1 Introduction
Orthogonal frequency-division multiplexing (OFDM) is the modulation technique for European standards such as the Digital Audio Broadcasting (DAB) [ETSI, 1997a] and the Digital Video Broadcasting (DVB) [ETSI, 1997b] systems. As such it has received much attention and has been proposed for many other applications, including local area networks [ETSI, 1998] and personal communication systems [ETSI, 1997c]. OFDM is a type of multichannel modulation that divides a given channel into many parallel subchannels or subcarriers, so that multiple symbols are sent in parallel. Earlier overviews of OFDM can be found in [Bingham, 1990; Zou and Wu, 1995].
The first multichannel modulation systems appeared in
the 1950's as military radio links, systems best characterized as
frequency-division multiplexed systems. The first OFDM schemes were presented
by Chang, [1966] and Saltzberg, [1967]. Actual use of OFDM was limited
and the practicability of the concept was questioned. However, OFDM was made
more practical through work of Chang and
Gibby [1968], Weinstein and Ebert, [1971], Peled and Ruiz, [1980],
and Hirosaki, [1981]. The type of OFDM that we will describe in this article uses
the discrete Fourier transform (DFT) [Weinstein
and Ebert, 1971] with a cyclic
prefix [Peled and Ruiz, 1980]. The DFT (implemented with a fast
Fourier transform (FFT)) and the cyclic prefix have made OFDM both practical
and attractive to the radio link designer. A similar multichannel modulation
scheme, discrete multitone (DMT) modulation, has been developed for static
channels such as the digital subscriber loop [ANSI, 1995]. DMT also
uses DFTs and the cyclic prefix but has the additional feature of bit-loading
which is generally not used in OFDM, although related ideas can be found in [
The choice for OFDM as transmission technique could be justified by comparative studies with single carrier systems. However, few such studies have been documented in the literature, see, e.g., [Gosh, 1996]. OFDM is often motivated by two of its many attractive features: it is considered to be spectrally efficient and it offers an elegant way to deal with equalization of dispersive slowly fading channels. We concentrate here on such channels.
Multiuser systems that use OFDM must be extended with a proper multiple-access scheme as must single carrier transmission systems. Compared to single carrier systems, OFDM is a versatile modulation scheme for multiple access systems in that it intrinsically facilitates both time-division multiple access and frequency-division (or ‘subcarrier-division’) multiple access [ETSI, 1997c]. In addition, considerable attention has been given to the combination of the OFDM transmission technique and code-division multiple access (CDMA) in multicarrier-CDMA systems, MC-CDMA, see [Hara and Prasad, 1997] and the references therein.
OFDM also has some drawbacks. Because OFDM divides a given spectral allotment into many narrow subcarriers each with inherently small carrier spacing, it is sensitive to carrier frequency errors. Furthermore, to preserve the orthogonality between subcarriers, the amplifiers need to be linear. OFDM systems also have a high peak-to-average power ratio or crest-factor, which may require a large amplifier power back-off and a large number of bits in the analog-to-digital (A/D) and digital-to-analog (D/A) designs. All these requirements can put a high demand on the transmitter and receiver design.
OFDM has been successfully used in the European DAB and DVB systems [ETSI, 1997a,b]. DAB is expected to be fully launched during 1998-2000 and is already broadcast on a trial basis in many countries. OFDM has also been a topic of research for use in wireless local-area networks and is a candidate for the European broadband radio access network standard [ETSI, 1998] currently being developed by the European Telecommunications Standards Institute (ETSI).
For the standardization of the European third generation personal communica-tions system within ETSI, the universal mobile telecommunications system (UMTS), OFDM was a technically promising candidate stressing versatility of services and resource allocation [ETSI, 1997c]. The OFDM-based proposal was, however, abandoned in favour of the two final CDMA-based proposals that were more thoroughly investigated and had broad support. OFDM is now under investigation for the fourth generation mobile communication system.
In addition to these radio systems, multicarrier techniques have also been used for broadband wired applications. Multicarrier modulation, in the form of the DMT modulation applied to the twisted copper-pair channel, has been adopted as the modulation technique for the asynchronous digital subscriber loop (ADSL) [ANSI, 1995] in the US and is now one of two candidates for the very high bit rate digital subscriber loop (VDSL) being standardized by the American National Standards Institute (ANSI) and by ETSI [VDSL Alliance, 1997].
This paper proceeds as follows. We describe the principles of the OFDM transmission technique in Section 2. In Section 3 we describe how to generate OFDM signals by means of an FFT, how to suppress out-of-band emission and how to reduce the dynamics of the transmitted signal. Section 4 describes receiver operations such as synchronization, channel estimation and equalization.
2 Principles of OFDM
OFDM is a block transmission technique. In the baseband, complex-valued data symbols modulate a large number of tightly grouped carrier waveforms. The transmitted OFDM signal multiplexes several low-rate data streams — each data stream is associated with a given subcarrier. The main advantage of this concept in a radio environment is that each of the data streams experiences an almost flat fading channel. In slowly fading channels, the intersymbol interference (ISI) and intercarrier interference (ICI) within an OFDM symbol can be avoided with a small loss of transmission energy using the concept of a cyclic prefix.
2.1 Signal characteristics
An OFDM signal consists of 
orthogonal subcarriers
modulated by parallel data streams.
Each baseband subcarrier is of the form
, (1)
where 
is the frequency of
the 
th subcarrier. One baseband OFDM symbol (without a cyclic
prefix) multiplexes 
modulated subcarriers:
(2)
where 
is the 
th complex data symbol (typically taken from a PSK or QAM
symbol constellation) and 
is the length of the
OFDM symbol. The subcarrier frequencies are equally spaced
(3)
which makes the subcarriers 
on 
orthogonal. The signal
(2) separates data symbols in frequency by overlapping subcarriers, thus using
the available spectrum in an efficient way. The left half of Figure 1
illustrates the quadrature component of some of the subcarriers of an OFDM
symbol. The right half of Figure 1 illustrates how the subcarriers are packed
in the frequency domain.
Figure 2 shows time and frequency characteristics of an OFDM signal with 1024 subcarriers. As the OFDM signal is the sum of a large number of independent, identically distributed components its amplitude distribution becomes approximately Gaussian due to the central limit theorem. Therefore, it suffers from large peak-to-average power ratios. In addition, OFDM symbols of the form (2) can have large out-of-band power as illustrated in Figure 2. Large peak-to-average power ratios also cause out-of-band emission because of amplifier non-linearities. Section 3 discusses ways to deal with high peak-to-average power ratios and out-of-band power.
