GRID MAPPING Z-TRANSFORMS
9-2-2011
By Laurence G. Hassebrook
1. Grid Mapping a Z-transform
As an example, let’s grid map a simple single pole Z-form
H (z ) =
1
1 − a z −1
(
)
The code used can be found in the appendix. We let a=0.4 with results in a pole on the real axis
at 0.4 units. A contour map is shown in Fig. 1. The term grid mapping indicates that to get Fig. 1
we looped through each pixel and then calculated the mapping back to H(z) a complex value.
The solution to the grid mapping is
%Rxout=axy*Nx+bxy, -Rxout=axy*1+bxy,
axy=2*Rxout/(Nx-1);
bxy=Rxout-axy*Nx;
where axy and bxy are the scale and offset to map from pixel coordinates {x,y} (sometimes
called “u,v”) and complex coordinates such that
alpha=axy*x+bxy;
sigma=axy*y+bxy;
where alpha is the Euclidian real coordinate and sigma is the imaginary coordinate.
Figure 1: Color encoded contour map of H(z).
The Discrete-Time Fourier Transform (DTFT) is defined on the “unit circle” in the Z-plane so
we can sample the DTFT by grid mapping between a unit circle and H(z). The resulting
amplitude and phase are plotted in Fig. 2. The grid mapping is done with a single loop such that
the index n sequences through an angle from 0 to 2π radians.
1 for n=1:Ndtft
alpha=cos(2*pi*(n-1)/Ndtft);
sigma=sin(2*pi*(n-1)/Ndtft);
x=round((alpha-bxy)./axy);
y=round((sigma-bxy)./axy);
dtft(n)=Zmap(y,x);
end; % n
Figure 2: Amplitude and Phase along unit circle.
If we set the suppression radius Rxin=1 for the code in the appendix, a 3-D plot using
GL3Dview would clearly show the relationship between the unit circle and the response as
shown in Fig. 3.
Figure 3: 3D representation of H(z) with |z|<1 suppressed to 0.
VISUALIZATION: Reproduce the example results and visualization with the 2-pole Z-form:
2 H (z ) =
1 − r z −1 cos(ω0 )
1 − 2 r z −1 cos(ω0 ) + r 2 z − 2
For r = 0.8 and ω0 = π/4.
APPENDIX:
A. Sample Code for a single pole Z-form
clear all;
% Grid Mapping 2-D signals
% H(z)=1/(1-az^(-1));
a=.4;
Rxin=0;
Rxout=2;
Zmax=20;
Nx=256;
My=Nx;
% grid map coefficients Rxin
%Rxout=axy*Nx+bxy, -Rxout=axy*1+bxy,
axy=2*Rxout/(Nx-1);
bxy=Rxout-axy*Nx;
%
Zmap=zeros(My,Nx)+i*zeros(My,Nx);
for x=1:Nx
for y=1:My
alpha=axy*x+bxy;
sigma=axy*y+bxy;
r2=alpha*alpha+sigma*sigma;
z=alpha-i*sigma;
if r2>=Rxin.^2
if z ~= a
Zmap(y,x)=1/(1-a/z);
else
Zmap(y,x)=0+i*0;
end;
else
Zmap(y,x)=0+i*0;
end;
if Zmap(y,x) >=Zmax
Zmap(y,x)=Zmap(x,y)/Zmax;
end;
end; % y
end; % x
zmax=max(max(abs(Zmap)));
zmin=min(min(abs(Zmap)));
figure(1);
imagesc(real(Zmap));
%colormap gray;
colormap VGA;
axis image
print -djpeg90 figure1_1.jpg % display Figure 1
% store mat5file
3 Albedo8=uint8(round( 255*(abs(Zmap)-zmin)./(zmax-zmin) ) );
imageC=uint8(zeros(My,Nx,3));
imageC(:,:,1)=Albedo8;
imageC(:,:,2)=Albedo8;
imageC(:,:,3)=Albedo8;
Albedo8=uint8(20*ones(My,Nx));
imageI=zeros(My,Nx,3);
imageI(:,:,1)=Albedo8;
imageI(:,:,2)=Albedo8;
imageI(:,:,3)=Albedo8;
index=1;
xw=zeros(My,Nx);
yw=xw;
zw=yw;
for x=1:Nx
for y=1:My
xw(index)=x-1;
yw(index)=y-1;
zw(index)= abs( Zmap(y,x)*Nx/(2*(zmax-zmin)) );
index=index+1;
end;
end;
mat5prefix = 'SinglePoleZform';
result= mat5CIXYZwrite2011(mat5prefix,imageC,imageI,xw,yw,zw);
% get the DTFT
Ndtft=512;
k=1:Ndtft;
dtft=zeros(1,Ndtft)+i*zeros(1,Ndtft);
for n=1:Ndtft
alpha=cos(2*pi*(n-1)/Ndtft);
sigma=sin(2*pi*(n-1)/Ndtft);
x=round((alpha-bxy)./axy);
y=round((sigma-bxy)./axy);
dtft(n)=Zmap(y,x);
end; % n
figure(2);
plot(k,abs(dtft),k,angle(dtft));
xlabel('frequency');
ylabel('Amplitude & Phase of DTFT');
axis([0 N 0 1.1]);print -djpeg90 figure1_2.jpg % display Figure 2
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