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orthogonal frequency-division multiplexing

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**orthogonal frequency-division multiplexing**(

**OFDM**) is a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital communication, used in applications such as digital television and audio broadcasting, DSL internet access, wireless networks, power line networks, and 4G mobile communications.In coded orthogonal frequency-division multiplexing (COFDM), forward error correction (convolutional coding) and time/frequency interleaving are applied to the signal being transmitted. This is done to overcome errors in mobile communication channels affected by multipath propagation and Doppler effects. COFDM was introduced by Alard in 1986 PATENT, WO, 8800417, WEB, EBU Technical Review nÂ°224, p.187, August 1987, Principles of modulation and channel coding for digital broadcasting for mobile receivers,weblink B. LeFloch, M. Alard, C. Berrou, "Coded Orthogonal Frequency Division Multiplex", Proc. IEEE, vol. 83, pp. 982-996, June 1995. for Digital Audio Broadcasting for Eureka Project 147. In practice, OFDM has become used in combination with such coding and interleaving, so that the terms COFDM and OFDM co-apply to common applications.WEB,weblink Yang, James Ching-Nung, October 10, 2001, Department of Computer Science and Information Engineering National Dong Hwa University, Shoufeng, Hualien 974, Taiwan, What is OFDM and COFDM?, 2017-04-16, OFDM is a frequency-division multiplexing (FDM) scheme used as a digital multi-carrier modulation method. OFDM was introduced by Chang of Bell Labs in 1966.WEB, The history of orthogonal frequency-division multiplexing, IEEE Communications Magazine ( Volume: 47, Issue: 11, November 2009 ),weblink Numerous closely spaced orthogonal sub-carrier signals with overlapping spectra are emitted to carry data.webe.org - 2GHz BAS Relocation Tech-Fair, COFDM Technology Basics. 2007-03-02 Demodulation is based on Fast Fourier Transform algorithms. OFDM was improved by Weinstein and Ebert in 1971 with the introduction of a guard interval, providing better orthogonality in transmission channels affected by multipath propagation. Each sub-carrier (signal) is modulated with a conventional modulation scheme (such as quadrature amplitude modulation or phase-shift keying) at a low symbol rate. This maintains total data rates similar to conventional single-carrier modulation schemes in the same bandwidth.The main advantage of OFDM over single-carrier schemes is its ability to cope with severe channel conditions (for example, attenuation of high frequencies in a long copper wire, narrowband interference and frequency-selective fading due to multipath) without complex equalization filters. Channel equalization is simplified because OFDM may be viewed as using many slowly modulated narrowband signals rather than one rapidly modulated wideband signal. The low symbol rate makes the use of a guard interval between symbols affordable, making it possible to eliminate intersymbol interference (ISI) and use echoes and time-spreading (in analog television visible as ghosting and blurring, respectively) to achieve a diversity gain, i.e. a signal-to-noise ratio improvement. This mechanism also facilitates the design of single frequency networks (SFNs) where several adjacent transmitters send the same signal simultaneously at the same frequency, as the signals from multiple distant transmitters may be re-combined constructively, sparing interference of a traditional single-carrier system.

## Example of applications

The following list is a summary of existing OFDM-based standards and products. For further details, see the Usage section at the end of the article.### Wired version mostly known as Discrete Multi-tone Transmission (DMT)

- ADSL and VDSL broadband access via POTS copper wiring
- DVB-C2, an enhanced version of the DVB-C digital cable TV standard
- Power line communication (PLC)
- ITU-T G.hn, a standard which provides high-speed local area networking of existing home wiring (power lines, phone lines and coaxial cables)BOOK, Ben-Tovim, Erez, February 2014, ITU G.hn - Broadband Home Networking, MIMO Power Line Communications, 457â€“472, CRC Press, Devices, Circuits, and Systems, Berger, Lars T., Schwager, Andreas, Pagani, Pascal, Schneider, Daniel M., 9781466557529, 10.1201/b16540-16,
- TrailBlazer telephone line modems
- Multimedia over Coax Alliance (MoCA) home networking
- DOCSIS 3.1 Broadband delivery

