Implementing a discrete analogue to Gaussian function Given a Gaussian function of the form $$g(x) = ae^{-(x-b)^2/(2c^2)}$$
I am interested in a discrete analogue to this, which deals with the case where $x$ is discrete. As I understand there are two ways of doing this, namely, the Sampled Gaussian kernel and the Discrete Gaussian kernel. 
Does anyone know how to explain the implementation of these in a simple way or has a good reference for a text or set of notes? 
I don't have a very strong background in statistics hence I am having difficulty understanding the protocol involved in these two methods. 
To give some context:
Consider an operator which takes the constants $a,b,c$ to a Gaussian function $$\{a,b,c\} \mapsto ae^{\frac{(x-b)^2}{2c^2}}~~~~~~~(1)$$
Given that $x'$ is discrete and possibly bounded $-N \leq x' \leq N$, is there a discrete case analogue to the operator (1) which gives some Gaussian type distribution given some parameters $a,b,c...$ which include the mean and the variance? 
 A: The most basic way to discretize the continuous probability distribution is to assumed its "rounded" form, i.e. if $X \sim \mathcal{N}(\mu, \sigma)$, then $\lfloor X \rfloor$ follows the discrete analogue. For discrete normal distribution, the probability mass function following from such procedure is
$$
f(x) = \Phi\left(\frac{x-\mu+1}{\sigma}\right) - \Phi\left(\frac{x-\mu}{\sigma}\right)
$$
where $\Phi$ is standard normal cumulative distribution function, as described in

Roy, D. (2003). The discrete normal distribution. Communications in
  Statistics-Theory and Methods, 32, 1871-1883.

Alternatively, you may consider rounding within the $\pm0.5$ interval (i.e. distribution of $\lfloor X +0.5\rfloor$).
A: Luc Devroye defines a discrete normal in his book (Non-Uniform
Random Variate Generation) (pg 117). Here is a link to chapter 3:
http://www.nrbook.com/devroye/Devroye_files/chapter_three.pdf
last page of chapter 3 is page 117.  
$$\Pr(X=i)= c\exp\left[\frac{(|x| +0.5)^2}{2\sigma^2}\right],$$ 
where $i$ is an integer. The book also gives an algorithm for generating a random variable from this distribution. This would apply for your question if $N\to\infty$. 
A: I don't know of any such distribution.  The normal distribution runs from negative infinity to positive infinity, so bounds (i.e., $-N$ to $N$) will weaken the analogy to the normal.  That said, if I wanted a distribution that seemed this way, for a simulation say, I would sample from a Binomial distribution with $n = 2N$ and $\pi=.5$.  Then I would subtract $N$ from all realized values.  If you wanted the mean to be anything else, you could use a different value of $\pi$, but you need to recognize that that would weaken the analogy to the normal even further.  The population SD of the generated distribution would necessarily be $.5\sqrt{2N}$.  You could make that wider if you wanted by sampling the $\pi$ parameter from a narrow, symmetrical distribution around .5, with the wider that distribution of $\pi$s, the higher the resulting SD could be.  The distribution then might be a beta binomial.  
