quantizing

Quantizing is the process of mapping a continuous range of values to a finite set of discrete levels. In digital systems, quantization occurs after sampling and conversion, producing a finite-precision representation of a signal or quantity. A quantizer assigns each input x to a representative level q, typically by rounding to the nearest level. If the levels are evenly spaced with step size Δ, a common form is q = Δ · round(x/Δ).

Types include uniform quantization (equal spacing) and nonuniform quantization (varying spacing, as in μ-law or A-law companding used in audio). Scalar quantization uses single samples; vector quantization operates on blocks of samples to exploit correlations.

Dithering adds controlled random noise before quantization to reduce distortion and avoid bias, especially for sparse signals. Quantization error e = x − q(x) is bounded in magnitude by Δ/2 for midrise or midtread quantizers, and can be treated as noise in many analyses, though it can be correlated with the input.

Performance measures such as signal-to-quantization-noise ratio (SQNR) depend on the input distribution and the step size; more bits increase resolution and data rate but require more storage or bandwidth. Applications include analog-to-digital converters, fixed-point arithmetic, color quantization in images and videos (reducing color space or palette), and neural networks for efficient inference (quantized weights and activations).

Quantization introduces distortion but is essential for digitization and compression; trade-offs include resolution, dynamic range, and data rate. Quantizing in physics is a related concept where quantities such as energy levels take discrete values; in signal processing, quantizing is an engineered mapping, not a fundamental property.