# A priori spectral efficiency of OFDM scheme

Of course I can compute the spectral efficiency as the ratio of the measured bit-rate (of data bits, no header, no preable, nor redundancy bits included) and the measured spectral bandwidth. This would be an a posteriori approach.

However, I wonder if there exists an analytical expression for a priori determination of the spectral efficiency of an OFDM communication system.

The system uses preable, header, forward error correction (in and out), cyclic redundancy check, cyclic prefix and taper window. And the modulation by subcarrier can also be changed, as well as the number of subcarriers, and which of them are nulled or pilots.

Is there some kind of formula that includes all of these parameters and returns the effective bit rate and spectral efficiency?

• Could you give an example? – AlexTP May 17 '18 at 18:17

Taking a look to the source code of the liquid-dsp project, and to the ofdmflexframegen.c file in particular, I found a partial answer to my question.

The spectral efficiency is: $$\beta = \frac{R_b}{W}~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $R_b$ is the bit rate and $W$ is the OFDM bandwidth.

The bit rate is: $$R_b = \frac{B}{T}~~~~~~~~~~~~~~~~~~~~~~~~~~~(2)$$ where $B$ is the data bits (of the decoded payload) and $T$ is the total transmission time.

Then, putting (2) into (1): $$\beta = \frac{B}{TW}~~~~~~~~~~~~~~~~~~~~~~~~~~~(3)$$ and the product $TW = N$ is the number of samples of the transmission.

As for $N$, we have:

$$N = (M + c)(N_{preamble} + N_{header} + N_{payload})~~~~~~~~~~~~~~~~~~~~~~~~~~~(4)$$ where $M$ the total number of subcarriers, $c$ the length of the cyclic prefix. $N_{preamble}$, $N_{header}$ and $N_{payload}$ are the number of OFDM symbols of the preamble, header and payload, respectively. Each OFDM symbol is comprised of $M + c$ samples.

Finally, the spectral efficiency for such a system would be: $$\beta = \frac{B}{(M + c)(N_{preamble} + N_{header} + N_{payload})}~~~~~~~~~~~~~~~~~~~~~~~~~~~(5)$$

However, practical measurements of the spectral efficiency of my communication system are a little higher of what this formula predicts. It is like if the measured bandwidth was actually smaller than the predicted one. The difference increases with the payload size. I would appreciate any extra idea in this regard.

EDIT 1: Taking a deeper look to my system, it is possible that the difference between the spectral efficiency predicted by Eq. 5 and what I am measuring is due to the presence of null subcarriers. Null subcarriers at both sides of the OFDM spectrum are not measured, but they are included in Eq. 5. I wonder if I may just substract the number of null subcarrier from the term $M+c$ in Eq. 5. Does that make any sense?