Properties of the Fourier Transform
Fundamental Rules of the Transform
Intermediate (undergraduate, years 2–3)
Introduction
The Fourier transform has many useful properties. Once understood, complicated transforms can be derived from known ones, and the properties can be applied to solving differential equations.
Linearity
$$\mathcal{F}[\alpha f + \beta g] = \alpha\mathcal{F}[f] + \beta\mathcal{F}[g]$$
The Fourier transform is a linear operator. This follows directly from the linearity of the integral.
Shift Theorems
Time shift
$$\mathcal{F}[f(x - a)] = e^{-i\omega a}\hat{f}(\omega)$$
A shift in the time domain corresponds to a phase rotation in the frequency domain. The amplitude spectrum $|\hat{f}(\omega)|$ is unchanged.
Frequency shift
$$\mathcal{F}[e^{i\omega_0 x}f(x)] = \hat{f}(\omega - \omega_0)$$
Modulation in the time domain (multiplication by a complex exponential) corresponds to a shift in the frequency domain.
Scaling
$$\mathcal{F}[f(ax)] = \dfrac{1}{|a|}\hat{f}\left(\dfrac{\omega}{a}\right) \quad (a \neq 0)$$
Compression in the time domain ($|a| > 1$) corresponds to expansion in the frequency domain, and vice versa.
Uncertainty principle
Making a signal shorter in time broadens its frequency band. A signal cannot be localized to a narrow range in both time and frequency simultaneously. This shares the same mathematical structure as the uncertainty principle of quantum mechanics.
Relation to Differentiation
Time derivative
$$\mathcal{F}[f'(x)] = i\omega\hat{f}(\omega)$$
$$\mathcal{F}[f^{(n)}(x)] = (i\omega)^n\hat{f}(\omega)$$
Differentiation corresponds to multiplication by $i\omega$. This turns a differential equation into an algebraic one.
Frequency derivative
$$\mathcal{F}[xf(x)] = i\dfrac{d\hat{f}}{d\omega}$$
$$\mathcal{F}[x^n f(x)] = i^n\dfrac{d^n\hat{f}}{d\omega^n}$$
Relation to Integration
$$\mathcal{F}\left[\displaystyle\int_{-\infty}^{x}f(t)\,dt\right] = \dfrac{\hat{f}(\omega)}{i\omega} + \pi\hat{f}(0)\delta(\omega)$$
When $\hat{f}(0) = 0$ (the integral of $f$ vanishes), this simplifies to:
$$\mathcal{F}\left[\displaystyle\int_{-\infty}^{x}f(t)\,dt\right] = \dfrac{\hat{f}(\omega)}{i\omega}$$
Complex Conjugate and Reflection
Complex conjugate
$$\mathcal{F}[\overline{f(x)}] = \overline{\hat{f}(-\omega)}$$
In particular, when $f(x)$ is real-valued:
$$\hat{f}(-\omega) = \overline{\hat{f}(\omega)}$$
This is called Hermitian symmetry. The Fourier transform of a real function is completely determined by its positive-frequency part.
Time reversal
$$\mathcal{F}[f(-x)] = \hat{f}(-\omega)$$
Parseval and Plancherel Theorems
Plancherel's theorem
$$\displaystyle\int_{-\infty}^{\infty}|f(x)|^2\,dx = \dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty}|\hat{f}(\omega)|^2\,d\omega$$
Energy is conserved between the time domain and the frequency domain.
Parseval's identity (general form)
$$\displaystyle\int_{-\infty}^{\infty}f(x)\overline{g(x)}\,dx = \dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty}\hat{f}(\omega)\overline{\hat{g}(\omega)}\,d\omega$$
Summary of Properties
| Time domain | Frequency domain |
|---|---|
| $f(x - a)$ | $e^{-i\omega a}\hat{f}(\omega)$ |
| $e^{i\omega_0 x}f(x)$ | $\hat{f}(\omega - \omega_0)$ |
| $f(ax)$ | $\dfrac{1}{|a|}\hat{f}(\omega/a)$ |
| $f'(x)$ | $i\omega\hat{f}(\omega)$ |
| $xf(x)$ | $i\dfrac{d\hat{f}}{d\omega}$ |
| $\overline{f(x)}$ | $\overline{\hat{f}(-\omega)}$ |
| $f(-x)$ | $\hat{f}(-\omega)$ |
Summary
- The Fourier transform is a linear operator.
- Time shift ↔ phase rotation; frequency shift ↔ modulation.
- Time compression ↔ frequency expansion (uncertainty principle).
- Differentiation ↔ multiplication by $i\omega$.
- The Fourier transform of a real function is Hermitian symmetric.
- Plancherel's theorem: energy conservation.
Frequently Asked Questions
Q1: What is the linearity of the Fourier transform?
A: The Fourier transform is a linear transform: $\mathcal{F}[af + bg] = a\mathcal{F}[f] + b\mathcal{F}[g]$. This lets the Fourier transform of a complex signal be computed as the sum of the Fourier transforms of its simple components.
Q2: How do a time shift and frequency modulation correspond?
A: A time shift gives $\mathcal{F}[f(t - t_0)] = e^{-i\omega t_0} F(\omega)$, changing only the phase while leaving the amplitude unchanged. Frequency modulation gives $\mathcal{F}[e^{i\omega_0 t} f(t)] = F(\omega - \omega_0)$, shifting the spectrum by $\omega_0$.
Q3: What is Parseval's identity?
A: Parseval's identity $\int_{-\infty}^{\infty} |f(t)|^2\,dt = \dfrac{1}{2\pi}\int_{-\infty}^{\infty} |F(\omega)|^2\,d\omega$ states that the energy in the time domain equals the energy in the frequency domain (energy conservation).