Keywords

OFDM; Multicarrier; Orthogonality; DFT; frequency-selective multipath channels; Cyclic prefix

2.05.1 Introduction

Orthogonal Frequency-Division Multiplexing (OFDM) is indubitably a milestone in wired and wireless communications, which pervades most of the telecommunication standards developed in the last two decades, such as DAB, DVB-T/H, WiFi (IEEE 802.11a/n), WiMAX (IEEE 802.16e), UMTS-LTE, ADSL, etc. [71–73,104,105,107,237]. However, the OFDM principle dates back to the early 1960s for military communication systems [69,268]. More generally, the roots of OFDM are at the end of the nineteenth century when frequency division multiplexing (FDM) was investigated to increase the efficiency of telegraph systems [209], and successively was widely employed during the twentieth century in analog telephone lines [209]. Thus, OFDM is the typical example of a technology, whose idea has evolved theoretically for a long time, and that had to wait for mature electronic and software technologies to be implemented, at reasonable costs, for mass-market applications. The interested readers can find in [256] an excellent historical perspective of OFDM evolution, and a recent survey of OFDM literature in [103]. Additional information about OFDM and multicarrier techniques can be found in tutorial articles and books such as [12,29,53,120,136,145,251,269].

The success of OFDM with respect to classical single-carrier (SC) communications is mainly due to its capability to enable wideband communications in time-dispersive (frequency-selective) channels with low-complexity channel equalization, and high spectral efficiency. The simpler equalization with respect to SC is somehow borrowed from classical FDM, where the whole data stream is split in multiple sub-streams with lower data-rate (bandwidth), each one modulated by a different carrier frequency. This way, each sub-stream experiences an almost frequency-flat channel, which requests a simple equalization.

Additionally, OFDM doubles the spectral efficiency with respect to FDM by employing properly separated (e.g., orthogonal) carriers, which let the spectrum of each sub-stream overlap with one another [48]. By exploiting orthogonality, the multiple sub-streams can be separated at the receiver side by a bank of correlation-based receivers, rather than by the bandpass filtering of FDM, where the spectra of different sub-streams cannot overlap. Although the orthogonality principle is quite intuitive and mathematically simple, its implementation in the analog OFDM systems of the 1960s was requesting a bank of perfectly tuned oscillators, both at the transmitter and the receiver side, making OFDM implementation costly and limited to a small number of parallel carriers [268]. Actually, at the beginning of the 1970s, Weinstein and Ebert recognized that OFDM signals can be generated and demodulated by Fourier-based synthesis [257], foreseeing the substitution of the bank of analog oscillators by digital Discrete Fourier Transform (DFT) processing together with digital-to-analog (D/A) [and analog-to-digital (A/D)] conversion. The introduction of a cyclic-prefix (CP) between each transmitted symbol has been another crucial step towards easy OFDM implementation in frequency-selective channels, as firstly proposed in [179]. Thanks to the CP, the almost frequency-flat channel experienced by each sub-stream in the continuous domain, is turned in a perfectly flat channel in the discrete (e.g., DFT) domain. Thus, by standard DFT processing, OFDM is capable to convert the transmission of multiple symbols through a frequency-selective channel, in a set of independent parallel transmissions through a set of frequency-flat channels, which can be easily and independently equalized. When at the end of the 1980s, the progress of electronic technology (A/D and D/A converters, CPU clock frequencies, memories, etc.) made real-time digital signal processor (DSP) a reality, the concurrent increasing request for high data-rate transmissions in frequency-selective channels suggested that the time for OFDM “has come” [29]. At the beginning of the 1990s European broadcasting manufacturers and operators were among the first to recognize the potentiality of OFDM, by producing DAB and DVB-T as the first wireless communication standards for digital audio and video broadcasting [71,72,184].

Similarly, in the same decades, telephone companies and operators recognized to OFDM [or discrete multi-tone (DMT)], the capability to convey high data-rate bit-streams by classical twisted-pairs wires, practically solving the so called last-mile problem for widespread connection of user homes with high-speed digital communications. Indeed, according to the water-filling principle, the use of wires for digital communications permitted also to exploit the almost optimality of OFDM from the capacity point of view, as predicted by the famous Shannon’s paper in 1948 [212]. The first pioneering work of Cioffi and his co-workers on the subject [58], found successively full application in the ADSL standard [107].

The smart use of multiple antennas is another important milestone of wireless communications in the last two decades, while historically multiple antennas were only employed as a source of diversity at the receiver side. Conversely, it has been recognized that multiple transmit and receive antennas can boost the capacity of wireless links by multiple-input multiple-output (MIMO) systems [79], and improve coverage and connection quality by space-time (ST) coding [229]. For instance, the Alamouti ST-coding [5], currently adopted in several communications standards [such as [105]], is a remarkable example of a quite simple idea, which greatly improves the performance of a wireless communication link by employing only two transmit antennas and one, or more, receive antennas.

