Code covered by the BSD License

# 2-dimensional Filter Design using McClellan transformation

by

### Iman (view profile)

12 Feb 2012 (Updated )

This function can be used to design 2D lowpass, highpass, bandpass, bandstop filters.

Filter_Design_2D_McClellan.m
```

function [Filter_1D,Filter_2D]=Filter_Design_2D_McClellan(Type,edges,Ap,Aa,transformation_vector)

%
%
% Written By:   Iman Moazzen
%               Dept. of Electrical Engineering
%               University of Victoria
%
% [Filter_1D,Filter_2D]=Filter_Design_2D_McClellan(Type,edges,Ap,Aa,transformation_vector)
%
% This function can be used to design a lowpass, highpass, bandpass, or bandstop
% two-dimensional filter which satisfies prescribed specifications.
%
% _ Type can be Lowpass, Highpass, Bandpass, or 'Bandstop'
% _ edges is a vector of normalized frequencies (rad/s) including passband and stopband edges.
%   (frequencies must be in an increasing order)
% _ Ap: peak to peak passband ripple (db)
% _ Aa: minimum stopband attenuation (db)
% _ transformation_vector: this is a vector with 4 elements which maps 1D
% space to 2D space [1]. These coefficients can be found based on the method
% presented in [2]. Some examples are as follows:
%
% 2D Filter with circularly symmetric spectrum=[-0.5 0.5 0.5 0.5];
% 2D Filter with elliptic spectrum=[-2.4973 2.9006 0.3127  0.2840];
% 2D Filter with fan shape spectrum=[0 0.5 -0.5 0];
%
% The algorithm first designs 1D filter based on kaiser method and
% Dig_Filter(http://www.mathworks.com/matlabcentral/fileexchange/30321-digfilter).
% Then by using the transformation vector and Chebyshev Polynomial
% (http://www.mathworks.com/matlabcentral/fileexchange/4913-chebyshevpoly-m)
% 2D filter will be designed.
%
% The amplitude response of 2D and 1D filters as well as contours of the
% transformation function will be shown at the output.
%
% [Filter_1D,Filter_2D] are 1D and 2D filters coefficients, respectively.
%
% Example:
% [Filter_1D,Filter_2D]=Filter_Design_2D_McClellan('lowpass',[0.1*pi,0.2*pi],0.5,30,[-0.5 0.5 0.5 0.5]);
% Which designs a lowpass 2D FIR filter with circularly symmetric spectrum using Kaiser Method.
% Passband edge is 0.1*pi, and stopband edge is 0.2*pi,
% Ap=0.5 (db) and Aa=30 (db).
%
%
% -------------------------------------------------------------------------
% References:
% [1] D.E. Dudgeon, R.M. Mersereau, "Multidimensional digital
% signal processing", Prentice-Hall.
% [2] R.M. Mersereau, W.F.G. Mecklenbrauker, T.F. Quatieri, "McClellan transformations
% for two-dimensional digital filtering: I-Design", IEEE transactions on circuit and systems, 1976.
%
%
%
% http://www.ece.uvic.ca/~imanmoaz/index_files/index1.htm

ws=2*pi;
error_flag=0;

if nargin<5
disp('Not enough input arguments')
error_flag=1;
end

if nargin==5
switch lower(Type)
case {'lowpass'}
if length(edges)~=2
disp('Lowpass filter needs exactly two edges')
error_flag=1;
end
case {'highpass'}
if length(edges)~=2
disp('Highpass filter needs exactly two edges')
error_flag=1;
end
case {'bandpass'}
if length(edges)~=4
disp('Bandpass filter needs exactly four edges')
error_flag=1;
end
case {'bandstop'}
if length(edges)~=4
disp('Bandstop filter needs exactly four edges')
error_flag=1;
end
otherwise
disp('Unknown Type')
error_flag=1;
end

if sum(sort(edges)~=edges)~=0
disp('The cut-off frequencies must be in an increasing order')
error_flag=1;
end

if length(transformation_vector)~=4
disp('transformation vector must have exactly 4 elements')
error_flag=1;
end

end

if error_flag==0;

[Num,Den]=Dig_Filter('FIR',Type,'Kaiser',edges,ws,Ap,Aa);

wx=linspace(-ws/2,ws/2,128);
wy=linspace(-ws/2,ws/2,128);

[wx,wy]=meshgrid(wx,wy);

Fp=transformation_vector(1)+transformation_vector(2)*cos(wx)+transformation_vector(3)*cos(wy)+transformation_vector(4)*cos(wx).*cos(wy);

F_max=max(max(Fp));
F_min=min(min(Fp));

c1=2/(F_max-F_min);
c2=c1*F_max-1;

F=c1*Fp-c2;

M=(length(Num)-1)/2;
a=[Num(M+1) 2*Num(M+2:end)];

[mm,nn]=size(F);

for i=1:mm
for j=1:nn

temp=0;

for n=0:M;
temp=a(n+1)*polyval(ChebyshevPoly(n),F(i,j))+temp;
end
H(i,j)=temp;
end
end

Filter_1D=Num;
Filter_2D=real(ifft2(H));

subplot(2,3,[1 2 4 5])
mesh(wx,wy,20*log10(abs(H)),'linewidth',2)
axis([-pi pi -pi pi -140 10])
xlabel('w_1','fontsize',10)
ylabel('w_2','fontsize',10)
zlabel('Amplitude Response (dB)','fontsize',10)
title('Frequency Response of 2D Filter','fontsize',10)

subplot(2,3,3)
contour(wx,wy,F,'linewidth',2);
title('Contours of Transfer Function','fontsize',10)
subplot(2,3,6)
[FF,WW]=freqz(Filter_1D);
plot(WW,20*log10(abs(FF)),'linewidth',2)
axis([0 pi -100 10])
xlabel('w','fontsize',10)
ylabel('Amplitude Response (dB)','fontsize',10)
title('Frequency Response of 1D Filter','fontsize',10)

else
Filter_1D=[];
Filter_2D=[];
return
end

```