Signal processing

Signal processing is a specialized topic in electrical engineering, with applications in other engineering disciplines, which is primarily concerned with the analysis and manipulation of signals. A signal here may be broadly thought of as a real or complex-valued function of time, with time being taken as some suitable subset $\mathbb {T}$ of the real numbers $\mathbb {R}$. For example, the function

$y(t)=A\cos(\omega t+\phi ),\quad A,\phi \in \mathbb {R}$

can be considered as a signal with $\mathbb {T} =\mathbb {R}$ (often referred to as continuous time), while the function

$y(n)=A\cos(\omega n+\phi ),\quad A,\phi \in \mathbb {R}$

is a signal with $\mathbb {T} =\{\ldots ,-2,1,0,1,2,\ldots \}$ (often referred to as discrete time).

Signals are of interest because they can be used to carry information and some important themes studied in signal processing include how to effectively impart information onto a signal (manipulation of a signal) and how to extract information from a signal (analysis of a signal). In everyday modern life such exploitation of signals are ubiquitous, for instance in the manipulation of electromagnetic waves to carry the information of our favourite radio and television programs.

Operatively, signal processing relies heavily on various tools of mathematics, especially statistics (such as sampling theory and hypothesis testing) and harmonic analysis (such as Fourier analysis). It is also closely related to the discipline of information theory

Special topics in signal processing

Below is a non-exhaustive list of important special topics which can be considered to belong under the general area of signal processing.

1. Coding and modulation: This topic is concerned with the problem of how to best represent certain information in a signal (coding) and ways of physically implementing that coding (modulation).
2. Detection: This topic is concerned with the problem of determining from available signals if one or more events of interest have occured, such as the event of a failure of one or more components in a system.
3. Estimation and filtering: This topic is concerned with the problem of estimating a certain signal of interest when it cannot be observed/measured directly but via another signal as a proxy. For example, suppose the signal of interest is x but it is only possible to observe or measure another signal y which is related to x in some way, such as via the relation $y(t)=x(t)+\eta (t)$ where $\eta (t)$ is some random disturbance signal which is independent of x.
4. System identification: This topic is concerned with the problem of estimating the parameters or coefficients of a dynamical system based on some signals measured from the system. For example, one may be interested in estimating the value of the resistance and capacitance of an RC circuit based on continuous measurements of some voltages and currents in the circuit.
5. Model selection: This topic is concerned with determining from a set of candidate models for a certain process the one that describes it the best with respect to some pre-determined criterion. For example, suppose that a signal y is assumed to be of the form:

$y(t)=A_{0}+\sum _{k=1}^{N}A_{k}\cos(\omega t+\phi _{k})+\eta (t)\quad (1),$

where N is an unknown integer, $A_{0},A_{1},\ldots ,A_{N}\in \mathbb {R}$ and $\phi _{1},\phi _{2},\ldots ,\phi _{N}\in (-\pi ,\pi ]$ are unknown coefficients, and $\eta$ is some random disturbance with known statistical properties. Now, suppose that based on measurements of the true signal y, some candidate set of values for N and the unknown coefficients of in (1) have been computed by some suitable methods. Then, rough speaking, model selection aims to determine which of these set of candidates would give the best fit to the true process with respect to some suitable criterion.

See also

Digital signal processing