Anyway, both MIMO and ST-coding approaches have been conceived, and are quite easy to deploy, for SC communications on frequency-flat channels. Thus, the capability to convert frequency-selective channels in a set of parallel frequency-flat channels made OFDM the perfect tool to enable both MIMO and ST-coding techniques. However, this topic is out of the scope of this paper, which for space constraint will address only the fundamental signal processing aspects of OFDM and multicarrier (MC) communications. The interested readers are redirected to [84,88,178], and references therein, for an overview of MIMO and ST-coding, and to [37,111,263], for their use in OFDM-based systems. For a thorough introduction and deep survey of MIMO techniques, the interested readers are redirected to the E-Reference contribution of Davidson [66].

2.05.2 Mathematical background

The paper is written targeting an electrical engineering student with adequate background on Fourier transforms (continuous and discrete), signal and systems theory, linear algebra basics (vector, matrices, eigenvalues, eigenvector, singular value decomposition, etc.), basics of probability and estimation theory, and basics of digital communications. In order to better motivate why OFDM emerged and obscured classical SC communications in the last decades, the background on classical SC digital communications is summarized in the next paragraph, which is also useful to clarify similarities and differences between SC and OFDM systems.

2.05.3 Single carrier background

The baseband equivalent of a radio-frequency (RF) modulated signal

(5.1)

which conveys a stream of digital information symbols by the RF carrier , has the general expression [183]

(5.2)

Equation (5.2) highlights that in general the waveform associated to the th symbol may depend on a state variable that takes into account some memory, i.e., dependence, on the previously emitted symbols, as it happens for instance for continuous-phase modulations [183]. Focusing on the simpler scenario of linear modulations without memory, (5.2) may be rewritten as

(5.3)

where is the symbol period, is the symbol rate, and is the pulse-shaping filter that imposes the spectral occupancy.¹ Specifically, for a given information stream, the spectrum of the transmitted signal is expressed by

(5.4)

where is the Fourier transform (FT) of , and is the discrete-time FT (DTFT) of the transmitted sequence . Thus, the information associated to each symbol spreads by throughout all the frequencies, it is mixed with the information associated to the other symbols , and it is shaped (and band-limited) by .

The signal received on an additive white Gaussian noise (AWGN) channel is expressed by

(5.5)

where and represent attenuation and delay, respectively, and stands for the thermal receiver noise. Thus, the AWGN channel is characterized by the propagation of the transmitted signal through a linear system with impulse response , where is the Dirac delta function. In order to recover the transmitted signal, and assuming and without restriction of generality, the receiver processes by a filter , which is responsible to maximize the signal-to-noise power-ratio (SNR), and to avoid inter-symbol interference (ISI) [183]. The SNR is maximized by a matched filter (MF), e.g., [183], leading to a filtered signal expressed by

(5.6)

In (5.6), stands for the linear convolution operator, is the complex-conjugate operator, is the pulse-shape autocorrelation function, and is the filtered noise. Thus, the MF is also known as correlator receiver [183]. If the pulse shaper is designed with unit energy and such that , the sample extracted at is expressed by

(5.7)

do not experience any ISI from adjacent symbols, and optimal reception can be performed on a symbol by symbol basis [183]. The ISI-free constraint on imposes a constraint on its FT , which has to be shaped such that [183]

(5.8)

The ISI-free constraint in (5.8) can be satisfied if has an anti-symmetrical shape with respect to the frequency and when (see Figure 5.1). This is known as the Nyquist criterion, and suggests that the positive (half) bandwidth associated with a linear digital modulation is , where . A typical prototype filter that respects such constraints is the root raised-cosine filter [183], whose shape is shown in Figure 5.1 together with the graphical interpretation of (5.8).

However, in most of wireless and wired communication systems, the propagation channel is not AWGN. Indeed, because of multipath propagation in wireless communications [222], and of frequency-dependent attenuation in wired communications [183], the propagation channel is modeled by a frequency-selective channel with a time-dispersive impulse response .

A typical baseband representation for a linear time-invariant (LTI) channel, with path is summarized by its time-dispersive impulse response

(5.9)

where and are the attenuation and delay, respectively, associated with the th path, and

(5.10)

is the frequency-selective channel spectrum, as shown in Figure 5.6.

In the presence of a time-dispersive (frequency-selective) channel, the received signal at the MF output is expressed by

(5.11)

where is the overall pulse shape that is perceived at the receiver side. The effect of the channel in the frequency-domain is captured by

(5.12)

where replaces the role of in (5.8).

Thus, if is not constant in the frequency support where , e.g., if when , the frequency selectivity of the physical channel destroys the ISI-free design of Figure 5.1. This means that, in the presence of multipath (frequency-selectivity), the time dispersion of the channel destroys the ISI-free design, because , and the received samples at the MF output become

(5.13)

where is the equivalent discrete-time impulse response of the overall channel, which is generally modeled as a finite impulse response (FIR) filter of order .

The presence of ISI from adjacent symbols requests both the estimation of the channel coefficients and more complex detection techniques. For equiprobable transmitted symbols, the optimal detection choice consists in a maximum likelihood (ML) receiver that can exploit the Viterbi algorithm [183], whose complexity grows exponentially with the discrete-time channel length . Alternative and simpler detection schemes, characterized by (sometimes significant) bit-error-rate (BER) performance degradation, first try to counteract the effect of the channel by a linear filter (equalizer), and successively resort to a single-symbol decision as in (5.7). The simplest choice for the linear equalizer is represented by a transversal FIR filter, whose coefficients (taps) can be determined according to different criteria [zero-forcing, minimum mean-squared error (MMSE), etc. [183]]. The channel equalization complexity for each transmitted symbol , without considering the channel estimation and the equalizer design, is linearly proportional to the number of taps of the FIR equalizer. Typically the FIR equalizer suffers from noise amplification and gives acceptable BER performance, when . Thus, we can conclude that the simplest linear equalization in SC is characterized by a complexity for transmitted symbol that grows linearly with the channel order , rather than exponentially as for ML equalizers.

