The Dirac Delta Distribution and Fourier Transform Interpretation

The Dirac delta function (denoted $ \delta(x) $) is a powerful mathematical concept used to model idealized point masses, charges, or frequencies in both theoretical and applied analysis. It plays a particularly significant role in Fourier analysis for representing isolated frequency components. Though it is often called a “function,” the Dirac delta is actually a distribution (or generalized function) in the context of measure theory.

1. Definition of the Dirac Delta Function

The Dirac delta function, $ \delta(x) $, is characterized by two main properties:

  • Localization: $ \delta(x) = 0 $ for all $ x \neq 0 $. This property means that the Dirac delta is zero everywhere except at $ x = 0 $.

  • Normalization: The integral of the delta function over all space equals 1:

    $$ \int_{-\infty}^{\infty} \delta(x) \, dx = 1 $$

  • Sifting Property: The Dirac delta extracts the value of a function $ f(x) $ at a specific point $ x_0 $:

    $$ \int_{-\infty}^{\infty} f(x) \delta(x - x_0) \, dx = f(x_0) $$

The delta function is often thought of as an “infinitely tall, infinitely narrow” spike at a particular point, but it is more rigorously defined as a distribution that acts on test functions, returning the value of the function at a given point.

2. Dirac Delta in Fourier Transforms

In Fourier analysis, the Dirac delta function is crucial for representing pure frequency components. The Fourier transform of a function $ f(x) $ is given by:

$$ \hat{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx $$

When the Fourier transform $ \hat{f}(k) $ includes terms like $ \delta(k - k_0) $, this indicates the presence of a pure frequency component at $ k_0 $.

Key Interpretation:

  • The delta function $ \delta(k - k_0) $ corresponds to a pure frequency oscillation at $ k_0 $.
  • The magnitude of any coefficient associated with $ \delta(k - k_0) $ represents the amplitude of the frequency component.
  • The phase of the frequency component is contained in the complex exponential associated with the delta function.

3. Amplitude and Phase in Fourier Transforms

A general Fourier transform result may include terms of the form:

$$ re^{i\theta} \cdot \delta(k - k_0) $$

Where:

  • $ r $ is the amplitude of the frequency component at $ k_0 $.
  • $ \theta $ is the phase shift in radians.
  • $ \delta(k - k_0) $ ensures that this component corresponds to the frequency $ k_0 $.

Using Euler’s formula $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $, we see that $ re^{i\theta} $ combines both the real (cosine) and imaginary (sine) components of the frequency at $ k_0 $.

Thus, the term $ re^{i\theta} \delta(k - k_0) $ tells us that:

  • The signal contains a frequency $ k_0 $ with amplitude $ r $ and phase shift $ \theta $.

4. Example: Fourier Transform of $ \sin(x) $

Let’s consider an example: the Fourier transform of $ \sin(x) $.

  1. Rewrite $ \sin(x) $ using Euler’s formula:

    $$ \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} $$

  2. Take the Fourier transform of each exponential term:

    The Fourier transform of $ e^{ix} $ is $ 2\pi \delta(k - 1) $, and the Fourier transform of $ e^{-ix} $ is $ 2\pi \delta(k + 1) $.

  3. Combine the results:

    $$ \hat{f}(k) = \frac{1}{2i} \cdot 2\pi \delta(k - 1) - \frac{1}{2i} \cdot 2\pi \delta(k + 1) $$

    Simplifying, we get:

    $$ \hat{f}(k) = \pi i (\delta(k + 1) - \delta(k - 1)) $$

This result tells us:

  • There are two frequency components: one at $ k = 1 $ and one at $ k = -1 $.
  • The coefficients $ \pi i $ indicate the relative phase and amplitude of these components.

Interpretation:

  • The delta functions $ \delta(k - 1) $ and $ \delta(k + 1) $ show that $ \sin(x) $ has pure frequencies at $ k = 1 $ and $ k = -1 $, corresponding to positive and negative frequencies, respectively.
  • The presence of $ i $ indicates a phase shift, as expected for a sine wave (which is 90 degrees out of phase with a cosine wave).

5. Numerical Fourier Transforms

In numerical computations (such as using the Fast Fourier Transform, or FFT), ideal delta functions $ \delta(k - k_0) $ do not appear explicitly, but instead, sharp peaks in the frequency spectrum approximate the behavior of delta functions.

For example, if you perform an FFT on a sampled $ \sin(x) $, you will observe peaks at $ k = \pm 1 $, corresponding to the positive and negative frequencies. The height of these peaks represents the amplitude, and the phase of the signal at these frequencies is encoded in the complex values associated with the peaks.

  • Amplitude: The magnitude of the complex value at each frequency gives the amplitude of that frequency component.
  • Phase: The argument (angle) of the complex value gives the phase shift of that frequency component.

6. Measure Theory Perspective on the Dirac Delta

In the context of measure theory, the Dirac delta is best understood as a Radon measure, which assigns values to subsets of a space. More formally, it is the Dirac measure $ \delta_{x_0} $, which is defined as:

$$ \delta_{x_0}(A) = \begin{cases} 1 & \text{if } x_0 \in A \\ 0 & \text{if } x_0 \notin A \end{cases} $$

This means that the Dirac measure concentrates all of its “mass” at the point $ x_0 $, similar to how the Dirac delta function acts in integrals to “pick out” the value of a function at a single point. In this sense, the Dirac delta is not a traditional function but rather a distribution that acts on test functions in a well-defined way.

Formal Definition Using Distributions

From the perspective of distributions, $ \delta(x - x_0) $ is defined by its action on smooth test functions $ \phi(x) $:

$$ \langle \delta(x - x_0), \phi(x) \rangle = \phi(x_0) $$

This formalizes the idea that the Dirac delta evaluates functions at specific points.

7. Summary

  • Dirac Delta Function: $ \delta(x - x_0) $ is a distribution that represents an idealized point mass or frequency at $ x_0 $. In Fourier analysis, it represents pure frequency components.

  • Amplitude and Phase: In Fourier transforms, complex coefficients of the form $ re^{i\theta} \cdot \delta(k - k_0) $ tell us that the signal contains a frequency $ k_0 $ with amplitude $ r $ and phase $ \theta $.

  • Numerical Interpretation: In practical Fourier transforms, sharp peaks in the spectrum approximate delta functions, and the magnitude and phase of complex values give the amplitude and phase of the frequency components.

  • Measure Theory: From a measure-theoretic perspective, the Dirac delta is a distribution that acts on test functions, picking out the value of the function at specific points, analogous to a point measure.

This framework is fundamental for understanding Fourier analysis in both continuous and discrete settings, as well as for interpreting the results of Fourier transforms in practical applications.