Frequency-Domain Design of Overcomplete Rational-Dilation Wavelet Transforms

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$Q$</tex></formula></emphasis>-factor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied. The new wavelet transform can attain higher <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$Q$</tex></formula></emphasis>-factors (desirable for processing oscillatory signals) or the same low <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$Q$</tex></formula></emphasis>-factor of the dyadic wavelet transform. The new wavelet transform is modestly overcomplete and based on rational dilations. Like the dyadic wavelet transform, it is an easily invertible 'constant-<emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$Q$</tex></formula></emphasis>' discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$L_{2}( {\BBR })$</tex></formula></emphasis>. The wavelet can be made to resemble a Gabor function and can hence have good concentration in the time-frequency plane. The construction of the new wavelet transform depends on the judicious use of both the transform's redundancy and the flexibility allowed by frequency-domain filter design. </para>