MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## How To Relate Poisson Summation formula to Fourier Transform

I am studying Fourier transforms by watching Stanford online videos on youtube. I am trying to solve the following problem but i do not know how to relate it to poisson summation formula

Let us represent an image by a function $y:\mathbb{R}^{2}\rightarrow\mathbb{R}$, which has an associated Fourier transform $Y:\mathbb{R}^{2}\rightarrow\mathbb{C}$ defined as:

$$Y\left(f_{1},f_{2}\right)=\iint_{t1,t2=-\infty}^{\infty}y\left(t_{1},t_{2}\right)\exp\left(-j2\pi\left(f_{1}t_{1}+f_{2}t_{2}\right)\right)\mathrm{d}t_{1}\mathrm{d}t_{2}.$$ Now consider a two dimensional digital image of size $\mathrm{N}\times\mathrm{N}$: $x[n_{1},n_{2}]$ with $n_{1},n_{2}\in{0,1,...,\mathrm{N-1}}$. The relation to its discrete Fourier transform $X[k_{1},k_{2}]$ with $k_{1},k_{2}\in{0,1,...\mathrm{,N-1}}$ is defined by:

$$X[k_{1},k_{2}]=\mathcal{F}\left(x[n_{1},n_{2}]\right)=\sum_{n_{1},n_{2}=0}^{\mathrm{N}-1}x[n_{1},n_{2}]\exp\left(-j2\pi\frac{\left(k_{1}n_{1}+k_{2}n_{2}\right)}{\mathrm{N}}\right).$$ If the image $x[n_{1},n_{2}]$ are actually samples from the physical object with sampling period $T_{s}$:

$$x[n_{1},n_{2}]=y\left(n_{1}\mathrm{T_{s}},n_{2}\mathrm{T_{s}}\right).$$

Develop a closed-form expression that relates $X[n_{1},n_{2}]$ to $Y\left(f_{1},f_{2}\right)$. Hint: look up the subject of the Poisson summation formula.

I tried to search but what i found did not help me that much.

-