Discretising

Discretising refers to the process of converting a continuous model, signal, or quantity into a discrete representation. This typically involves sampling at specific points, partitioning a range into finite intervals, or quantizing values into a limited set of levels. In British English, the terms discretising and discretisation are common; in American English, discretizing and discretization are used.

In numerical analysis, discretisation replaces continuous equations with discrete ones that a computer can solve. Time is often sliced into steps and space into a grid, giving rise to methods such as finite differences, finite elements, and finite volumes. The choice of step size or mesh quality affects accuracy, governed by discretisation error. Stability and convergence are central concerns as the discrete solution approximates the continuous model.

In signal processing, discretising a continuous-time signal yields a discrete-time signal by sampling at regular intervals. The sampling rate must satisfy the Nyquist criterion to avoid aliasing, and practical systems include anti-aliasing filters before sampling and quantization after sampling.

In statistics and data analysis, discretisation (or binning) turns continuous variables into discrete categories or intervals. This can simplify modeling and visualization (for example, histograms) but may also reduce information and introduce bias or artifacts if not done carefully.

Other applications include computer graphics, where continuous geometry is represented by discrete polygons, and numerical solvers for differential equations, which rely on grids or meshes. Overall, discretisation enables computation on continuous phenomena, trading some fidelity for tractability and efficiency.