Image Processing Reference
In-Depth Information
Fig. 12. Optimum SFG of decimating LP FIR HT (a) and its interpolating multirate transpose
(b)
R
→
C
C
→
R
C
→
C
C
→
C
Dec:
Int:
Dec:
Int:
+
+
n
mc
3
n
/4
1/2
n
2
N
M
(
n
+
2
)
/4
(
n
+
2
)
/2
N
A
n
/2
n
+
2
n
+
+
+
N
Op
3
n
/4
1/2
3
n
/2
3
3
n
/2
1
(
)
Table 3. Expenditure of linear-phase FIR CHBF;
n
:order,
n
mc
: McMillan degree,
N
M
N
A
:
number of multipliers (adders), operational clock frequency:
f
Op
=
f
n
/2
where
the associated stopband
cut-off frequency. Obviously, strict complementarity (7) is retained as follows
Ω
p
+
represents the upper passband cut-off frequency and
Ω
s
−
e
j
(Ω
∓
2
)
)+
e
j
(Ω
±
2
)
)=
H
(
H
(
1,
(23)
where (3) is applied in the frequency domain.
Efficient implementations
The optimum implementation of an
n
=
10th order LP FIR CHBF for twofold downsampling
is again based on the polyphase decomposition of (20) according to (12). Its SFG is depicted in
Fig. 12(a) that exploits the odd symmetry of the HT part of the system. Note that all imaginary
units are included deliberately. Hence, the optimal FIR CHBF interpolator according to Fig.
12(b), which is derived from the original decimator of Fig. 12(a) by applying the multirate
transposition rules [Göckler & Groth (2004)], performs the dual operation with respect to
the underlying decimator. Since, however, an LP FIR CHBF is strictly rather than power
complementary (cf. (23)), the inverse functionality of the decimator is only approximated
[Göckler & Groth (2004)].
In addition, Fig. 13 shows the optimum SFG of an LP FIR CHBF for decimation of a complex
signal by a factor of two. In essence, it represents a doubling of the SFG of Fig. 12(a). Again,
the dual interpolator is readily derived by transposition of multirate systems, as outlined in
Section 3.
The expenditure of the half- (
) CHBF decimators and
their transposes is listed in Table 3. A comparison of Tables 1 and 3 shows that the overall
numbers of operations
N
CFIR
R
C
) and the full-complex (
C
→
C
Op
of the half-complex CHBF sample rate converters (cf. Fig. 12)
are almost the same as those of the real FIR HBF systems depicted in Fig. 7. Only the number
of delays is, for obvious reasons, higher in the case of CHBF.
