Image Processing Reference
In-Depth Information
Fig. 8. Magnitude specification of minimum-phase IIR lowpass HBF;
(
2
s
− δ
p
)
+
δ
=
+ Ω
s
=
π
1
1,
Ω
p
Design outline
In order to compare MP IIR and LP FIR HBF, we subsequently consider elliptic filter designs.
Since an elliptic (minimax) HBF transfer function satisfies the conditions (6) and (13), the
design result is uniquely determined by specifying the passband
Ω
s
)cut-off
frequency and one of the three remaining parameters: the odd filter order
n
, allowed minimal
stopband attenuation
A
s
Ω
p
(stopband
=
−
(
δ
s
)
=
20log
or allowed maximum passband attenuation
A
p
−
(
− δ
p
)
.
There are two most common approaches to elliptic HBF design. The first group of
methods is performed in the analogue frequency domain and is based on classical analogue
20log
1
filter design techniques: The desired magnitude response
e
j
Ω
)
D
(
of the elliptic HBF
transfer function
H
to be designed is mapped onto an analogue frequency domain
by applying the bilinear transformation [Mitra (1998); Oppenheim & Schafer (1989)]. The
magnitude response of the analogue elliptic filter is approximated by appropriate iterative
procedures to satisfy the design requirements [Ansari (1985); Schüssler & Steffen (1998; 2001);
Valenzuela & Constantinides (1983)]. Finally, the analogue filter transfer function is remapped
to the
z
-domain by the bilinear transformation.
The other group of algorithms starts from an elliptic HBF transfer function, as given by (17).
The filter coefficients
a
i
,
i
(
z
)
n
−
1
are obtained by iterative nonlinear optimization
techniques minimizing the peak stopband deviation. For a given transition bandwidth, the
maximum deviation is minimized e.g. by the Remez exchange algorithm or by Gauss-Newton
methods [Valenzuela & Constantinides (1983); Zhang & Yoshikawa (1999)].
For the particular class of elliptic HBF with
minimal Q-factor
, closed-form equations for
calculating the exact values of stopband and passband attenuation are known allowing for
straightforward designs, if the cut-off frequencies and the filter order are given [Lutovac et al.
(2001)].
=
0, 1, ...,
(
−
1
)
2
Efficient implementation
In case of a monorate filter implementation, the McMillan degree
n
mc
is equal to the filter
order
n
. Having the same hardware prerequisites as in the previous subsection on FIR HBF,
the computational load of hardware operations per output sample is given in Table 2 (column
MoR). Note that multiplication by a factor of 0.5 does not contribute to the overall expenditure.
In the general decimating structure, as shown in Fig. 9(a), decimation is performed by an
input commutator in conjunction with a shimming delay according to Fig. 6(b). By the
underlying exploitation of the noble identities [Göckler & Groth (2004); Vaidyanathan (1993)],
the cascaded second order allpass sections of the transfer function (17) are transformed to
first order allpass sections:
z
−
1
d
z
−
2
a
i
+
a
i
+
0, 1, ...,
n
−
1
2
:
=
,
i
=
−
1, as illustrated in Fig. 9(b).
a
i
z
−
2
1
+
+
a
i
z
−
1
d
1
