Signal processing

Signal processing, a branch of electrical engineering, deals with analyzing, altering, and creating signals—examples include sound, images, magnetic fields, seismic data, altimetry readings, and scientific measurements. These methods help improve transmission quality, boost digital storage efficiency, fix distorted signals, enhance perceived video clarity, and identify specific elements within a measured signal.

The roots of signal processing trace back to 17th-century numerical analysis, as noted by Alan V. Oppenheim and Ronald W. Schafer. They point out that digital refinements of those techniques emerged in the digital control systems of the 1940s and 1950s. Claude Shannon’s 1948 paper, "A Mathematical Theory of Communication," published in the Bell System Technical Journal, established a foundation for later advances in communication systems and signal transmission. The field matured in the 1960s and 1970s, and by the 1980s, dedicated digital signal processor chips made digital signal processing widespread.

A signal in this context is a function of time, written as \( x(t) \). It can be either deterministic—meaning its behavior is fully predictable—or a realization of a stochastic process, denoted as \( (x_t)_{t \in T} \), which introduces randomness.

Signal processing is divided into several categories. Analog signal processing handles signals that haven’t been digitized, as seen in most 20th-century radio, telephone, and television systems. It uses both linear electronic circuits—like passive and active filters, additive mixers, integrators, and delay lines—and nonlinear ones, such as compandors, multipliers (frequency mixers, voltage-controlled amplifiers), voltage-controlled filters, voltage-controlled oscillators, and phase-locked loops.

Continuous-time signal processing works with signals that vary smoothly over time, without being broken into discrete samples. Its methods involve time domain, frequency domain, and complex frequency domain analysis. This area focuses on modeling linear time-invariant continuous systems, calculating the system’s zero-state response through integrals, setting up system functions, and filtering deterministic signals continuously. For instance, in the time domain, a continuous signal \( x(t) \) passing through a linear time-invariant filter \( h(t) \) yields an output \( y(t) = \int_{-\infty}^{\infty} h(\tau) x(t-\tau) \, d\tau \). Here, \( h(t) \) is often called the impulse response, and the operation is a convolution between input and system.

Discrete-time signal processing deals with sampled signals—defined only at specific time points and quantized in time but not in magnitude. Analog discrete-time processing relies on electronic components like sample-and-hold circuits, analog time-division multiplexers, analog delay lines, and analog feedback shift registers. It preceded digital signal processing and remains useful for handling gigahertz-range signals. The term also refers to a theoretical framework that underpins digital signal processing, ignoring quantization errors.

Digital signal processing processes digitized, discrete-time sampled signals using general-purpose computers or digital circuits like ASICs, field-programmable gate arrays, or specialized digital signal processors. Common arithmetic operations include fixed-point and floating-point, real and complex multiplication and addition. Hardware often supports circular buffers and lookup tables. Key algorithms include the fast Fourier transform (FFT), finite impulse response (FIR) filters, infinite impulse response (IIR) filters, and adaptive filters such as the Wiener and Kalman filters.

Nonlinear signal processing analyzes and processes signals from nonlinear systems, operating in time, frequency, or spatiotemporal domains. Nonlinear systems can generate complex behaviors—bifurcations, chaos, harmonics, and subharmonics—that linear methods cannot produce or analyze. Polynomial signal processing, a subtype of nonlinear processing, treats polynomial systems as straightforward extensions of linear ones.