# Dirac delta function fourier transform

Fourier Transform Mathematics. Previous: Fourier Transform Home. Previous: Intro to Complex Math. The Dirac-Delta function, also commonly known as the impulse function, is described on this page.

This function technically a functional is one of the most useful in all of applied mathematics. To understand this function, we will several alternative definitions of the impulse function, in varying degrees of rigor. The larger n gets, the narrower the pulse is in time, but the amplitude increases such that the total area is the same for all values of n.

The unit step function. Dirac-Delta: The Sifting Functional Probably the most useful property of the dirac-delta, and the most rigorous mathematical defintion is given in this section. This property is extremely useful in signal processing, communication systems theory, quantum physics, etc. This is known as the 'sifting property' of the impulse function. Finally, as a further note on notation, the impulse shifted to the right by 1, given in equation [8] is plotted as shown in Figure 3.

The graph of the Dirac-Delta Impulse Function.In the following we present some important properties of Fourier transforms. These results will be helpful in deriving Fourier and inverse Fourier transform of different functions.

The following results are particularly useful when applying Fourier transforms to differential equations as seen later this term and next year in the context of partial differential equations. Proof We prove a and c and leave the others as exercises. The convolution theorem suggests that convolution is commutative. This can also be shown easily from the definition by using a change of variable in the integration. A similar convolution theorem holds for the inverse functions. Geometrically this means that the area under the curve is equivalent to that of a rectangle with length equal to the interval of integration.

The key property however, is that its integral area under the curve is one. Figure 2. Sifting property of the delta function The delta function is most useful in how it interacts with other functions. Using our definition of the delta-function we can rewrite this as.

We finish by recommending this video on a very intuitive visual introduction to Fourier transform from the popular 3Blue1Brown YouTube channel in mathematics education. Do check it out and also the additional videos on related topics such as uncertainty principle.

Calculus and Applications - Part II. Chapter 2 Properties of Fourier Transforms In the following we present some important properties of Fourier transforms. Example 2. Theorem 2.Let's continue our study of the following periodic force, which resembles a repeated impulse force:.

It's straightforward to find the Fourier series for this force, but we don't have to because Taylor already worked it out as his example 5.

This is straightforward to do in the Fourier coefficients themselves: we have. But now we run into a problem when we try to actually use the Fourier series, which takes the approximate form. So our usual approach of truncation won't be useful at all! This suggests that we try to replace the sum with an integral. Let's define the quantity. Let's take this and rewrite the Fourier sum slightly:.

Once again, just like the Fourier series, this is a representation of the function. There is also no restriction about periodicity - we can use the Fourier transform for any function at all, periodic or not.

One note: there are several equivalent but slightly different-looking ways to define the Fourier transform! More importantly, the cosine version that we're using is actually not as commonly used, in part because it can only work on functions that are even; in general we need both sine and cosine.

A more common definition of the Fourier transform is in terms of complex exponentials:. This is because both representations are functions here, instead of trying to match a function onto a sum.

To understand inverting the Fourier transform, we can take these formulas and plug them in to each other. Let's use the complex exponential version:.

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Now, I can do some rearranging of these integrals:. This is sort of familiar, actually; it resembles the orthogonality relation we had for the cosines and sines in the Fourier series, where the integral would be zero except for the special case where the two cosines match in frequency. In other words, if we define. Once again, this isn't technically a function, so it's a little dangerous to try to interpret it unless it's safely inside of an integral.

If you're worried about mathematical rigor, it is possible to define the delta function as a limit of a regular function, for example a Gaussian curve, as its width goes to zero. From the plot, we can visually observe that it's important that the delta function spike is located within the integration limits, or else we won't pick it up and will just end up with zero. In other words.

There are some other useful properties of the delta function that we could derive you can look them up in a number of math references, including on Wikipedia or Mathworld. Let's prove a simple one, to get used to delta-function manipulations a bit.

### Dirac delta function

As always, to make sense of this we really need to put it under an integral:. So we have the full result. One more comment on the delta function, bringing us back towards physics. We use the delta function often to represent "point particles", which we imagine as being concentrated at a single point in space - not totally realistic, but often a useful approximation.

In fact, one example where this is realistic is if we want to describe the charge density of a fundamental charge like an electron. The one-dimensional charge density describing this would be. This might seem unphysical, but it's the only consistent way to define density for an object that has finite charge but no finite size. If we ask sensible questions like "what is the total charge", we get good answers back:.

This makes sense, because of the identity. Let's go back to our non-periodic driving force example, the impulse force, and apply the Fourier transform to it.It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. In engineering and signal processingthe delta function, also known as the unit impulse symbol, [6] may be regarded through its Laplace transformas coming from the boundary values of a complex analytic function of a complex variable.

