Mendations Are Those of the Authors and Are Not Necessarily Endorsed by the United States Air Force or the United States

A lterbank-based method of time-scale modi cation is analyzed for elemental signals including clicks, sines, and AM-FM sines. It is shown that with the use of some basic properties of linear systems, as well as FM-to-AM lter transduction, \perfect reconstruction" time-scaling lterbanks can be constructed for these elemental signal classes under certain conditions on the lterbank. Conditions for perfect reconstruction time-scaling are shown analytically for the uniform lterbank case, while empirically for the nonuniform constant-Q (gammatone) case. Extension of perfect reconstruction to multi-components signals is shown to require both lterbank and signal-dependent conditions and indicates the need for a more complete theory of \perfect reconstruction" time-scaling lterbanks. 1 FILTERBANK FRAMEWORK Consider a discrete-time signal x(n) passed through a bank of lters hk(n) where each lter is given by a modulated version of a baseband prototype lter h(n), i.e., hk(n) = h(n)exp[j(2 =R)kn] where h(n) is assumed to lie over a duration N=2 n < N=2 (N even without loss of generality), and R, the frequency sampling factor, is the number of lters. In the context of this paper, the lters are zero phase in time and frequency, e.g., a Gaussian or Hamming function. The lters are designed to satisfy a perfect reconstruction constraint in frequency, i.e, P k hk(n) = (n), where (n) is the unit sample sequence. One condition for perfect reconstruction is that the length of h(n) be less than twice the frequency sampling factor, i.e., N < 2R [4]. Each lter output yk(n) = x(n) hk(n) is complex [each lter response hk(n) is complex] so that the temporal envelope of the output of the kth channel is ak(n) = jyk(n)j and the phase of each bandpass output is k(n) = tan (Im[yk(n)]=Re[yk(n)]). Thus the output of each lter can be viewed as an amplitude and phase modulated (complex) sine wave yk(n) = ak(n)exp[j k(n)] and reconstruction of the signal can be viewed as a sum of complex exponentials