The OFDM symbol (2) could typically be received using a bank of matched filters. However, an alternative demodulation is used in practice. T-spaced sampling of the in-phase and quadrature components of the OFDM symbol yields (ignoring channel impairments such as additive noise or dispersion)
, (4)
which is the inverse discrete Fourier transform (IDFT) of the
constellation symbols . Accordingly, the sampled data is demodulated with a DFT.
This is one of the key properties of OFDM, first proposed by Weinstein and Ebert, [1971]. The DFT, typically implemented
with an FFT, actually realizes a sampled matched-filter receiver in systems
without a cyclic prefix.
2.2 OFDM with a cyclic prefix
Two difficulties arise when the signal in (2) is transmitted over a dispersive channel. One difficulty is that channel dispersion destroys the orthogonality between subcarriers and causes intercarrier interference (ICI). In addition, a system may transmit multiple OFDM symbols in a series so that a dispersive channel causes intersymbol interference (ISI) between successive OFDM symbols. The insertion of a silent guard period between successive OFDM symbols would avoid ISI in a dispersive environment but it does not avoid the loss of the subcarrier orthogonality. Peled and Ruiz [1980] solved this problem with the introduction of a cyclic prefix. This cyclic prefix both preserves the orthogonality of the subcarriers and prevents ISI between successive OFDM symbols. Therefore, equalization at the receiver is very simple. This often motivates the use of OFDM in wireless systems.
The cyclic prefix, illustrated in Figure 3, works as
follows. Between consecutive OFDM signals a guard period is inserted that
contains a cyclic extension of the OFDM symbol. The OFDM signal (2) is extended
over a period 
so that
(5)
The signal then passes through a channel, modeled by a finite-length
impulse response limited to the interval 
. If the length of the cyclic prefix is chosen such that 
the received OFDM
symbol evaluated on the interval 
, ignoring any noise effects, becomes
(6)
where
(7)
is the Fourier transform of 
evaluated at the
frequency . Note that within this interval the received signal is
similar to the original signal except that 
modulates the th subcarrier instead of . In this way the cyclic prefix preserves the orthogonality
of the subcarriers.
Equation (6) suggests that the OFDM signal can be
demodulated as described in the previous section, taking an FFT of the sampled
data over the interval , ignoring the received signal before and after 
. The received data (disregarding additive noise) then has
the form
(8)
The received data in Equation (8) can be recovered with parallel one-tap
equalizers. This simple channel equalization motivates the use of a cyclic
prefix and often the use of OFDM itself. Because we ignore the signal within
the cyclic prefix this prefix also acts as the above mentioned silent guard
period preventing ISI between successive OFDM symbols.
The use of a cyclic prefix in the transmitted signal has the disadvantage of requiring more transmit energy. The loss of transmit energy (or loss of signal-to-noise ratio (SNR)) due to the cyclic prefix is
(9)
This is also a measure of the bit rate reduction required by a
cyclic prefix. That is, if each subcarrier can transmit 
bits, the overall bit
rate in an OFDM system is 
bits per second as
compared to the bit rate of 
in a system without a
cyclic prefix. If latency requirements allow, these losses can be made small by
choosing a symbol period much longer than the
length of the cyclic prefix .
2.3 Channel noise and Doppler spread
In this paper we devote little space to the radio channel as it is described in length and detail in other parts of this book. However, we mention a few channel impairments that are important for OFDM. OFDM systems often experience not only channel dispersion as addressed above, but also additive white Gaussian noise (AWGN), Doppler spreading and synchronization errors. Many of these impairments can be modeled as AWGN if they are relatively small. Synchronization errors such as carrier frequency offsets, carrier phase noise, sample clock offsets and symbol timing offsets are discussed in Section 4.
The inclusion of Gaussian noise in the signal model
(2) yields a received OFDM signal 
and Equation (8)
extended with a noise term becomes
(10)
where 
is the FFT of the
sampled noise terms 
. If the received noise 
is white, the noise after the FFT will
also be white.
In a fading channel the channel variations affect the
performance of the OFDM system. For a fixed sampling period, the OFDM symbol
length increases with the number of subcarriers and so does its sensitivity to
channel variations. To illustrate the effects, consider an OFDM system in a
flat-fading channel, a channel with a time-varying one-tap impulse response 
. The transmitted OFDM signal is multiplied with this
time-varying scalar which yields the received 
. The multiplication appears as a convolution in the
frequency domain causing spreading of the subcarriers and, consequently, ICI.
The sampled signal after the DFT is of the form [Cimini, 1985]
(11)
where 
is the DFT of the now
time-varying channel tap 
.
In some cases the above spreading may be desirable as it is a way to introduce diversity [Cimini, 1985]. A frequency domain channel equalizer can exploit such diversity. Other systems requiring orthogonality between subcarriers may suffer from the spreading. For a fixed sampling time the ICI due to the Doppler spreading increases with the number of carriers. Russell and Stüber [1995], using a central limit theorem argument, characterize the effect of the ICI as an additive Gaussian noise with a variance that increases with the number of subcarriers and with the maximum Doppler frequency. This noise is correlated in time, but white across subcarriers. The ICI leads to an error floor which may be unacceptable. Antenna diversity or coding are suggested to reduce this error floor [Russell and Stüber , 1995].
2.4 Design of OFDM signals
The number of subcarriers , the bandwidth of each subcarrier 
, the bandwidth of the system
, and the length of the cyclic prefix are all important
parameters in the design of an OFDM system.
First, the length of the cyclic prefix should be
chosen to be a small fraction of the OFDM symbol length to minimize the loss of
SNR (or data rate) in (9). Because the size of the cyclic prefix is directly
related to the delay spread 
of the channel a rule
of thumb is that the length of the OFDM symbol 
or, equivalently, the
number of subcarriers 
. However, if the OFDM symbol length is too long the ICI
caused by Doppler spreading in the fading channel can become performance
limiting. If the intercarrier spacing is chosen much larger
than the maximum Doppler frequency 
, the system is relatively insensitive to the Doppler spread
and the associated ICI. Therefore, the number of subcarriers should satisfy 
or equivalently 
. The above two constraints result in the following
restriction on the number of subcarriers
(12)
Equation (12) also states a requirement on the delay-
and Doppler-spread of the physical channel for proper design of an OFDM system.
The far left hand side and the far right hand side also lead to 
, which means that the more the channel is underspread, i.e.,
the more correlated the channel is in either time or frequency, the easier it
is to find a suitable number of subcarriers .
