Convolution - Rectangular Kernel

 

Convolution would be the one of the most important operations in engineering math. If you don't have any prior study and understanding on how Convolution works, it would not be easy to make sense out of it. For those who is not familiar with Convolution, I would suggest you to go through my descriptive note here.

 

The operation in this tutorial can be illustrated as follows. The value at each and every point in the covoluted plot (the bottom plot) is calculated by this operation (i.e, element by element multiplication and summation).

 

 

 

 

Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.

 

< Code 1 >

 

x = [0 0.4 0.4 0.4 0.4 0.4 0 0 0];

 

sf = 11; % samples per bit

 

p_x = x;

 

if sf > 1

   for i = 2:sf

     p_x= [p_x ; x];

   end

   p_x = reshape(p_x,[],1);

 

end

 

chan = 0.3*[0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0];

%chan(1) = 1;

%chan(30) = 0.5;

 

cs = 123;

 

hFig = figure(1,'Position',[300 300 700 600]);

 

subplot(4,2,1);

stem(p_x,'MarkerFaceColor',[0 0 1],'color','blue');

axis([1 length(p_x) -1.0 1.0]);

title("data");

box on;

set(gca,'xticklabel',[]);

set(gca,'yticklabel',[]);

set(gca,'xtick',[]);

set(gca,'ytick',[]);

 

subplot(4,2,2);

stem(chan,'MarkerFaceColor',[1 0 0],'MarkerEdgeColor',[1 0 0],'color','red');

axis([1 length(p_x) -1.0 1.0]);

title("kernel");

box on;

set(gca,'xticklabel',[]);

set(gca,'yticklabel',[]);

set(gca,'xtick',[]);

set(gca,'ytick',[]);

 

subplot(4,2,[3 8]);

hold on;

s = p_x;

for i = 0:100

    if i < length(p_x)

       line([i i],[5 5+s(i+1)],'color','blue');

       plot(i,5+s(i+1),'bo');

    end;   

end;

line([-40 140],[5 5],'color','green');

 

k = flip(chan);

ks = length(k);

 

for i = 0:100

    if i < length(k)

       line([i-ks+cs i-ks+cs],[3 3+k(i+1)],'color','red');

       plot(i-ks+cs,3+k(i+1),'ro');

    end;   

end;  

line([-40 140],[3 3],'color','green');

 

y = conv(p_x(1:min([length(p_x) cs])),chan(1:min([length(chan) cs])));

y = y(1:cs);

for i = 0:200

    if i < length(y)

       line([i i],[1 1+y(i+1)],'color','black');

       plot(i,1+y(i+1),'ko');

    end;   

end;

line([-40 140],[1 1],'color','green');

 

rdx = min(cs,length(k));

rdy = 3.0;

rx = cs-rdx;

ry = 2.9;

rectangle('Position',[rx ry rdx rdy]);

 

line([cs cs],[1 1+y(end)],'color','red','LineWidth',3, 'Marker','o');

line([cs cs],[1 1+2],'color','red','LineWidth',0.5);

 

hold off;

axis([-50 150 0 6]);

tStr = sprintf("step = %d",cs);

title(tStr);

box on;

set(gca,'xticklabel',[]);

set(gca,'yticklabel',[]);

set(gca,'xtick',[]);

set(gca,'ytick',[]);