# Looking for a analytical formula to compute the central frequency of a signal analyzed by a discrete wavelet at a given scale

I am reading this paper by Han et al. (2014). In this article, the authors extract detailed information from geomagnetic data sampled at 1Hz using a daubechies 5 (db5) wavelet: they reconstruct the signal using the coefficients from the 6th level only. The authors explain that the central frequency of the resulting reconstructed signal is 0.01Hz.

I have reprocessed similar data following the same steps and have convinced myself that the central frequency of the signal reconstructed using only coefficients from the 6th level is indeed around 0.01Hz. Below are the PSD plots for both the original and reconstructed signals, generated using Python:

But is there a way to analytically determine the central frequency of the reconstructed signal?

• On one hand, I have found explanations stating that, considering a SR of 1Hz, the central frequency of the signals reconstructed using the 6th level should lie within [0.5, 1]/2**6, i.e. [0.0078125, 0.015625]Hz, which is the case. Is this the correct answer? If this was the case, it would mean that the central frequency of the analyzed signal would be "wavelet-independent"...
• On the other hand, I know about the formula from Abry, 1997, stating that the pseudo-frequency of a wavelet at a given scale Fa is the central frequency of the wavelet Fc divided by the scale a: Fa=Fc/a. But my undestanding is that this formula yields the pseudo frequency of the wavelet, not the central frequency of the analyzed signal!

Any clarification would be appreciated. Thanks!

They do mean the pseudo-frequency of the wavelet which is not dependent on the signal being analyzed. The misleading terminology that they use seems to come from from one of the references, Han, P. (2013), Investigation of ULF seismo-magnetic phenomena in Kanto, Japan during 2000–2010, PhD thesis, Chiba University, Chiba, Japan. Quoting an article of a similar name and where Han is one of the authors, Hattori et al., Investigation of ULF Seismo-Magnetic Phenomena in Kanto, Japan During 2000–2010: Case Studies and Statistical Studies, Surv Geophys (2013) 34:293–316, DOI 10.1007/s10712-012-9215-x, directly answers your question:

In this paper, we yield discrete wavelet transform (DWT) and decompose the original signature into six levels. The central frequency ($$F_a$$) of signals in each level is given as

$$F_a = \frac{F_c}{a\Delta} \tag{4}$$

where $$F_c$$ is the frequency of basic wavelet; $$a$$ is the scale of the level, and $$\Delta$$ is the sampling rate of original data. After evaluating different kinds of wavelets, we adopted Daubechies 5 (db5) as the mother wavelet. Moreover, the db5 wavelet has been proved to be effective in the previous study (Jach et al. 2006). According to Eq. 4, for db5 wavelet, the central frequency of signals in the sixth level is 0.01 Hz.

• Thanks for your well-documented reply, Olli! I am now convinced that the authors are indeed referring to the pseudo-frequency... Which still leaves wondering how to analytically determine the central frequency of the reconstructed signal. – Sheldon Nov 5 '19 at 8:45
• Perhaps the closest thing would be numerical argmax of the power spectrum. – Olli Niemitalo Nov 5 '19 at 10:59
• Correct! This is the approach that I tried as discussed in this question that I recently posted on Stack Overflow. – Sheldon Nov 6 '19 at 12:22