Bandwidth compression of speech by analytic-signal rooting

If s(t) is a real, band-limited signal, the corresponding analytic signal is defined as s(t)+js
^{^}
(t), where s
^{^}
(t) is the Hilbert transform of s(t). For signals whose spectral width is due primarily to large-index frequency modulation, the "square-rooted" signal, defined as s
½
(t) = Re [s(t) + js
^{^}
(t)]
^½
, has approximately only half the bandwidth of s(t). A case of practical interest of a signal having approximately this property is a speech signal filtered to remove all but one formant. In such a case, a close replica of the original signal can be recovered by squaring the analytic signal corresponding to a band-limited version s
^~
½
(t) of s
½
(t): s(t) ≈ Re [s
^~
½
(t) + js
^~
½
(t)]
²
. Application of these two processes to the transmission of speech signals over channels of reduced bandwidth is described. Results of computer simulation for a 2-to-1 bandwidth compression are encouraging and suggest that even higher compression factors, using higher roots of the analytic signal, may be feasible.