Frequently Asked Questions on Wavelets

The wavelet transform (in the signal processing context) is a method to decompose an input signal of interest into a set of elementary waveforms, called “wavelets,” and provides a way to analyze the signal by examining the coefficients (or weights) of these wavelets. The Fourier transform also decomposes signals into elementary waveforms; however, in this case, they are global oscillations, i.e., sines and cosines. Thus, when one wants to analyze the local property of the input signals such as edges or transients, the Fourier transform is not an efficient analysis tool. Operationally, the wavelet transform essentially partitions the frequency axis smoothly and recursively and analyzes each segment (i.e., frequency band) with a resolution matched to its scale. An input signal is first decomposed into low and high frequency bands by convolution-subsampling operations with a pair consisting of a “lowpass” filter {hk} and a “highpass” filter {gk} directly on the discrete time domain. Let H and G be the convolution-subsampling operators using these filters. These are called quadrature mirror filters (QMFs) if they satisfy the following orthogonality (or perfect reconstruction) conditions: HG∗ = GH∗ = 0 and H∗H + G∗G = I , where H∗ and G∗ are the adjoint (i.e., upsampling-anticonvolution) operators and I is the identity operator. Various design criteria (concerning regularity, symmetry etc.) on the lowpass filter coefficients {hk} can be found in [4]. In the wavelet transform, this decomposition (also known as expansion or analysis) process is iterated only on the low frequency bands and each time the high frequency coefficients are retained intact. Each high frequency subband is spanned by a set of translated versions of a single elementary waveform with a specific scaling parameter. This most elementary waveform is called “mother wavelet.” Each low