Another possibility to contrast the effect of the ISI, and avoid the noise amplification associated with linear equalization is to use decision feedback equalization, which is a nonlinear data-aided approach that represents a trade-off between BER performance and computational complexity [183].

It is important to observe that any physical channel is characterized by its own frequency selectivity and coherence bandwidth , i.e., a band where is almost constant. If the frequency support of the transmitted signal is within the channel coherence bandwidth, i.e., if the bandwidth of is , the channel will not significantly alter the ISI-free design of Eq. (5.8) and Figure 5.1, which will be almost respected by . On the contrary, the effect of ISI becomes critical if the system symbol rate increases, leading to a proportional increase of the system bandwidth (see Figure 5.1), which may become greater than the coherence bandwidth . In the discrete-time domain, the increase of the symbol rate would correspond to an increased number of channel coefficients [222], and consequently an increased complexity for the channel equalizer.

Just to give an idea, a wireless SC system in a urban environment is typically characterized by a multipath channel with a maximum delay spread [60]: if the requested system capacity can be granted by a bandwidth , this means that the channel length is of the order , and a linear transversal FIR equalizers with at least taps is request to obtain acceptable BER performance. Needless to say that in this scenario it is impossible to think about ML equalization/detection, whose complexity is exponential in .

Thus, the ever increasing demand for high data-rate communications in the last decades has posed tremendous challenges to channel estimation and equalization of classical SC wideband communications. This was the main motivation, even if not the single one, to look for an alternative communication scheme that could more easily handle the propagation through frequency-selective (multipath) channels.

2.05.4 The path to OFDM

This section is dedicated to show how the OFDM principle can naturally evolve from SC communications, through the classical concept of FDM.

2.05.5 OFDM in frequency-selective multipath channels

Note that, the architecture in Figure 5.5 has solved only the problem of efficiently separating parallel streams in AWGN channels. In the presence of frequency-selective (multipath) channels, each of the orthogonal single (sub-) carrier systems would suffer from the classical problem of ISI. However, the reduction of the symbol rate of each stream to leads to consequent reduction, by a factor , of the length of the discrete-time impulse response associated with each of the parallel systems. Thus, the OFDM system could be equalized by parallel SC equalizers. As we already mentioned at the end of Section 2.05.1, the complexity grows linearly and exponentially with the channel length for time-domain linear and ML equalization, respectively [183]: thus we can immediately conclude that the equalization complexity of OFDM is significantly reduced with respect to a SC system with the same bandwidth. Furthermore, we will show that the smart use of time guards further reduces OFDM equalization, by a frequency-domain approach.

In order to identify how to recover the transmitted data also in the presence of a frequency-selective channel, let us investigate the expression for the received signal in case of a typical time-invariant multipath propagation, as shown in Figure 5.6 and summarized by

(5.35)

where has been defined in (5.9), and captures the low-pass filtering included in the A/D and D/A converters, as well as the baseband (BB) equivalent of the band-pass filtering stages in the RF-to-BB and the BB-to-RF conversion.

To better understand the effect of the channel, and the potential resistance of OFDM to multipath propagation, first it is convenient to consider the spectrum of the received signal, which is given by

(5.36)

Focusing on a single OFDM symbol, the useful part of the spectrum (i.e., ignoring the noise) is expressed by

(5.37)

which clarifies that a frequency-selective channel does not destroy the orthogonality of the transmitted data on the equispaced frequency grid , as shown in Figure 5.1, and summarized by

(5.38)

Thus, the receiver can effectively separate and detect the data transmitted during a single OFDM block, still by sampling the OFDM spectrum of the received signal . However, in time-dispersive channels, because of the convolution with the channel impulse response (CIR) (5.35), the duration of each single received OFDM block will be longer than . If the CIR has a time support (duration) , the received OFDM block will have duration , as shown in Figure 5.7.

Thus, sampling the OFDM spectrum of the 0th block requests to compute

(5.39)

which can be obtained by the DTFT of the sequence , evaluated at , as expressed by

(5.40)

Note that (5.40) is quite similar to a DFT processing: the only difference is that the summation index exceeds the DFT period . However, because of the periodicity of the discrete Fourier exponentials, i.e., , it is evident that

(5.41)

Equation (5.41) highlights that the sampling can still be performed by an -point DFT processing, on a new -length sequence obtained adding to the received sequence its replica anticipated by discrete-time indexes, by an overlap-and-add strategy [173]. This way, the tail of that exceeds the transmitted length is added at the beginning of the original sequence, as shown in Figure 5.8, making (5.41) equivalent to (5.40).

