Definition of the Fourier Transform

Introduction

The Fourier series is a tool for periodic functions, yet in practice many signals are non-periodic. The Fourier transform extends the Fourier series to non-periodic functions, yielding a continuous frequency spectrum.

Derivation from the Fourier Series

The complex Fourier series of a function of period $2L$:

$$f(x) = \displaystyle\sum_{n=-\infty}^{\infty}c_n e^{in\pi x/L}, \quad c_n = \dfrac{1}{2L}\displaystyle\int_{-L}^{L}f(x)e^{-in\pi x/L}\,dx$$

Setting $\omega_n = \dfrac{n\pi}{L}$ (the discrete frequencies) and $\Delta\omega = \dfrac{\pi}{L}$:

$$f(x) = \displaystyle\sum_{n=-\infty}^{\infty}\dfrac{\Delta\omega}{2\pi}\left[\displaystyle\int_{-L}^{L}f(t)e^{-i\omega_n t}\,dt\right]e^{i\omega_n x}$$

As $L \to \infty$, we have $\Delta\omega \to d\omega$ and the sum passes into an integral:

$$f(x) = \dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty}\left[\displaystyle\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt\right]e^{i\omega x}\,d\omega$$

Definition of the Fourier Transform

On Notation

• $\hat{f}(\omega)$ or $F(\omega)$: the Fourier transform of $f(x)$
• $\omega$: the angular frequency ($\omega = 2\pi\nu$, where $\nu$ is the ordinary frequency)

Existence Conditions

Sufficient conditions for the Fourier transform to exist:

Absolute integrability

$$\displaystyle\int_{-\infty}^{\infty}|f(x)|\,dx < \infty$$

A function satisfying this condition is said to belong to $L^1(\mathbb{R})$.

Square integrability

$$\displaystyle\int_{-\infty}^{\infty}|f(x)|^2\,dx < \infty$$

A function satisfying this condition is said to belong to $L^2(\mathbb{R})$. By Plancherel's theorem, the Fourier transform of an $L^2$ function is also well defined.

Continuous Spectrum

For the Fourier series, amplitudes are given at the discrete frequencies $\omega_n = \dfrac{n\pi}{L}$. For the Fourier transform, a spectrum is defined for every continuous frequency $\omega$.

Power spectrum

$$S(\omega) = |\hat{f}(\omega)|^2$$

This represents the energy density of each frequency component.

Phase spectrum

$$\phi(\omega) = \arg(\hat{f}(\omega))$$

This represents the phase of each frequency component.

Worked Examples

Example 1: One-sided exponential decay

$$f(x) = \begin{cases} e^{-ax} & (x \geq 0, a > 0) \\ 0 & (x < 0) \end{cases}$$

$$\hat{f}(\omega) = \displaystyle\int_0^{\infty}e^{-ax}e^{-i\omega x}\,dx = \displaystyle\int_0^{\infty}e^{-(a+i\omega)x}\,dx = \dfrac{1}{a + i\omega}$$

Example 2: Two-sided exponential decay

$$f(x) = e^{-a|x|} \quad (a > 0)$$

$$\hat{f}(\omega) = \displaystyle\int_{-\infty}^{\infty}e^{-a|x|}e^{-i\omega x}\,dx = \displaystyle\int_{-\infty}^{0}e^{ax}e^{-i\omega x}\,dx + \displaystyle\int_0^{\infty}e^{-ax}e^{-i\omega x}\,dx$$

$$= \dfrac{1}{a - i\omega} + \dfrac{1}{a + i\omega} = \dfrac{2a}{a^2 + \omega^2}$$

Conditions for Inversion

For the inverse Fourier transform to recover the original function:

• At a point where $f(x)$ is continuous, it recovers $f(x)$.
• At a discontinuity, it recovers $\dfrac{f(x^+) + f(x^-)}{2}$.

This is the transform analogue of the convergence theorems for the Fourier series.

Summary

• Fourier transform: $\hat{f}(\omega) = \displaystyle\int_{-\infty}^{\infty}f(x)e^{-i\omega x}\,dx$
• Inverse Fourier transform: $f(x) = \dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty}\hat{f}(\omega)e^{i\omega x}\,d\omega$
• It gives a continuous spectrum for non-periodic functions.
• It applies to absolutely integrable or square-integrable functions.

Frequently Asked Questions