### Wireless

- The wireless LAN (WLAN) radio interfaces IEEE 802.11a, g, n, ac, ah and HIPERLAN/2
- The digital radio systems DAB/EUREKA 147, DAB+, Digital Radio Mondiale, HD Radio, T-DMB and ISDB-TSB
- The terrestrial digital TV systems DVB-T and ISDB-T
- The terrestrial mobile TV systems DVB-H, T-DMB, ISDB-T and MediaFLO forward link
- The wireless personal area network (PAN) ultra-wideband (UWB) IEEE 802.15.3a implementation suggested by WiMedia Alliance

- The mobility mode of the wireless MAN/broadband wireless access (BWA) standard IEEE 802.16e (or Mobile-WiMAX)
- The mobile broadband wireless access (MBWA) standard IEEE 802.20
- The downlink of the 3GPP Long Term Evolution (LTE) fourth generation mobile broadband standard. The radio interface was formerly named
*High Speed OFDM Packet Access*(HSOPA), now named Evolved UMTS Terrestrial Radio Access (E-UTRA) - WLAN IEEE 802.11ax

## Key features

The advantages and disadvantages listed below are further discussed in the Characteristics and principles of operation section below.### Summary of advantages

- High spectral efficiency as compared to other double sideband modulation schemes, spread spectrum, etc.
- Can easily adapt to severe channel conditions without complex time-domain equalization.
- Robust against narrow-band co-channel interference
- Robust against intersymbol interference (ISI) and fading caused by multipath propagation
- Efficient implementation using fast Fourier transform
- Low sensitivity to time synchronization errors
- Tuned sub-channel receiver filters are not required (unlike conventional FDM)
- Facilitates single frequency networks (SFNs) (i.e. transmitter macrodiversity)

### Summary of disadvantages

- Sensitive to Doppler shift
- Sensitive to frequency synchronization problems
- High peak-to-average-power ratio (PAPR), requiring linear transmitter circuitry, which suffers from poor power efficiency
- Loss of efficiency caused by cyclic prefix/guard interval

## 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 sub-carrier frequencies are chosen so that the sub-carriers are orthogonal to each other, meaning that cross-talk 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; unlike conventional FDM, a separate filter for each sub-channel is not required.The orthogonality requires that the sub-carrier spacing is scriptstyleDelta f ,=, frac{k}{T_U} Hertz, where*T*U seconds 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*sub-carriers, the total passband bandwidth will be

*B*â‰ˆ

*N*Â·Î”

*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

OFDM requires very accurate frequency synchronization between the receiver and the transmitter; with frequency deviation the sub-carriers will no longer be orthogonal, causing *T*U = 1 ms would require a sub-carrier spacing of scriptstyleDelta f ,=, frac{1}{1,mathrm{ms}} ,=, 1,mathrm{kHz} (or an integer multiple of that) for orthogonality.*N*= 1,000 sub-carriers would result in a total passband bandwidth of*N*Î”f = 1 MHz. For this symbol time, the required bandwidth in theory according to Nyquist is scriptstylemathrm{BW}=R/2=(N/T_U)/2 = 0.5,mathrm{MHz} (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 MHz, which would require a baseband bandwidth of 0.5 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 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.*inter-carrier interference*(ICI) (i.e., cross-talk between the sub-carriers). 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, as reflections will appear at various frequency offsets, which is much harder to correct. This effect typically worsens as speed increases,Robertson, P.; Kaiser, S. "The effects of Doppler spreads in OFDM(A) mobile radio systems", Vehicular Technology Conference, 1999. VTC 1999 - Fall. IEEE VTS. Link 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 sub-carrier in order to minimize the interference resulting in a non-orthogonal subcarriers overlapping.JOURNAL, A Time-Frequency Well-localized Pulse for Multiple Carrier Transmission, Haas, R., Belfiore, J.C., Wireless Personal Communications, 1997, 5, 1, 1â€“18, 10.1023/A:1008859809455, . 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.JOURNAL, Performances of Weighted Cyclic Prefix OFDM with Low-Complexity Equalization, Roque, D., Siclet, C., IEEE Communications Letters, 2013, 17, 3, 439â€“442, 10.1109/LCOMM.2013.011513.121997,weblink . Other ICI suppression techniques usually increase drastically the receiver complexity.JOURNAL, An equalization technique for orthogonal frequency-division multiplexing systems in time-variant multipath channels, Jeon, W.G., Chang, K.H., Cho, Y.S., IEEE Transactions on Communications, 1999, 47, 1, 27â€“32, 10.1109/26.747810, 10.1.1.460.4807, .

