![](bg3.png)
The corresponding LC circuit is shown in figure
6. Discrete realization of the wave digital filter is ob-
tained from this LC prototype of the Chebyshev filter,
see Fig. 8. High-pass filter in the filter-bank synthesis
has the same structure and values as high-pass filter
in the filter-bank analysis, only the multiplier -1 at the
end of each delay elements must be added, see Fig. 9,
(B. Psenicka and Rodriguez, 2006).
0.9941
2.0235 2.0235
Figure 6: LC prototype of Chebyshev low-pass filter.
LC ladder network in Fig.6 can be redrawn net-
work in Fig. 7. The elements are connected with the
parallel and serial adapters together. Parallel depen-
dent adapter have to be used in the end of the struc-
ture.
R
0
R
3
L = 0.9941
C = 2.0235 C = 2.0235
R
1
R
2
Figure 7: LC prototype of Chebyshev low-pass filter – re-
drawn diagram.
To obtain the multiplier coefficients of the discrete
Chebyshev low-pass filter the conductances G
1
and
G
2
must be calculated first.
R
0
= 1 R
3
= 1 (12)
G
1
= G
0
+C = 3.0235 R
1
= 0.33074 (13)
R
2
= R1 + L = 1.32484 G
2
= 0.754806 (14)
Finally the multiplier coefficients of the discrete
low-pass Chebychev filter can be calculated by fol-
lowing relations:
A
1
=
G
0
G
0
+C
= 0.3307 B
2
=
R
1
R
1
+L
= 0.2496
A
31
=
2G
2
G
2
+C+G
3
= 0.3995 A
32
=
2G
3
G
2
+C+G
3
= 0.5293
(15)
Attenuation of the low-pass Chebyshev filter can
be obtained by following script. The script was writ-
ten according the structure in Fig. 8. The attenuation
of the low-pass and high-pass filter are presented in
the Fig. 10. Attenuation of the high-pass filter can
be obtained by changing the signs of variables in the
script: N2 = −N1, N4 = −N3 and N6 = −N5.
A1=0.3307; B2=0.2496; A31=0.3995; A32=0.5293;
N2=0; N4=0; N6=0; XN=1;
for i=1:1:200
XN1=N2-N2*A1+XN*A1;
XN2=XN1+N4;
BN2=XN2-A31*XN2+2*N6-N6*A31-N6*A32;
BN1=XN1-B2*XN2-B2*BN2;
N1=XN*A1-A1*N2+BN1;
N3=BN1+BN2;
N5=N6-XN2*A31-N6*A31-N6*A32;
YN(i)=2*N6-N6*A31-N6*A32-XN2*A31;
N2=N1; N4=N3; N6=N5; XN=0;
end
[h,w]=freqz(YN,1,50)
plot(w,20*log10(abs(h)))
axis([0.5184 2.623-40.3 0])
Minimum attenuation of the high-pass Inverse
Chebyshev filter in the analysis part of filter-bank
must be calculated from the ripple factor of the low-
pass filter in the analysis part of the filter-bank by re-
lation (16), (Weinberg, 1962).
A
min
= 10log
10
(
1
10
A
max
10
−1
+ 1) (16)
Transfer function of inverse Chebyshev filter in s
domain is then given by:
H(s) =
s
2
+ 1.3333
0.655s
3
+ 1.6512s
2
+ 1.3177s + 1.3333
=
k(s)
h(s)
(17)
Relevant characteristic function to the relation
(17) is
F(s) =
s
2
+ 1.3333
0.6550s
2
=
k(s)
f (s)
(18)
Input impedance z
11
and transfer impedance z
12
can be calculated from equation (17) and (18). Inverse
Chebyshev low-pass filter presented in Fig. 11 can
be obtained from these impedances (z
11
, z
12
), (Storer,
1957).
Discrete realization of the parallel LC circuit from
Fig. 11 is demonstrated in the Fig. 12, where K is
given by equation (19), (Kammeyer and K, 1992).
K =
1 − LC
1 + LC
(19)
The coefficients of the serial and parallel adapters
of the discrete filter in figure 18 obtained by equation
(5) are A
1
= 0.6693, B
2
= 0.2496, A
31
= 0.7032 A
32
=
0.5293 and K = 0.1428. Discrete Chebychev high-
pass filter is obtained from the discrete low-pass filter
by changing the sings as is seen in Fig. 19.
Attenuation of the Inverse Chebyshev filter can be
calculated by the following Matlab script. The script
is written according the structure in Fig. 18.
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