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# OFDM-Orthogonal Frequency-Division Multiplexing

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, whether wireless or over copper wires, used in applications such as digital television and audio broadcasting, DSL broadband internet access, wireless networks, and 4G mobile communications.

OFDM is essentially identical to coded OFDM (COFDM) and discrete multi-tone modulation (DMT), and is a frequency-division multiplexing (FDM) scheme used as a digital multi-carrier modulation method.

A large number of closely spaced orthogonal sub-carrier signals are used to carry data. The data is divided into several parallel data streams or channels, one for each sub-carrier. Each sub-carrier is modulated with a conventional modulation scheme (such as quadrature amplitude modulation or phase-shift keying) at a low symbol rate, maintaining total data rates similar to conventional single-carrier modulation schemes in the same bandwidth.

The primary 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 utilize echoes and time-spreading (that shows up as ghosting on analogue TV) 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 combined constructively, rather than interfering as would typically occur in a traditional single-carrier system.

# Orthogonality

Conceptually, OFDM is a specialized FDM, the additional constraint being: all the carrier signals are orthogonal to each other.

FDM Non-Overlapping Carriers (Guard Band)

Spaced apart in such a way that signals can be received with conventional filters and demodulators

OFDM Technique uses Overlapping Carriers

Carriers can be received without Crosstalk. The word Orthogonal indicates that there is a precise mathematical relationship between the carriers the demodulation is performed in the digital domain using special DSP techniques

## Carrier Spacing

What Carrier Spacing & Data Rate = Orthogonal relationship?

The duration Δt of the rectangular pulse can change the spacing Δf between the zero points in the spectrum

## Orthongonality (Frequency Domain)

Orthogonality condition: Δf = 1/Δt

## Orthongonality (Time Domain)

Note each carrier is an Integer number of cycles in Δt. Deference of number of cycles between adjacent carriers = 1

The kth carrier (at baseband) can be written as:

and the orthogonality condition that the carriers satisfy is:

The orthogonality requires that the sub-carrier spacing is Hertz, where TU seconds is the useful symbol duration (the receiver side window size), and k is a positive integer, typically equal to 1. Therefore, with N sub-carriers, the total passband bandwidth will be B ≈ N•Δf (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 utilized.

OFDM generally has a nearly ‘white’ spectrum, giving it benign electromagnetic interference properties with respect to other co-channel users.

OFDM requires very accurate frequency synchronization between the receiver and the transmitter; with frequency deviation the sub-carriers will no longer be 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. 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, 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.

# Propagation Models

### Additive White Gaussian Noise (AWGN):

When there is only a single RF path to the receiver, the system can be viewed as operating over an AWGN channel

### Rician Fading Channel:

Consists of a direct RF path and one or more indirect paths that may be static or dynamic in nature. Most urban and indoor reception environments qualify.

### Rayleigh Fading Channel:

When there is no direct RF path to the receiver, only echoes (static or time varying) are received. Applies to urban outdoor and all indoor sites when no direct RF path to the receiver exists. Also, a portable or mobile receiver if used, would most likely exhibit Rayleighian channel characteristics.

The use of multiple carriers follows from the presence of significant levels of multipath. Suppose we modulate a carrier with digital information. During each symbol, we transmit the carrier with a particular phase and amplitude which is chosen from the constellation in use. Each symbol conveys a number of bits of information, equal to the logarithm (to the base 2) of the number of different states in the constellation.

Now imagine that this signal is received via two paths, with a relative delay between them. Taking transmitted symbol n as an example, the receiver will attempt to demodulate the data that was sent in this symbol by examining all the received information relating to symbol n – both the directly-received information and the delayed information.

When the relative delay is more than one symbol period – the signal received via the second path acts purely as interference, since it only carries information belonging to a previous symbol or symbols. Such inter-symbol interference (ISI) implies that only very small levels of the delayed signal can be tolerated (the exact level depending on the constellation in use and the acceptable loss of noise margin).

When the relative delay is less than one symbol period, part of the signal received via the second path acts purely as interference, since it only carries information belonging to the previous symbol. The rest of it carries the information from the wanted symbol – but may add constructively or destructively to the main-path information.

This tells us that, if we are to cope with any appreciable level of delayed signals, the symbol rate must be reduced sufficiently so that the total delay spread (between the first- and last received paths) is only a modest fraction of the symbol period. The information that can be carried by a single carrier is thus limited in the presence of multipath. If one carrier cannot then carry the information rate we require, this leads naturally to the idea of dividing the high-rate data into many low-rate parallel streams, each conveyed by its own carrier – of which there are a large number.

