Biorthogonal Signal Expansions

A set of signals $\{h_k,f_k\}_{k=1}^N$ is said to be a biorthogonal basis set if any signal $x$ can be represented as

$$x = \sum_{k=1}^N \alpha_k\left<x,h_k\right>f_k$$

where $\alpha_k$ is some normalizing scalar dependent only on $h_k$ and/or $f_k$ . Thus, in a biorthogonal system, we project onto the signals $h_k$ and resynthesize in terms of $f_k$ .