### Implementation using the FFT algorithm

The orthogonality allows for efficient modulator and demodulator implementation using the 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 transform has to take less than the time for each symbol,THESIS, B.E.,weblink The suitability of OFDM as a modulation technique for wireless telecommunications, with a CDMA comparison, Eric Lawrey, October 1997, {{rp|84}} which for example for DVB-T {{nowrap|(FFT 8k)}} means the computation has to be done in {{nowrap|896 Âµs}} or less.For an {{gaps|8|192}}-point FFT this may be approximated to:{{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}}
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}
The computational demand approximately scales linearly with FFT size so a double size FFT needs double the amount of time and vice versa.{{rp|83}}As a comparison an Intel Pentium III CPU at 1.266 GHz is able to calculate a {{nowrap|8192 point}} FFT in {{nowrap|576 Âµs}} using FFTW.WEB,weblink fftw.org, 1.266 GHz Pentium 3, 2006-06-20, Intel Pentium M at 1.6 GHz does it in {{nowrap|387 Âµs.}}WEB,weblink fftw.org, 1.6 GHz Pentium M (Banias), GNU compilers, 2006-06-20, Intel Core Duo at 3.0 GHz does it in {{nowrap|96.8 Âµs}}.WEB,weblink fftw.org, 3.0 GHz Intel Core Duo, Intel compilers, 32-bit mode, 2006-10-09, &= 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

### 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, 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 sub-carrier (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 less operations, which translate to less 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., scriptstyle N ,=, 1000 complex multiplications per OFDM symbol; i.e., one million multiplications per second, at the receiver). The FFT algorithm requires scriptstyle N log_2 N ,=, 10,000 [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 sub-carrier, 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 sub-carriers in some of the OFDM symbols may carry pilot signals for measurement of the channel conditionsJOURNAL, Coleri S, Ergen M, Puri A, Bahai A, Channel estimation techniques based on pilot arrangement in OFDM systems, IEEE Transactions on Broadcasting, Sep 2002,weblink CONFERENCE, Hoeher P, Kaiser S, Robertson P,, Two-dimensional pilot-symbol-aided channel estimation by Wiener filtering, IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-97, 1997,weblink (i.e., the equalizer gain and phase shift for each sub-carrier). Pilot signals and training symbols (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.JOURNAL, Zemen T, Mecklenbrauker CF, Time-Variant Channel Estimation Using Discrete Prolate Spheroidal Sequences, IEEE Transactions on Signal Processing, 53, 9, 3597â€“3607, Sep 2005, 10.1109/TSP.2005.853104, 10.1.1.60.9526, JOURNAL, Tang Z, Cannizzaro RC, Leus G, Banelli P, Pilot-Assisted Time-Varying Channel Estimation for OFDM Systems, IEEE Transactions on Signal Processing, 55, 5, 2226â€“2238, May 2007, 10.1109/TSP.2007.893198, 10.1.1.418.2386, JOURNAL, Hrycak T, Das S, Matz G, Feichtinger HG, Low Complexity Equalization for Doubly Selective Channels Modeled by a Basis Expansion, IEEE Transactions on Signal Processing, 58, 11, 5706â€“5719, Aug 2010, 10.1109/TSP.2010.2063426, ### Channel coding and interleaving