Example 1. The UMTS has been assigned frequencies in the 2.2 GHz band. Operators expect to be assigned 5 MHz for uplink and 5 MHz for downlink transmission and therefore in the following we assume a sample frequency of 5 MHz. A proper design of a radio interface based on OFDM depends on the characteristics of the radio environment in these bands. For the evaluation of the UMTS, ETSI has developed a set of channel models that describe the environment for different transmission situations (indoor, pedestrian, vehicular) [ETSI, 1997d]. The system should typically be used in all these environments and therefore we base our design on the worst values for the Doppler frequencies and channel delay spreads.
The vehicular channel models adopt a mobile speed of
120 km/h. This speed corresponds to a maximum Doppler frequency of about 250
Hz. Furthermore, if we want our system to accommodate echoes up to about 10 
sec, we require (12)
(13)
Since the effects on the system performance of violating either the
left or right hand requirement are different, it is not obvious how to choose . One possible design could be, for instance, the geometric
mean of the boundary values: 
. A typical UMTS radio interface thus could use 1024
subcarriers implemented with an 1024-point FFT.
The length of the cyclic prefix must be larger than
the channel impulse response, that is 
sec. A possible design
example of a 64-sample cyclic prefix (12.8 sec) leads to a reduction of the data rate and transmission
power of 6%. If we modulate each subcarrier with a QPSK symbol, we have 
bits per subcarrier
giving us an overall raw data rate of 9.4 Mbits/sec. Similarly, an extension to
16-QAM symbols would give 18.8 Mbits/sec raw data rate.
2.5 Coding
Error control coding is an essential part of an OFDM system for mobile communication. OFDM in a fading environment is almost always used with coding to improve its performance and as such is often referred to as Coded OFDM or COFDM [Zou and Wu, 1995]. For an uncoded OFDM system in a frequency selective Rayleigh-fading environment, each OFDM subcarrier has a flat-fading channel. Accordingly, the average probability of error for an uncoded OFDM system is the same as that for a flat-fading single-carrier system with the same average geometric mean of the subcarriers’ SNRs.
Just as we can introduce time diversity through coding and interleaving in a flat-fading single-carrier system, we can introduce frequency diversity through coding and interleaving across subcarriers in an OFDM system [ETSI, 1997c; Sandell, et al., 1997]. However, since OFDM in itself does not increase the system bandwidth it can never introduce frequency diversity on flat fading channels.
With coding and interleaving across subcarriers, the
strong subcarriers help the weak ones. Over a single OFDM signal, we cannot
guarantee that the interleaved subcarriers are all independent. In [Wilson, 1994; Chini, 1994] we find that the achievable
frequency diversity in a COFDM system is limited by the number of resolvable
independent paths in the channel impulse response. One heuristic explanation is
as follows. Coding diversity requires independent SNRs on each path to get the
full diversity of the code. In an OFDM system the SNRs of different subcarriers
are usually correlated because the channel length is small compared to the OFDM
symbol length, . How correlated the subcarriers are depends on the number of
resolvable taps. In the extreme case of a one-tap channel, the SNR on all the
subcarriers is the same. So regardless of the interleaving or power of the code
interleaving in frequency will not increase the diversity. As the number of
taps in the channel increases the correlation between subcarriers decreases but
they will never be totally independent.
3 OFDM transmission
3.1 Signal generation
Equation (2) shows an ideal OFDM signal, which could be generated by a bank of oscillators. Such an implementation could, however, become prohibitively complex as the number of subcarriers becomes large. Similar to the demodulation of the data samples with the DFT, the baseband signal can be generated digitally by means of an IDFT [Weinstein and Ebert, 1971]. Figure 4 shows such an OFDM transmitter. This has paved the way for practical use of OFDM. Consider the signal
(14)
where
(15)
is the oversampled IDFT of the constellation symbols , the integer 
, and 
is an interpolating
filter. The output signal from the D/A, 
, can be made very close to the ideal signal 
as defined in (2). The
quality of the approximation depends on, for example, the characteristics of
the D/A including the interpolation filter and the IDFT.
A spectrum of an OFDM signal is shown in Figure 2. The
spectrum decays with 
and spectral leakage
into neighboring bands is sometimes too large to meet regulation requirements.
Several approaches have been taken to combat this out-of-band emission. The
most straightforward is perhaps to use a large number of subcarriers to narrow
the spectra, however at the cost of increased complexity, increased sensitivity
to Doppler effects, and higher demands on the accuracy of frequency
synchronization.
Another approach is to use pulse shaping of the OFDM symbol to change the spectral occupancy. Pulse shaping can be done either by applying a time window [ETSI, 1997c] to the OFDM symbol or by passing the OFDM signal through a filter [Pollet and Moeneclaey, 1995], typically combined with the interpolation filter above. Pulse shaping has to be applied with care since orthogonality between the subcarriers is rarely maintained.
An example of using a time-window to shape the symbol is shown in the left half of Figure 5. Compared with the rectangular window, the symbol has been cyclically extended with a number of extra samples shaped with a raised cosine window [ETSI, 1997c]. The original symbol is left unchanged in the center. The associated power spectra for the two signals are shown in the right half of Figure 5.
Whereas the application of pulse shaping can improve the spectral occupancy of an OFDM system, its effect on other system characteristics has also motivated some recent studies towards design of suitable pulse shapes [Le Floch, et al., 1995; Sandberg and Tzannes, 1995; Haas and Belfiore, 1997]. Much of this work focuses on the choice of new sets of basis functions for the modulation of parallel channels and on how such pulses improve an OFDM system’s robustness to Doppler effects, carrier frequency variations, and time synchronization. The application of wavelet theory to multicarrier communications and pulse shaping is documented in, e.g., [Wornell, 1996].
3.2 Techniques to reduce the peak-to-average power ratio
An OFDM signal has an approximately Gaussian amplitude distribution when the number of subcarriers is large. Therefore, very high peaks in the transmitted signal can occur. This property is often measuerd via the signal's peak-to-average power ratio. To be able to transmit and receive these peaks without clipping the signal, the A/D and D/A need to be designed with high demands on range and precision.
If the dynamic ranges of the A/D and D/A are increased, the resolution also needs to be increased in order to maintain the same quantization noise level. Therefore, an OFDM signal may require expensive A/Ds and D/As compared to many other modulation formats, and for some applications suitable A/Ds and D/As may not be available at all. Also, a large power back-off of the amplifiers is necessary. Intentional or accidental clipping of the OFDM signal often occurs in practice. The clipping of a received sample affects all subcarriers in the system. The sensitivity to clipping effects is investigated by, e.g., Gross and Veeneman, [1993].
