Update 20240918: I have found an absolutely amazing YouTube video on the linear algebraic perspective of this topic: The Linear Algebra behind sound

I also discovered this video: The math behind music | Linear Algebra episode 3 in a playlist on linear Algebra. I’ll have to watch the playlist for sure.

Fourier Transforms: Two Perspectives

Fourier transforms are essential in analyzing and understanding signals, particularly in terms of how they can be broken down into constituent frequencies. This analysis is crucial in fields ranging from engineering to quantum physics, and the transform is named after the French mathematician Jean-Baptiste Joseph Fourier, who introduced the concept in the early 19th century.

Motivation via Problem Statement & Historical Context

  • Problem Statement: Given a complex signal (e.g., a sound wave or image), how can we analyze its frequency content? More specifically, how can we represent it as a sum of simple sinusoidal waves (sine and cosine functions)?
  • Historical Context: Joseph Fourier, in the early 1800s, worked on the problem of heat conduction. He proposed that any periodic function could be written as a sum of sine and cosine waves, leading to the development of what we now call Fourier series. This idea was later generalized to non-periodic functions, leading to the Fourier transform.

Real-World Application: Audio Signal Analysis

  • In audio processing, sound waves can be analyzed by converting them into the frequency domain using Fourier transforms. This allows sound engineers to isolate specific frequencies (e.g., removing background noise or enhancing bass).
  • The integral formulation is crucial for analyzing continuous signals, allowing us to directly compute the frequency content of any function over time.

Intuitive Definition and Practical Interpretation

  • The Fourier Transform decomposes a function (or signal) into a set of basic building blocks: sine and cosine waves of varying frequencies and amplitudes. It answers the question: “What frequencies are present in my signal?”
  • Why it matters: Many signals, such as sounds or electrical signals, are complex. Fourier transforms allow us to understand these signals by converting them into simpler, more manageable components—frequencies.

Analogy

  • Think of the Fourier transform as a prism splitting white light into its constituent colors. Just as a prism reveals the spectrum of frequencies in light, the Fourier transform reveals the spectrum of frequencies in a function or signal.

1. Concrete/Informal Approach

Here, we’ll define the Fourier transform as an improper integral and explore its intuitive and practical significance.

Fourier Transform as an Improper Integral

  • The Fourier Transform converts a signal (a function of time) into a function of frequency. It tells us the amount of each frequency present in the signal.
  • The concrete approach defines the Fourier transform as an integral that accumulates the contributions of all frequencies present in the signal.

For a function $ f(t) $, its Fourier transform $ F(w) $ is given by:

$$ F(w) = \int_{-\infty}^{\infty} f(t) e^{-iwt} \, dt $$
  • This formula represents a weighted sum (in the sense of an integral) of the signal $ f(t) $, weighted by complex exponential functions $ e^{-iwt} $ (which are oscillating functions in the form of sine and cosine waves).
  • Each frequency $ w $ corresponds to a sinusoidal wave of that frequency, and the integral captures how much of that frequency is present in $ f(t) $.

Step-by-Step Example: Simple Sine Wave

Let’s compute the Fourier transform of a simple sine wave $ f(t) = \sin(\omega_0 t) $:

  1. First, express the sine function using Euler’s formula: $$ \sin(\omega_0 t) = \frac{e^{i\omega_0 t} - e^{-i\omega_0 t}}{2i} $$
  2. Apply the Fourier transform integral: $$ F(w) = \int_{-\infty}^{\infty} \sin(\omega_0 t) e^{-iwt} \, dt $$
  3. Substitute $ \sin(\omega_0 t) $: $$ F(w) = \frac{1}{2i} \int_{-\infty}^{\infty} \left(e^{i\omega_0 t} - e^{-i\omega_0 t}\right) e^{-iwt} \, dt $$
  4. Solve the integrals: $$ F(w) = \pi \delta(w - \omega_0) - \pi \delta(w + \omega_0) $$

This shows that the sine wave is made up of two frequency components, $ \omega_0 $ and $ -\omega_0 $, as expected for a sine function.