OFDM is invariably used in conjunction with channel coding (forward error correction), and almost always uses frequency and/or time interleaving.Frequency (subcarrier) 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 coding, often concatenated with 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 codes and LDPC codes, which perform close to the Shannon limit for the 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 codes (on the DVB-S2 system) to improve upon an error floor inherent to these codes at high signal-to-noise ratios.BOOK, MIMO Power Line Communications: Narrow and Broadband Standards, EMC, and Advanced Processing, Devices, Circuits, and Systems, February 2014, Berger, Lars T., Schwager, Andreas, Pagani, Pascal, Schneider, Daniel M, CRC Press, 25, 10.1201/b16540-1, 978-1-4665-5753-6, Introduction to Power Line Communication Channel and Noise Characterisation,### 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 sub-carriers, or individually to each sub-carrier. 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 sub-carriers.The term**{{visible anchor|discrete multitone modulation}}**(

**DMT**) denotes OFDM-based communication systems that adapt the transmission to the channel conditions individually for each sub-carrier, 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

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 sub-carriers to different users in a similar fashion as in CDMA, and thus complex packet scheduling or Media 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 now defunct Qualcomm/3GPP2 Ultra Mobile Broadband (UMB) project, intended as a successor of CDMA2000, but replaced by LTE.

*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).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 sub-carriers, 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 networks (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 (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 sub-carriers.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 â€” for instance, a guard interval of 200 microseconds would allow transmitters to be spaced 60 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 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 peak-to-average power ratio (PAPR) because the independent phases of the sub-carriers 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

- Raises the noise floor
- May cause inter-carrier interference
- Generates out-of-band spurious radiation

*n*uncorrelated sub-carriers isWEB,weblink The Crest Factor in DVB-T (OFDM) Transmitter Systems and its Influence on the Dimensioning of Power Components, Bernhard Kaehs, January 2007, Rohde & Schwarz,weblink" title="web.archive.org/web/20140705103705weblink">weblink 2014-07-05, yes,

CF = 10 log ( n ) + CF_c

where CFc is the crest factor (in dB) for each sub-carrier.(CFc is 3.01 dB for the sine waves used for BPSK and QPSK modulation).For example, the DVB-T signal in 2K mode is composed of 1705 sub-carriers that are each QPSK-modulated, giving a crest factor of 35.32 dB.Many crest factor reduction techniques have been developed.The dynamic range required for an FM receiver is {{nowrap|120 dB}} while DAB only require about {{nowrap|90 dB.}}BOOK, Digital Audio Broadcasting: Principles and Applications of DAB, DAB + and DMB,weblink 333, 2009, Hoeg, Wolfgang, Lauterbach, Thomas, John Wiley & Sons, 3rd, 2013-07-04, 9780470746196, As a comparison, each extra bit per sample increases the dynamic range with {{nowrap|6 dB.}}## Efficiency comparison between single carrier and multicarrier

The performance of any communication system can be measured in terms of its power efficiency and bandwidth efficiency.The power efficiency describes the ability of communication system to preserve bit error rate (BER) of the transmitted signal at low power levels.Bandwidth efficiency reflects how efficiently the allocated bandwidth is used and is defined as the throughput data rate per Hertz in a given bandwidth.If the large number of subcarriers are used, the bandwidth efficiency of multicarrier system such as OFDM with using optical fiber channel is defined asBOOK, William Shieh, Ivan Djordjevic., "OFDM for Optical Communications"., 525 B Street, Suite 1900, San Diego, California 92101-4495, USA, Academic Press, 2010,- eta=2 cdot frac{R_{s}}{B_{OFDM}}

