Continuetime

Continuetime, often written as continuous time, refers to models, signals, and systems defined over a continuum of time, typically the real numbers. A continuous-time signal x(t) assigns a value to every real time t, and a continuous-time system maps input signals to outputs with respect to that same continuum. In contrast to discrete-time signals, which are defined only at discrete instants, continuous-time signals can change at any moment.

Common representations include differential equations for dynamical systems, integral equations, and impulse-response descriptions via convolution. For example, a linear time-invariant continuous-time system is characterized by y(t) = ∫ h(τ)x(t−τ) dτ, where h(t) is the impulse response. Transform methods such as the Laplace transform and Fourier transform are standard tools for analysis in the continuous-time domain.

Examples of continuous-time signals include x(t) = sin(ωt), exponentials, and physical measurements like voltage over time. Applications span electrical engineering, control theory, physics, and communications. The continuous-time framework aligns closely with real physical processes, while digital computation typically processes discretized samples.

Discrete-time becomes relevant via sampling, converting a continuous-time signal into a sequence by x[n] = x(nT). The sampling theorem, aliasing, and discretization are central concerns when bridging continuous and discrete representations.