2.05.6 A vector-matrix representation for OFDM

Figure 5.5 shows the discrete-time equivalent model of an OFDM system that exploiting a serial-to-parallel (S/P) converter, selects a block of data symbols every seconds, then generates the discrete-time samples to be transmitted by IDFT processing, and finally, after a parallel-to-serial (P/S) conversion, sends the samples to a D/A converter. In a vector-matrix notation equivalent to (5.34), the OFDM symbol transmitted during the th period is obtained by

(5.50)

where is a zero-mean uncorrelated data vector with covariance , and is the unitary DFT matrix of size . For notational convenience, in (5.50) we have normalized the transmitted samples with respect to . In AWGN channels, the received vector is simply expressed by

(5.51)

where , is an i.i.d. zero-mean Gaussian vector with covariance , whose elements are the samples of the noise generated by low-pass filtering of in the A/D converter. Thus, the transmitted data can be recovered by a DFT processing that generates the soft estimate of the data vector , as expressed by

(5.52)

where is the th sample of the unitary DFT of the received vector , and is a vector containing the frequency-domain samples of the noise, which is still AWGN.

The presence of an LTI multipath channel induces a dispersion in time of the symbols transmitted during the th block, as expressed by

(5.53)

where is the CIR order, and for OFDM transmissions. In a vector-matrix notation, this time-domain spreading is captured by the Toeplitz convolution matrix of the channel , which is expressed by

(5.54)

such that a transmitted vector of size , generates a received vector , of size , as expressed by

(5.55)

However, because of the channel delay spread , the received blocks associated to different OFDM blocks overlap in time, thereby generating IBI (see Figure 5.9). As detailed in Section 2.05.3, a simple way to contrast IBI is the introduction of a time guard between OFDM blocks, with a duration , which can be exploited to insert either zeros as in ZP-OFDM, or the CP as in CP-OFDM. Thus, as shown in Figure 5.12, each transmitted block , of size , is expressed by

(5.56)

where is the -size OFDM block before the insertion of the time-guard, with is the CP insertion matrix for CP-OFDM, and is the ZP insertion matrix for ZP-OFDM. Assuming for simplicity that , and that the delay spread of the discrete-time channel is shorter than the OFDM block length , the received samples associated to the th transmitted symbol can be collected in a vector of size , as expressed by

(5.57)

In (5.57) is the matrix that shifts up a vector, inserting zeros in the empty positions, to mimic the time advance associated with the transmission of consecutive OFDM blocks, is the matrix that shifts down a vector, inserting zeros in the empty positions, to represent the time delay, , and is the AWGN term.

2.05.7 Symbol error rate (SER) performance analysis

The SER analysis for ML detection of CP-OFDM is quite simple, because of the frequency-domain channel diagonalization induced by the CP. Indeed, (5.65) and (5.45) show that the data transmitted on each separate subcarrier are impaired by a flat-fading channel, as expressed by

(5.81)

If the discrete-time channel taps are zero-mean Gaussian distributed, then it is straightforward to verify that in (5.63) is also zero-mean Gaussian, is Rayleigh distributed with a probability density function (pdf) expressed by , with , and

(5.82)

where is the autocorrelation matrix of the discrete-time channel, and is the AWGN in the frequency-domain. Thus, the symbol error rate (SER) performance on a frequency-flat Rayleigh fading channel for each subcarrier is expressed by [183]

(5.83)

where and depend on the complex alphabet of . Defining the average SNR for each subcarrier as , if is drawn from a QPSK alphabet, (5.83) leads to [183]

(5.84)

and the total is the average of (5.84) over all the subcarriers, as expressed by

(5.85)

Equation (5.84), together with (5.82), highlights that the SER performance depends on the sum of the second order moments of the time-domain channel paths rather than on the number of the paths. For instance, in the simpler case when the channel paths are independents (i.e., , (5.82) becomes

(5.86)

which highlights the two following observations:

The last observation indicates that an uncoded CP-OFDM system is not capable to exploit the potential diversity offered by a multipath fading channel. Indeed, while it is highly improbable that all the time-domain paths fade to zero simultaneously, the single frequency-domain path associated to each data in (5.81) has a greater probability to fade towards zero, making the data unrecoverable also in the presence of a low noise value . Actually, each single path of the CIR is a potential source of diversity,² because the receiver collects multiple (independent) copies of the transmitted signal. This is similar to the space diversity offered by multi-antenna receiving systems [183]. However, the CP insertion and removal, which leads to the frequency-domain channel diagonalization, destroys the multipath diversity and sacrifices uncoded SER performance for a simpler per-subcarrier equalization. Such loss of diversity for uncoded SER performance can be recovered by channel coding (see Section 2.05.6), or by linear precoding (see Section 2.05.7), which spreads each uncoded data over all (or several) different subcarriers [167]. The SER analysis for ML decoding of ZP-OFDM is not as simple as for CP-OFDM, because the ML detection based on (5.73) leads to

(5.87)

which does not have an equivalent (diagonal) formulation in the frequency-domain, because of the banded Toeplitz structure of . Consequently, differently from CP-OFDM, the ML detector for ZP-OFDM is not based on PSE. SER and diversity analysis for ML problems like in (5.87) are typically addressed by the probability that a given transmitted data vector is confused at the receiver side with another possible transmitted vector . This leads to the so-called pairwise-error probability (PEP) [218,219,229,236], which is defined as

(5.88)

and dominates the SER. As detailed in [250], the minimum PEP for all the possible couples is granted if the observation matrix has a full column rank for any channel realization. This is equivalent to state that, in the absence of noise, is not possible to confuse with at the receiver, or equivalently, to the symbol detectability condition [251]

(5.89)

Actually, (5.89) is granted because both and are always full column rank, and consequently . The same analysis for CP-OFDM highlights that the observation matrix in (5.66) is , which is not full rank for those channel realizations that contain a zero in one of its elements. Indeed, in CP-OFDM, if the discrete-time channel induces a frequency-domain channel with a zero on the th subcarrier, i.e., , the data on that subcarrier would be not recoverable even in the absence of noise, as clarified by (5.81). Thus, at the receiver side, those data vectors that differ from only on the th element would be undistinguishable from , causing a degradation of the average PEP and SER.

