Editing Orthogonal frequency-division multiplexing (section)

== Characteristics and principles of operation ==

=== 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>

===Implementation using the FFT algorithm===
The orthogonality allows for efficient modulator and demodulator implementation using the [[fast Fourier transform|FFT]] algorithm on the receiver side, and inverse FFT on the sender side. Although the principles and some of the benefits have been known since the 1960s, OFDM is popular for wideband communications today by way of low-cost [[digital signal processing]] components that can efficiently calculate the FFT.

The time to compute the inverse-FFT or FFT has to take less than the time for each symbol,<ref name=ce883>{{cite thesis |type=B.E. |url=http://ce.sharif.ir/courses/85-86/2/ce883/resources/root/Lectures/About-OFDM.pdf |title=The suitability of OFDM as a modulation technique for wireless telecommunications, with a CDMA comparison |author=Eric Lawrey |date=October 1997 |access-date=2012-08-28 |archive-date=2012-09-14 |archive-url=https://web.archive.org/web/20120914014043/http://ce.sharif.ir/courses/85-86/2/ce883/resources/root/Lectures/About-OFDM.pdf |url-status=dead }}</ref>{{rp|84}} which for example for [[DVB-T]] {{nowrap|(FFT 8k)}} means the computation has to be done in {{nowrap|896 µs}} or less.

For an {{gaps|8192}}-point [[fast Fourier transform|FFT]] this may be approximated to:<ref name=ce883/><!-- extrapolated from the FFT 2048 example -->{{clarify|reason=Normally complexity is said to be proportional to N log2(N) per block for FFT; i.e., 8192×13 when N=8192.|date=September 2012}}

:<math>\begin{align}
  \mathrm{MIPS}
   &= \frac {\mathrm{computational\ complexity}}{T_\mathrm{symbol}} \times 1.3 \times 10^{-6} \\
   &= \frac{147\;456 \times 2}{896 \times 10^{-6}} \times 1.3 \times 10^{-6} \\
   &= 428
\end{align}</math>
* MIPS: [[Instructions per second#Million instructions per second|Million instructions per second]]

The computational demand approximately scales linearly with FFT size so a double size FFT needs double the amount of time and vice versa.<ref name=ce883/>{{rp|83}}<!--someone deduce this directly from n log(n) which needs ×4 to match ? --> As a comparison an [[Pentium III|Intel Pentium III]] CPU at 1.266&nbsp;GHz is able to calculate a {{nowrap|8192 point}} FFT in <!--$n=8192; $mflops=925; 5*$n*(log($n)/log(2))/$mflops=575.654054-->{{nowrap|576 µs}} using [[FFTW]].<!--double-precision complex, 1d transforms FFTW3 out-of-place--><ref>{{cite web |url=http://www.fftw.org/speed/Pentium3-1.266GHz/ |website=fftw.org |title=1.266 GHz Pentium 3 |date=2006-06-20}}</ref> [[Pentium M|Intel Pentium M]] at 1.6&nbsp;GHz does it in {{nowrap|387 µs.}}<ref>{{cite web |url=http://www.fftw.org/speed/PentiumM-1.6GHz-gcc/ |website=fftw.org |title=1.6 GHz Pentium M (Banias), GNU compilers |date=2006-06-20}}</ref> [[Core Duo|Intel Core Duo]] at 3.0&nbsp;GHz does it in {{nowrap|96.8 µs}}.<ref>{{cite web |url=http://www.fftw.org/speed/CoreDuo-3.0GHz-icc/ |website=fftw.org |title=3.0 GHz Intel Core Duo, Intel compilers, 32-bit mode |date=2006-10-09}}</ref>

=== Guard interval for elimination of intersymbol interference ===
One key principle of OFDM is that since low symbol rate modulation schemes (i.e., where the symbols are relatively long compared to the channel time characteristics) suffer less from [[intersymbol interference]] caused by [[multipath propagation]], it is advantageous to transmit a number of low-rate streams in parallel instead of a single high-rate stream. Since the duration of each symbol is long, it is feasible to insert a [[guard interval]] between the OFDM symbols, thus eliminating the intersymbol interference.

The guard interval also eliminates the need for a [[pulse-shaping filter]], and it reduces the sensitivity to time synchronization problems.

