Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 210

Fig. 3.27 PDF of the variable X

Fig. 3.28 Convolution of PDFs

Answer The characteristic function of the sum of independent variables is equal to
the product of their characteristic functions:

f X ðoÞ¼f X 1 ðoÞf X 2 ðoÞ; (3.327)

where

ð ð

e jox 1 f X 1 ðx 1 Þ d x 1 ; e jox 2 f X 2 ðx 2 Þ d x 2 :

f X 1 ðoÞ¼ f X 2 ðoÞ¼ (3.328)

x 1 x 2

The PDF from Fig. 3.27 is obtained by the convolution of the densities of the
variables X 1 and X 2 , as shown in Fig. 3.28 .
The characteristic functions are:

ð ð

1 1

1
2 e jox 1 d x 1 ¼ f X 2 ðoÞ¼

1
2 e jox 2 d x 2 ¼

sin o
o :

f X 1 ðoÞ¼ (3.329)

1 1

Finally, from ( 3.327 ) and ( 3.329 ) we have:

2

sin o
o

f X ðoÞ¼ : (3.330)