At least three concepts for reducing the peak-to-average power ratio have been proposed [Müller and Huber, 1997; Narahashi and Nojima, 1997; Jones, et al., 1996; van Nee, 1996; Tellado and Cioffi, 1997]. In the first concept one signal with a low peak-to-average power ratio out of a set of signals is transmitted. In [Narahashi and Nojima, 1997], for instance, it is observed that by appropriately choosing the phase of each subcarrier the peak-to-average power ratio can be reduced.
The second concept reduces the peak-to-average power ratio by coding. Block codes can accomplish this, see, e.g., [Jones, et al., 1996], and in [van Nee, 1996] it is shown that complementary codes have good properties to combine both peak-to-average power reduction and forward error correction.
Finally, in [Tellado and Cioffi, 1997] impulse-like time-domain functions are iteratively subtracted from the original signal to reduce the peaks. These time-domain signals are generated by a set of reserved, unused symbols in the DFT domain. Subcarriers which are not used to transmit data symbols are used to transmit symbols, chosen to generate a transmitted signal with low peak-to-average power ratio.
4 OFDM reception
In Section 2 we have discussed aspects of the demodulation of data in OFDM. Two other important parts of the processing of the received OFDM signal are synchronization and channel estimation. Here we address these topics.
4.1 Synchronization
At the front-end of the receiver OFDM signals are subject to synchronization errors due to oscillator impairments and sample clock differences. The demodulation of the received radio signal to baseband, possibly via an intermediate frequency, involves oscillators whose frequencies may not be perfectly aligned with the transmitter frequencies. This results in a carrier frequency offset. Figure 6 illustrates the front end of an OFDM receiver where these errors can occur. Also, demodulation (in particular the radio frequency demodulation) usually introduces phase noise acting as an unwanted phase modulation of the carrier wave. Carrier frequency offset and phase noise degrade the performance of an OFDM system.
When the baseband signal is sampled at the A/D, the sample clock frequency at the receiver may not be the same as that at the transmitter. Not only may this sample clock offset cause errors, it may also cause the duration of an OFDM symbol at the receiver to be different from that at the transmitter. If the symbol clock is derived from the sample clock this generates variations in the symbol clock. Since the receiver needs to determine when the OFDM symbol begins for proper demodulation with the FFT, a symbol synchronization algorithm at the receiver is usually necessary. Symbol synchronization also compensates for delay changes in the channel.
The effects of synchronization errors are investigated in, among others, [Pollet and Moeneclaey, 1995; Pollet, et al., 1995; Moose, 1994; Pollet, et al., 1994; Gudmundson and Anderson, 1996; Muschallik, 1995; Wei and Schlegel, 1995; Garcia Armada and Calvo, 1998]. Table 1 summarizes some of the important results.
The most important effect of a frequency offset
between transmitter and receiver is a loss of orthogonality between the
subcarriers resulting in ICI. The characteristics of this ICI are similar to
white Gaussian noise and lead to a degradation of the SNR [Pollet, et al., 1995]. For both AWGN and fading channels, this
degradation increases with the square of the number of subcarriers. Table 1
illustrates this degradation as a function of the frequency offset normalized
to the intercarrier spacing, 
.
Like frequency offsets, phase noise and sample clock
offsets cause ICI and thus a degradation of the SNR. The extent of these
degradations is indicated in Table 1. The table shows the phase noise
degradation for an phase noise bandwidth 
. This bandwidth is normalized to the intercarrier spacing 
. The degradation due to a sample clock frequency offset 
, also normalized to the sample clock frequency and denoted
in parts per million (ppm), is shown, too. For a DVB-like OFDM system Muschallik, [1997], concludes that phase noise is not performance limiting in
properly designed consumer receivers for OFDM. Pollet, et al., [1994],
show that the degradation due to a sample clock frequency offset differs from
subcarrier to subcarrier, the highest subcarrier experiencing the largest
SNR-loss.
Finally, the degradation due to symbol timing errors is not graceful. If the length of the cyclic prefix exceeds the length of the channel impulse response a receiver can capture an OFDM symbol anywhere in a region where the symbol appears cyclic, without sacrificing orthogonality. A small error only appears as pure phase-rotations of the data symbols and may be compensated by the channel equalizer, still preserving the system's orthogonality. A large error resulting in capturing a symbol outside this allowable interval, on the other hand, causes ISI, ICI, and a performance degradation.
Example 2. For the UMTS system in Example 1 the distortion expressions from
Table 1 are illustrated by the curves in Figure 7. A typical sample clock
offset of 
ppm results in an SNR
degradation of less than 0.1 dB. A phase noise bandwidth of 50 Hz (
) results in an SNR degradation of about 1 dB. In [Garcia Armada and Calvo, 1998] it is shown that a channel
equalizer may compensate for a part of this degradation.
The sensitivity to frequency offsets in the UMTS scenario determines the performance requirements of a frequency tracking scheme. In order to keep the distortion less than 1 dB, the frequency offset may not exceed 2% of the subcarrier spacing. Finally, the sensitivity to symbol time offsets determines the performance requirements of a time tracking scheme. Due to the presence of the cyclic prefix that is 64 samples long, time offsets on the order of 10 samples will not affect our system's performance. The remaining samples of the cyclic prefix are needed to keep the system’s orthogonality in dispersive channels.
Summarizing, oscillator phase noise and sample-clock variations generate ICI but seldom limit system performance. Frequency offsets and symbol clock offsets, however, generally need to be tracked at the receiver. We now give a brief review of some recently proposed frequency and timing estimators for OFDM and then describe one of these methods, based on the cyclic prefix, in more detail.
We model the uncertainty about the start of the baseband OFDM symbol as an unknown delay and the uncertainty about the transmitter carrier frequency as a rotation of the complex-valued transmitted signal. We ignore the channel dispersion, isolating the offset problem. The model for the received signal becomes
(16)
where 
is the unknown
integer-valued delay and is the unknown carrier
frequency offset relative to the intercarrier spacing. The received signal now
contains unknown time and frequency offsets and, of course, the unknown data
symbols (via 
).