:A simple example: If one sends a million symbols per second using conventional single-carrier modulation over a wireless channel, then the duration of each symbol would be one microsecond or less. This imposes severe constraints on synchronization and necessitates the removal of multipath interference. If the same million symbols per second are spread among one thousand sub-channels, the duration of each symbol can be longer by a factor of a thousand (i.e., one millisecond) for orthogonality with approximately the same bandwidth. Assume that a guard interval of 1/8 of the symbol length is inserted between each symbol. Intersymbol interference can be avoided if the multipath time-spreading (the time between the reception of the first and the last echo) is shorter than the guard interval (i.e., 125 microseconds). This corresponds to a maximum difference of 37.5 kilometers between the lengths of the paths.

The [[cyclic prefix]], which is transmitted during the guard interval, consists of the end of the OFDM symbol copied into the guard interval, and the guard interval is transmitted followed by the OFDM symbol. The reason that the guard interval consists of a copy of the end of the OFDM symbol is so that the receiver will integrate over an integer number of sinusoid cycles for each of the multipaths when it performs OFDM demodulation with the FFT.
[[File:OFDMCyclicPrefixInsertion.svg|800px|center]]

In some standards such as [[Ultrawideband]], in the interest of transmitted power, cyclic prefix is skipped and nothing is sent during the guard interval. The receiver will then have to mimic the cyclic prefix functionality by copying the end part of the OFDM symbol and adding it to the beginning portion.

=== 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>

=== Channel coding and interleaving ===
OFDM is invariably used in conjunction with [[channel coding]] ([[forward error correction]]), and almost always uses frequency and/or time [[Bit interleaving|interleaving]].

Frequency (subcarrier) [[Bit interleaving|interleaving]] increases resistance to frequency-selective channel conditions such as [[fading]]. For example, when a part of the channel bandwidth fades, frequency interleaving ensures that the bit errors that would result from those subcarriers in the faded part of the bandwidth are spread out in the bit-stream rather than being concentrated. Similarly, time interleaving ensures that bits that are originally close together in the bit-stream are transmitted far apart in time, thus mitigating against severe fading as would happen when travelling at high speed.

However, time interleaving is of little benefit in slowly fading channels, such as for stationary reception, and frequency interleaving offers little to no benefit for narrowband channels that suffer from flat-fading (where the whole channel bandwidth fades at the same time).

The reason why interleaving is used on OFDM is to attempt to spread the errors out in the bit-stream that is presented to the error correction decoder, because when such decoders are presented with a high concentration of errors the decoder is unable to correct all the bit errors, and a burst of uncorrected errors occurs. A similar design of audio data encoding makes compact disc (CD) playback robust.

A classical type of error correction coding used with OFDM-based systems is [[convolutional code|convolutional coding]], often [[concatenated error correction codes|concatenated]] with [[Reed–Solomon error correction|Reed-Solomon]] coding. Usually, additional interleaving (on top of the time and frequency interleaving mentioned above) in between the two layers of coding is implemented. The choice for Reed-Solomon coding as the outer error correction code is based on the observation that the Viterbi decoder used for inner convolutional decoding produces short error bursts when there is a high concentration of errors, and Reed-Solomon codes are inherently well suited to correcting bursts of errors.

Newer systems, however, usually now adopt near-optimal types of error correction codes that use the turbo decoding principle, where the decoder iterates towards the desired solution. Examples of such error correction coding types include [[turbo code]]s and [[LDPC]] codes, which perform close to the [[Shannon limit]] for the Additive White Gaussian Noise ([[Additive white Gaussian noise|AWGN]]) channel. Some systems that have implemented these codes have concatenated them with either Reed-Solomon (for example on the [[MediaFLO]] system) or [[BCH code]]s (on the [[DVB-S2]] system) to improve upon an [[error floor]] inherent to these codes at high [[signal-to-noise ratio]]s.<ref name="bookMPLC">{{cite book|title= MIMO Power Line Communications: Narrow and Broadband Standards, EMC, and Advanced Processing |series= Devices, Circuits, and Systems |date=February 2014|editor= Berger, Lars T. |editor2=Schwager, Andreas |editor3=Pagani, Pascal |editor4=Schneider, Daniel M|publisher= CRC Press |page=25|doi=10.1201/b16540-1|isbn=978-1-4665-5753-6|chapter= Introduction to Power Line Communication Channel and Noise Characterisation }}</ref>

=== Adaptive transmission ===
The resilience to severe channel conditions can be further enhanced if information about the channel is sent over a return-channel. Based on this feedback information, adaptive [[modulation]], channel coding and power allocation may be applied across all subcarriers, or individually to each subcarrier. In the latter case, if a particular range of frequencies suffers from interference or attenuation, the carriers within that range can be disabled or made to run slower by applying more robust modulation or [[error coding]] to those subcarriers.