10 Gbit/s}} | {{nowrap | -37.3 dBm}} | 6.0000 |

10 Gbit/s}} | {{nowrap | -36.3 dBm}} | 10.6022 |

## Idealized system model

This section describes a simple idealized OFDM system model suitable for a time-invariant AWGN channel.### Transmitter

(File:OFDM transmitter ideal.png|center|754px)An OFDM carrier signal is the sum of a number of orthogonal sub-carriers, with baseband data on each sub-carrier being independently modulated commonly using some type of quadrature amplitude modulation (QAM) or phase-shift keying (PSK). This composite baseband signal is typically used to modulate a main RF carrier.scriptstyle s[n] is a serial stream of binary digits. By inverse multiplexing, these are first demultiplexed into scriptstyle N parallel streams, and each one mapped to a (possibly complex) symbol stream using some modulation constellation (QAM, PSK, etc.). Note that the constellations may be different, so some streams may carry a higher bit-rate than others.An inverse FFT is computed on each set of symbols, giving a set of complex time-domain samples. These samples are then quadrature-mixed to passband in the standard way. The real and imaginary components are first converted to the analogue domain using digital-to-analogue converters (DACs); the analogue signals are then used to modulate cosine and sine waves at the carrier frequency, scriptstyle f_c, respectively. These signals are then summed to give the transmission signal, scriptstyle s(t).### Receiver

(File:OFDM receiver ideal.png|center|800px)The receiver picks up the signal scriptstyle r(t), which is then quadrature-mixed down to baseband using cosine and sine waves at the carrier frequency. This also creates signals centered on scriptstyle 2 f_c, so low-pass filters are used to reject these. The baseband signals are then sampled and digitised using analog-to-digital converters (ADCs), and a forward FFT is used to convert back to the frequency domain.This returns scriptstyle N parallel streams, each of which is converted to a binary stream using an appropriate symbol detector. These streams are then re-combined into a serial stream, scriptstyle {hat s}[n], which is an estimate of the original binary stream at the transmitter.## Mathematical description

If scriptstyle N sub-carriers are used, and each sub-carrier is modulated using scriptstyle M alternative symbols, the OFDM symbol alphabet consists of scriptstyle M^N combined symbols.The low-pass equivalent OFDM signal is expressed as:
nu(t)=sum_{k=0}^{N-1}X_k e^{j2pi kt/T}, quad 0le t

COFDM is also used for other radio standards, for Digital Audio Broadcasting (DAB), the standard for digital audio broadcasting at VHF frequencies, for Digital Radio Mondiale (DRM), the standard for digital broadcasting at shortwave and medium wave frequencies (below 30 MHz) and for DRM+ a more recently introduced standard for digital audio broadcasting at VHF frequencies. (30 to 174 MHz)The USA again uses an alternate standard, a proprietary system developed by iBiquity dubbed *HD Radio*. However, it uses COFDM as the underlying broadcast technology to add digital audio to AM (medium wave) and FM broadcasts.Both Digital Radio Mondiale and HD Radio are classified as in-band on-channel systems, unlike Eureka 147 (DAB: Digital Audio Broadcasting) which uses separate VHF or UHF frequency bands instead.

#### BST-OFDM used in ISDB

The*band-segmented transmission orthogonal frequency division multiplexing*(

*BST-OFDM*) system proposed for Japan (in the ISDB-T, ISDB-TSB, and ISDB-C broadcasting systems) improves upon COFDM by exploiting the fact that some OFDM carriers may be modulated differently from others within the same multiplex. Some forms of COFDM already offer this kind of hierarchical modulation, though BST-OFDM is intended to make it more flexible. The 6 MHz television channel may therefore be "segmented", with different segments being modulated differently and used for different services.It is possible, for example, to send an audio service on a segment that includes a segment composed of a number of carriers, a data service on another segment and a television service on yet another segmentâ€”all within the same 6 MHz television channel. Furthermore, these may be modulated with different parameters so that, for example, the audio and data services could be optimized for mobile reception, while the television service is optimized for stationary reception in a high-multipath environment.