Time and frequency offset estimators have been addressed in a number of publications, see [Moose, 1994; Wei and Schlegel, 1995; Warner and Leung, 1993; Classen and Meyr, 1994; Schmidl and Cox, 1997a; Tourtier, et al.; 1993; Daffara and Adami, 1995; Sandell, et al., 1995; van de Beek, et al., 1997; Lee and Cheon, 1997; Seki, et al., 1997; Nogami and Nagashima, 1995; Sheu, et al., 1997; Schmidl, 1997b]. We divide these estimators conceptually into two groups. The first group [Moose, 1994; Warner and Leung, 1993; Classen and Meyr, 1994; Schmidl and Cox, 1997a] assumes that transmitted data symbols are known at the receiver. This can in practice be accomplished by transmitting known pilot symbols according to some protocol. The unknown symbol timing and carrier frequency offset may then be estimated from the received signal. The insertion of pilot symbols usually implies a reduction of the data rate. An example of such a pilot-based algorithm is found in [Warner and Leung, 1993], joint time and frequency offset estimators based on this concept are described in [Classen and Meyr, 1994; Schmidl and Cox, 1997], and in [Moose, 1994] the repetition of an OFDM symbol supports the estimation of a frequency offset.
A second approach [Tourtier,
et al., 1993; Daffara and Adami, 1995; Sandell, et al., 1995; van de Beek, et al., 1997;
Lee and Cheon, 1997] uses statistical redundancy in the received signal. The
transmitted signal is modeled as a
Gaussian process. The offset values are then estimated by exploiting the
intrinsic redundancy provided by the L samples constituting the cyclic prefix. The basic idea behind these
methods is that the cyclic prefix of the transmitted signal (5) yields
information about where an OFDM symbol is likely to start. Moreover, the
transmitted signal's redundancy also contains useful information about the
carrier frequency offset. In [Tourtier,
et al., 1993 ] the authors
recognize that the statistic
(17)
contains information about the time offset . This statistic, implemented with a sliding sum, identifies
samples of the cyclic prefix by the sum of 
consecutive
differences. The statistic is likely to become small when the index 
is close to the start
of the OFDM symbol. Therefore, Tourtier,
et al., [1993], propose the time
offset estimator
(18)
Sandell, et al., [1995], van de Beek, et al., [1997], and later Lee and Cheon, [1997], use the statistic
(19)
to estimate the time offset. This statistic is the sum of consecutive
correlations and its magnitude is likely to become large when the index is close to the start
of the OFDM symbol. Furthermore, the phase of the statistic 
at the time 
is related to the
frequency offset. In [Lee and Cheon, 1997] this offset is estimated by
(20)
where 
denotes the angle of
its complex-valued argument. From (16), (19) and (20), the frequency estimate
is the argument of a sum of complex numbers. Without additive noise and correct
time estimate 
, each term 
has the same argument:
. Hence, each term contributes coherently to the sum, while
the additive noise contributes incoherently. Because of this implicit averaging
the variance of the frequency estimator is usually low. The concept of using
statistics based on the pairwise correlation in the received signal due to the
cyclic prefix for the estimation of time and frequency offsets is patented in [Seki, et al., 1997 ].
The joint maximum likelihood (ML) estimator of the
time and frequency offsets for the AWGN channel is derived by van de Beek, et al., [1997]. This optimal estimator is shown
to be based on the sufficient statistic in (19) together with
the additional statistic
(21)
The joint ML estimate is
(22)
, (23)
where 
is the magnitude of
the correlation coefficient at the receiver
between a sample in the tail of the OFDM symbol and its copy in the
cyclic prefix.
Example 3. Consider the downlink of the UMTS system in Examples 1 and 2 for
the “Vehicular B” channel for UMTS [ETSI,
1997d]. Each mobile can track its
time and frequency offsets with the ML estimator (22) and (23). Figure 8 shows
the “eye-diagram” of the statistics 
and 
. The indexes at which the upper statistic peak provide the
most likely start of the OFDM symbols, while the values of the lower statistic
at these indexes give the most likely frequency offsets.
The distribution of these estimates is shown in the right half of Figure 8. The time offset estimates are within about 10 samples of the true time instant and the frequency offset estimates are within about 2% of the true offset. See Figure 7 and Example 2 for the consequences of this estimation for the system performance.
Synchronization schemes are discussed in, e.g., [Tourtier, et al., 1993; Nogami and Nagashima,
1995; Sheu, et al., 1997].
Figure 9 shows a possible receiver structure that compensates for time and
frequency offsets. In addition to controlling a voltage controlled oscillator
in the analog receiver front-end, frequency correction can also be performed
digitally by multiplying the received signal in front of the FFT with the estimate-based
signal 
. Timing correction is typically performed in conjunction
with the removal of the cyclic prefix and the FFT.
4.2 Channel estimation
Some OFDM systems (as, for instance, the DAB standard [ETSI, 1997a]) modulate the subcarriers differentially [Engels and Rohling, 1995]. The information symbols may be encoded differentially from one OFDM symbol to the next within one subcarrier, or differentially between adjacent subcarriers within one OFDM symbol. In a fading channel environment, such a modulation does not need to track the subcarrier attenuations (tracking of the carrier frequencies has still to be done). The performance sacrifice associated with this modulation scheme compared with coherent modulation schemes is often motivated by its simple receiver structure and its avoidance of pilot symbols. However, if the subcarriers are coherently modulated as in the DVB standard [ETSI, 1997b], estimation of the channel's attenuations of each subcarrier is necessary. These estimates are used in the channel equalizer, which, in an OFDM receiver, may consist of one complex multiplication for each subcarrier in an OFDM symbol. Figure 9 shows the receiver structure for a coherent OFDM receiver.
Channel estimation in OFDM is usually performed with the aid of pilot symbols. Since each subcarrier is flat fading, techniques from single-carrier flat fading systems are directly applicable to OFDM. For such systems pilot-symbol assisted modulation (PSAM) on flat fading channels [Moher and Lodge, 1989; Cavers, 1991] involves the sparse insertion of known pilot symbols in a stream of data symbols. The attenuation of the pilot symbols is measured and the attenuations of the data symbols in between these pilot symbols are typically estimated/interpolated using time-correlation properties of the fading channel.
The concept of PSAM in OFDM systems also allows the use of the frequency correlation properties of the channel. The time-frequency grid in Figure 10 illustrates three ways of inserting pilots symbols among data symbols. The first pilot pattern inserts entirely known OFDM symbols in the OFDM signal. The second modulates pilot symbols on a particular set of subcarriers. The third pattern uses scattered pilot symbols, as in the DVB standard.
