Question

Consider the signal space representation of a 4-ary modulation scheme as shown in the figure below:



If the symbols $s_0,s_1,s_2$ and $s_3$ are occurring with probabilities 0.3,0.2,0.4 and 0.1 respectively, then the average symbols energy of the modulation scheme will be equal to

• A. $2.2A^2$
• B. $2A^2$
• C. $1.6A^2$
• D. $A^2$

Answer 1

The correct option is A $2.2A^2$



Let the energy associated with the symbols $s_0,s_1,s_2,$ and $s_3$ are $E_0,E_1,E_2$ and $E_3$ respectively

$E_i=(d_i{)}^2;i=0,1,2,3$

From the above diagram,

$d_0=0$

$d_1=d_3=\sqrt{A^2+A^2}=\sqrt{2A^2}$

$d_2=2A$

So, $E_0=0$

$E_1=E_3=2A^2$

$E_2=4A^2$

The average symbol energy of the modulation scheme can be given as,

$E_s=\sum _{i=0}^3{E}_{i}P\left(s_i\right);$
$P\left(s_i\right)$ = probability of occurrence of the symbol $s_1$

$=0\left(0.3\right)+2A^2\left(0.2\right)+4A^2\left(0.4\right)+2A^2\left(0.1\right)$

$=(0.4+1.6+0.2)A^2$