Editing Orthogonal frequency-division multiplexing (section)

=== Orthogonality ===
Conceptually, OFDM is a specialized [[frequency-division multiplexing]] (FDM) method, with the additional constraint that all subcarrier signals within a communication channel are orthogonal to one another.

In OFDM, the subcarrier frequencies are chosen so that the subcarriers are [[orthogonality#Telecommunications|orthogonal]] to each other, meaning that [[crosstalk]] between the sub-channels is eliminated and inter-carrier guard bands are not required. This greatly simplifies the design of both the [[transmitter]] and the [[receiver (radio)|receiver]]; unlike conventional FDM, a separate filter for each sub-channel is not required.

The orthogonality requires that the '''subcarrier spacing''' is <math>\scriptstyle\Delta f \,=\, \frac{k}{T_U}</math> [[Hertz]], where ''T''<sub>U</sub> [[second]]s is the useful symbol duration (the receiver-side window size), and ''k'' is a positive integer, typically equal to 1. This stipulates that each carrier frequency undergoes ''k'' more complete cycles per symbol period than the previous carrier. Therefore, with ''N'' subcarriers, the total passband bandwidth will be ''B'' ≈ ''N''·Δ''f'' (Hz).

The orthogonality also allows high [[spectral efficiency]], with a total symbol rate near the [[Nyquist rate]] for the equivalent baseband signal (i.e., near half the Nyquist rate for the double-side band physical passband signal). Almost the whole available frequency band can be used. OFDM generally has a nearly 'white' spectrum, giving it benign electromagnetic interference properties with respect to other co-channel users.

:A simple example: A useful symbol duration ''T''<sub>U</sub> = 1&nbsp;ms would require a subcarrier spacing of <math>\scriptstyle\Delta f \,=\, \frac{1}{1\,\mathrm{ms}} \,=\, 1\,\mathrm{kHz}</math> (or an integer multiple of that) for orthogonality. ''N'' = 1,000 subcarriers would result in a total passband bandwidth of ''N''Δf = 1&nbsp;MHz. For this symbol time, the required bandwidth in theory according to Nyquist is  <math>\scriptstyle\mathrm{BW}=R/2=(N/T_U)/2 = 0.5\,\mathrm{MHz}</math> (half of the achieved bandwidth required by our scheme), where ''R'' is the bit rate and where ''N'' = 1,000 samples per symbol by FFT. If a guard interval is applied (see below), Nyquist bandwidth requirement would be even lower. The FFT would result in ''N'' = 1,000 samples per symbol. If no guard interval was applied, this would result in a base band complex valued signal with a sample rate of 1&nbsp;MHz, which would require a baseband bandwidth of 0.5&nbsp;MHz according to Nyquist. However, the passband RF signal is produced by multiplying the baseband signal with a carrier waveform (i.e., double-sideband quadrature amplitude-modulation) resulting in a passband bandwidth of 1&nbsp;MHz. A single-side band (SSB) or vestigial sideband (VSB) modulation scheme would achieve almost half that bandwidth for the same symbol rate (i.e., twice as high spectral efficiency for the same symbol alphabet length). It is however more sensitive to multipath interference.

OFDM requires very accurate frequency synchronization between the receiver and the transmitter; with frequency deviation the subcarriers will no longer be orthogonal, causing ''inter-carrier interference'' (ICI) (i.e., cross-talk between the subcarriers). Frequency offsets are typically caused by mismatched transmitter and receiver oscillators, or by [[Doppler shift]] due to movement. While Doppler shift alone may be compensated for by the receiver, the situation is worsened when combined with [[Multipath interference|multipath]], as reflections will appear at various frequency offsets, which is much harder to correct. This effect typically worsens as speed increases,<ref>{{cite book|doi=10.1109/vetecf.1999.797150|chapter=The effects of Doppler spreads in OFDM(A) mobile radio systems|title=Gateway to 21st Century Communications Village. VTC 1999-Fall. IEEE VTS 50th Vehicular Technology Conference|pages=329–333|volume=1|year=1999|last1=Robertson|first1=P.|last2=Kaiser|first2=S.|isbn=0-7803-5435-4|s2cid=2052913}}</ref> and is an important factor limiting the use of OFDM in high-speed vehicles. In order to mitigate ICI in such scenarios, one can shape each subcarrier in order to minimize the interference resulting in a non-orthogonal subcarriers overlapping.<ref>{{cite journal |title=A Time-Frequency Well-localized Pulse for Multiple Carrier Transmission | last1=Haas | first1=R. | last2=Belfiore | first2=J.C. |journal= Wireless Personal Communications |year= 1997 |volume= 5 |number= 1 |pages= 1–18 |doi= 10.1023/A:1008859809455 | s2cid=5062251 }}</ref> For example, a low-complexity scheme referred to as WCP-OFDM (''Weighted Cyclic Prefix Orthogonal Frequency-Division Multiplexing'') consists of using short filters at the transmitter output in order to perform a potentially non-rectangular pulse shaping and a near perfect reconstruction using a single-tap per subcarrier equalization.<ref>{{cite journal |title=Performances of Weighted Cyclic Prefix OFDM with Low-Complexity Equalization | last1=Roque | first1=D. | last2=Siclet | first2=C. |journal= IEEE Communications Letters |year= 2013 |volume= 17 |number= 3 |pages= 439–442 |doi= 10.1109/LCOMM.2013.011513.121997 | s2cid=9480706 |url=https://hal.archives-ouvertes.fr/hal-01260517/file/wcp-ofdm-equalization-ieee-sp-letter-submitted.pdf }}</ref> Other ICI suppression techniques usually drastically increase the receiver complexity.<ref>{{cite journal |title=An equalization technique for orthogonal frequency-division multiplexing systems in time-variant multipath channels | last1=Jeon | first1=W.G. | last2=Chang | first2=K.H. | last3=Cho | first3=Y.S. |journal= IEEE Transactions on Communications |year= 1999 |volume= 47 |number= 1 |pages= 27–32 |doi= 10.1109/26.747810 |citeseerx=10.1.1.460.4807 }}</ref>