IIRfilters

IIR filters, or Infinite Impulse Response filters, are a class of digital filters whose current output depends on present and past inputs as well as past outputs. They are described by a difference equation y[n] = sum_{k=0}^M b_k x[n-k] - sum_{l=1}^N a_l y[n-l], with a0 typically 1. In the z-domain, H(z) = Y(z)/X(z) = B(z)/A(z), where B and A are polynomials in z^{-1}. The poles of H lie inside the unit circle for stability.

IIR filters are implemented using feedback, unlike FIR filters which have finite impulse responses and depend only on current and past inputs. They can achieve sharp frequency selectivity with lower order than FIR but may introduce non-linear phase unless designed for linear phase. Their impulse response is, in general, infinite in duration.

Design methods commonly start from analog prototypes and apply transformations such as bilinear or impulse invariant to obtain digital implementations. Common filter families include Butterworth, Chebyshev, and Elliptic designs, as well as Bessel when phase linearity is important. Realizations are typically implemented as cascades of second-order sections or as Direct Form I/II or transposed forms. Numerical issues arise from finite word length, including coefficient quantization and potential instability; thus, careful structure selection and scaling, or using cascaded sections, help improve robustness.

Applications span audio processing (equalization and effects), communications receivers and transmitters, control systems, and sensor data pre-processing. IIR filters offer efficient sharp filtering with lower order than FIR counterparts but require attention to stability and phase characteristics.

Example: a first-order IIR with y[n] = b0 x[n] + b1 x[n-1] - a1 y[n-1], has transfer function H(z) = (b0 + b1 z^{-1})/(1 + a1 z^{-1}).