LTIfiltre

LTIfiltre is a term used in signal processing to denote a class of linear time-invariant filters. Such filters have an input-output relationship that is linear and does not change over time, meaning their response to any input can be predicted by convolution with a fixed impulse response. In continuous time, the output is y(t) = (h * x)(t); in discrete time, y[n] = sum_k h[k] x[n-k]. The impulse response h completely characterizes a LTIfiltre.

Mathematically, LTIfiltres are described by transfer functions. In analog form, the transfer function is H(s) = Y(s) / X(s) in the Laplace domain, with poles and zeros determining stability and frequency behavior. In discrete time, the transfer function is H(z) = Y(z) / X(z) in the Z-transform domain, where pole locations relative to the unit circle indicate stability and the frequency response describes filtering characteristics. Causality and stability are central properties: in continuous time, a stable system has poles with negative real parts; in discrete time, poles must lie inside the unit circle.

Common realizations include finite impulse response (FIR) and infinite impulse response (IIR) filters. FIR filters have a finite h[n] and are inherently stable with linear phase when designed symmetrically. IIR filters use feedback and can achieve sharp frequency selectivity with fewer coefficients but require careful stability analysis and suitable realizations, such as direct form or lattice structures.

Applications of LTIfiltres span audio processing, communications, control systems, and image processing. Design methods range from time-domain windowing and frequency-domain techniques to transformations like the bilinear transform or impulse invariance for digital implementation, all aiming to achieve the desired amplitude and phase response while ensuring stability and efficient realization.