The Dirac delta function $ \delta(x) $ is a special mathematical object used to represent a concentrated impulse at a specific point. It is defined by two key properties:

  1. It is zero everywhere except at $ x = 0 $.
  2. Its integral over all space is 1: $$ \int_{-\infty}^{\infty} \delta(x) \, dx = 1 $$

However, the Dirac delta function cannot be considered a proper function in the traditional sense because, in order to satisfy these properties, the delta function would need to be infinite at $ x = 0 $. In classical analysis, functions are expected to have finite values at every point, but the delta function instead “concentrates” all its value at a single point. This means it is formally treated as a generalized function or distribution, a concept used in advanced mathematical analysis to handle such objects.

The Dirac delta function is primarily defined through its action under an integral. For example, it satisfies the sifting property:

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

This means that $ \delta(x - x_0) $ “picks out” the value of a function $ f(x) $ at the point $ x_0 $, effectively acting like a spike at that location, but we don’t directly evaluate the delta function at any point.

Dirac Delta Function in Fourier Transforms

In the context of Fourier transforms, the delta function appears when the original signal contains pure sinusoidal components. For example, the Fourier transform of a sine wave $ \sin(\omega_0 t) $ results in:

$$ F(w) = \pi \delta(w - \omega_0) - \pi \delta(w + \omega_0) $$

This indicates that the sine wave contains only two specific frequencies: $ \omega_0 $ and $ -\omega_0 $, represented by delta functions. In the frequency domain, these delta functions act as spikes or impulses that show the presence of precise frequencies. The delta function “picks out” the contribution of these frequencies in the signal.

Common Misconceptions and Corrections

  • Misconception: The Fourier transform gives you the frequencies directly.
    • Correction: The Fourier transform gives a function in the frequency domain, which shows how strongly different frequencies contribute to the signal.
  • Misconception: The Fourier transform can be directly applied to any function.
    • Correction: The Fourier transform, as an improper integral, may not converge for all functions. A function needs to satisfy certain conditions (e.g., being integrable or belonging to the space $ L^2(\mathbb{R}) $) for the Fourier transform to exist.
  • Misconception: Fourier transforms only apply to periodic functions.
    • Correction: They can be applied to non-periodic functions as well.
  • Misconception: Fourier transforms always produce a continuous spectrum.
    • Correction: Discrete Fourier transforms exist for discrete data sets.

2. Abstract/Formal Approach

In this approach, the Fourier transform is viewed as a linear functional or operator acting on a space of functions, mapping them from the signal domain (time or space) to the frequency domain.

Formal Mapping of the Fourier Transform

  • For general signals $ f(t) $, the Fourier transform is well-defined for functions in the space $ L^1(\mathbb{R}) $, which is the set of functions for which: $$ \int_{-\infty}^{\infty} |f(t)| dt < \infty $$
  • The Fourier transform of a function $ f \in L^1(\mathbb{R}) $ maps it to a function $ F(w) $ that is continuous and bounded, meaning: $$ F(w) \in C(\mathbb{R}) \cap L^\infty(\mathbb{R}) $$ This means that the Fourier transform results in a function that is both continuous and essentially bounded (has a finite maximum value).

Why This Matters

  • For functions in $ L^1(\mathbb{R}) $, the Fourier transform is well-behaved. The resulting function in the frequency domain is bounded (does not blow up) and continuous (smooth in the frequency variable $ w $).
  • When you move to other function spaces like $ L^2(\mathbb{R}) $, which is the space of square-integrable functions, the Fourier transform still exists but behaves differently. Specifically, in $ L^2(\mathbb{R}) $, the Fourier transform is an isometry (it preserves the norm of the function), but this requires a more abstract discussion involving functional analysis.

