Can a Gaussian distribution be represented as a sum of Dirac Deltas?

In summary, the Dirac Delta function is not a function in the traditional sense, but can be represented as a limiting case of the Gaussian distribution. It is also possible to construct a Gaussian spectrum using a weighted sum of Dirac Delta functions, where the weights determine the distribution. However, this can only be done for certain functions and can be approximated by using small x increments.
  • #1
106
26
We know that Dirac Delta is not a function. However, I just talk about the numerical version of it that we use every day. We can simply represent the Dirac delta function as a limiting case of Gaussian distribution when the width of the distribution ##\sigma->0##.

$$
\delta(x - \mu) = lim_{\sigma -> 0} \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x - \mu)^2}{2\sigma^2}}
$$

Is it possible to also say the reverse with a weighted sum of Dirac Deltas to construct a Gaussian spectrum?

$$
\frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x - mu)^2}{2\sigma^2}} = \sum_{i} w_i \delta(x - i)
$$

Where, somehow the weights ##w_i## constitute how it is distributed (##\sigma##). If yes, how do we decide these weights?
 
Mathematics news on Phys.org
  • #2
Any function can be represented as a sum of Dirac delta functions:

Let ##f(x)## be an arbitrary function of ##x##. Then you can represent it as:

##\int f(y) \delta(x-y) dy##

So that's a weighted sum (well, integral) of delta functions.
 
  • Like
Likes BvU
  • #3
stevendaryl said:
Any function can be represented as a sum of Dirac delta functions:

Let ##f(x)## be an arbitrary function of ##x##. Then you can represent it as:

##\int f(y) \delta(x-y) dy##

So that's a weighted sum (well, integral) of delta functions.

If you really want a discrete sum, instead of an integral, then it can't be done for most functions. But I guess for some purposes, you can approximate a function by delta functions: Pick a small positive x increment ##\Delta x## and define ##\tilde{f}(x, \Delta x)## by:

##\tilde{f}(x, \Delta x) = \sum_j f(j \Delta x) \delta(x- j\Delta x) \Delta x##

where ##\Delta x## is some small real number. This approximation works in an integration sense: For any other smooth function ##g(x)##, we have:

##lim_{\Delta x \Rightarrow 0} \int \tilde{f}(x, \Delta x) g(x) dx = \int f(x) g(x) dx##
 
  • Like
Likes tworitdash

1. Can a Gaussian distribution be represented as a sum of Dirac Deltas?

Yes, a Gaussian distribution can be represented as a sum of Dirac Deltas. This is known as the Dirac comb or the sampling function.

2. What is a Gaussian distribution?

A Gaussian distribution, also known as a normal distribution, is a type of probability distribution that follows a bell-shaped curve. It is commonly used in statistics to describe many natural phenomena.

3. What are Dirac Deltas?

Dirac Deltas, also known as the delta function, is a mathematical concept that represents an infinitely narrow spike or impulse at a specific point. It is often used in physics and engineering to model point sources or idealized systems.

4. How is a Gaussian distribution represented as a sum of Dirac Deltas?

A Gaussian distribution can be represented as a sum of Dirac Deltas by taking the limit of an infinite number of Dirac Deltas, each with a smaller magnitude and located closer to the mean of the Gaussian distribution. This creates a continuous curve that approximates the Gaussian distribution.

5. What are the applications of representing a Gaussian distribution as a sum of Dirac Deltas?

This representation has many applications in mathematics and physics, including signal processing, Fourier analysis, and quantum mechanics. It allows for easier calculations and simplifies the analysis of complex systems.

Suggested for: Can a Gaussian distribution be represented as a sum of Dirac Deltas?

Replies
1
Views
703
Replies
12
Views
1K
Replies
2
Views
1K
Replies
6
Views
722
Replies
5
Views
882
Back
Top