### Ultra-wideband

Ultra-wideband (UWB) wireless personal area network technology may also use OFDM, such as in Multiband OFDM (MB-OFDM). This UWB specification is advocated by the WiMedia Alliance (formerly by both the Multiband OFDM Alliance [MBOA] and the WiMedia Alliance, but the two have now merged), and is one of the competing UWB radio interfaces.### FLASH-OFDM

*Fast low-latency access with seamless handoff orthogonal frequency division multiplexing*(Flash-OFDM), also referred to as F-OFDM, was based on OFDM and also specified higher protocol layers. It was developed by Flarion, and purchased by Qualcomm in January 2006.NEWS, Qualcomm and Exoteq Sign OFDM/OFDMA License Agreement, August 1, 2007, News release, Qualcomm,weblink July 23, 2011, NEWS, Qualcomm Completes Acquisition Of WiMAX Competitor, January 19, 2006, Network Computing,weblink July 23, 2011, Flash-OFDM was marketed as a packet-switched cellular bearer, to compete with GSM and 3G networks. As an example, 450 MHz frequency bands previously used by NMT-450 and C-Net C450 (both 1G analogue networks, now mostly decommissioned) in Europe are being licensed to Flash-OFDM operators.{{citation needed |date= July 2011}}In Finland, the license holder Digita began deployment of a nationwide "@450" wireless network in parts of the country since April 2007. It was purchased by Datame in 2011.WEB, Briefly in English, @450-Network web site, Datame,weblink July 23, 2011, In February 2012 Datame announced they would upgrade the 450 MHz network to competing CDMA2000 technology.JOURNAL, @450 siirtyy cdma2000-tekniikkaan - jopa puhelut mahdollisia, Tietoviikko, Aleksi Kolehmainen, February 8, 2012,weblink Finnish, Slovak Telekom in Slovakia offers Flash-OFDM connectionsWEB, Mapy pokrytia, Slovak Telekom web site,weblink Slovak, May 30, 2012, with a maximum downstream speed of 5.3 Mbit/s, and a maximum upstream speed of 1.8 Mbit/s, with a coverage of over 70 percent of Slovak population.{{citation needed |date= July 2011}} The Flash-OFDM network was switched off in the majority of Slovakia on 30 September 2015.NEWS, Slovak Telekom closed Flash-OFDM network, November 5, 2015, ceeitandtelecom,weblink T-Mobile Germany used Flash-OFDM to backhaul Wi-Fi HotSpots on the Deutsche Bahn's ICE high speed trains between 2005 and 2015, until switching over to UMTS and LTE.NEWS, Ins Netz bei Tempo 300, December 23, 2014, heise online,weblink December 20, 2016, American wireless carrier Nextel Communications field tested wireless broadband network technologies including Flash-OFDM in 2005.NEWS, Nextel Flash-OFDM: The Best Network You May Never Use, March 2, 2005, PC Magazine,weblink July 23, 2011, Sprint purchased the carrier in 2006 and decided to deploy the mobile version of WiMAX, which is based on Scalable Orthogonal Frequency Division Multiple Access (SOFDMA) technology.NEWS, Sprint Nextel Goes To The WiMax, August 8, 2006, Sascha Segan, PC Magazine,weblink July 23, 2011, Citizens Telephone Cooperative launched a mobile broadband service based on Flash-OFDM technology to subscribers in parts of Virginia in March 2006. The maximum speed available was 1.5 Mbit/s.NEWS, Citizens Offers First "Truly Mobile" Wireless Internet in Christiansburg and other parts of the New River Valley, News release, Citizens Wireless, March 28, 2006,weblink July 23, 2011, The service was discontinued on April 30, 2009.WEB, Thank you for supporting Citizens Mobile Broadband, 2009, Citizens Wireless,weblink July 23, 2011,

