# A sequence OFDM modulated with normalization factor

The following sequence is used to modulate 52 subcarriers. Twelve out of 52 are non zero. I was wondering

$S_{[–26, 26]} = \sqrt{13/6} \{0, 0, 1+j, 0, 0, 0, –1–j, 0, 0, 0, 1+j, 0, 0, 0, –1–j, 0, 0, 0, –1–j, 0, 0, 0, 1+j, 0, 0, 0, 0, 0, 0, 0, –1–j, 0, 0, 0, –1–j, 0, 0, 0, 1+j, 0, 0, 0, 1+j, 0, 0, 0, 1+j, 0, 0, 0, 1+j, 0,0\}$

The book says that The multiplication by a factor of $\sqrt{13/6}$ is in order to normalize the average power of the resulting OFDM symbol, which utilizes 12 out of 52 subcarriers.

I tried to compute it..

My guess is that the average energy in such sequence before normalization would be $$12\times \frac{\sqrt{2}}{52}=\frac{3\sqrt{2}}{13}$$

where $\sqrt{2} = \sqrt{1^2+1^2}$ from the complex symbols withing the sequence.

Then why is the normalization factor $\frac{\sqrt{13}}{\sqrt{6}}$?

Thanks

The power of the non-zero elements is $|1+j|^2=2$, so the average power is
$$\overline{P}=\frac{12\cdot 2}{52}=\frac{24}{52}=\frac{6}{13}$$
So in order to get unity average power, you have to normalize by $\sqrt{13/6}$.