Block Diagrams / Lohkokaavio

A causal LTI system can be written with a differential equation

$$\sum_{k=0}^N a_k \, y[n-k] = \sum_{k=0}^M b_k \, x[n-k]$$

There are three basic functions in LTI systems.

FIR (finite impulse response) systems have an impulse response h[n]of finite length. For example $h[n] = \delta[n] - 0.5\delta[n-1]$is zero when n<0 or n>1. FIRs are therefore always BIBO-stable. Transfer function H(z) consists only of numerator polynomial.

IIR (infinite impulse response) systems have an impulse response h[n]of infinite length. For example h[n] = 0.5ⁿ u[n]is nonzero when $n \ge 0$. IIRs contain feedback loops. Transfer function H(z) consists of numerator and denominator polynomials

$$H(z) = \frac{b_0 + b_1 \, z^{-1} + ... + b_M \, z^{-M}}{a_0 + a_1 \, z^{-1} + ... + a_N \, z^{-N}}$$

Stability can be derived from positions of poles (roots of denominator). If poles are strictly inside unit circle, then system is stable.