Harmonic wavelet transform

In the mathematics of signal processing, the harmonic wavelet transform, introduced by David Edward Newland in 1993, is a wavelet-based linear transformation of a given function into a time-frequency representation. It combines advantages of the short-time Fourier transform and the continuous wavelet transform. It can be expressed in terms of repeated Fourier transforms, and its discrete analogue can be computed efficiently using a fast Fourier transform algorithm. In the mathematics of signal processing, the harmonic wavelet transform, introduced by David Edward Newland in 1993, is a wavelet-based linear transformation of a given function into a time-frequency representation. It combines advantages of the short-time Fourier transform and the continuous wavelet transform. It can be expressed in terms of repeated Fourier transforms, and its discrete analogue can be computed efficiently using a fast Fourier transform algorithm. The transform uses a family of 'harmonic' wavelets indexed by two integers j (the 'level' or 'order') and k (the 'translation'), given by w ( 2 j t − k ) {displaystyle w(2^{j}t-k)!} , where These functions are orthogonal, and their Fourier transforms are a square window function (constant in a certain octave band and zero elsewhere). In particular, they satisfy: where '*' denotes complex conjugation and δ {displaystyle delta } is Kronecker's delta. As the order j increases, these wavelets become more localized in Fourier space (frequency) and in higher frequency bands, and conversely become less localized in time (t). Hence, when they are used as a basis for expanding an arbitrary function, they represent behaviors of the function on different timescales (and at different time offsets for different k). However, it is possible to combine all of the negative orders (j < 0) together into a single family of 'scaling' functions φ ( t − k ) {displaystyle varphi (t-k)} where The function φ is orthogonal to itself for different k and is also orthogonal to the wavelet functions for non-negative j: In the harmonic wavelet transform, therefore, an arbitrary real- or complex-valued function f ( t ) {displaystyle f(t)} (in L2) is expanded in the basis of the harmonic wavelets (for all integers j) and their complex conjugates: or alternatively in the basis of the wavelets for non-negative j supplemented by the scaling functions φ: