Regarding the scaling function used for the generalized morse wavelet in CWT function

Hello,

I am looking for a reference on the basis of the scaling function used in the "cwt" function for the generalized morse wavelet. I have reviewed the referenced literature on the generalized morse wavelets [1-4] below, as well as the matlab code for the "cwt" and "morsescalingfuction" MATLAB functions. It appears that a gamma function is used to define the fourier transform of this scaling function. In addition, I have reviewed this response by Wayne King to a question on the cwt: https://www.mathworks.com/matlabcentral/answers/352355-about-using-cwt-function#answer_278078?s_tid=prof_contriblnk

It is not clear to me what literature reference is used to define the scaling function, i.e. what is the theoretical basis. I did not find any referenced literature that explicitly defines the scaling function for the generalized morse wavelet. Is it possible that you can provide a reference for the basis of the scaling function used in "cwt" for the generalized morse wavelet?

Morse wavelet references: [1] Olhede, S. C., and A. T. Walden. “Generalized morse wavelets.” IEEE Transactions on Signal Processing, Vol. 50, No. 11, 2002, pp. 2661-2670. [2] Lilly, J. M., and S. C. Olhede. “Higher-order properties of analytic wavelets.” IEEE Transactions on Signal Processing, Vol. 57, No. 1, 2009, pp. 146-160. [3] Lilly, J. M., and S. C. Olhede. “On the analytic wavelet transform.” IEEE Transactions on Information Theory, Vol. 56, No. 8, 2010, pp. 4135–4156. [4] Lilly, J. M., and S. C. Olhede. “Generalized Morse wavelets as a superfamily of analytic wavelets.” IEEE Transactions on Signal Processing Vol. 60, No. 11, 2012, pp. 6036-6041.