I am using the continuous wavelet transform to extract periodic components of an original signal. I'm looking for a simple, digestible way to describe how 'much' of the original signal is comprised of a specific periodic component. At the moment I am using wavelet power, but this is a very abstract concept and I would like to be able to describe the signal presence in a more direct manner; as a % of the original signal perhaps.
I have the option of using either a reconstructed time series, or wavelet coefficients somehow.
Cheers, Will
Once you have computed the signal series representation $\sum_k c_k \phi_k$ on a given basis $\{\phi_k\}$ (e.g., wavelet, fourier, etc), it is up to you on how to interpret the result (i.e., expansion coefficients values $\{c_k\}$) according to your needs and application. There is no general approach.
If you are interested for a specific component (basis element) with a coefficient, say, $c_l$, one common single-valued measure that may be useful in your case is the "singal-to-noise (power) ratio", defined as (assuming that $\{\phi_k\}$ is orthonormal) $$ \frac{|c_l|^2}{\sum_{k\neq l} |c_k|^2}, $$ i.e., the ratio of the power of the (periodic) component of interest to the sum of powers of other components (which are not of interest and regarded as additive "noise"). Clearly, the greater this ratio is, the more the signal is similar to the component of interest.