OFDM BER vs Eb/N0 in AWGN channel

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

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

Effect of cyclic prefix

The last N G MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBaaaleaacaWGhbaabeaaaaa@2FE4@ symbols of the OFDM symbol of length N MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@2EEC@ are copied and added as cyclic prefix. As the receiver removes the cyclic prefix its energy is lost. This results in a loss of Eb/N0:

ß 1 = N N+ N G MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43amaaBaaaleaacaaIXaaabeaakiabg2da9maalaaabaGaamOtaaqaaiaad6eacqGHRaWkcaWGobWaaSbaaSqaaiaadEeaaeqaaaaaaaa@35D7@

Where

N MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@2EEC@ Number of subcarriers

N G MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBaaaleaacaWGhbaabeaaaaa@2FE4@ 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 Eb/N0:

ß 2 = N SD N SD + N SP MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43amaaBaaaleaacaaIYaaabeaakiabg2da9maalaaabaGaamOtamaaBaaaleaacaWGtbGaamiraaqabaaakeaacaWGobWaaSbaaSqaaiaadofacaWGebaabeaakiabgUcaRiaad6eadaWgaaWcbaGaam4uaiaadcfaaeqaaaaaaaa@3A67@

Where

N SD MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBaaaleaacaWGtbGaamiraaqabaaaaa@30B9@ Number of data subcarriers

N SP MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBaaaleaacaWGtbGaamiuaaqabaaaaa@30C5@ Number of pilot subcarriers


Overall loss

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

ß= N ( N+ N G ) N SD ( N SD + N SP ) MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9maalaaabaGaamOtaaqaamaabmaabaGaamOtaiabgUcaRiaad6eadaWgaaWcbaGaam4raaqabaaakiaawIcacaGLPaaaaaWaaSaaaeaacaWGobWaaSbaaSqaaiaadofacaWGebaabeaaaOqaamaabmaabaGaamOtamaaBaaaleaacaWGtbGaamiraaqabaGccqGHRaWkcaWGobWaaSbaaSqaaiaadofacaWGqbaabeaaaOGaayjkaiaawMcaaaaaaaa@40FE@

Where

ß=1 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9iaaigdaaaa@313E@ for single carrier systems

ß1 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabgsMiJkaaigdaaaa@31ED@ for OFDM systems


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

p b =0.5erfc( ß E b N 0 ) MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBaaaleaacaWGIbaabeaakiabg2da9iaaicdacaGGUaGaaGynaiaadwgacaWGYbGaamOzaiaadogadaqadaqaamaakaaabaGaam43amaalaaabaGaamyramaaBaaaleaacaWGIbaabeaaaOqaaiaad6eadaWgaaWcbaGaaGimaaqabaaaaaqabaaakiaawIcacaGLPaaaaaa@3DC7@

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

ß= N ( N+ N G ) N SD ( N SD + N SP ) MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaad+nacqGH9aqpdaWcaaqaaiaad6eaaeaadaqadaqaaiaad6eacqGHRaWkcaWGobWaaSbaaSqaaiaadEeaaeqaaaGccaGLOaGaayzkaaaaamaalaaabaGaamOtamaaBaaaleaacaWGtbGaamiraaqabaaakeaadaqadaqaaiaad6eadaWgaaWcbaGaam4uaiaadseaaeqaaOGaey4kaSIaamOtamaaBaaaleaacaWGtbGaamiuaaqabaaakiaawIcacaGLPaaaaaaaaa@45AA@

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 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaaaa@2EF0@ N BPSC MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBaaaleaacaWGcbGaamiuaiaadofacaWGdbaabeaaaaa@3254@ N SD MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBaaaleaacaWGtbGaamiraaqabaaaaa@30B9@ N SP MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBaaaleaacaWGtbGaamiuaaqabaaaaa@30C5@ Data rate (Mb/s)
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 SS =1 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBaaaleaacaWGtbGaam4uaaqabaGccqGH9aqpcaaIXaaaaa@3293@

For the 400 ns GI setup the discrete length of the guard interval N G MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBaaaleaacaWGhbaabeaaaaa@2FE4@ is 32. The number of subcarriers N MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@2EEC@ is 256. The loss of Eb/N0 yields to

ß= N ( N+ N G ) N SD ( N SD + N SP ) = 256 ( 256+32 ) 234 ( 234+8 ) =0,8595=0,658dB. MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@6438@

The effective energy per bit to noise power-spectral-density ratio is ß E b / N 0 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacaWGFdGaamyramaaBaaaleaacaWGIbaabeaaaOqaaiaad6eadaWgaaWcbaGaaGimaaqabaaaaaaa@3333@ . This loss results in a BER degeneration as shown in the following table.

E b / N 0 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacaWGfbWaaSbaaSqaaiaadkgaaeqaaaGcbaGaamOtamaaBaaaleaacaaIWaaabeaaaaaaaa@31CF@ in dB SC BER ß E b / N 0 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacaWGFdGaamyramaaBaaaleaacaWGIbaabeaaaOqaaiaad6eadaWgaaWcbaGaaGimaaqabaaaaaaa@3333@ 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 Eb/N0. The Monte Carlo simulation starts with E b / N 0 =0dB MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacaWGfbWaaSbaaSqaaiaadkgaaeqaaaGcbaGaamOtamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdacaWGKbGaamOqaaaaaaa@3549@ and approximates the corresponding analytical bit error probability.

Let's first check the Eb/N0 setting.


Measure the bit error rate in the simulation!

BER Bit error rate meter

Adjust Eb/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 Eb/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 Eb/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 =0dB MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacaWGfbWaaSbaaSqaaiaadkgaaeqaaaGcbaGaamOtamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdacaWGKbGaamOqaaaaaaa@3549@ 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)

settings

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

Simulation - Setup (F12)

Setup

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