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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
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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:
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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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)
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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)
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
11
S KHI CTHC HIN
Tn hiu vo l nhiu trng vi = 0 v variance 2 = 1
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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');
xlabel('Frequency (Hz)');
ylabel('Power Spectral Density');
% 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');
xlabel('Frequency (Hz)');
ylabel('Power Spectral Density');
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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');
xlabel('Frequency (Hz)');
ylabel('Power Spectral Density');
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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Code Correlogram
Kt qu
clear all
clc
load Project.mat YY;
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');
xlabel('Frequency (Hz)');
ylabel('Power Spectral Density');
%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');
xlabel('Frequency (Hz)');
ylabel('Power Spectral Density');
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
load Project.mat YY;
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]
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
load Project.mat YY;
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));
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
load Project.mat YY;
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));
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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.
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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)
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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;
% Xac dinh P[exp(j2*Pi*f)] voi cua so chu nhat
[Pper,f]= periodogram(XX,window,f,1);
plot(f,Pper);
title('Periodogram LS with Rectangular Window');
xlabel('Frequency (Hz)');
ylabel('Power Spectral Density');
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Bo co thc hnh Matlab Trn Nht Hoi Bo 41100249
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;
% Xac dinh P[exp(j2*Pi*f)] voi cua so chu nhat
[Pper,f]= periodogram(XX,window,f,1);
plot(f,Pper);
title('Periodogram RLS with Rectangular Window');
xlabel('Frequency (Hz)');
ylabel('Power Spectral Density');
clear all
clc
load ProjectJW.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);
plot(f,Pper);
title('Periodogram JW with Rectangular Window');
xlabel('Frequency (Hz)');
ylabel('Power Spectral Density');