Frequency-Domain Design of Overcomplete Rational-Dilation Wavelet Transforms
İlker Bayram, Ivan Selesnick · IEEE Transactions on Signal Processing · 2009
The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low$Q$-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$Q$-factors (desirable for processing oscillatory signals) or the same low$Q$-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-$Q$' discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for$L_{2}( {\BBR })$. 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.