Orthogonal Frequency Multiplexing Division

Orthogonal Frequency Division Multiplexing ( OFDM or Orthogonal Frequency Division Modulation ) is a modulation of numeric signals per distribution in orthogonal frequencies. DMT (Discrete Multi Thunders) and COFDM (Coded Orthogonal Frequency Division Multiplexing) uses a similar principle.

The OFDM is a numerical process of modulation of the signals which is used inter alia for the mobile systems of transmissions to high banc of data. The OFDM is particularly well adapted to the transmission channels radio on long distances without multiple transmissions of wave (echoes), it then makes it possible to reduce the intersymbol interferences appreciably. On the other hand it can become unusable if the echoes are strong, it is then necessary to use COFDM.

The OFDM (or a comparable technique) is used in

• Radiodiffusion for digital terrestrial television (DVB-T, DVB-H) and the regional DAB and world terrestrial numerical radio DRM.
• telegraphic connections: ADSL, VDSL, modem on transmission current (Homeplug), cable modem (Standard Docsis).
• the networks wireless telegraphies based on the standards 802.11a, 802.11g (Wifi), 802.16 (Wimax) and HiperLAN.
• mobile networks of new generation (4G).

Principle

The principle of the OFDM consists in dividing on a great number of carrying the numeric signal which one wants to transmit. As if one combined the signal to be transmitted on a great number of systems of transmission (example: transmitters) independent and at different frequencies. So that the frequencies of carrying are closest possible and thus to transmit the maximum of information on a given portion of frequencies, the OFDM uses the carrying orthogonal ones between them. The signals of different carrying overlap but thanks to orthogonality do not interfere between them. The signal to be transmitted is generally repeated on various carrier frequencies. Thus in a transmission channel with multiple ways where certain frequencies will be destroyed because of the destructive combination of ways, the system will be all the same able to recover the information lost on other carrier frequencies which will not have been destroyed. Each carrying is modulated independently by using numerical modulations: QPSK, QAM-16, QAM-64,…

This principle makes it possible to limit the interference between symbols. To eliminate it, one can add a interval of guard (i.e. one period during which there is no transmission) after each very large symbol emitted in front of the time of transmission (the speed of light multiplied by the distance separating the transmitter er the receiver).

Mathematical description

The low-pass equivalent of a signal OFDM is expressed as follows:

where $\backslash \; \{I\_k\; \backslash \}$ is the symbols of data, $N$ is the number of subcarriers and $T$ the duration of block OFDM. Spacing between carrying $1/T$ Hz makes the subcarriers orthogonal between them; this property is expressed as follows:

$\backslash \; frac\; \{1\}\; \{T\}\; \backslash \; int\_0^\; \{T\}\; \backslash \; left\; (e^\; \{i2\; \backslash \; pi\; k\_1t/T\}\; \backslash \; right)\; ^*\; \backslash \; left\; (e^\; \{i2\; \backslash \; pi\; k\_2t/T\}\; \backslash \; right)\; dt=\; \backslash \; frac\; \{1\}\; \{T\}\; \backslash \; int\_0^\; \{T\}\; e^\; \{i2\; \backslash \; pi\; (k\_2-k\_1)\; t/T\}\; dt$

\ begin {boxes} 1, & k_1=k_2, \ \ 0, & k_1 \ k_2, \ end {boxes}

où $(\backslash \; cdot)\; ^*$ corresponds to the complex conjugé operator.

To avoid the interference intersymbol in an environment of propagation multichemins, an interval of guard , where $T\_\; \backslash \; mathrm\; \{G\}$ is the period of guard, is inserted before block OFDM. During this interval, a préfixe cyclique is transmitted. This cyclic prefix is equal to the last $T\_\; \backslash \; mathrm\; \{G\}$ of block OFDM. Signal OFDM with the cyclic prefix is thus:

The low-pass signal above can is to be made up of actual value or complex. For the signal with actual values this one is generally transmitted in baseband and expressed as follows:

$S\; (T)\; =\; \backslash \; Re\; \backslash \; naked\; left\; \backslash \; \{\backslash \; (T)\; e^\; \{i2\; \backslash \; pi\; f\_c\; T\}\; \backslash \; right\; \backslash \}.$

The signal in baseband with complex values is on the other hand modulated at a higher frequency $f\_c$. In general the signal is represented as follows:

$S\; (T)\; =\; \backslash \; sum\_\; \{k=0\}\; ^\; \{N-1\}|I\_k|\backslash \; cos\; \backslash \; left\; (2\; \backslash \; pi\; +\; k/Tt\; +\; \backslash \; arg\; \backslash \; right).$

See too

Internal bonds

• TNT
• WiMAX
• Audio DIGITAL Broadcasting
• DIGITAL Radio World
• ADSL

External bond

• Article in the technical review of UER on the OFDM: J.H Stott, It why and it how COFDM

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