## Wavelet-OFDM

OFDM has become an interesting technique for power line communications (PLC). In this area of research, a wavelet transform is introduced to replace the DFT as the method of creating orthogonal frequencies. This is due to the advantages wavelets offer, which are particularly useful on noisy power lines.BOOK, S. Galli, H. Koga, N. Nodokama, Advanced Signal Processing for PLCs: Wavelet-OFDM, 2008 IEEE International Symposium on Power Line Communications and its Applications, pp. 187â€“192, May 2008, 10.1109/ISPLC.2008.4510421,weblink 978-1-4244-1975-3, Instead of using an IDFT to create the sender signal, the wavelet OFDM uses a synthesis bank consisting of a textstyle N-band transmultiplexer followed by the transform function
F_n(z) = sum_{k=0}^{L-1} f_n(k) z^{-k}, quad 0 leq n < N

On the receiver side, an analysis bank is used to demodulate the signal again. This bank contains an inverse transform
G_n(z) = sum_{k=0}^{L-1} g_n(k) z^{-k}, quad 0 leq n < N

followed by another textstyle N-band transmultiplexer. The relationship between both transform functions is
f_n(k) = g_n(L - 1 - k)

F_n(z) = z^{-(L-1)} G_n * (z-1)

An example of W-OFDM uses the Perfect Reconstruction Cosine Modulated Filter Bank (PR-CMFB) and Extended Lapped Transform (ELT) is used for the wavelet TF. Thus, textstyle f_n (k) and textstyle g_n (k) are given as
f_n (k) = 2 p_0(k) cos left[ frac{pi}{N}(n+frac{1}{2})(k-frac{L-1}{2}) - (-1)^{n} frac{pi}{4} right]

g_n (k) = 2 p_0(k) cos left[ frac{pi}{N}(n+frac{1}{2})(k-frac{L-1}{2}) + (-1)^{n} frac{pi}{4} right]

P_0(z) = sum_{k=0}^{N-1} z^{-k} Y_k(z^{2N})

These two functions are their respective inverses, and can be used to modulate and demodulate a given input sequence. Just as in the case of DFT, the wavelet transform creates orthogonal waves with textstyle f_0, textstyle f_1, ..., textstyle f_{N-1}. The orthogonality ensures that they do not interfere with each other and can be sent simultaneously. At the receiver, textstyle g_0, textstyle g_1, ..., textstyle g_{N-1} are used to reconstruct the data sequence once more.### Advantages over standard OFDM

W-OFDM is an evolution of the standard OFDM, with certain advantages.Mainly, the sidelobe levels of W-OFDM are lower. This results in less ICI, as well as greater robustness to narrowband interference. These two properties are especially useful in PLC, where most of the lines aren't shielded against EM-noise, which creates noisy channels and noise spikes.A comparison between the two modulation techniques also reveals that the complexity of both algorithms remains approximately the same.## History