So, if we take the more general mapping into account, the Fourier transform is typically defined as:

$$ \mathcal{F}: L^1(\mathbb{R}) \to C(\mathbb{R}) \cap L^\infty(\mathbb{R}) $$
  • $ L^1(\mathbb{R}) $: The space of integrable functions (functions where the absolute value can be integrated over $ \mathbb{R} $).
  • $ C(\mathbb{R}) $: The space of continuous functions.
  • $ L^\infty(\mathbb{R}) $: The space of bounded functions (functions whose values have a finite supremum or maximum).

For functions in $ L^1(\mathbb{R}) $, this mapping ensures that the Fourier transform will produce continuous and bounded functions in the frequency domain.

  • The Fourier transform of a function $ f(t) \in L^1(\mathbb{R}) $ maps to a continuous and bounded function $ F(w) \in C(\mathbb{R}) \cap L^\infty(\mathbb{R}) $.

For square-integrable functions $ f(t) \in L^2(\mathbb{R}) $, the Fourier transform still exists, but it maps between two different function spaces: $ L^2(\mathbb{R}) \to L^2(\mathbb{R}) $. In this case, the Fourier transform preserves the norm, known as Plancherel’s theorem.

  • The Fourier transform can be treated as a linear operator $ \mathcal{F} $ that acts on the space of square-integrable functions $ L^2(\mathbb{R}) $, which are functions where the integral of their square is finite: $$ \int_{-\infty}^{\infty} |f(t)|^2 dt < \infty $$
  • This means that the Fourier transform takes a function from $ L^2(\mathbb{R}) $ (the time domain) to another function in $ L^2(\mathbb{R}) $ (the frequency domain).

Properties of the Fourier Transform as an Operator

  • Linearity: The Fourier transform is a linear operator, meaning that for any functions $ f(t) $ and $ g(t) $ and scalars $ a $, $ b $:

    $$ \mathcal{F}\{a f(t) + b g(t)\} = a \mathcal{F}\{f(t)\} + b \mathcal{F}\{g(t)\} $$

  • Invertibility: The Fourier transform is invertible, meaning that if we know the Fourier transform $ F(w) $, we can recover the original function $ f(t) $ using the inverse Fourier transform:

    $$ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(w) e^{iwt} \, dw $$

  • Plancherel’s Theorem: For $ f(t) \in L^2(\mathbb{R}) $, the Fourier transform preserves the norm of the function:

    $$ \int_{-\infty}^{\infty} |f(t)|^2 dt = \int_{-\infty}^{\infty} |F(w)|^2 dw $$

    This shows that the energy of the signal is conserved when moving between the time and frequency domains.

  • Operator Duality: The Fourier transform transforms differentiation into multiplication by $ iw $:

    $$ \mathcal{F}\left\{ \frac{d}{dt} f(t) \right\}(w) = iw \mathcal{F}\{f(t)\}(w) $$

    This operator property simplifies solving many differential equations by transforming them into algebraic equations in the frequency domain.

Theoretical Significance in Function Spaces

  • Fourier Transform as a Basis Change: The Fourier transform can be viewed as a change of basis in a function space. Just as vectors in $ \mathbb{R}^n $ can be represented in different bases, functions can be represented in the time domain or the frequency domain. The Fourier transform maps between these two representations.
  • Connection to Harmonic Analysis: The Fourier transform is fundamental in harmonic analysis, the study of functions through their frequency components. It reveals how a function can be broken down into harmonics (sine and cosine functions), which is useful in many branches of mathematics and physics.

Summary: Connecting the Two Perspectives

  • Concrete/Improper Integral Perspective: Here, the Fourier transform is seen as an improper integral that decomposes a signal into its frequency components. This approach emphasizes practical computation and understanding in real-world applications, like audio processing and image compression.

  • Abstract/Operator Perspective: In this view, the Fourier transform is a linear operator acting on a function space, mapping functions from the time domain to the frequency domain. This perspective highlights its theoretical significance, including its role in functional analysis, differential equations, and harmonic analysis.