diagonalization of matrices -...
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Diagonalization of Matrices
• Motivation• Eigenvalue decomposition• Singular-value decomposition• Discrete Fourier transform • Applications : FEXT cancellation
Wireless applications
Eigenvalue decomposition
• Definition: Let A be any square matrix. A scalar λ is called an Eigenvalue of A if there exists a nonzero (column) vector v such that Av =λv
• Any vector satisfying this relation is called an eigenvector of A belonging to the eigenvalue λ
EVD cont.
• An n × n matrix A is similar to D=diag(d1,d2,……..dn) iff A has n linearly independent eigenvectors.
• The (d1,d2……..dn ) are the corresponding eigenvalues
P is the matrix whose columns are the eigenvectors
**
APPD 1−=
1−= PDPA
Algorithm to solve EVD
1) Find the eigenvalues of the matrix2) Find n linearly independent eigenvectors.3) Construct P from the vectors in step 24) Construct D from the eigenvalues
Singular-value decomposition
• Any m by n matrix A may be factored such that A = UΣVT
• U: m by m, orthogonal, columns are the eigenvectors of • V: n by n, orthogonal, columns are the eigenvectors of • Σ: m by n, diagonal, the singular values are the square roots of the
eigenvalues of both and
SVD of Aunitarily (orthogonally) equivalent to the
diagonal matrix Σ.
TAAAAT
TAA AAT
TVUA Σ=
Algorithm to find SVD
1) Find the eigenvalues of the matrix ATA and arrange them in descending order
2) Find the number of nonzero eigenvalues of the matrix
1) Find the Orthogonal vectors of corresponding to the eigenvalues above ( arrange the same order to form the V matrix.)
2) Form the diag matrix ∑.3) Find the first column-vectors of the matrix U(mxm)
AAT
AAT
Algorithm cont.
6) Add to the matrix U the rest of vectors (they must be orthogonal to the r vectors)
• use the Gram_Schmidt Orthogonalization process.
rm−
Toeplitz matrix
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−−
=
..)2()1(...
........
.)0()1()2()1()0()1(
...)2()1()0(
NhNh
hhhnhhhNhNhh
T
DFT Matrix
• WW*=I unitary
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
−−−− )1)(1()1(21
642
321
.....1...1...1...1
...1
...1
.....1111
1
NNN
NN
NN
NNN
NNN
WWW
WWWWWW
NW
Y=XH ,(DFT domain)
• Wy=WTx=DX=DWx• WTx=DWx• WT=DW
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
=
)1(0...
........
.....00.0)2(0.00)1(0
.....000)0(
NHo
HH
H
D
DWTW =−1
MIMO systemsA possibility to increase data rates without
boosting the power
1a2a
1−MaMa
1c2c
1−McMc
1b2b
1−LbLb
...
CabinetCentral office
FEXT
NEXT
ka kc.........
cablebundle
I
cablebundle
II
11,kj
FEXT
K Lmjk kk k
mj jk
Nk
m
EXT
c bha a gh== ≠
= ∗ + ∗ + ∗∑ ∑14442444314444244443
MIMO systemsSingular-value decomposition
MIMO equationsfor each frequency: ( ) ( ) ( ), 1,...,n n n n N= ⋅ =y A x
system matrices ( )no diagonalare for aFEX ll , .,T 1..n n N=⇔ A
Singular-value decompositions (SVDs):
( ) ( ) ( ), 1,( ) ...,Hn n n nn N= ⋅ ⋅ =ΛA Q P
( ), ( ) ... unitary matrices ... ( ) diagonalreal matrices with elements 0n
n n K KK K× −× − ≥
P QΛ
MIMO-SystemeSingular-value decomposition
( ) ( ) ( ), 1,( ) ...,Hn n n nn N= ⋅ ⋅ =Λ P
Inverse: ( )nP
Inverse: ( )H nQ
A Q
( ), ( ) ... unitary matrices ... ( ) diagonalreal matrices with elements 0n
n n K KK K× −× − ≥
P QΛ
MIMO systemsBlock diagram of the procedure
T-Part 1
T-Part KR-Part K
R-Part 1
MIMO
channel
-
K loops
(1)P(2)P
( )NP
(1)HQ
(1)t
( )Nt
(1)r
( )Nr (2)HQ
( )H NQ
( ) ( ), ( 1,...,)nn n n N= ⋅ =Λr t
MIMO systemsGain in capacity
0 500 1000 1500 20000
20
40
60
80
100
120
140
160
Distance CO to cabinet [m]
Bitr
ate
per l
oop
[Mbi
t]
MIMO
Non-MIMO
Reference:• http://www.coastal.edu/~jbernick/• http://www.cs.ut.ee/~toomas_l/linalg/lin2/node14.html• http://web.mit.edu/be.400/www/SVD/Singular_Value_Decomposition.htm• http://www.cs.utk.edu/~dongarra/etemplates/node43.html