Digital Signal Processing Reference
In-Depth Information
¼
;
cos
ð
0
Þ
cos
ð
p
=
6
Þ
10
:
866
0
:
5
h
2
¼
sin
ð
0
Þ
sin
ð
p
=
6
Þ
cos
ð
0
Þ
cos
ð
p
=
6
Þ
... cos
ð
4p
=
6
Þ
cos
ð
5p
=
6
Þ
h
6
¼
sin
ð
0
Þ
sin
ð
p
=
6
Þ
...
sin
ð
4p
=
6
Þ
sin
ð
5p
=
6
Þ
;
10
:
866
...
0
:
5
0
:
866
¼
0
:
5
... 0
:
866
0
:
5
cos
ð
0
Þ
cos
ð
p
=
6
Þ
... cos
ð
10p
=
6
Þ
cos
ð
11p
=
6
Þ
h
12
¼
sin
ð
0
Þ
sin
ð
p
=
6
Þ
...
sin
ð
10p
=
6
Þ
sin
ð
11p
=
6
Þ
10
:
866
...
0
:
5
:
866
¼
0
:
5
...
0
:
866
0
:
5
thus the matrix P
N
and the estimation result X become:
;
;
1
0
0
:
707
0
:
707
X
¼
P
2
x
2
¼
P
2
¼
1
:
732
2
;
;
0
:
333
0
:
289
...
0
:
167
0
:
289
0
:
707
0
:
707
X
¼
P
6
x
6
¼
P
6
¼
0
:
167
0
:
289
0
:
167
...
;
:
0
:
333
0
:
144
...
0
:
083
0
:
144
0
:
707
0
:
707
X
¼
P
12
x
12
¼
P
12
¼
0
:
083
...
0
:
144
0
:
083
From the above one can see that the estimates of signal parameters are identical
and equal to correct values irrespective of the applied window length. The situ-
ation changes, however, when the signal contains some distortion, here—10% of
the
second
harmonic.
The
results
of
signal
magnitude
calculation
(X
1m
¼
X
1C
þ
X
1
p
) for such a case are illustrated in Fig.
8.5
b (for pure cosine
wave—Fig.
8.5
a). One can see that the impact of distortion component decreases
with the increase in window length (time from estimation start) and the results are
accurate when N reaches full-cycle length N
1
.
Some peculiar advantages of the LSE algorithm are obtained when one assumes
additional probabilistic features of the error or when the number of samples N is
adequately adjusted to the signal period, what leads to the DFT or the correlation
algorithms, respectively.
Considering the special case when N = N
1
the transformation matrix P
N
becomes:
P
N
¼
2
N
1
h
N
:
ð
8
:
21
Þ
Thus, for the h
N
matrix form as above the expressions for particular signal
model parameters are:
Search WWH ::
Custom Search