In OFDM systems where Doppler effects are kept small (that is, the OFDM symbol is short compared with the coherence time of the channel) the time correlation between the channel attenuations of consecutive OFDM symbols is high. Furthermore, in a properly designed OFDM system the subcarrier spacing is small compared with the coherence bandwidth of the channel (see Section 2). Therefore, there is also a substantial frequency correlation between the channel attenuations of adjacent subcarriers. Both the time and frequency correlation can be exploited by a channel estimator. The choice of pilot pattern determines the form of the channel estimator.
Most documented channel estimation concepts consist of two steps, one or both of which use the correlation of the channel. First, the attenuations at the pilot positions are measured and possibly smoothed using the channel correlation. These measurements then serve to estimate (interpolate) the complex-valued attenuations of the data symbols in the second step. This second step uses the channel correlation properties either with interpolation filters or with a decision-directed scheme. Depending on the pilot pattern (see Figure 10) the estimation strategies diverge in this second step.
Höher, [1991], for instance, proposes a scattered pilot pattern. The interpolation in the presented scheme uses channel measurements with two FIR Wiener filters. The first Wiener filter interpolates and smoothes the channel attenuations in frequency. A second Wiener filter then interpolates and smoothes the channel attenuations in time. This scheme exploits the channel correlation properties in the design of the interpolating Wiener filters. In general the correlation properties and the SNR needed to design the estimator are not known. Therefore, Höher, [1991], proposes to design the estimator for fixed, assumed values of the channel correlations and SNR.
Recent publications by Edfors, et al., [1998a,b]
and Li, et al., [1998], focus on the
first pilot pattern of Figure 10, where completely known OFDM symbols are
sparsely inserted in the stream of OFDM symbols. The channel attenuations in
between these OFDM symbols are then either interpolated using the channel time
correlation or the estimator is applied on consecutive OFDM symbols in a
decision-directed scheme [
We first adopt a matrix formulation of Equation (10)
and collect the channel attenuations of one OFDM symbol (the Fourier transform
of evaluated at the
frequencies ) in the vector 
. The observed symbols after the receiver DFT become
, (24)
where the diagonal matrix 
contains the
transmitted symbols on its diagonal (either known pilot symbols or receiver
decisions of information symbols which we in the following assume are correct),
and the vector 
contains the observed
outputs of the DFT. In this matrix notation the least-squares (LS) channel
estimate (minimizing 
for all possible 
) becomes
(25)
This estimator simply divides the received symbol on each subcarrier by the transmitted symbol to obtain an estimate of the channel attenuation. From the system property (10), this is an estimator that intuitively makes sense.
The frequency correlation can now be used to smooth
and improve the LS channel estimate. Various strategies can be adopted to use
the frequency correlation. The optimal linear minimum mean-squared error
(LMMSE) estimate of (minimizing 
for all possible
linear estimators ) becomes
(26)
where
(27)
and 
is the channel
autocorrelation matrix, that is, the matrix containing the correlations of the
channel attenuations of the subcarriers, see [Edfors, et al., 1998a,b].
Similarly, 
denotes the
correlation matrix between the channel attenuations and their LS-estimates, and
denotes the
autocorrelation matrix of the LS estimates.
This LMMSE estimator is, for complexity reasons, of
little practical value. Not only does Equation (26) assume knowledge of the
channel correlation and the SNR, it also requires multiplications per
estimated attenuation, and the dependency on the pilots or decisions may require frequent
recalculation of the matrix 
. However, the LMMSE estimator (or any other high-performance
and complex estimator) can be used as a basis for the design of more feasible
estimators. In [Chini, 1994; Edfors, et al., 1998a,b; Li, et al., 1998] generic low-complexity approximations of (26) are developed.
Their performances can be made very close to that of the optimal LMMSE
estimator. They are generic in the sense that they use assumed (fixed) channel
correlation and SNR for the design of . They are low-complexity in the sense that they require
significantly fewer than multiplications per
estimated attenuation.
Example 4. For the UMTS system developed in Examples 1-3, we now adopt a multiple access scheme that separates users both in time and in frequency. As a minimal access entity each user modulates 22 adjacent subcarriers during one OFDM symbol. Let us consider how we could design a channel estimator for one user with a 64 kbit/s downlink.
This user is assigned 22 particular adjacent
subcarriers every third OFDM symbol. Of the 22 subcarriers, four are used for
transmission of pilot symbols, as shown in Figure 11. The other subcarriers
carry QPSK data symbols, yielding an uncoded data rate of 165.44 kbit/s. First,
we estimate the channel attenuations on the pilot positions using the LS
estimator (25). Because the number of pilots is small, we choose a channel
estimator based on (27) that assumes a fixed channel correlation. The channel
estimation requires 4 multiplications per transmitted data symbol.
5 Summary
This paper reviews the OFDM transmission technique. We have described its key properties, the FFT and the cyclic prefix, and we have addressed many of the recent research developments associated with OFDM. This paper discusses receiver operations such as synchronization, channel estimation and equalization. It also outlines the problems of out-of-band emission, and how to reduce the dynamics of the transmitted signal. For further reading, the reference list below covers many important recent contributions.
6 Acknowledgements
We would like to gratefully acknowledge the helpful
comments and suggestions of Ove Edfors,
7 Bibliography
ANSI [1995], “Network and Customer Installation Interfaces — Asymmetric Digital Subscriber Line (ADSL) Metallic Interface,” ANSI standard T1.413-1995.
J.J. van de Beek, M. Sandell, and P.O. Börjesson [1997], “ML Estimation of Time and Frequency Offsets in OFDM Systems,” IEEE Transactions on Signal Processing, 45, 7, pp. 1800-1805.
J.A.C. Bingham [1990], “Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come,” IEEE Communications Magazine, 28, 5, pp. 5-14.
J.K. Cavers [1991], “An Analysis of Pilot-Symbol Assisted Modulation for Rayleigh-Fading Channels,” IEEE Transactions on Vehicular Technology, 40, 4, pp. 686-693.
R.W. Chang [1966], “Synthesis of Band-Limited Orthogonal Signals for Multichannel Data Transmission,” Bell System Technical Journal, 45, pp. 1775-1796.
R.W. Chang, and R.A. Gibby [1968], “Theoretical Study of Performance of an Orthogonal Multiplexing Data Transmission Scheme,” IEEE Transactions on Communications, 16, 4, pp. 529-540.