The term '''{{visible anchor|discrete multitone modulation}}''' ('''DMT''') denotes OFDM-based communication systems that adapt the transmission to the channel conditions individually for each subcarrier, by means of so-called ''bit-loading''. Examples are [[ADSL]] and [[VDSL]].

The upstream and downstream speeds can be varied by allocating either more or fewer carriers for each purpose. Some forms of [[rate-adaptive DSL]] use this feature in real time, so that the bitrate is adapted to the co-channel interference and bandwidth is allocated to whichever subscriber needs it most.

=== OFDM extended with multiple access ===
{{Main|Orthogonal frequency-division multiple access}}
OFDM in its primary form is considered as a digital modulation technique, and not a multi-user [[channel access method]], since it is used for transferring one bit stream over one communication channel using one sequence of OFDM symbols. However, OFDM can be combined with [[multiple access]] using time, frequency or coding separation of the users.

In [[orthogonal frequency-division multiple access]] (OFDMA), [[frequency-division multiple access]] is achieved by assigning different OFDM sub-channels to different users. OFDMA supports differentiated [[quality of service]] by assigning different number of subcarriers to different users in a similar fashion as in [[CDMA]], and thus complex packet scheduling or [[medium access control]] schemes can be avoided. OFDMA is used in:
* the mobility mode of the [[IEEE 802.16]] Wireless MAN standard, commonly referred to as WiMAX,
* the [[IEEE 802.20]] mobile Wireless MAN standard, commonly referred to as MBWA,
* the [[3GPP Long Term Evolution]] (LTE) fourth generation mobile broadband standard downlink. The radio interface was formerly named High Speed OFDM Packet Access (HSOPA), now named Evolved UMTS Terrestrial Radio Access ([[E-UTRA]]).
* the [[5G NR|3GPP 5G NR]] (New Radio) fifth generation mobile network standard downlink and uplink. 5G NR is the successor to LTE. 
* the now defunct [[Qualcomm]]/[[3GPP2]] [[Ultra Mobile Broadband]] (UMB) project, intended as a successor of [[CDMA2000]], but replaced by LTE.
OFDMA is also a candidate access method for the [[IEEE 802.22]] ''Wireless Regional Area Networks'' (WRAN). The project aims at designing the first [[cognitive radio]]-based standard operating in the VHF-low UHF spectrum (TV spectrum).
* the most recent amendment of [[802.11]] standard, namely [[802.11ax]], includes OFDMA for high efficiency and simultaneous communication.

In [[multi-carrier code-division multiple access]] (MC-CDMA), also known as OFDM-CDMA, OFDM is combined with CDMA spread spectrum communication for coding separation of the users. Co-channel interference can be mitigated, meaning that manual [[fixed channel allocation]] (FCA) frequency planning is simplified, or complex [[dynamic channel allocation]] (DCA) schemes are avoided.

=== Space diversity ===
In OFDM-based wide-area broadcasting, receivers can benefit from receiving signals from several spatially dispersed transmitters simultaneously, since transmitters will only destructively interfere with each other on a limited number of subcarriers, whereas in general they will actually reinforce coverage over a wide area. This is very beneficial in many countries, as it permits the operation of national [[single-frequency network]]s (SFN), where many transmitters send the same signal simultaneously over the same channel frequency. SFNs use the available spectrum more effectively than conventional multi-frequency broadcast networks ([[Multi-frequency network|MFN]]), where program content is replicated on different carrier frequencies. SFNs also result in a [[diversity gain]] in receivers situated midway between the transmitters. The coverage area is increased and the outage probability decreased in comparison to an MFN, due to increased received signal strength averaged over all subcarriers.

Although the guard interval only contains redundant data, which means that it reduces the capacity, some OFDM-based systems, such as some of the broadcasting systems, deliberately use a long guard interval in order to allow the transmitters to be spaced farther apart in an SFN, and longer guard intervals allow larger SFN cell-sizes. A rule of thumb for the maximum distance between transmitters in an SFN is equal to the distance a signal travels during the guard interval — for instance, a guard interval of 200 microseconds would allow transmitters to be spaced 60&nbsp;km apart.

A ''single frequency network'' is a form of transmitter [[macrodiversity]]. The concept can be further used in ''[[dynamic single-frequency networks]]'' (DSFN), where the SFN grouping is changed from timeslot to timeslot.

OFDM may be combined with other forms of [[space diversity]], for example [[Phased array|antenna arrays]] and [[MIMO]] channels. This is done in the [[IEEE 802.11]] [[Wireless LAN]] standards.

