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From the expression in a) and b), we see that <math> p_0(e^{jw}) </math> and <math> p_1(e^{jw}) </math> are only slices of the DSFT. It lost the information when <math> \mu </math> and <math> \nu <math> are not zero. | From the expression in a) and b), we see that <math> p_0(e^{jw}) </math> and <math> p_1(e^{jw}) </math> are only slices of the DSFT. It lost the information when <math> \mu </math> and <math> \nu <math> are not zero. | ||
A simple example would be: | A simple example would be: | ||
| − | Let <math> | + | Let |
| + | <math> | ||
| + | x(m,n) = | ||
| + | \left[ {\begin{array}{*{20}{c}} | ||
| + | 1 2 \\ | ||
| + | 3 4\\ | ||
| + | \end{array}} \right] </math> | ||
Revision as of 20:39, 10 November 2014
a) Since
$ X(e^{j\mu},e^{j\nu}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n)e^{-j(m\mu+n\nu)} $
and
$ p_0(e^{jw}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n)e^{-jnw} $，
we have:
$ p_0(e^{jw}) = X(e^{j\mu},e^{jw}) |_{\mu=0} $
b) Similarly to a), we have:
$ p_1(e^{jw}) = X(e^{jw},e^{j\nu}) |_{\nu=0} $
c)
$ \sum_{n=-\infty}^{\infty} p_0(n) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n) = X(e^{j\mu}, e^{j\nu}) |_{\mu=0, \nu=0} $
which is the DC point of the image.
d) No, it can't provide sufficient information. From the expression in a) and b), we see that $ p_0(e^{jw}) $ and $ p_1(e^{jw}) $ are only slices of the DSFT. It lost the information when $ \mu $ and $ \nu <math> are not zero. A simple example would be: Let <math> x(m,n) = \left[ {\begin{array}{*{20}{c}} 1 2 \\ 3 4\\ \end{array}} \right] $