The price paid by ZP-OFDM with respect to CP-OFDM, for better uncoded SER performance with ML detection, is the higher detection complexity induced by (5.87), which grows exponentially with the vector size and the cardinality of the data alphabet . Conversely, if ZP-OFDM is detected by exploiting the overlap-and-add (OLA) approach of (5.75), the detection complexity is significantly reduced. Indeed, the OLA operation of (5.75) induces a circulant channel at the receiver side: in this case, a simpler PSE can be employed by compensating for the term in (5.78). Therefore, OLA-based per-subcarrier detection of ZP-OFDM presents a decoding complexity that is similar to CP-OFDM. In addition, using OLA-based per-subcarrier detection of ZP-OFDM, also the SER performance would be somewhat similar to CP-OFDM (see Figure 5.13). Indeed, with respect to CP-OFDM, OLA-based ZP-OFDM presents a noise power penalty caused by the extra noise term in (5.78), and an equivalent advantage on the average (useful) signal power due to the absence of CP transmission. Each of these two effects compensates with one another, as shown in Figure 5.13.

2.05.8 Coded—OFDM

As highlighted in Section 2.05.5, and summarized by (5.81) and (5.83), CP-OFDM is not capable to exploit the potential diversity offered by a multipath fading channel, whose taps fade in an independent fashion. This is a direct consequence of transmitting each single data onto a single subcarrier: if the th frequency channel fades to zero, there is no way to recover , because is not contained in any other diversely received sample. Thus, in highly frequency-selective channels, the SER performance is significantly degraded because of the loss of the data on those subcarriers that are impaired by an almost null channel transfer function. For instance, in multipath Rayleigh fading conditions, the SER performance is equivalent to that of a single-carrier system impaired by a Rayleigh flat-fading channel, with diversity order equal to one.

Possible counter-measures to channel fading are typically known as diversity techniques and historically emerged in telecommunications by providing the receiver with multiple copies of the same information. These multiple copies of the same information are transmitted in the frequency-domain [or in the space (antenna)-domain or in the time-domain], and possibly are impaired by independent fading distortions [183]. The correct combination of multiple information manifests in an increase of the average SNR, a better SNR statistic, and consequently a SER performance that tends to the SER in AWGN channels, when the number of independent information copies is high enough [183]. Anyway, a pure diversity technique cannot improve performance over the SER threshold represented by the AWGN scenario.

Actually, Forward Error Correcting codes (FECs), or channel codes, were historically designed to contrast the errors in AWGN scenarios, by introducing algebraically-structured redundancy [183]. This form of redundancy, introduced at the transmitter side, distributes the information data on almost all the subcarriers: consequently, by channel decoding at the receiver side, it is possible to exploit this redundancy to jointly contrast the errors induced by fading channels and collect part of the channel diversity. Some examples of popular FEC schemes [183] exploited in standard Coded-OFDM (C-OFDM) systems include block-codes (e.g., Reed-Solomon codes), convolutional codes (CC), and concatenated codes, which were exploited since the beginning of the OFDM era in commercial systems such as DAB [71], DVB-T/S [72], WiFi [104], WiMAX [105], while BCH codes, turbo codes and low-density parity-check (LDPC) codes [183] have been more recently exploited in DVB-T2/S2 [73] and UMTS-LTE [237].

The basic idea to use FEC to contrast fading is to make the errors statistic induced by fading looks similar to the error statistic introduced by the noise in AWGN channels: this is obtained by equipping the OFDM system with appropriate interleavers, which scramble the coded data through the subcarriers to collect the channel frequency diversity, and through consecutive OFDM blocks to collect the channel time diversity. The price for such a joint resistance to fading and AWGN is the increase of decoding delay, because the interleaver depth may span several OFDM blocks in order to be effective. Thus, especially for time-sensitive communications, it could be preferable to have some sort of diversity resistance that does not require coding over several consecutive OFDM blocks.

This possibility has been explored more recently, by exploiting the possibility to transmit (possibly redundant) linear combinations of the data by the so called linearly precoded-OFDM (see Section 2.05.9), or by antenna-diversity through space-time coding (see [66,84,88,178] ad references therein for further details).

2.05.9 Linearly precoded-OFDM (LP-OFDM)

FEC are typically designed to contrast AWGN channels and therefore are designed in order to maximize the Euclidean distance among codewords in order to be resistant to an additive distortion. This Euclidean maximization is not the optimal strategy in a fading channel, where the Hamming distance is the key element to minimize the PEP among codewords [229,250], in order to be resistant to a multiplicative Distortion.