=== Linear transmitter power amplifier ===
An OFDM signal exhibits a high [[Crest factor|peak-to-average power ratio (PAPR)]] because the independent phases of the subcarriers mean that they will often combine constructively. Handling this high PAPR requires:
* A high-resolution [[digital-to-analog converter]] (DAC) in the transmitter
* A high-resolution [[analog-to-digital converter]] (ADC) in the receiver
* A linear [[signal chain]]

Any non-linearity in the signal chain will cause [[intermodulation distortion]] that
* Raises the noise floor
* May cause inter-carrier interference
* Generates out-of-band spurious radiation

The linearity requirement is demanding, especially for transmitter RF output circuitry where amplifiers are often designed to be non-linear in order to minimise power consumption. In practical OFDM systems a small amount of peak clipping is allowed to limit the PAPR in a judicious trade-off against the above consequences. However, the transmitter output filter which is required to reduce out-of-band spurs to legal levels has the effect of restoring peak levels that were clipped, so clipping is not an effective way to reduce PAPR.

Although the spectral efficiency of OFDM is attractive for both terrestrial and space communications, the high PAPR requirements have so far limited OFDM applications to terrestrial systems.

The crest factor CF (in dB) for an OFDM system with ''n'' uncorrelated subcarriers is<ref name="Kaehs">{{cite web|url=https://karriere.rohde-schwarz.de/career-cdn-pull/rs-common/fileadmin/customer/downloads/PDF/Huellkurve_eng.pdf|title=The Crest Factor in DVB-T (OFDM) Transmitter Systems and its Influence on the Dimensioning of Power Components|author=Bernhard Kaehs|date=January 2007|publisher=Rohde & Schwarz|archive-url=https://web.archive.org/web/20140705103705/http://cdn.rohde-schwarz.com/dl_downloads/dl_application/application_notes/7ts02/7TS02_2E.pdf|archive-date=2014-07-05|url-status=dead}}</ref>

:<math> CF = 10 \log_{10} ( n ) + CF_c </math>

where CF<sub>c</sub> is the crest factor (in dB) for each subcarrier.
(CF<sub>c</sub> is 3.01&nbsp;dB for the sine waves used for BPSK and QPSK modulation).

For example, the DVB-T signal in 2K mode is composed of 1705 subcarriers that are each QPSK-modulated, giving a crest factor of 35.32&nbsp;dB.<ref name="Kaehs" />

Many PAPR (or [[crest factor]]) reduction techniques have been developed, for instance, based on iterative clipping.<ref>{{cite journal |last1=Wang |first1=Y.-C. |last2=Luo |first2=Z.-Q. |title=Optimized Iterative Clipping and Filtering for PAPR Reduction of OFDM Signals |journal=IEEE Transactions on Communications |date=January 2011 |volume=59 |issue=1 |pages=33–37 |doi=10.1109/TCOMM.2010.102910.090040|s2cid=2487860 }}</ref>
Over the years, numerous model-driven approaches have been proposed to reduce the PAPR in communication systems. In recent years, there has been a growing interest in exploring data-driven models for PAPR reduction as part of ongoing research in end-to-end communication networks. These data-driven models offer innovative solutions and new avenues of exploration to address the challenges posed by high PAPR effectively. By leveraging data-driven techniques, researchers aim to enhance the performance and efficiency of communication networks by optimizing power utilization.
<ref>{{cite conference |last1=Huleihel |first1=Yara |last2=Ben-Dror|first2=Eilam |last3=Permuter |first3=Haim H. |title=Low PAPR Waveform Design for OFDM Systems Based on Convolutional Autoencoder |conference=2020 IEEE International Conference on Advanced Networks and Telecommunications Systems (ANTS) |date=2020 |pages=1–6}}</ref>


The [[dynamic range]] required for an FM receiver is {{nowrap|120 dB}} while DAB only require about {{nowrap|90 dB.}}<ref name=dabpr>{{cite book|title=Digital Audio Broadcasting: Principles and Applications of DAB, DAB + and DMB |url=https://books.google.com/books?id=T0KSa6w3qH4C&pg=PA333 |page=333 |year=2009 |last1=Hoeg |first1=Wolfgang |last2=Lauterbach |first2=Thomas |publisher=John Wiley & Sons |edition=3rd |access-date=2013-07-04|isbn=9780470746196 }}</ref> As a comparison, each extra bit per sample increases the dynamic range by {{nowrap|6 dB.}}