Demystify Dirac delta function for data representation on discrete space

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Dirac delta function is an important tool in Fourier Analysis. It is used specially in electrodynamics and signal processing routinely.  A function over set of data points
is often shown with a delta function representation. A novice reader relying on integral properties of the delta function may found this notation quite confusing.  Probably, the notation itself is an example of  abuse of notation.

One dimensional function/distribution: Sum of delta functions

Let’s define a one dimensional function, $f(x)$ as follows, $x \in \mathbb{R}$ and $a$ being constant:

$ f(x) = a \sum_{i=-n}^{n} \delta(x – x_{i})$

This representation is inspired from Dirac comb and used in spike trains.  Note that set of data points in one dimension $\{x_{i} \}$ will determine the graph of this function. Using the shifting property of delta function, the value of the function will be zero every where except on data points. The constant $a$ will simply be the height of the graph at the data point.

Figure: A spike train.

Numeric Example

Let’s plot $f(x)$ for some specific values of the set $\{x_{i} \} = {-0.5, -0.2, -0.1, 0.2, 0.4}$ and $a=0.5$. Here is the R code for plotting this spike train. 

x_i = c(-0.5, -0.2, -0.1, 0.2, 0.4)
a   = c(0.5, 0.5, 0.5, 0.5, 0.5)

Representing Histograms: One dimensional example

Particularly convenient representation of histograms can be developed similarly. Consider set of points $\{x_{i}\}_{i=1}^{n}$ where we would like to establish a histogram out of this set, let’s say $h(x)$. If we set our histogram intervals as $\{x_{j}\}_{j=1}^{m}$. The histogram $h(x)$ then can be written as

$h(x_{j}) =  \sum_{i=1}^{n} \sum_{j=1}^{m} \delta(x_{j}- x_{i}^{min})$
where set $x_{i}^{min}$  represents the the the value from set $\{x_{j}\}_{j=1}^{m}$ that is closest to given $x_{i}$. Where as, second sum determines the height at a given point, i.e., frequency. This is just a confusing mathematical representation and practical implementation only counts the frequency of $x_{i}^{min}$ directly.


However it is quite trivial, the above usage of sum of delta functions appear in mathematical physics as well, not limited to statistics.

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