Synthesis and analysis of Band pass from LPF and HPF Jan 2015
We can first build the band stop filter . For this ,first design the LPF , say hseqa(i). Then use this to convert to HPF using highpass i.e hseqh(i) =
-hseqa(i)*(-1)^(i) ; . Next the band stop is hseqb = hseqa+hseqb .
We can see that every other hseqb(i) of band stop is zero and the other values are double the lpf hseqa or hpf hseqb . So for analysis of the band stop we need to take say for lpf alternate values as hseqb/2 and expand it to include zeros like an expander with L=2.This has to be filled with interpolated values . The hpf is then obtained by changing signs of alternate hseqa(i).
1. Eigen lp filter type 1
// set sigma values passband sig1 sig2 in radians
// stop band sig3 sig4 sig3 =.5 *%pi // stop band sig4 = 1*%pi // stop band sig1 = 0*%pi // passband sig2 =.3*%pi // pass band
1.1. lpf hseqa = [ 0.5197220
0.7810746
0.2383787
- 0.1455732
- 0.1698201
- 0.0035968
0.0906716
0.0448063
- 0.0291368
- 0.0377701
- 0.0018667
0.0180220
0.0086882
- 0.0038613
- 0.0045153
- 0.0004323 ]
2 hpf hseqb =[ 0.5197220
0.7810746
0.2383787
- 0.1455732
- 0.1698201
- 0.0035968
0.0906716
0.0448063
- 0.0291368
- 0.0377701
- 0.0018667
0.0180220
0.0086882
- 0.0038613
- 0.0045153
- 0.0004323 ]
3. band stop
hseqb =[ 1.039444
0.
0.4767574
0.
- 0.3396402
0.
0.1813432
0.
- 0.0582736
0.
- 0.0037334
0.
0.0173764
0.
- 0.0090306
0. ]
4. Scilab 5.3.3 code for plotting
//plot lpf using result of spec function npt =100 dw= %pi/npt ; omg= zeros(1:npt); n1=length(hseqa)

for k=1:npt xm(k)= 0; omg(k)=omg(k)+(k-1)* dw; for i=1:n1 xm(k) = hseqa(i)*cos(omg(k)*(i-1) ) +xm(k) ;
end end xm =real(xm) ; figure(2); plot(omg, xm); // plot the magn vs freq figure(3); mx= max(xm) xml = 20*log10( abs(xm/mx)); oma=omg/2/%pi ; plot(oma, xml);
// lpf to hpf for i=1:n1 hseqh(i) = -hseqa(i)*(-1)^(i) ;
end
//plot hpf using result of spec function npt =100 dw= %pi/npt ; omg= zeros(1:npt); n1=length(hseqa)
for k=1:npt xm(k)= 0; omg(k)=omg(k)+(k-1)* dw; for i=1:n1 xm(k) = hseqh(i)*cos(omg(k)*(i-1) ) +xm(k) ;
end end xm =real(xm) ; figure(2); plot(omg, xm); // plot the magn vs freq figure(3); mx= max(xm) xml = 20*log10( abs(xm/mx)); oma=omg/2/%pi ; plot(oma, xml);
// add lp+hpf hseqb=hseqa+hseqh ;
//plot bpf using result of spec function npt =100 dw= %pi/npt ; omg= zeros(1:npt); n1=length(hseqb)
for k=1:npt xm(k)= 0; omg(k)=omg(k)+(k-1)* dw; for i=1:n1 xm(k) = hseqb(i)*cos(omg(k)*(i-1) ) +xm(k) ;
end end xm =real(xm) ; figure(2); plot(omg, xm); // plot the magn vs freq figure(3); mx= max(xm) xml = 20*log10( abs(xm/mx)); oma=omg/2/%pi ; plot(oma, xml);
5. Plots of the filters.
Below the plots of the lpf , hpf and band pass filters are shown.

lpf
lpf

hpf
hpf

band stop