Page 67 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

2. Probability and Random Process 55

Kurtosis is the statistical parameter used to measure the Gaussian nature
of the signal. Kurtosis is inversely proportional to the Gaussianity of the
signal. Note that the kurtosis values are maximum for independent signals
compared to the mixed signals. (i.e.) Independent signals are more non-
Gaussian compared with mixed signals. Mathematically kurtosis is
computed using the formula as displayed below, where E[X] is the
expectation of the vector X

ICA problem is to obtain the matrices [X] and [A] such that column
T
vectors of the matrix [X] are independent to each other. (i.e.) Kurtosis
values computed for the column vectors are maximum.

T
T
T
[Y] = [X] [A]
( i.e.) [Y] = [A] [X]

-1
[X] = [A] [Y]

[X]=[B][Y]

X1 and X2 are the two column vectors of the matrix [X] T

T
(i.e.) [X] = [X1 X2]

Kurtosis measured for the Column vector [X1] is given as

2
2
4
E [X1 ]-3{E [X1 ]}
Similarly kurtosis measured for the column vector [X2] is given as

2
4
2
E [X2 ]-3{E [X2 ]}