Digital Signal Processing Reference
In-Depth Information
(a)
(b)
A
(
j
Ω
)
1
ak
c
()
A
(
j
Ω
)
0
1
k
0
0.8
a
S
-1
N
0
21
−
N
0
1
−
0
0.6
a
C
1
ak
s
()
0.4
k
0
0.2
-1
0
N
0
21
−
0
N
0
1
−
0
2
4
6
8
Ω
Ω
0
(a)
(b)
A
(
j
Ω
)
1
A
(
j
Ω
)
ak
c
()
0
1.2
k
0
1
a
C
-
1
0.8
0
N
0
1
−
N
0
21
−
a
S
0.6
1
ak
s
()
0.4
k
0
0.2
-
1
0
0
N
0
1
−
0
2
4
6
8
N
0
21
−
Ω
Ω
0
Fig. 6.8
Characteristics of half cycle sine/cosine filters: a impulse responses, b frequency
spectra
parameters the difference between their arguments equal always 90, however, for
specific parameter relationships only the filters have identical gains for chosen
frequencies. These identical gains are especially important for frequency equal to
frequency of fundamental component of current and voltage, allowing to get
simplified algorithms of measurement of criterion values. Assuming that the filters
are designed to get identical gains for this frequency one arrives at the conclusion
that it is obtainable when the second terms of Eqs.
6.30
and
6.31
equal zero. It is
reached when: NX
0
¼
lp, where l
¼
1
;
2...
:;
and then:
N
¼
lN
0
=
2
:
ð
6
:
32
Þ
It means that the gain of the filters are identical for frequency equal to natural
filters frequency when the filter window length is equal to a multiple of half period
of their impulse response. One can also notice that in such cases the filter gains are
equal to a half of filter window length, i.e. N
=
2 for frequency X
¼
X
0
:
Two cases are important in practice and are widely applied: filters having full
cycle window
ð
N
¼
N
0
Þ
and having half cycle window
ð
N
¼
N
0
=
2
Þ
. Normalized
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