(New page: a) Since <math>X(e^{j\mu},e^{j\nu}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n)e^{-j(m\mu+n\nu)}</math> and <span class="texhtml"><math>p_0(e^{jw}) = \sum_{m=-\infty...) |
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| − | a) Since <math>X(e^{j\mu},e^{j\nu}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} | + | a) Since |
| − | x(m,n)e^{-j(m\mu+n\nu)}</math> and | + | |
| − | x(m,n)e^{-jnw}</math> </span> | + | <math>X(e^{j\mu},e^{j\nu}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} |
| + | x(m,n)e^{-j(m\mu+n\nu)}</math><br> | ||
| + | |||
| + | and | ||
| + | |||
| + | <span class="texhtml"><math>p_0(e^{jw}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} | ||
| + | x(m,n)e^{-jnw}</math>， </span> | ||
| + | |||
| + | we have: | ||
| + | |||
| + | <math> p_0(e^{jw}) = X(e^{j\mu},e^{jw}) |_{\mu=0} </math> | ||
| + | |||
| + | b) Similarly to a), we have: | ||
| + | |||
| + | <math> p_1(e^{jw}) = X(e^{jw},e^{j\nu}) |_{\nu=0}</math> | ||
| + | |||
| + | c) <br> | ||
| + | <math> \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} </math> | ||
| + | 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 <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: | ||
| + | Let <br> | ||
| + | <math> | ||
| + | x(m,n) = | ||
| + | \left[ {\begin{array}{*{20}{c}} | ||
| + | 1 ~ 2 \\ | ||
| + | 3 ~ 4\\ | ||
| + | \end{array}} \right] </math>, | ||
| + | so<br> | ||
| + | <math> p_0(n) =[4~6], p_1(m) = [3 ~7]^T </math>. | ||
| + | With the above the information of the projection, the original form of the 2D signal cannot be determined. For example, | ||
| + | <math> | ||
| + | x(m,n) = | ||
| + | \left[ {\begin{array}{*{20}{c}} | ||
| + | 2 ~ 1 \\ | ||
| + | 2 ~ 5\\ | ||
| + | \end{array}} \right] </math> gives the same projection. | ||
| + | |||
| + | Related problem: | ||
| + | Let <math> g(x,y) = sinc(x/2, y/2) </math>, and let <math> s(m,n) = g(mT, nT) </math> where T = 1. | ||
| + | |||
| + | a) Calculate <math> G(\mu, \nu) </math> the CSFT of <math>g(x,y) </math>. | ||
| + | |||
| + | |||
| + | b) Calculate <math> S(e^{j\mu}, e^{j\nu}) </math> the DSFT of <math> s(m,n) </math>. | ||
Latest revision as of 20:48, 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 $ are not zero.
A simple example would be:
Let
$ x(m,n) = \left[ {\begin{array}{*{20}{c}} 1 ~ 2 \\ 3 ~ 4\\ \end{array}} \right] $,
so
$ p_0(n) =[4~6], p_1(m) = [3 ~7]^T $.
With the above the information of the projection, the original form of the 2D signal cannot be determined. For example,
$ x(m,n) = \left[ {\begin{array}{*{20}{c}} 2 ~ 1 \\ 2 ~ 5\\ \end{array}} \right] $ gives the same projection.
Related problem: Let $ g(x,y) = sinc(x/2, y/2) $, and let $ s(m,n) = g(mT, nT) $ where T = 1.
a) Calculate $ G(\mu, \nu) $ the CSFT of $ g(x,y) $.
b) Calculate $ S(e^{j\mu}, e^{j\nu}) $ the DSFT of $ s(m,n) $.
