Image Processing Reference
In-Depth Information
Subsequently, all comparisons are based on equiripple designs obtained by minimization of
the maximum deviation max
Ω
e j Ω )
e j Ω )
on the region of support according to
[McClellan et al. (1973)]. To this end, we briefly recall the clever use of this minimax design
procedure in order to obtain the exact values of the predefined (centre and zero) coefficients of
(9), as proposed in [Vaidyanathan & Nguyen (1987)]: To design a two-band HBF of even order
n
H
(
D
(
=
N
1
=
4 m
2, as specified in Fig. 5, start with designing i
)
a single-band zero-phase
FIR filter g
(
k
) ←→
G
(
z
)
of odd order n /2
=
2 m
1forapassbandcut-offfrequency
of 2
Ω p which, as a type II filter [Mitra & Kaiser (1993)], has a centrosymmetric zero-phase
frequency response about G
e j π )=
(
)
(
)
by two by
inserting between any pair of coefficients an additional zero coefficient (without actually
changing the sample rate), which yields an interim filter impulse response h (
0, ii
upsample the impulse response g
k
H (
z 2
) ←→
)
k
of the desired odd length N with a centrosymmetric frequency response about H (
e j π /2
)=
0
lift the passband (stopband) of H (
e j Ω )
)
[Göckler & Groth (2004); Vaidyanathan (1993)], iii
to
(
)=
)
2 (0) by replacing the zero centre coefficient with 2 h
0
1, and iv
scale the coefficients of
with 2 .
(
) ←→
(
)
the final impulse response h
k
H
z
Efficient implementations
Monorate FIR filters are commonly realized by using one of the direct forms [Mitra (1998)]. In
our case of an LP HBF, minimum expenditure is obtained by exploiting coefficient symmetry,
as it is well known [Mitra & Kaiser (1993); Oppenheim & Schafer (1989)]. The count of
operations or hardware required, respectively, is included below in Table 1 (column MoR).
Note that the “multiplication” by the central coefficient h 0 does not contribute to the overall
expenditure.
The minimal implementation of an LP HBF decimator (interpolator) for twofold
down(up)sampling is based on the decomposition of the HBF transfer function into two (type
1) polyphase components [Bellanger (1989); Göckler & Groth (2004); Vaidyanathan (1993)]:
z 2
z 1 E 1 (
z 2
(
)=
E 0 (
)+
)
H
z
.
(12)
In the case of decimation, downsampling of the output signal (cf. upper branch of Fig. 1) is
shifted from filter output to system input by exploiting the noble identities [Göckler & Groth
(2004); Vaidyanathan (1993)], as shown in Fig. 6(a). As a result, all operations (including delay
and its control) can be performed at the reduced (decimated) output sample rate f d =
f n /2:
z 2
E i (
0, 1. In Fig. 6(b), the input demultiplexer of Fig. 6(a) is replaced with
a commutator where, for consistency, the shimming delay z 1/2
d
)
:
=
E i (
z d )
, i
=
z 1 must be introduced
=
:
[Göckler & Groth (2004)].
As an example, in Fig. 7(a) an optimum, causal real LP FIR HBF decimator of n
=
10th order
and for twofold downsampling is recalled [Bellanger et al. (1974)]. Here, the odd-numbered
coefficients of (9) are assigned to the zeroth polyphase component E 0
(
z d
)
of Fig. 6(b), whereas
the only non-zero even-numbered coefficient h 0 belongs to E 1
.
For implementation we assume a digital signal processor as a hardware platform. Hence, the
overall computational load of its arithmetic unit is given by the total number of operations
N Op =
(
z d
)
N A , comprising multiplication (M) and addition (A), times the operational
clock frequency f Op [Göckler & Groth (2004)]. All contributions to the expenditure are listed
in Table 1 as a function of the filter order n , where the McMillan degree includes the
shimming delays. Obviously, both coefficient symmetry ( N M <
N M +
n /2) and the minimum
memory property ( n mc
<
n [Bellanger (1989); Fliege (1993); Göckler & Groth (2004)]) are
Search WWH ::




Custom Search