d liu cho cc cu v ph pha sau.

File ARmodel.m cha code matlab cc cu c lng bng 3 phng php khc nhau.

File Correlogram.m cha code matlab cho v ph cng sut bng Correlogram.

File Periodogram.m cha code matlab cho v ph cng sut bng Periodogram.

Cc file PeriodogramLS.m, PeriodogramRLS.m v PeriodogramJW.m cha code matlab cho v

ph cng sut bng Periodogram cho cc tn hiu c lng h s.

clear all

clc

load ProjectRLS.mat YY;

TT=YY(1,:);

XX=YY(2,:);

length=size(XX,2);

f=0:1/1000:1;



plot(f,Pper);

title('Periodogram RLS with Rectangular Window');



clear all

clc

load ProjectJW.mat YY;

TT=YY(1,:);

XX=YY(2,:);

length=size(XX,2);

f=0:1/1000:1;



plot(f,Pper);

title('Periodogram JW with Rectangular Window');



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