On the asymptotic performance of hierarchical transforms

Explicit formulas are derived for the coding gain of hierarchical transforms for a given number of stages, and the asymptotic gain as this number goes to infinity. The intuitive result that hierarchical transforms are not asymptotically optimal, that is, their coding gains do not approach the inverse of the spectral flatness measure as the number of stages goes to infinity, is confirmed. Examples comparing the limits for hierarchical transforms and M-band parallel systems (filter banks) for AR(1) signals are presented