The formal rules obeyed by this function are part of the operational calculusa standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit a weak limit of a sequence of functions having a tall spike at the origin in theory of distributions, this is a true limit. The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.

The graph of the delta function is usually thought of as following the whole x -axis and the positive y -axis. For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a delta function.

In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the ball by only considering the total impulse of the collision without a detailed model of all of the elastic energy transfer at subatomic levels for instance.

To be specific, suppose that a billiard ball is at rest. The exchange of momentum is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous. That is. The delta function allows us to construct an idealized limit of these approximations.

To make proper sense of the delta function, we should instead insist that the property. In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit a weak limit of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.

Despite its name, the delta function is not truly a function, at least not a usual one with range in real numbers.

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A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions. Later, Augustin Cauchy expressed the theorem using exponentials: [10] [11]. Cauchy pointed out that in some circumstances the order of integration in this result is significant contrast Fubini's theorem. A rigorous interpretation of the exponential form and the various limitations upon the function f necessary for its application extended over several centuries.

The problems with a classical interpretation are explained as follows: [14]. Further developments included generalization of the Fourier integral, "beginning with Plancherel's pathbreaking L 2 -theorycontinuing with Wiener's and Bochner's works around and culminating with the amalgamation into L. Schwartz's theory of distributions An infinitesimal formula for an infinitely tall, unit impulse delta function infinitesimal version of Cauchy distribution explicitly appears in an text of Augustin Louis Cauchy.

Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussianswhich also corresponded to Lord Kelvin 's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse.

The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite. This is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.

Formally, the Lebesgue integral provides the necessary analytic device. Consequently, the delta measure has no Radon—Nikodym derivative with respect to Lebesgue measure — no true function for which the property. As a probability measure on Rthe delta measure is characterized by its cumulative distribution functionwhich is the unit step function [22].

Thus in particular the integral of the delta function against a continuous function can be properly understood as a Riemann—Stieltjes integral : [23]. In particular, characteristic function and moment generating function are both equal to one.

In the theory of distributionsa generalized function is considered not a function in itself but only in relation to how it affects other functions when "integrated" against them.We begin with a brief review of Fourier series.

## 2.1: Fourier Series and Integrals, the Dirac Function

Any periodic function of interest in physics can be expressed as a series in sines and cosines—we have already seen that the quantum wave function of a particle in a box is precisely of this form. In particular, it turns out that step discontinuities are never handled perfectly, no matter how many terms are included.

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Just as in matrix algebra the eigenstates of the unit matrix are a set of vectors that span the space, and the unit matrix elements determine the set of dot products of these basis vectors, the delta function determines the generalized inner product of a continuum basis of states. It plays an essential role in the standard formalism for continuum states, and you need to be familiar with it!

The number of terms of the series necessary to give a good approximation to a function depends on how rapidly the function changes. As we include more and more terms, the function becomes smoother but, surprisingly, the initial overshoot at the step stays at a finite fraction of the step height. It is also clear why convoluting this curve with a step function gives an overshoot and oscillations.

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That gives the overshoot. Therefore, any reasonably smooth initial wavefunction describing the electron can be represented as a Fourier series. The time development can then be found be multiplying each term in the series by the appropriate time-dependent phase factor. In the previous lecture, we discussed briefly how a Gaussian wave packet in x -space could be represented as a continuous linear superposition of plane waves that turned out to be another Gaussian wave packet, this time in k -space.

The first step is a trivial one: we need to generalize from real functions to complex functions, to include wave functions having nonvanishing current.

Therefore, we must take the limit N going to infinity before taking L going to infinity. This is not what we want. This is the Dirac delta function. This hand-waving approach has given a result which is not clearly defined.

This integral over x is linearly divergent at the origin, and has finite oscillatory behavior everywhere else. To make any progress, we must provide some form of cutoff in k -space, then perhaps we can find a meaningful limit by placing the cutoff further and further away.

That is to say. This is still a rather pathological function, in that it is oscillating more and more quickly as the infinite limit is taken. It is straightforward to verify the following properties from the definition as a limit of a Gaussian wavepacket:. There is no unique way to define the delta function, and other cutoff procedures can give useful insights. This representation of the delta function will prove to be useful later. Exponential Fourier Series In the previous lecture, we discussed briefly how a Gaussian wave packet in x -space could be represented as a continuous linear superposition of plane waves that turned out to be another Gaussian wave packet, this time in k -space.

This is what we do in the rest of this section. Yet Another Definition, and a Connection with the Principal Value Integral There is no unique way to define the delta function, and other cutoff procedures can give useful insights. Exercises 1.Thank you for sharing that. Where are you located. My dentist and doctors just wear regular shoes. The Clinic is huge. More than 200 patients are treated per month and they must wear those shoes in the clinic.

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fourier transform 01 : FT of delta function

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### Fourier transforms and the delta function

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