LTIsysteemit

LTIsysteemit, or linear time-invariant systems, are a fundamental class of systems in signal processing and control theory. They map input signals to output signals and are characterized by two properties: linearity and time invariance. Linearity means the principle of superposition holds: scaling an input scales the output by the same factor, and summing inputs yields the sum of the outputs. Time invariance means that if the input is shifted in time, the output is shifted by the same amount.

A central concept is the impulse response, the system’s output for a Dirac delta input. For continuous time, the output is the convolution y(t) = ∫ x(τ) h(t−τ) dτ. For discrete time, y[n] = Σ x[k] h[n−k]. The impulse response fully characterizes an LTI system.

In the frequency domain, LTI systems multiply spectra: Y(s) = H(s) X(s) in Laplace domain for continuous time, or Y(z) = H(z) X(z) in the Z-domain for discrete time. The transfer function H encapsulates the system dynamics and can be derived from the impulse response via the Laplace or Z-transform.

Stability and causality are important considerations. BIBO (bounded-input, bounded-output) stability requires the impulse response to be absolutely integrable (continuous time) or summable (discrete time). Causality means the impulse response is zero for times t < 0 (continuous) or n < 0 (discrete).

LTIsystem representations include differential equations with constant coefficients, state-space models, and digital filters (FIR and IIR). Common examples are RC circuits and mass–spring–damper systems, which motivate applications in audio processing, communications, and control. Real systems are not always LTI; many are nonlinear or time-varying, and LTI models often serve as useful linear approximations.