LTIsystem

An LTI system, or linear time-invariant system, is a mathematical model used in signal processing and control theory to describe systems whose output is a linear function of the input and whose properties do not change over time. Such a system can be completely characterized by its impulse response, h(t) for continuous time or h[n] for discrete time.

Linearity means that for any inputs x1, x2 and any scalars a, b, the system satisfies S[a x1 + b x2] = a S[x1] + b S[x2]. Time invariance means that if input x(t) produces output y(t), then input x(t − t0) produces output y(t − t0) for any time shift t0.

Because of these properties, the output of an LTI system is the convolution of the input with the impulse response: y(t) = x(t) ∗ h(t) in continuous time, or y[n] = x[n] ∗ h[n] in discrete time. The impulse response fully describes the system’s behavior.

Equivalently, the system is described by a transfer function H(s) in the Laplace domain for continuous time, or H(jω) in the frequency domain, and by H(z) in the Z-transform for discrete time. The Fourier transform of the impulse response gives the frequency response, which indicates how different frequency components are amplified or attenuated.

Stability, particularly BIBO stability, requires that bounded inputs produce bounded outputs. LTI systems include many practical filters, such as RC networks and digital FIR filters, and are contrasted with nonlinear or time-varying systems where superposition or shift invariance does not hold.