4th order cumulant of signal

I'm trying to implement some code for watermarking on audio based on a scientific paper. I'm stuck in the part of the pseudo code where they calculate the fourth order cumulant of the approximation coefficients of DWT. I have no idea what a cumulant is even when I tried to understand the theory.

I found the following MATLAB code in this book

Cum4 = mean(P.^4) - 4*mean(P)*mean(P.^3) - 3*mean(P.^2)^2 +12*mean(P)^2*mean(P.^2)- 6*mean(P)^4;

Which outputs a single value, meaning the 4th cumulant. However the paper is expecting more than one value (at least that is what I understand).

This the part I don't get

For each AHC, calculate is fourth-order cumulant, denoted as Ci(k)

$C_{i(k)} = \left \{ c_{i(k)}(n) | n = 0,...,\frac{L_{f}}{2^{H-1}} \right \}$

Where AHC are the approximation coefficients of the DWT H level and Lf are the samples where the DWT was applied.

Is the calculation for the 4th cumulant correct? Am I missing something?

Hope someone can help me, thanks

• did you resolve the problem? I'm having a similar problem. I need to compute the 2,3 and 4th order cumulant function from a 1D signal and then select a few values corresponding to several positions from the 1D signal, referenced as time lag in the paper. So I also understand that a cumulant is obtained for each value of the signal. – Victor Mondejar-Guerra Jun 13 '17 at 17:49
• without looking at the paper, I have no clue about the AHC. When it come to cumulants, mathworld.wolfram.com/k-Statistic.html. Your equation is cut off so, you have to verify that yourself – Stanley Pawlukiewicz Jun 13 '17 at 19:47