Saturday, August 29, 2009

Discrete convolution properties - 1

I received a comment here. And, this post is about discrete convolution properties.

The continuous form of the convolution is given by:

So, the discrete form is given by:

If x[n] and h[n] are limited in the time ({x[n] = 0 if n <> Xn} and {h[n] = 0 if n <> Hn}), then the convolution is implemented as follow:

Let's suppose Xn > Hn (if Hn > Xn, the analysis is the same).

The equation is separated in three cases:

Case 1
Case 2
Case 3
Any other situation, y[n] = 0.

If you make an analysis in y[n], you'll see that y[n] = 0, if n <> Xn + Hn - 1.

Some other properties of the discrete convolution are the same than the continuous convolution:
  • x[n] * h[n] = h[n] * x[n];
  • x[n]*(h1[n]*h2[n]) = (x[n]*h1[n])*h2[n];
  • x[n]*(h1[n] + h2[n]) = x[n]*h1[n] + x[n]*h2[n];
  • a(x[n]*h[n]) = (ax[n])*h[n] = x[n]*(ah[n]), a is real;
  • x[n]*d[n] = x[n], d[n] = 1 if n = 0 and d[n] = 0 if n is not null;
  • x[n]*xi[n] = d[n], xi[n] is the inverse of x[n].
Ok, now you can make your own script for the convolution and try the properties.

Say me what happens.

3 comments:

sivaprasad84 said...

sir!
i need a code for gabor filter for texture analysis in image processing..pls send me code and suggestions..its very urgent as im supposed 2 finish my project..thanx in advance..

Poormanphysics said...

How does one extend the idea of 1D convolution to a 2D convolution involving for example an image and a convolution filter?

Sheep said...

1D convolution uses 1D kernel function (h[n]) and a simple integration, for extending it to 2D it's only to define the 2D function (h[m,n]) and to use a 2D integration as 1D convolution.

It's alike this:
y[m,n] = sum(x[u,v]*h[m - u,n - v])