- 1957: Kineplex, multi-carrier HF modem (R.R. Mosier & R.G. Clabaugh)
- 1966: Chang, Bell Labs: OFDM paperJOURNAL, Chang, R. W., 1966, Synthesis of band-limited orthogonal signals for multi-channel data transmission, Bell System Technical Journal, 45, 10, 1775â€“1796, 10.1002/j.1538-7305.1966.tb02435.x, and patentPATENT, US, 3488445,
- 1971: Weinstein & Ebert proposed use of FFT and guard intervalS. Weinstein and P. Ebert, Data transmission by frequency-division multiplexing using the discrete Fourier transform,
*IEEE Transactions on Communication Technology*, vol. 19, no. 5, pp. 628â€“634, October 1971. - 1985: Cimini described use of OFDM for mobile communications
- 1985: Telebit Trailblazer Modem introduced a {{nowrap|512 carrier}} Packet Ensemble Protocol ({{nowrap|18 432 bit/s}})
- 1987: Alard & Lasalle: COFDM for digital broadcasting
- 1988: In September TH-CSF LER, first experimental Digital TV link in OFDM, Paris area
- 1989: OFDM international patent application PCT/FR 89/00546, filed in the name of THOMSON-CSF, Fouche, de Couasnon, Travert, Monnier and alweblink
- October 1990: TH-CSF LER, first OFDM equipment field test, 34 Mbit/s in an 8 MHz channel, experiments in Paris area
- December 1990: TH-CSF LER, first OFDM test bed comparison with VSB in Princeton USA
- September 1992: TH-CSF LER, second generation equipment field test, 70 Mbit/s in an 8 MHz channel, twin polarisations. Wuppertal, Germany
- October 1992: TH-CSF LER, second generation field test and test bed with BBC, near London, UK
- 1993: TH-CSF show in Montreux SW, 4 TV channel and one HDTV channel in a single 8 MHz channel
- 1993: Morris: Experimental 150 Mbit/s OFDM wireless LAN
- 1995: ETSI Digital Audio Broadcasting standard EUreka: first OFDM-based standard
- 1997: ETSI DVB-T standard
- 1998: Magic WAND project demonstrates OFDM modems for wireless LAN
- 1999: IEEE 802.11a wireless LAN standard (Wi-Fi)
- 2000: Proprietary fixed wireless access (V-OFDM, FLASH-OFDM, etc.)
- 2002: IEEE 802.11g standard for wireless LAN
- 2004: IEEE 802.16 standard for wireless MAN (WiMAX)
- 2004: ETSI DVB-H standard
- 2004: Candidate for IEEE 802.15.3a standard for wireless PAN (MB-OFDM)
- 2004: Candidate for IEEE 802.11n standard for next generation wireless LAN
- 2005: OFDMA is candidate for the 3GPP Long Term Evolution (LTE) air interface E-UTRA downlink.
- 2007: The first complete LTE air interface implementation was demonstrated, including OFDM-MIMO, SC-FDMA and multi-user MIMO uplinkWEB,weblink Nortel 3G World Congress Press Release,

## See also

- ATSC standards
- Carrier interferometry
- N-OFDM
- Single-carrier FDMA (SC-FDMA)
- Single-carrier frequency-domain-equalization (SC-FDE)

## References

{{Reflist|30em}}## Further reading

- M. Bank. System free of channel problems inherent in changing mobile communication systems. Electronics Letters, 43(7), 2007 (401-402)
- M. Bank, B. Hill, Miriam Bank. A wireless mobile communication system without pilot signals Patent PCT/Il N 2006000926, Patent PCT International Application N0 PCT/IL 2006000926. Patent No. 7,986,740, Issue date: 26 July 2011

## External links

{{Commons category}}- Numerous useful links and resources for OFDM - WCSP Group - University of South Florida (USF)
- WiMAX Forum, WiMAX, the framework standard for 4G mobile personal broadband
- Stott, 1997 weblink Technical presentation by J H Stott of the BBC's R&D division, delivered at the 20 International Television Symposium in 1997; this URL accessed 24 January 2006.
- Page on Orthogonal Frequency Division Multiplexing atweblink accessed on 24 September 2007.
- A tutorial on the significance of Cyclic Prefix (CP) in OFDM Systems.
- weblink" title="web.archive.org/web/20060525045608weblink">Siemens demos 360 Mbit/s wireless
- An Introduction to Orthogonal Frequency Division Multiplex Technology
- Ali Imran Khan(2009), Student at National University of Computer and Emerging Sciences did efficient research with help of Dr. Khawar Khokhar (Technical Member PTA)
- weblink" title="web.archive.org/web/20091229213334weblink">Short Introduction to OFDM - Tutorial written by Prof. Debbah, head of the Alcatel-Lucent Chair on flexible radio.
- Short free tutorial on COFDM by Mark Massel formerly at STMicroelectronics and in the digital TV industry for many years.
- A popular book on both COFDM and US ATSC by Mark Massel
- OFDM transmission step-by-step â€“ online experiment
- Simulation of optical OFDM systems

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