A. Chini [1994], Multicarrier
Modulation in Frequency Selective Fading Channels, PhD thesis,
L.J. Cimini [1985], “Analysis and Simulation of a Digital
F. Classen, and H. Meyr [1994], “Frequency Synchronization
Algorithms for OFDM Systems Suitable for Communication over Frequency-Selective
Fading Channels,” in Proceedings of the
IEEE Vehicular Technology Conference (VTC’94), pp. 1655-1659,
F. Daffara, and O. Adami [1995], “A New Frequency Detector for Orthogonal Multicarrier Transmission Techniques,” in Proceedings of the Vehicular Technology Conference (VTC'95), pp. 804-809.
O. Edfors, M. Sandell, J.J. van de Beek, S.K. Wilson, and P.O. Börjesson [1998a], “OFDM Channel Estimation by Singular Value Decomposition,” IEEE Transactions on Communications,” 46, 7, pp. 931-939.
O. Edfors, M. Sandell, J.J. van de Beek, S.K. Wilson, and P.O. Börjesson [1998b], “Analysis of DFT-Based Channel Estimators for OFDM,” submitted to Wireless Personal Communications, Kluwer Academic Publishers.
V. Engels, and H. Rohling [1995], “Multilevel Differential Modulation Techniques (64-DAPSK) for Multicarrier Transmission Systems,” European Transactions on Telecommunications, 6, 6, pp. 633-640.
ETSI [1997a], “Radio Broadcasting Systems; Digital Audio
Broadcasting (DAB) to
ETSI [1997b], “Digital Video Broadcasting (DVB); Framing Structure, Channel Coding and Modulation for Digital Terrestrial Television,” European Telecommunications Standards Institute, ETS EN 300 744 v 1.1.2.
ETSI [1997c], “OFDMA Evaluation Report - The Multiple Access Proposal for the UMTS Terrestrial Radio Air Interface (UTRA),” European Telecommunications Standards Institute , Tdoc/SMG 896/97, ETSI SMG meeting no. 24, Madrid, Spain.
ETSI [1997d], “Selection Procedures for the Choice of Radio
Transmission Technologies for the UMTS,” European Telecommunications Standards
Institute, ETR/SMG 30.03, v.3.0.0,
ETSI [1998], “Broadband Radio Access Networks (BRAN); Inventory of Broadband Radio Technologies and Techniques,” European Telecommunications Standards Institute, ETSI Technical Report, DTR/BRAN-030001.
B. Le Floch, M. Alard, and C. Berrou [1995], “Coded Orthogonal Frequency Division Multiplex,” Proceedings of the IEEE, 83, 6, pp. 982-996.
A. Garcia Armada, and M. Calvo [1998], “Phase Noise and Sub-Carrier Spacing Effects on the Performance of an OFDM Communication System,” IEEE Communications Letters, 2, 1, pp. 11-13.
M. Gosh [1996], “Analysis of the Effect of Impulse Noise on Multicarrier and Single-Carrier QAM Systems,” IEEE Transactions on Communications, 44, 2, pp. 145-147.
R. Gross, and D. Veeneman [1993], “Clipping Distortion in DMT ADSL Systems,” Electronics Letters, 29, 24, pp. 2080-2081.
M. Gudmundson, and P.-O.
R. Haas, and J.C. Belfiore [1997], “A Time-Frequency Well-Localized Pulse for Multiple Carrier Transmission,” Wireless Personal Communications, 5, pp. 1-18.
S. Hara, and R. Prasad [1997], “Overview of Multicarrier CDMA,” IEEE Communications Magazine, 35, 12, pp. 126-133.
B. Hirosaki [1981], “An Orthogonally Multiplexed QAM System Using the Discrete Fourier Transform,” IEEE Transactions on Communications, 29, 7, pp. 982-989.
P. Höher [1991], “TCM on Frequency-Selective Land-Mobile Fading Channels,” in Proceedings of the Tirrenia International Workshop on Digital Communications, pp. 317-328.
A.J. Jones, T.A. Wilkinson, and S.K. Barton [1994], “Block coding scheme for reduction of peak to mean envelope power ratio of multicarrier transmission schemes,” in Electronics Letters, 30, 25, pp. 2098-2099.
D. Lee, and K. Cheon [1997], “A New Symbol Timing Recovery Algorithm for OFDM Systems,” IEEE Transactions on Consumer Electronics, 43, 3, pp. 767-775.
Y. Li, L.J. Cimini, Jr., and N.R. Sollenberger [1998], “Robust Channel Estimation for OFDM Systems with Rapid Dispersive Fading Channels”, IEEE Transactions on Communications, 46, 7, 902-915.
V. Mignone, and A. Morello [1996], “CD3-OFDM: A Novel Demodulation Scheme for Fixed and Mobile Receivers,” IEEE Transactions on Communications, 44, 9, 1144-1150.
M.L. Moher, and J.H. Lodge [1989], “TCMP - A Modulation and Coding Strategy for Rician-Fading Channels”, IEEE Journal on Selected Areas in Communications, 7, 9, pp. 1347-1355.
P.H. Moose [1994], “A Technique for Orthogonal Frequency Division Multiplexing Frequency Offset Correction,” IEEE Transactions on Communications, 42, 10, pp. 2908-2914.
S.H. Müller, and J.B. Huber [1997], “A Comparison of Peak Power Reduction Schemes for OFDM,” in Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM'97), pp. 1-5, Phoenix, Arizona, USA.
C. Muschallik [1995], “Influence of RF Oscillators on an OFDM Signal,” IEEE Transactions on Consumer Electronics, 41, 3, pp. 592-603.
S. Narahashi, and T. Nojima [1997], “A New Phasing Scheme for
Multitone Signal Systems to Reduce Peak-to-Average Power Ratio,” Electronics and Communications in
R.D.J. van Nee [1996], “OFDM Codes for Peak-to-Average Power
Reduction and Error Correction,” in Proceedings
of the IEEE Global Teleommunications Conference (GLOBECOM'96), pp. 740-744,
H. Nogami, and T. Nagashima [1995], “A Frequency and Timing Period Acquisition Technique for OFDM Systems,” in Proceedings of the Personal, Indoor, and Mobile Radio Communications (PIMRC'95), pp. 1010-1015.
A. Peled, and A. Ruiz [1980], “Frequency Domain Data Transmission Using Reduced Computational Complexity Algorithms,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’80), pp. 964-967.
