value of $k_{\psi}$ is a satisfactory compromise between a value small-enough that we have good resolution of temporal structures, and large-enough that the admissibility condition is satisfied, at least to machine accuracy [@far92]. The wavelet, being continuous and complex, permits a rendering in transform space that highlights temporally localized, periodic activity – oscillatory behavior in the real part and a smooth distribution of power in the modulus – and being progressive (zero power at negative frequency), is optimal for the study of causal signals. We have deliberately avoided any form of weighting, such as that introduced by @foster to allow for uneven sampling, or @johnson to rescale within the cone of influence, in order to facilitate our interpretation of the cross wavelet, and to allow the use of existing methods of significance analysis.

From this mother wavelet, we generate a set of translated ($t'$) and dilated ($l$) wavelets $$\psi_{lt^{'}}(t)=\frac{1}{\sqrt{l}} \psi (\frac{t-t^{'}}{l}),  l \in \Re^{+},  t\in \Re$$ and we then take the inner product with the signal $F(t)$ to obtain the wavelet coefficients $$\label{coefficients}
\widetilde f (l,t^{'})= \int_{\Re}f(t)\psi^{*}_{lt'}(t) dt .$$ The wavelet coefficients are later mapped in wavelet space which has as coordinates translation and dilation, and so periodic behavior shows 