OFDM loss of E_b/N₀ resulting from cyclic prefix and pilot subcarriers

OFDM systems show a loss of E_b/N₀ due to the cyclic prefix and pilot carriers.

Effect of cyclic prefix

The last $N_{G}$ symbols of the OFDM symbol of length $N$ are copied and added as cyclic prefix. As the receiver removes the cyclic prefix its energy is lost. This results in a loss of E_b/N₀:

$ß_{1} = \frac{N}{N + N_{G}}$

Where

$N$ Number of subcarriers

$N_{G}$ Discrete length of the guard interval

Effect of pilot subcarriers

Pilot carriers form an overhead and do not contribute to the user bit rate.

This results in a loss of E_b/N₀:

$ß_{2} = \frac{N_{S D}}{N_{S D} + N_{S P}}$

Where

$N_{S D}$ Number of data subcarriers

$N_{S P}$ Number of pilot subcarriers

Overall loss

The overall loss of both cyclic prefix and pilot subcarriers is given by

$ß = \frac{N}{(N + N_{G})} \frac{N_{S D}}{(N_{S D} + N_{S P})}$

Where

$ß = 1$ for single carrier systems

$ß \leq 1$ for OFDM systems

This applies to any modulation scheme. For BPSK and QPSK the BER for AWGN channel is given by:

$p_{b} = 0.5 e r f c (\sqrt{ß \frac{E_{b}}{N_{0}}})$

OFDM systems show a loss of E_b/N₀ due to cyclic prefix and pilot carriers, see tutorial.

ß = \frac{N}{(N + N_{G})} \frac{N_{S D}}{(N_{S D} + N_{S P})}

Example

In this experiment we analyze a specific IEEE 802.11ac OFDM setup. The theoretical BER for AWGN channel is calculated and compared with simulation results.

The following setting of IEEE 802.11ac is used:

VHT- MCS Index	Modulation	$R$	$N_{B P S C}$	$N_{S D}$	$N_{S P}$	Data rate (Mb/s)
VHT- MCS Index	Modulation	$R$	$N_{B P S C}$	$N_{S D}$	$N_{S P}$	800 ns GI	400 ns GI
1	QPSK	1/2	2	234	8	58.5	65.0

Physical layer parameters IEEE 802.11ac, 80 MHz,

N_{S S} = 1

For the 400 ns GI setup the discrete length of the guard interval $N_{G}$ is 32. The number of subcarriers $N$ is 256. The loss of E_b/N₀ yields to

ß = \frac{N}{(N + N_{G})} \frac{N_{S D}}{(N_{S D} + N_{S P})} = \frac{256}{(256 + 32)} \frac{234}{(234 + 8)} = 0,8595 = - 0,658 d B .

The effective energy per bit to noise power-spectral-density ratio is $ß E_{b} / N_{0}$ . This loss results in a BER degeneration as shown in the following table.

$E_{b} / N_{0}$ in dB	SC BER	$ß E_{b} / N_{0}$ in dB	OFDM BER
-20	4,438E-01	-20,658	4,478E-01
-15	4,007E-01	-15,658	4,078E-01
-10	3,274E-01	-10,658	3,392E-01
-5	2,132E-01	-5,658	2,305E-01
0	7,865E-02	-0,658	9,491E-02
5	5,954E-03	4,342	9,863E-03

OFDM shows increased BER compared to single carrier systems (SC) in AWGN channel

Start

OFDM BER vs E<sub>b</sub>/N<sub>0</sub> block diagram

OFDM bit-error rate performance in AWGN channel - block diagram. Implementation of bit error analysis for adjustable E_b/N₀. The Monte Carlo simulation starts with

E_{b} / N_{0} = 0 d B

and approximates the corresponding analytical bit error probability.

Let's first check the E_b/N₀ setting.

The uncoded bitrate is $R_{B} = 130 M b / s$ . That's the simulation's bitrate as it doesn't consider coding.
The transmission power after the symbol mapper is $1 V^{2}$ . Unused subcarriers reduce the transmission power: $S_{S} = \frac{N_{S D} + N_{S P}}{N} 1 V^{2} = \frac{234 + 8}{256} = 0,9453 V^{2}$
$\to$ Measure the transmission power in the simulation!
The energy per bit yields to $E_{b} = \frac{S_{S}}{R_{B}} = \frac{0,9453 V^{2}}{130 M b / s} = 7,2716 n V^{2} s$
Noise power spectral density must be set to $N_{0} = 7,2716 n V^{2} / H z$ for $E_{b} / N_{0} = 1$ .
$\to$ Check the value of Power-spectral-density N0 of the block Noise in the simulation (click on Noise).

Measure the bit error rate in the simulation!

Adjust E_b/N₀ (press F11) and measure the corresponding bit error rate.

Dialog to adjust E<sub>b</sub>/N<sub>0</sub>

Next steps

Now select a different OFDM setup: Simulation - Setup (F12). When the guard interval (GI) is increased to 800 ns the loss of E_b/N₀ and BER will increase. Calculate and measure the BER analogous to the table above.

Select specific IEEE 802.11ac OFDM setup

This simulation app implements an OFDM transmission using AWGN channel. The bit error rate is measured for variable E_b/N₀.

OFDM transmission using AWGN channel - block diagram. Bit error rate is measured for variable E<sub>b</sub>/N<sub>0</sub>.

The Monte Carlo simulation starts with

E_{b} / N_{0} = 0 d B

and shows the resulting bit error rate. The parameters of Wi-Fi 80 MHz, 400 ns GI, MCS 1, 2 are used. Different IEEE 802.11ac OFDM schemes can be selected.

Key	Action
Simulation - Settings (F11)	Adjust E_b/N₀ and measure the corresponding bit error rate.
Simulation - Setup (F12)	Modify the OFDM config and press OK to restart the simulation.

Key

Action

Simulation - Settings (F11)

settings

Adjust E_b/N₀ and measure the corresponding bit error rate.

Simulation - Setup (F12)

Setup

Modify the OFDM config and press OK to restart the simulation.

OFDM BER vs Eb/N0 in AWGN channel

OFDM loss of Eb/N0 resulting from cyclic prefix and pilot subcarriers

Effect of cyclic prefix

Effect of pilot subcarriers

Overall loss

Example

Start

Next steps

OFDM loss of E_b/N₀ resulting from cyclic prefix and pilot subcarriers