T. Pollet, and M. Moeneclaey [1995], “Synchronizability of OFDM
Signals,” in Proceedings of the IEEE
Global Telecommunications Conference (GLOBECOM'95), pp. 2054-2058,
T. Pollet, M. van Bladel, and M. Moeneclaey [1995], “BER Sensitivity
of OFDM Systems to Carrier Frequency Offset and Wiener Phase Noise,” IEEE Transactions on Communications, 43,
T. Pollet, P. Spruyt, and M. Moeneclaey [1994], “The BER Performance of OFDM Systems Using Non-Synchronized Sampling,” in Proceedings of the IEEE GLOBECOM'94, pp. 253-257.
M. Russell, and G. Stüber [1995], “Interchannel Interference
Analysis of OFDM in a
B.R. Saltzberg [1967], “Performance of an Efficient Parallel Data Transmission System,” IEEE Transactions on Communications, 15, 6, pp. 805-811.
S.D. Sandberg, and M.A. Tzannes [1995], “Overlapped Discrete Multitone Modulation for High Speed Copper Wire Communications,” IEEE Journal on Selected Areas in Communications, 13, 9, pp. 1571-1585.
M. Sandell, S.K. Wilson, and P.O. Börjesson [1997], “Performance Analysis of Coded OFDM on Fading Channels with Non-Ideal Interleaving and Channel Knowledge,” in Proceedings of the IEEE Vehicular Technology Conference (VTC’97), pp. 1380-1384.
M. Sandell, J.J. van de Beek, and P.O. Börjesson [1995], “Timing and Frequency Synchronization in OFDM Systems Using the Cyclic Prefix,” in Proceedings of the IEEE International Symposium on Synchronization, pp. 16-19.
T.M. Schmidl [1997b], Synchronization
Algorithms for Wireless Data Transmission Using Orthogonal Frequency Division
Multiplexing (OFDM), PhD thesis,
T.M. Schmidl, and C. Cox [1997a], “Robust Frequency and Timing Synchronization for OFDM,” IEEE Transactions on Communications, 45, 12, pp. 1613-1621.
T. Seki, Y. Sugita, and T. Ishikawa [1997], “OFDM Synchronization
Demodulation Unit,”
C.-R. Sheu, Y.-L. Huang, and C.-C. Huang [1997], ”Joint Symbol Frame,
and Carrier Synchronization for
J. Tellado, and J.M. Cioffi [1997], “PAR Reduction in Multicarrier Transmission Systems,” ANSI T1E1.4/97-367.
P.J. Tourtier, R. Monnier, and P. Lopez [1993], “Multicarrier Modem for Digital HDTV Terrestrial Broadcasting,” Signal Processing: Image Communication, 5, 5/6, pp. 379-403.
VDSL
W.D. Warner, and C. Leung [1993], “OFDM/FM Frame Synchronization for
L. Wei, and C. Schlegel [1995], “Synchronization Requirements for Multiuser OFDM on Satellite Mobile and Two-Path Rayleigh Fading Channels,” IEEE Transactions on Communications, 43, 2/3/4, pp. 887-895.
S.B. Weinstein, and P.M. Ebert [1971], “Data Transmission by Frequency-Division Multiplexing Using the Discrete Fourier Transform,” IEEE Transactions on Communications, 19, 5, pp. 628-634.
R.
S.K. Wilson [1994], Digital
Audio Broadcasting in a Fading and Dispersive Channel, PhD thesis,
S.K. Wilson, R. Ellen Khayata, and J.M. Cioffi, “16-QAM Modulation with Orthogonal Frequency-Division Multiplexing in a Rayleigh-Fading Environment,” in Proceedings of the IEEE Vehicular Technology Conference (VTC'94), pp. 1660-1664.
G. Wornell [1996], “Emerging Applications of Multirate Signal Processing and Wavelets in Digital Communications,” Proceedings of the IEEE, 84, 4, pp. 586-603.
W.Y. Zou, and Y. Wu [1995], “COFDM: an Overview,” IEEE Transactions on Broadcasting, 41, 1, pp. 1-8.
Figure 1. The real parts of three of the basis functions (with indexes 1,2, and 7) that constitute a baseband OFDM signal (left) and the concept of densely packed subcarriers in OFDM (right).
Figure 2. Time and frequency characteristics of an OFDM signal with 1024 subcarriers.
Figure 3. The concept of a cyclic prefix: the last part of the OFDM symbol is
copied as a prefix of duration .
Figure 4. A block diagram of an OFDM transmitter.
Figure 5. The OFDM signal shaped with cosine roll-off edges (solid) and with a
rectangular window (dashed). The edge of the OFDM symbol for both pulses (left)
and the OFDM power spectrum (right).
Figure 6. Example of the front end of an OFDM receiver employing one intermediate frequency.
Figure 7. SNR degradation for three operating SNRs versus 1) a frequency
offset , normalized to the intercarrier spacing (upper left), 2)
phase-noise bandwidth , normalized to the intercarrier spacing (upper right), 3) a
sample clock offset for the 
th subcarrier, normalized to the sample clock frequency, in
ppm (lower left), and 4) a symbol clock offset, normalized to the OFDM symbol
length 
(lower right).
Figure 8. Statistics for the joint ML-estimation of and in Example 3. The
statistic , whose maximizing argument points at the start of an OFDM
symbol (top) and the statistic , whose value at these instants is proportional to the
frequency offset (bottom).
Figure 9. Time/frequency offset correction and channel estimation/equalization in an OFDM receiver.
Figure 10. Time-frequency grid for an OFDM system with three pilot patterns:
entirely known OFDM symbols (left), pilot subcarriers (middle) and scattered
pilots (right).
Figure 11. OFDM symbols from the multiple-access example scenario in a
time-frequency grid. The pilot symbols are black, the data symbols are grey,
and unused symbols (by this user) are white.
Table 1. SNR degradation
in OFDM because of various synchronization impairments: 1) a normalized carrier
frequency offset , 2) carrier phase noise with normalized bandwidth , 3) a normalized sample-clock offset (in ppm) for the th subcarrier and 4) a normalized symbol-timing offset.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
|
Impairment |
SNR loss, D (dB), at
operating SNR= |
|
|
Carrier
frequency offset1 [Pollet, et al., 1995] |
|
|
|
Carrier
frequency offset1 [Moose, 1994] |
|
|
|
Phase noise1
with bandwidth [Pollet, et al., 1995] |
|
|
|
Sample clock
frequency offset2 [Pollet, et al., 1994] |
|
|
|
Symbol timing offset3 [Gudmundson and Andersson, 1996] |
Composite equations |
|
1normalized
to the intercarrier spacing
2normalized to the sample clock frequency, in ppm
3normalized
to the OFDM symbol length
Table 1