Image Processing Reference
In-Depth Information
decimators. Both for FIR and IIR CHBF, the number of operations has to be substituted:
N CHBF
Op
N HBF
Op
:
=
1.
2.3 Complex Offset Halfband Filters (COHBF)
A complex offset HBF, a Hilbert-Transformer with a frequency offset of
f n /8 relative
to an RHBF, is readily derived from a real HBF according to Subsection 2.1 by applying the zT
modulation theorem (3) with c
Δ
f
= ±
∈ {
1, 3, 5, 7
}
,asintroducedin(2):
c 4 )+
c 4 )= ±
1
j
2 .
±
e jc 4
e j 2 π f c / f n
=
=
=
(
(
z c
cos
j sin
(28)
As a result, the real prototype HBF is shifted to a passband centre frequency of
f c
8 . In the sequel, we predominantly consider the case f c
f n
8 ,
3 f n
±
±
=
f 1 1 = π
)
/4
.
2.3.1 Linear-Phase (LP) FIR filters
Again, the frequency shift operation (3) is applied in the time domain. However, in order
to get the smallest number of full-complex COHBF coefficients, we introduce an additional
complex scaling factor of unity magnitude. As a result, the modulation of a carrier of
frequency f c according to (28) by the impulse response (9) of any real LP FIR HBF yields
the complex-valued COHBF impulse response:
e jc 4 h
e j ( k + 1 ) c 4
z c =
j c ( k + 1 ) /2 ,
h k =
(
k
)
h
(
k
)
=
h
(
k
)
(29)
n
2
n
2 and c
=
=
where
k
1, 3, 5, 7. By directly equating (39) for c
1, and relating the result
to (9), we get:
1
j
2
+
2
k
=
0
h k =
=
=
(
)
(30)
0
k
2 l
l
1,2,...,
n
2
/4
j ( k + 1 ) /2 h
(
k
)
k
=
2 l
1 l
=
1,2,...,
(
n
+
2
)
/4
where, in contrast to (21), the impulse response exhibits the symmetry property:
j ck h k
=
>
h
k
0.
(31)
k
Note that the centre coefficient h 0 is the only truly complex-valued coefficient where,
fortunately, its real and imaginary parts are identical. All other coefficients are again either
purely imaginary or real-valued. Hence, the symmetry of the impulse response can still be
exploited, and the implementation of an LP FIR COHBF requires just one multiplication more
than that of a real or complex HBF [Göckler (1996b)].
Specification and properties
All properties of the real HBF are basically retained except of those which are subjected to the
frequency shift operation according to (28). This applies to the filter specification depicted in
Fig. 5 and, hence, (6) modifies to
c 4 + Ω s
c 4 = Ω p + + Ω s = π +
c 2 .
Ω p
+
+
(32)
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