This is a form of FDM – the first step towards COFDM. Even when the delay spread is less than one symbol period, a degree of ISI from the previous symbol remains. This could be eliminated if the period for which each symbol is transmitted were made longer than the period over which the receiver integrates the signal – a first indication that adding a guard interval may be a good thing.

## 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.

In practice, our carriers are modulated by complex numbers which change from symbol to symbol. If the integration period spans two symbols, not only will there be same-carrier ISI, but in addition there will be inter-carrier interference (ICI) as well. This happens because the beat tones from other carriers may no longer integrate to zero if they change in phase and/or amplitude during the period.

We avoid this by adding a guard interval, which ensures that all the information integrated comes from the same symbol and appears constant during it. The following figure shows this addition of a guard interval.

The symbol period is extended so it exceeds the receiver integration period . Since all the carriers are cyclic within, so too is the whole modulated signal. Thus the segment added at the beginning of the symbol to form the guard interval is identical to the segment of the same length at the end of the symbol.

As long as the delay of any path with respect to the main (shortest) path is less than the guard interval, all the signal components within the integration period come from the same symbol and the orthogonality criterion is satisfied.

ICI and ISI will only occur when the relative delay exceeds the guard interval. The guard interval length is chosen to match the level of multipath expected. It should not form too large a fraction of Tu, otherwise too much data capacity (and spectral efficiency) will be sacrificed.

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.

Note: Cyclic Extension Preserves Orthogonality

# Use of FFT

We’ve avoided thousands of filters, thanks to orthogonality – what about implementing all the demodulating carriers, multipliers and integrators? In practice, we work with the received signal in sampled form (sampled above the Nyquist limit, of course). The process of integration then becomes one of summation, and the whole demodulation process takes on a form which is identical to the Discrete Fourier Transform (DFT). Fortunately, efficient Fast Fourier Transform (FFT) implementations of this already exist (the integrated circuits are already available).

Common versions of the FFT operate on a group of time samples (corresponding to the samples taken in the integration period) and deliver the same number of frequency coefficients. These correspond to the data demodulated from the many carriers.

The inverse FFT is similarly used in the transmitter to generate the OFDM signal from the input data.

## Choice of basic modulation

In each symbol, each carrier is modulated (multiplied) by a complex number taken from a constellation set. The more states there are in the constellation, the more bits that can be conveyed by each carrier during one symbol – but the closer become the constellation points, assuming constant transmitted power. Thus there is a well- known trade-off of ruggedness versus capacity.

At the receiver, the corresponding demodulated value (the frequency coefficient from the receiver FFT) has been multiplied by an arbitrary complex number (the response of the channel at the carrier frequency). The constellation is thus rotated and changed in size. How can we then determine which constellation point was sent?

One simple way is to use differential demodulation, such as the DQPSK used in DAB. Information is carried by the change of phase from one symbol to the next. As long as the channel changes slowly enough, its response does not matter. Using such a differential (rather than a coherent) demodulation process causes some loss in thermal noise performance – but DAB is nevertheless a very rugged system.

When higher capacity is needed (as in DVB-T) there are advantages in coherent demodulation. In this, the response of the channel for each carrier is somehow determined, and the received constellation is appropriately equalized before determining which constellation point was transmitted, and hence what bits were transmitted.

To do this in DVB-T, some pilot information is transmitted (so-called scattered pilots) so that, in some symbols on some carriers, known information is transmitted from which a sub-sampled version of the frequency response is measured. This is then interpolated, using a 1-D or 2-D filter, to fill in the unknown gaps, and is used to equalize all the constellations which carry data.

## Coherent Demodulation Process

- Receiver Synchronization ( Time & Frequency)
- Perform FFT (Convert to Freq Domain), Locally generate a carrier equal in frequency & phase to the first carrier, mix with received COFDM symbol Integrate over the period Tu. The first carrier will be shifted vertically (beat down zero dc) and hence separated, Modulation recovered, Other carriers Integrate to Zero
- Very rapidly repeat step two above (1704 times 2K mode) for each carrier frequency in turn until all carriers have been effectively separated

# Transmitter

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 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 pass band 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, , respectively. These signals are then summed to give the transmission signal.

# Receiver

The receiver picks up the signal , 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 digitized using analog-to-digital converters (ADCs), and a forward FFT is used to convert back to the frequency domain.

This returns 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.

# 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.

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, 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.

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 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.

# 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 the bitrate is adapted to the co-channel interference and bandwidth is allocated to whichever subscriber needs it most.

## 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-analogue converter (DAC) in the transmitter
- a high-resolution analogue-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 minimize 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.

# 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 utilize 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 — 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 utilized 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 IEEE802.11n Wireless LAN standard.