- #1
Number2Pencil
- 208
- 1
Homework Statement
Calculate, by hand, the DFT of cos(2*pi(fx*m + fy*n)). It should simplify to something simple. It should NOT be left as a summation.
Homework Equations
2D DFT Formula:
[tex]
\tilde{s(k,l)} = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} s(m,n) e^{-j 2 \pi (\frac{1}{M}mk + \frac{1}{N}ln)}
[/tex]
The Attempt at a Solution
[tex]
\tilde{s(k,l)} = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} cos(2 \pi (f_x m + f_y n)) e^{-j 2 \pi (\frac{1}{M}mk + \frac{1}{N}ln)}
[/tex]
One of Euler's Formulas:
[tex]
cos(u+v) = \frac{1}{2}(e^{ju + jv} + e^{-ju -jv})
[/tex]
After much algebra crunching, I wound up with this:
[tex]
\tilde{s(k,l)} = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} \left[ \frac{1}{2}e^{-j2 \pi
\frac
{-f_x m M N + m k N + l n M - f_y n M N}
{MN}
} + \frac{1}{2} e^{-j2 \pi
\frac
{f_x m M N + f_y n M N + m k N + l n M}
{MN}} \right]
[/tex]
I am really not sure how to simplify this double geometric sum. Does it look recognizable? Was there an earlier simplification I could take?