ISSN: 1693-6930
TELKOMNIKA Vol. 14, No. 3A, September 2016 : 47 – 55
48 described estimation strategy of the signal subspace ranks. Section 6 discussed the method to
obtain the harmonic amplitude and phase. Section 7 verified performance of the new algorithm we proposed through simulation experiment. Finally a conclusion is given in Section 8.
2. Signal Model
Hypothesize the harmonic signal in power system can be described as
1
cos 2
M i
i i
i
x n A
f n e n
1 Where
i
A
,
i
f
,
i
are the amplitude, normalized harmonic frequency, Initial phase, respectively,
e n
is the zero-mean wide-sense stationary Gaussian white noise, M is the order of harmonic. Using complex representation for the mathematical simplicity, formula 1
can be expressed as:
2 1
i i
M j
n i
i
x n A e
e n
Where
2 f
,
1 j
2 Denote
[ , -1, ,
1]
T
x n x n x n
K
as
Xn
and
[ 1, ,
, 2]
T
x n x n
x n K
as
Yn
,
Xn
can be expressed as:
AS n E n
Xn
3
Where
1 2
1 2
1 1
1
M M
j j
j jK
jK jK
K M
e e
e A
e e
e C
1 1
2 2
1 1
2
1 1
1
M M
T j n
j n j n
M M
T M
E n e n
e n e n
M C
S n A e
A e A e
C
Yn
can be expressed as:
1 = 1
1 =A
1 1
AS n E n
S n E n
Yn Xn
4
Where
1 2
{ ,
, }
j j
j M
M M
diag e e
e C
3. Frequency Estimation Method based Esprit
Denoted
X
R
as the covariance matrix of
Xn
,
Y X
R
as the cross-correlation matrix of
Xn
and
Yn
:
2
{ }
{[ ][
] } AE{ }A
{ }
H H
X H
H H
H
R E X n X
n E
AS n E n
AS n E n
S n S n E E n E n
APA I
5
TELKOMNIKA ISSN: 1693-6930
Harmonic Estimation Algorithm based on ESPRIT and Linear Neural … Xiangwen Sun
49
2
{ }
{[ ][A
1 1] }
H H
XY H
H
R E X n Y
n E AS n
E n S n
E n AP
A Z
6
{ }
H
P E S n S
n
,
2
{ }
H
E E n E n
1 Z=
1
Where, H is the Hermitian transpose operator.
Structure matrix pencils
2 2
- ,
-
XX XY
D R
I R Z
, it can prove that
i
j
e
is generalized eigenvalue [5] of D. Therefore, harmonic frequency estimation can be obtained by
request of generalized eigenvalue of
, C
C
XX XY
at unit circle. As mentioned above, ESPRIT algorithm need to compute eigen decomposition of
sample autocorrelation matrix and matrix pencils D. In which the computational complexity is high. Therefore, find a method with a smaller expense of computational complexity to obtain
signal eigen subspace and thereby reduce time cost makes sense.In order to do this, many researchers have conducted lots of studies on this area, and achieved considerable valuable
results, including QR iteration, power method, Lanczos algorithm and so on. By which the course of eigen decomposition can be preventd in some degree.however, these methods still
need calculation of sample covariance matrix and the complexity is still relatively high.
4. Rapid Subspace Decomposition Based MSWF
