Fourier & Laplace

The Fourier transform converts a function of time into a function of frequency. Any signal, no matter how complex, can be expressed as a sum of pure sine and cosine waves of different frequencies, amplitudes, and phases. This is arguably the most important mathematical tool in signal processing. The transform takes a function f of t and produces F of omega, where omega is the angular frequency. The inverse transform reconstructs the original signal from its frequency components. The key insight is that operations that are complicated in the time domain — like convolution — become simple multiplication in the frequency domain.

A Fourier series represents a periodic function as an infinite sum of sines and cosines. If a function repeats with period T, it can be written as a constant plus a sum of sine and cosine terms at frequencies that are multiples of the fundamental frequency (one over T). The coefficients are computed by integrating the function against the sine and cosine basis functions — this is essentially projecting the function onto the orthogonal set of trigonometric functions. The Gibbs phenomenon is a fascinating artifact: near a jump discontinuity, the Fourier series overshoots by about nine percent, no matter how many terms you include.

The Laplace transform generalizes the Fourier transform by introducing a complex frequency variable s equal to sigma plus i omega. The exponential damping factor e to the minus sigma t allows the Laplace transform to handle functions that the Fourier transform cannot, such as signals that grow without bound. The Laplace transform converts differential equations into algebraic equations — derivatives become multiplication by s. This is why electrical engineers love the Laplace transform electrical engineers and control theorists love the Laplace transform: it turns circuit analysis (solving integro-differential equations) into simple algebra with impedances.

Fourier and Laplace transforms are everywhere in modern technology. MP3 and JPEG compression use frequency-domain representations to discard information that human ears and eyes cannot perceive. Magnetic Resonance Imaging (MRI) reconstructs images from frequency-domain data. Radar and sonar detect targets by analyzing reflected frequency shifts. Control systems use Laplace-domain transfer functions to design stable controllers. In quantum mechanics, the position and momentum representations are Fourier transforms of each other, leading directly to the Heisenberg uncertainty principle.

Adjust the parameter to see how changing a signal's frequency content affects its shape. This is the essence of Fourier analysis.