Fundamental Mode

A Fundamental Mode is the geometric sequence $p_k^{n}u[n]$ associated with a Pole $p_k$ of a system’s Transfer Function $H(z)$. The Impulse Response of any LTI system with a rational $H(z)$ is a superposition of fundamental modes weighted by their residues $A_k$:

$$h[n]=\sum _{k=1}^N{A}_k{p}_k^{n}u[n]$$

The residues are obtained from the partial fraction decomposition of $H(z)$:

$$H(z)=\sum _{k=1}^N\frac{A_k}{1-p_k{z}^{-1}},A_k=\mathrm{Res}(H(z),z=p_k)$$

The behavior of each mode is determined solely by $|p_k|$. See Convergence.