Timediscrete

Timediscrete refers to systems, signals, or processes that exist or are defined only at discrete time instants, typically t = nT, where n is an integer and T>0 is the sampling period. In this view, signals are sequences x[n] rather than continuous functions x(t). Timediscrete models can arise by sampling a continuous-time signal or from inherently discrete processes such as digital control, computer simulations, or event-driven systems.

Discrete-time linear time-invariant (LTI) systems are commonly described by difference equations or by impulse response. A typical input-output relation is y[n] = sum_{k=0}^{M} b_k x[n-k] - sum_{k=1}^{N} a_k y[n-k], or equivalently y = x * h, where h is the system's impulse response. Causality means y[n] depends only on current and past inputs. Stability is commonly defined as BIBO: every bounded input produces a bounded output.

Analysis of discrete-time systems often uses the Z-transform, H(z) = Y(z)/X(z) = sum h[n] z^{-n}, with a region of convergence in the complex plane. For frequency-domain intuition, the discrete-time Fourier transform (DTFT) characterizes responses on the unit circle when the sequence is suitably well-behaved.

Sampling and aliasing are central for time-discrete models derived from continuous signals. If a continuous signal x_c(t) is sampled at rate Fs = 1/T, the resulting x[n] = x_c(nT). If sampling frequency exceeds twice the maximum frequency (Nyquist), reconstruction is possible; otherwise aliasing occurs.

Applications include digital signal processing, where discrete-time signals are filtered and analyzed; digital control, where controllers operate in discrete time; and numerical simulation or data processing over sampled data.