Image Processing Reference
In-Depth Information
Fig. 16. Optimum SFG of decimating LP FIR COHBF (a) and its transpose for interpolation
(b)
where
the associated stopband
cut-off frequency. Obviously, strict complementarity (7) reads as follows
Ω
p
+
represents the upper passband cut-off frequency and
Ω
s
−
e
j
(Ω
−
c
4
)
)+
e
j
(Ω
−π
(
1
+
c
/4
))
)=
H
(
H
(
1.
(33)
Efficient implementations
The optimum implementation of an
n
10th order LP FIR COHBF for twofold
downsampling is again based on the polyphase decomposition of (40). Its SFG is depicted
in Fig. 16(a) that exploits the coefficient symmetry as given by (41).
The optimum FIR COHBF interpolator according to Fig. 16(b) is readily derived from the
original decimator of Fig. 16(a) by applying the multirate transposition rules, as discussed
in Section 3. As a result, the overall expenditure is again retained (c.f. invariant property of
transposition [Göckler & Groth (2004)]).
In addition, Fig. 17 shows the optimum SFG of an LP FIR COHBF for decimation of a complex
signal by a factor of two. It represents essentially a doubling of the SFG of Fig. 16(a). The dual
interpolator can be derived by transposition [Göckler & Groth (2004)].
The expenditure of the half- (
=
C
→
C
) LP COHBF decimators
and their transposes is listed in Table 5 in terms of the filter order
n
. A comparison of Tables
3 and 5 shows that the implementation of any type of COHBF requires just two or four extra
R
C
) and the full-complex (
