A wavelet shall possess the properties:
\[\int_{-\infty}^{\infty} \psi(x) dx = 0\]
\[another \_condition \_to\_ add\]
where the \(\psi(x)\) above is mother wavelet. The one can generate wavelets using mother wavelets by dilation and translation as follows:
\[\psi_{j,k}(x) = 2^{j/2}\psi(2^jx-k)\]
as such we can express time series or functions in general in form of
\[f(x) = \sum_{j=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}d_{j,k}\psi_{j,k}(x)\]
where the coefficients, due to orthogonality, are
\[d_{j,k} = \int_{-\infty}^{\infty}f(x)\psi_{j,k}(x)dx\]
The wavelets overcome the disadvantages, that fourier transform has, that is ignoring time domain.
This is one of the core issue
Wavelets can be seens as a filter bank which is an array of band-pass filters that separates the input signal into multiple components.
If one of these filters shows strong resonance, while the others show little or no activity, there is presumably a strong cycle in the market. An entry is generated by looking at the pair of filter outputs and buying at the next bar, if the cycle phase is such that a cyclic bottom will occur on that bar, or selling at the next bar
— from book encyclopedia of trading
Wavelet approach can detect cycles which is exactly what make momentum strategy work. However, Momentum based on one fixed empirical number of months as holding period, the wavelet approach may be better at predicting entry and exit point of investment.
Why choose momentum based strategy as benchmark?