Editing Orthogonal frequency-division multiplexing (section)

=== Simplified equalization ===
The effects of frequency-selective channel conditions, for example fading caused by multipath propagation, can be considered as constant (flat) over an OFDM sub-channel if the sub-channel is sufficiently narrow-banded (i.e., if the number of sub-channels is sufficiently large). This makes frequency domain equalization possible at the [[receiver (radio)|receiver]], which is far simpler than the time-domain equalization used in conventional single-carrier modulation. In OFDM, the equalizer only has to multiply each detected subcarrier (each Fourier coefficient) in each OFDM symbol by a constant [[complex number]], or a rarely changed value. On a fundamental level, simpler digital equalizers are better because they require fewer operations, which translates to fewer round-off errors in the equalizer. Those round-off errors can be viewed as numerical noise and are inevitable.

:Our example: The OFDM equalization in the above numerical example would require one complex valued multiplication per subcarrier and symbol (i.e., <math>\scriptstyle N \,=\, 1000</math> complex multiplications per OFDM symbol; i.e., one million multiplications per second, at the receiver). The FFT algorithm requires <math>\scriptstyle N \log_2 N \,=\, 10,000</math> [this is imprecise: over half of these complex multiplications are trivial, i.e. = to 1 and are not implemented in software or HW]. complex-valued multiplications per OFDM symbol (i.e., 10 million multiplications per second), at both the receiver and transmitter side. This should be compared with the corresponding one million symbols/second single-carrier modulation case mentioned in the example, where the equalization of 125 microseconds time-spreading using a [[FIR filter]] would require, in a naive implementation, 125 multiplications per symbol (i.e., 125 million multiplications per second). FFT techniques can be used to reduce the number of multiplications for an [[FIR filter]]-based time-domain equalizer to a number comparable with OFDM, at the cost of delay between reception and decoding which also becomes comparable with OFDM.

If differential modulation such as [[DPSK]] or [[DQPSK]] is applied to each subcarrier, equalization can be completely omitted, since these non-coherent schemes are insensitive to slowly changing amplitude and [[phase distortion]].

In a sense, improvements in FIR equalization using FFTs or partial FFTs leads mathematically closer to OFDM,{{Citation needed|date=January 2011}} but the OFDM technique is easier to understand and implement, and the sub-channels can be independently adapted in other ways than varying equalization coefficients, such as switching between different [[QAM]] constellation patterns and error-correction schemes to match individual sub-channel noise and interference characteristics.{{Clarify|date=January 2011}}

Some of the subcarriers in some of the OFDM symbols may carry [[pilot signal]]s for measurement of the channel conditions<ref>{{cite journal |vauthors=Coleri S, Ergen M, Puri A, Bahai A |title=Channel estimation techniques based on pilot arrangement in OFDM systems |journal=IEEE Transactions on Broadcasting |volume=48 |issue=3 |pages=223–229 |date=Sep 2002 |doi=10.1109/TBC.2002.804034 }}</ref><ref>{{cite conference |vauthors=Hoeher P, Kaiser S, Robertson P |title=1997 IEEE International Conference on Acoustics, Speech, and Signal Processing |chapter=Two-dimensional pilot-symbol-aided channel estimation by Wiener filtering |conference=IEEE [[International Conference on Acoustics, Speech, and Signal Processing]], ICASSP-97 |year=1997 |volume=3 |pages=1845–1848 |doi=10.1109/ICASSP.1997.598897|isbn=0-8186-7919-0 }}</ref> (i.e., the equalizer gain and phase shift for each subcarrier). Pilot signals and training symbols ([[Preamble (communication)|preambles]]) may also be used for time synchronization (to avoid intersymbol interference, ISI) and frequency synchronization (to avoid inter-carrier interference, ICI, caused by Doppler shift).

OFDM was initially used for wired and stationary wireless communications. However, with an increasing number of applications operating in highly mobile environments, the effect of dispersive fading caused by a combination of multi-path propagation and [[doppler shift]] is more significant. Over the last decade, research has been done on how to equalize OFDM transmission over doubly selective channels.<ref>{{cite journal |vauthors=Zemen T, Mecklenbrauker CF |title=Time-Variant Channel Estimation Using Discrete Prolate Spheroidal Sequences |journal=IEEE Transactions on Signal Processing |volume=53 |issue=9 |pages=3597–3607 |date=Sep 2005 |doi=10.1109/TSP.2005.853104 |citeseerx=10.1.1.60.9526 |bibcode=2005ITSP...53.3597Z |s2cid=16493970 }}</ref><ref>{{cite journal |vauthors=Tang Z, Cannizzaro RC, Leus G, Banelli P |title=Pilot-Assisted Time-Varying Channel Estimation for OFDM Systems |journal=IEEE Transactions on Signal Processing |volume=55 |issue=5 |pages=2226–2238 |date=May 2007 |doi=10.1109/TSP.2007.893198 |citeseerx=10.1.1.418.2386 |bibcode=2007ITSP...55.2226T |s2cid=570753 }}</ref><ref>{{cite journal |vauthors=Hrycak T, Das S, Matz G, Feichtinger HG |title=Low Complexity Equalization for Doubly Selective Channels Modeled by a Basis Expansion |journal=IEEE Transactions on Signal Processing |volume=58 |issue=11 |pages=5706–5719 |date=Aug 2010 |doi=10.1109/TSP.2010.2063426 |bibcode=2010ITSP...58.5706H |s2cid=17077919 }}</ref>