What is Fourier Analysis?
Fourier analysis is the study of representing functions and signals as combinations of sinusoids (sines and cosines) or complex exponentials. By moving between the time/space domain and the frequency domain, it becomes easier to analyze, filter, compress, and solve problems involving oscillations, waves, and diffusion.
Core Ideas
- Decomposition: Break a function into basic building blocks (sinusoids).
- Dual domains: The same object can be described in time/space or in frequency.
- Linearity: Superposition makes linear operations especially simple in the frequency domain.
- Energy and orthogonality: Sinusoids form orthogonal bases; energy conservation appears as Parseval/Plancherel identities.
Fourier Series (Periodic Signals)
For a T-periodic function f, the Fourier series represents f as a sum of harmonics at integer multiples of the fundamental frequency 1/T:
f(x) ≈ a0/2 + Σ (a_n cos(2π n x / T) + b_n sin(2π n x / T)) for n = 1, 2, ...
Equivalently (complex form), f(x) ≈ Σ c_k e^{i 2π k x / T} over all integers k, with coefficients c_k determined by inner products with e^{i 2π k x / T}.
- Coefficients capture how much of each frequency is present.
- Convergence depends on smoothness; discontinuities lead to Gibbs phenomenon (overshoot near jumps).
- Even/odd symmetry simplifies coefficients (cosine-only or sine-only series).
Fourier Transform (Aperiodic Signals)
The Fourier transform extends Fourier series to nonperiodic functions, replacing sums with integrals. Given f(t), its transform F(ω) measures the amplitude/phase of frequency ω. The inverse transform recovers f from F. Different fields use slightly different conventions for scaling factors.
- Time shifts ↔ phase factors.
- Scaling in time ↔ inverse scaling in frequency.
- Differentiation in time ↔ multiplication by iω in frequency.
- Convolution in time ↔ multiplication in frequency (convolution theorem).
Discrete-Time, DFT, and FFT
Digital signals are sampled and finite-length. The Discrete-Time Fourier Transform (DTFT) describes infinite sequences; the Discrete Fourier Transform (DFT) maps N samples to N discrete frequencies. The Fast Fourier Transform (FFT) is an efficient algorithm to compute the DFT in O(N log N) time.
- Windowing finite data reduces spectral leakage but widens main lobes.
- Zero-padding interpolates the spectrum (increases display resolution) but does not add new frequency content.
- Sampling and the Nyquist rate: To avoid aliasing, sample at least twice the highest frequency present.
Energy, Norms, and Orthogonality
In L2 spaces, sinusoids form an orthonormal basis (for periodic domains) or an integral basis (for transforms). Plancherel/Parseval identities state that total energy is preserved between domains.
The Uncertainty Principle
A function cannot be simultaneously localized in both time and frequency. The product of spreads has a lower bound. Gaussians minimize this bound and are their own Fourier transforms (up to scaling).
Solving Differential Equations
Fourier methods turn differential operators into algebraic multipliers. For example, the heat equation and the wave equation can be solved by transforming, solving in frequency space, and inverting the transform or by using Fourier series with boundary conditions.
Convolution Theorem and Filtering
Convolution in time corresponds to multiplication in frequency. This underlies filter design: specify a desired frequency response, then realize it as a time-domain filter. In images, this explains blurring (low-pass), edge detection (high-pass), and sharpening.
Common Pitfalls
- Aliasing: Undersampling causes high frequencies to masquerade as low frequencies.
- Spectral leakage: Finite windows smear energy across frequencies.
- Gibbs phenomenon: Persistent overshoot near discontinuities does not vanish with more terms, though it narrows.
- Units and conventions: Be careful with 2π factors and normalization constants.
Typical Workflow
- Model or sample your signal.
- Choose an appropriate transform (series, continuous FT, DFT).
- Apply windowing or pre-processing if needed.
- Compute spectrum (analytically or via FFT).
- Analyze, filter, or solve in frequency domain.
- Transform back and interpret results.
Applications
- Signal and image processing: denoising, compression, feature extraction.
- Communications: modulation, equalization, channel estimation.
- Scientific computing: solving PDEs, spectral methods.
- Statistics and time-series: spectral density, periodograms.
- Physics and engineering: optics, acoustics, quantum mechanics.
Quick Examples
- Square wave: only odd harmonics with amplitudes decreasing like 1/n.
- Gaussian: transforms to a Gaussian; narrower in time means wider in frequency.
- Heat equation on a rod: expand initial temperature in a sine series; each mode decays exponentially over time.