Orthogonal frequency-division Multiplexing
Orthogonal Frequency-Division Multiplexing (OFDM) — essentially identical to Coded OFDM (COFDM) — is a digital multi-carrier modulation scheme, which uses a large number of closely-spaced orthogonal sub-carriers. Each sub-carrier is modulated with a conventional modulation scheme (such as quadrature amplitude modulation) at a low symbol rate, maintaining data rates similar to conventional single-carrier modulation schemes in the same bandwidth. In practice, OFDM signals are generated using the Fast Fourier transform algorithm.
The primary advantage of OFDM over single-carrier schemes is its ability to cope with severe channel conditions — for example, multipath and narrowband interference — 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 orthogonality of the sub-carriers results in zero cross-talk, even though they are so close that their spectra overlap. Low symbol rate helps manage time-domain spreading of the signal (such as multipath propagation) by allowing the use of a guard interval between symbols. The guard interval also eliminates the need for a pulse-shaping filter.
Key features
Summary of advantages
• Can easily adapt to severe channel conditions without complex equalization
• Robust against narrow-band co-channel interference
• Robust against Intersymbol interference (ISI) and fading caused by multipath propagation
• High spectral efficiency
• Efficient implementation using FFT
• Low sensitivity to time synchronization errors
• Tuned sub-channel receiver filters are not required (unlike conventional FDM)
• Facilitates Single Frequency Networks, i.e. transmitter macrodiversity.
Summary of disadvantages
• Sensitive to Doppler shift.
• Sensitive to frequency synchronization problems.
• Inefficient transmitter power consumption, due to linear power amplifier requirement.

Characteristics and principles of operation
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 also allows high spectral efficiency, near the Nyquist rate. Almost the whole available frequency band can be utilized. OFDM generally has a nearly 'white' spectrum, giving it benign electromagnetic interference properties with respect to other co-channel users.
The orthogonality allows for efficient modulator and demodulator implementation using the FFT algorithm. 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.
OFDM requires very accurate frequency synchronisation between the receiver and the transmitter; any deviation and the sub-carriers are no longer orthogonal, causing 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. Whilst 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, and is an important factor limiting the use of OFDM in high-speed vehicles. Several techniques for ICI suppression are suggested, but they may increase the receiver complexity.
Guard interval for elimination of inter-symbol 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, 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 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 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 last 125 microseconds of each symbol are copied and sent in advance of the symbol as a cyclic prefix.
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.
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 km apart.
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 equalization far simpler at the receiver in OFDM in comparison to conventional single-carrier modulation. The equalizer only has to multiply each sub-carrier by a constant value, or a rarely changed value.
Our example: The OFDM equalization in the above numerical example would require N = 1000 complex multiplications per OFDM symbol, i.e. one million multiplications per second, at the receiver. The FFT algorithm requires Nlog2N = 10.000 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 125 multiplications per symbol, i.e. 125 million multiplications per second.
Some of the sub-carriers in some of the OFDM symbols may carry pilot signals for measurement of the channel conditions, i.e. the equalizer gain for each sub-carrier. Pilot signals may also be used for synchronization.
If differential modulation such as DPSK or DQPSK is applied to each sub-carrier, equalization can be completely omitted, since these schemes are insensitive to slowly changing amplitude and phase distortion.
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 is faded, 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 is faded 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 common type of error correction coding used with OFDM-based systems is convolutional coding, which is often concatenated with Reed-Solomon coding. Convolutional coding is used as the inner code and Reed-Solomon coding is used for the outer code — usually with additional interleaving (on top of the time and frequency interleaving mentioned above) in between the two layers of coding. The reason why this combination of error correction coding is used is that the Viterbi decoder used for convolutional decoding produces short errors 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 the near-optimal types of error correction coding 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. These codes only perform close to the Shannon limit for the Additive White Gaussian Noise (AWGN) channel, however, and some systems that have adopted 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 performance further over the wireless channel.

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 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 bandwidth is allocated to whichever stream 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 technique, since it is utilized 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 uplink of the IEEE 802.16 Wireless MAN standard, commonly referred to as WiMAX.
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 against, 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 (SFNs), where many transmitters send the same signal simultaneously over the same channel frequency. SFNs utilise 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 scheme 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.
Single-frequency networks is a form of transmitter macrodiversity.
OFDM may be combined with other forms of space diversity, for example antenna arrays and MIMO channels. This is done in the IEEE802.11n Wireless LAN standard.
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.
Any non-linearity in the signal chain will cause intermodulation distortion that
• raises the noise floor
• may cause intersymbol 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 tradeoff against the above consequences. However the transmitter output filter which is required to reduce out-of-band spurrii 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.

Ideal system model
This section describes a simple idealized OFDM system model suitable for a time-invariant AWGN channel.
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
s[n] is a serial stream of binary digits. By inverse multiplexing, these are first demultiplexed into 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, fc, respectively. These signals are then summed to give the transmission signal, s(t).

The receiver picks up the signal 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 2fc, so low-pass filters are used to reject these. The baseband signals are then sampled and digitised using analogue-to-digital converters (ADCs), and a forward FFT is used to convert back to the frequency domain.
This returns 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, , which is an estimate of the original binary stream at the transmitter.
Mathematical description
If K sub-carriers are used, and each sub-carrier is modulated using M alternative symbols, the OFDM symbol alphabet consists of MK combined symbols.
The low-pass equivalent OFDM signal is expressed as:

where {Xk} are the data symbols, N is the number of sub-carriers, and T is the OFDM symbol time. The sub-carrier spacing of 1 / T makes them orthogonal over each symbol period; this property is expressed as:

where denotes the complex conjugate operator.
To avoid intersymbol interference in multipath fading channels, a guard interval , where Tg is the guard period, is inserted prior to the OFDM block or OFDM symbol. During this interval, a cyclic prefix is transmitted. The cyclic prefix is equal to the last Tg of the OFDM symbol. The OFDM signal with cyclic prefix is thus:

The low-pass signal above can be either real or complex-valued. Real-valued low-pass equivalent signals are typically transmitted at baseband—wireline applications such as DSL use this approach. For wireless applications, the low-pass signal is typically complex-valued; in which case, the transmitted signal is up-converted to a carrier frequency fc. In general, the transmitted signal can be represented as:


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