In an OFDM system, which conveys information on parallel carriers, the transmitter could spread a group of data over several of these carriers, by exploiting a linear precoding (LP) matrix . The aim is to introduce frequency diversity and potentially be capable to recover all the transmitted data (exploiting the finite alphabet of even if one of the carriers fades to zero. Thus, an LP transmitter generates a (potentially redundant) vector by performing a linear combination of the original data, as shown in Figure 5.12 and expressed by

(5.93)

where , is the (possibly redundant) precoding matrix³ of size . Thus, with LP, a CP-OFDM system transmits through a multipath fading channel , and (5.65) becomes

(5.94)

where the number of subcarriers (i.e., the size of and ) is . From (5.94), ML detection requests to compute

(5.95)

which involves the non-diagonal channel-precoding matrix . Thus, the easy PSE (and easy ML detection) of OFDM is lost, but the PEP (and SER) performance can be boosted by a proper design of the precoding matrix .

As clarified in [172], in order to exploit all the diversity offered by a multipath fading channel, the overall observation matrix should guarantee that in the absence of noise any couple of transmitted data vectors are observable at the receiver side with a minimum Hamming distance greater than or equal to . To this end, it is sufficient (but not necessary) to guarantee the symbol detectability condition, i.e., that has full column rank. Note that the discrete-time frequency response of a time-domain channel with non-zero paths may have at most zeros: therefore, can cancel-out up to rows of . Thus, it would be enough to redundantly design with linearly independent rows, such that any set of rows of are linearly independent. If this design condition is verified, the row cancellation in still guarantee that has (full-column) rank . Thus, by the out of observations in (5.94) that are different from zero, it would be always possible in the absence of noise to distinguish any transmitted data vector from any other . A possible choice for such precoding matrices are Vandermonde matrices where [251]

(5.96)

or cosine matrices with

(5.97)

When and , the Vandermonde matrix becomes the DFT matrix , while, when represents the discrete cosine transform (DCT) matrix. Moreover, note that the precoding matrix should be normalized such that all the columns have a unit norm, i.e., [250] in order to not decrease the minimum Euclidean distance between the precoded vectors and consequently penalize the SER performance in AWGN channels where .

An interesting choice for the redundant precoding matrix is , i.e., the Vandermonde matrix obtained by selecting the first columns of an IDFT matrix of size . By this choice, the precoded vector to be transmitted, is expressed by

(5.98)

which is nothing else than a ZP-SC system, which does not need the use of a CP. Thus, the redundancy introduced by ZP in a block SC system grants not only IBI suppression, but also symbol detectability and, consequently, the capability by ML decoding to exploit the maximum diversity offered by the channel [253,254].

In practice, LP-OFDM systems (including ZP-SC), can capture (part of) the diversity [236] also by exploiting suboptimal linear equalization/detection schemes by producing the soft estimate

(5.99)

where the equalization/detection matrix can be computed by an LS or MMSE approach [117], by substituting with in (5.91) and (5.92), respectively. Further details can be found in [250].

However, the sufficient condition guarantees that any couple of vectors could be distinguished at the receiver side in the absence of noise, without exploiting the fact that the elements of belong to a finite alphabet . Thus, by exploiting the structure imposed on the possible data vectors by the finite alphabet and algebraic number theory, it is possible to design also non-redundant (square) precoding matrices that are capable to capture all the diversity offered by the channel, with . As detailed in [148], such a matrix is proved to always exist: it is a square Vandermonde matrix, whose elements are expressed by

(5.100)

where is the Euler’s totient function, which identifies the number of integers that are lower, and relative prime, with respect to . Although for a given size the solution of (5.100) is not unique, for the special but important case where is a power of two, the precoding matrix can be always written as

(5.101)

which highlights that the precoding matrix is unitary, i.e., . Moreover, (5.101) practically states that the diversity-optimum CP-aided block transmission, is a CP-SC system equipped by a simple diagonal non-redundant precoder, as expressed by the transmitted data vector

(5.102)

It is important to highlight that the non redundant precoding based on (5.100) or (5.101) is not only capable to capture all the diversity, but can also maximize the coding gain⁴ in SER performance, by selecting the lowest that satisfies (5.100). For further details on optimal constellation rotations, the interested readers are redirected to [24,32,87].

The price to be paid for the exploitation of the channel diversity is the computational complexity of ML decoding, which grows exponentially with the size of the data vector . Quasi-ML decoding performance can be obtained by algorithms such as sphere decoding [250] characterized by reduced complexity for moderate precoder sizes and many SNR values [108].

