Image Processing Reference
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+ Ω s
= π
Fig. 5. Specification of a zero-phase FIR HBF;
Ω p
Hence, the non-causal impulse response of a real zero-phase FIR HBF is characterized by
[Bellanger et al. (1974); Göckler & Groth (2004); Mintzer (1982); Schüssler & Steffen (1998)]:
1
2
=
k
0
h k =
=
=
=
(
)
h
0
k
2 l
l
1,2,...,
n
2
/4
(9)
k
(
)
=
=
(
+
)
h
k
k
2 l
1 l
1,2,...,
n
2
/4
giving rise to efficient implementations. Note that the name Nyquist(2)filter is justified
by the zero coefficients of the impulse response (9). Moreover, if an HBF is used as an
anti-imaging filter of an interpolator for upsampling by two, the coefficients (9) are scaled
by the upsampling factor of two replacing the central coefficient with h 0
=
1 [Fliege (1993);
Göckler & Groth (2004); Mitra (1998)]. As a result, independently of the application this
coefficient does never contribute to the computational burden of the filter.
Design outline
Assuming an ideal lowpass desired function consistent with the specification of Fig.
5
with a cut-off frequency of
/2 and zero transition bandwidth,
and minimizing the integral squared error, yields the coefficients [Göckler & Groth (2004);
Parks & Burrus (1987)] in compliance with (9):
Ω t
=( Ω p
+ Ω s
)
/2
= π
k 2 )
k 2
(
h k = Ω t
π
sin
(
k
Ω t
)
1
2
sin
1, 2, . . . , n
=
|
| =
,
k
2 .
(10)
k
Ω t
This least squares design is optimal for multirate HBF in conjunction with spectrally
white input signals since, e.g in case of decimation, the overall residual power aliased by
downsampling onto the usable signal spectrum is minimum [Göckler & Groth (2004)]. To
master the Gibbs' phenomenon connected with (10), a centrosymmetric smoothed desired
function can be introduced in the transition region [Parks & Burrus (1987)]. Requiring, for
instance, a transition band of width
ΔΩ = Ω s Ω p
>
0 and using spline transition
e j Ω )
(
functions for D
, the above coefficients (10) are modified as follows [Göckler & Groth
(2004); Parks & Burrus (1987)]:
sin
β ,
k Δ 2 β )
k Δ 2 β
(
k 2 )
k 2
sin
(
1
2
1,2,..., n
h k =
|
k
| =
2 ,
β R
.
(11)
Least squares design can also be subjected to constraints that confine the maximum deviation
from the desired function: The Constrained Least Squares (CLS) design [Evangelista (2001);
Göckler & Groth (2004)]. This approach has also efficiently been applied to the design of
high-order LP FIR filters with quantized coefficients [Evangelista (2002)].
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