However, it can be observed that, in order to not sacrifice too much the system efficiency, the CP or ZP length is in general , and consequently, in order to collect the diversity offered by the channel, it would be enough to spread each symbol on a number of subcarriers , where is an integer greater than 1. Therefore, instead of a large precoder of size equal to the number of subcarriers, many precoders can be applied to different groups of subcarriers. This corresponds to an overall precoded matrix that can be written as

(5.103)

where is a permutation matrix that establishes how the subcarriers are grouped into the groups, and is a non-redundant precoding matrix designed according to (5.101). Since the benefit in SER performance tends to reduce for increasing diversity [183], suboptimal designs based on (5.103) with can be employed: this way, the ML decoding in (5.95) leads to parallel ML decoding problems for the data observed (and transmitted) on the orthogonal sets, with a complexity instead of . The parallel ML decoding is expressed by

(5.104)

The data sub-vectors , as well as the other quantities in (5.104), are just the sub-vectors selected from the corresponding ones in (5.95), by means of (5.103). This approach has been proposed in [148], where it is called grouped linear precoding (GLP). In order to collect the highest possible diversity for all the subgroups, and thus optimize the average SER, the best strategy is to design the permutation matrix such that each subgroups of data is assigned to a different set of equispaced subcarriers, which, being separated, are most likely different (more diverse) than contiguous subcarriers [148]. In the context of non-redundant GLP-OFDM, the performance of ML and other suboptimal equalization/detection schemes, such as ZF and MMSE, is shown in Figure 5.14.

Linear precoding approaches for multicarrier transmissions have been discussed also in [38], which compares linear precoders based on rotated (Fourier or Walsh-Hadamard) transforms, and in [146], which derives the conditions for minimum-SER precoding design for a linear detector.

A fundamental contribution to redundant LP can be found in [201,202]. The first of these two companion papers generalizes OFDM, DMT, CDMA, and TDMA communications, and establishes sufficient conditions for a precoded transmission scheme to perfectly equalize any FIR channel (channel identifiability), independently of its zeros locations. Additionally, a joint ZF design of the transmitter and receiver filterbanks is proposed, under MMSE and maximum SNR criteria. The second paper, by a ZP approach, derives also blind channel estimators, block synchronizers, and direct-equalizers for the proposed filterbank structure. Note that, also multiuser MC systems, such those addressed in the next section, fall within the general filterbank formulation in [201,202].

As a final remark, it should be noted that LP and FEC are not alternative with one another, but they are both valuable tools to protect an OFDM system from errors induced by a fading channel. As already explained, FECs have been historically introduced to protect digital communications from AWGN, they introduce redundancy, and can be effective also to combat channel fading by properly scrambling and interleaving the coded data. Conversely, LP techniques are designed to exploit the channel diversity (if it is present), cannot improve performance in AWGN, and they do not necessarily request redundancy. Thus, a proper design could combine LP with FEC, in order to collect the channel diversity (if any) to make the equivalent channel almost AWGN, and then exploit FEC mainly to protect from AWGN. In severe multipath fading channels, this approach can guarantee SER performance equivalent to those of a pure FEC-OFDM system, with lower redundancy, lower complexity, and lower decoding delay. Readers interested on a comparison of LP with FEC, and on their joint design, are redirected to [252–254].

2.05.10 Multiuser multicarrier systems

Section 2.05.2 has clarified that OFDM can be seen as a special case of CDMA, where parallel data flows are simultaneously transmitted in time and distinguished by the use of different codes , as expressed by

(5.105)

where is the symbol period and is the symbol duration, less than or equal to the symbol period. For instance, a CDMA system with parallel transmissions, can be represented during the useful th symbol by a transmitted data vector expressed by

(5.106)

where , with , is the th spreading sequence, also known as spreading code, and is the code matrix. Additionally, to handle IBI as in OFDM transmissions, each transmitted block in (5.105) could be separated in time by , leading to a CP-CDMA when , or to a ZP-CDMA when . Further considerations on CDMA are left for the end of this section where CDMA systems are briefly compared with multiuser MC systems.

In the case of OFDM, the codes are , which are orthogonal on the useful duration by (21), , and the overall transmitted block is

(5.107)

with for CP-OFDM and for ZP-OFDM. When orthogonal frequencies are employed to distinguish different users, the system is identified as an OFDM access (OFDMA) systems, which has the nice property to preserve users’ orthogonality in LTI channels. Moreover, different data rates can be easily handled in OFDMA by reserving different groups of subcarriers to different users, as it happens for instance in WiMAX and LTE [105,237].

Researchers have historically tried to combine the easy equalization of OFDM, with desirable properties of classical CDMA systems, such as resistance to narrowband interference, and capability to exploit diversity in multipath fading channels [183,247].

A possibility is to use a code in the frequency-domain, to spread a single data on all the sub-carries, such that, after spreading, the data vector for the th user is , and the overall data vector for all the users is expressed by

(5.108)

which is transmitted by OFDM through

(5.109)

and where the number of orthogonal codes (users) is .

The transmission mode with is typically called multicarrier CDMA (MC-CDMA) [95,251], whose implementation principle is shown in Figure 5.15.

Note that the last line in (5.109) highlights that MC-CDMA corresponds to a CP-CDMA system, which transmits the inverse spectra of the original codes , plus a CP to handle IBI and to guarantee easy frequency-domain equalization. This way, MC-CDMA grants both easy PSE and multipath diversity exploitation, since each data symbol is transmitted (spread) by the code matrix on several (potentially all) subcarriers. Indeed, the frequency-domain spreading operation expressed in (5.109) is performed on all the subcarriers, and therefore tries to achieve a diversity gain of . However, the frequency-domain channel coefficients on the subcarriers are correlated, because they are obtained by DFT of nonzero time-domain channel coefficients in (5.63), with . Therefore, the maximum achievable diversity gain is only . For this reason, similarly to GLP in (5.103), multicarrier CDMA systems can employ an alternative frequency-domain spreading operation that uses only subcarriers out of . In order to minimize the correlation, the subcarriers are chosen as maximally separated, i.e., equispaced. Therefore, using spreading sequences of length , we can accommodate up to users into the subcarriers. If we assume that is integer, there are different groups of equispaced subcarriers, and therefore groups of users can be accommodated. This scheme is usually known as group-orthogonal MC-CDMA [42], because each group behaves as a reduced-dimension MC-CDMA systems, while the different groups are orthogonal in the frequency-domain, similarly to OFDMA.

The above mentioned group approach is also suitable for multirate transmissions. For instance, the subcarriers of a given group, and all the associated spreading codes, can be assigned to the same user. In this case, only users can be accommodated, but the data rate of each user can be times higher than conventional MC-CDMA. This multirate scheme is basically a combination of linear precoding and OFDMA, since users, which are orthogonal in the frequency-domain, can apply a linear precoder onto the assigned subcarriers, as detailed in Section 2.05.7. Alternatively, multirate transmissions may be obtained by multicode MC-CDMA, as detailed in [187].

Another possibility to mix CDMA with OFDM is to apply the CDMA spreading principle to each of the independent SC transmissions embedded in OFDM, as shown in Figure 5.16. Indeed, by (5.105), each SC data flow can accommodate different users by expressing

(5.110)

where, ignoring the potential presence of time guards to prevent IBI, the pulse shaper associated to each data is spread by the chip sequence through the signal , as expressed by

(5.111)

where the code duration is . The time spreading of each waveform in (5.111) induces a spectrum

(5.112)

and an overall spectrum support on the th subcarrier expressed by

(5.113)

which is centered on and has equispaced zeros on the frequency grid .

Thus, similarly to classical OFDM, the spectra on different subcarriers will preserve orthogonality if the subcarriers are chosen such that the subcarrier separations are multiple of , e.g., when . This transmission mode is typically called multicarrier direct-sequence CDMA (MC-DS-CDMA) [95]: this corresponds to an OFDM system that transmits a chip of duration in each OFDM block, i.e., in MC-DS-CDMA the OFDM block duration is , and the symbol duration is (see Figure 5.16). Thus, assuming a fixed number of subcarriers, if we want to compare MC-DS-CDMA (with block duration with respect to single-user OFDM (with block duration , two cases are of interest. First, if each MC-DS-CDMA user wants to preserve the same data rate with respect to single-user OFDM systems, the spreading operation imposes a shorter OFDM block duration , and, consequently, a bandwidth overexpansion of a factor . Alternatively, if each MC-DS-CDMA user wants to preserve the same bandwidth with respect to single-user OFDM systems, the spreading operation leads to a data rate reduction of a factor , which is a direct consequence of the OFDM block duration and of the increased symbol duration . However, it should be emphasized that CDMA systems allows for a maximum of orthogonal users: therefore, in the second case, the aggregate data rate of the MC-DS-CDMA users would be equal to that of single-user OFDM systems.

Moreover, in MC-DS-CDMA, in order to prevent IBI (i.e., inter-chip interference) and grant easy equalization in LTI channels, CP or ZP has to be inserted between each OFDM block (i.e., between each chip). In a vector-matrix notation, this corresponds to a coded (row) vector for each subcarrier expressed by

(5.114)

which leads to an overall spread data-matrix for each user

(5.115)

whose columns are sequentially transmitted by OFDM. In (5.115), stands for the Kronecker product. The vector transmitted during each chip period is expressed by

(5.116)

and the overall transmitted OFDM chip, i.e., OFDM block, is

(5.117)

where is the column-wise data matrix containing the data vector associated to each user, and is the column vector containing the th chips of all the users, i.e., the th row of the code matrix . Equation (5.117) highlights that MC-DS-CDMA grants the easy separation (and equalization) of the th flow on the subcarriers by preserving the subcarrier orthogonality through a frequency-selective channel, and the separation among different users by orthogonal design of the code matrix, which requests to collect all the chips associated with a single data matrix .

Alternatively, as also shown in Figure 5.16, it is possible to renounce to easy equalization and separation of subcarriers by resorting to the so-called multi-tone CDMA (MT-CDMA) [95]. This system corresponds to choose subcarriers that are not orthogonal on the chip duration, and which would be orthogonal (in the absence of spreading) on the symbol duration , that is, . Thus, the spectra in (5.112), for different , are not centered on the zeros of the other spectra as shown in Figure 5.17.

This way, ICI would emerge at the receiver side, independently of the presence or absence of CP or ZP. For this reason, MT-CDMA systems typically avoid the use of a CP [244]. MT-CDMA corresponds to an OFDM system, without CP, on the time interval , followed by a chip spreading (bandwidth expansion) to distinguish different users [244]. This signal could be generated by a classical OFDM modulator at the symbol rate, followed by up-sampling to describe the signal at the chip-rate and by the spreading operation, which is just a chip-by chip multiplication. Due to the fact that a time-domain up-sampling can be easily generated by DFT processing, through a zero-filled spectrum [173], a possible matrix representation for the overall data vector transmitted by the th user is

(5.118)

where is a selection of the DFT matrix , containing its first and last rows, and and are two vectors of size obtained by splitting into two parts.