CHANNEL ESTIMATION FOR WIRELESS OFDM SYSTEMS

Transcription

1 2ND QUARTER 27, VOLUME 9, NO. 2 CHANNEL ESTIMATION FOR WIRELESS OFDM SYSTEMS MEHMET EMAL OZDEMIR, LOGUS BROADBAND WIRELESS SOLUTIONS, INC. AND HUSEYIN ARSLAN, UNIVERSITY OF SOUTH FLORIDA ABSTRACT Orthogonal frequency division multiplexing (OFDM) is a special case of multi-carrier transmission and it can accommodate high data rate requirement of multimedia based wireless systems. Since channel estimation is an integral part of OFDM systems, it is critical to understand the basis of channel estimation techniques for OFDM systems so that the most appropriate method can be applied. In this article, an extensive overview of channel estimation techniques employed in OFDM systems are presented. In addition, the advantages, drawbacks, and relationship of these estimation techniques with each other are analyzed and discussed. As the combination of multiple input multiple output (MIMO)-OFDM systems promises higher data rates, estimation techniques are further investigated for these systems. Although the existing proposed techniques differ in terms of computational complexity and their mean squared error (MSE) performance, it has been observed that many channel estimation techniques are indeed a subset of LMMSE channel estimation technique. Hence, based on a given system s resources and specifications, a suitable method among the presented techniques can be applied. Driven by multimedia based applications, anticipated future wireless systems will require high data rate capable technologies. Novel techniques such as OFDM and MIMO stand as promising choices for future high data rate systems [, 2]. OFDM divides the available spectrum into a number of overlapping but orthogonal narrowband subchannels, and hence converts a frequency selective channel into a nonfrequency selective channel [3]. Moreover, ISI is avoided by the use of CP, which is achieved by extending an OFDM symbol with some portion of its head or tail [4]. With these vital advantages, OFDM has been adopted by many wireless standards such as DAB, DVB, WLAN, and WMAN [5, 6]. MIMO, on the other hand, employs multiple antennas at the transmitter and receiver sides to open up additional subchannels in spatial domain. Since parallel channels are established over the same time and frequency, high data rates without the need of extra bandwidth are achieved [7, 8]. Due to this bandwidth efficiency, MIMO is included in the standards of future BWA [9]. Overall, these benefits have made the combination of MIMO-OFDM an attractive technique for future high data rate systems [ 2]. As in many other coherent digital wireless receivers, channel estimation is also an integral part of the receiver designs in coherent MIMO-OFDM systems [3]. In wireless systems, transmitted information reaches to receivers after passing through a radio channel. For conventional coherent receivers, the effect of the channel on the transmitted signal must be estimated to recover the transmitted information [4]. As long as the receiver accurately estimates how the channel modifies the transmitted signal, it can recover the transmitted information. Channel estimation can be avoided by using differential modulation techniques, however, such systems result in low data rate and there is a penalty for 3 4 db SNR [5 9]. In some cases, channel estimation at user side can be avoided if the base station performs the channel estimation and sends a pre-distorted signal [2]. However, for fast varying channels, the pre-distorted signal might not bear the current channel distortion, causing system degradation. Hence, systems with a channel estimation block are needed for the future high data rate systems. Channel estimation is a challenging problem in wireless systems. Unlike other guided media, the radio channel is highly dynamic. The transmitted signal travels to the receiver by undergoing many detrimental effects that corrupt the signal X IEEE Communications Surveys & Tutorials 2nd Quarter 27

2 CIR CFR Coefficients DFT/IDFT Coefficients Tap index Subcarrier index nfigure. Time and frequency domain channels representation for OFDM based systems. and often place limitations on the performance of the system. Transmitted signals are typically reflected and scattered, arriving at receivers along multiple paths. Also, due to the mobility of transmitters, receivers, or scattering objects, the channel response can change rapidly over time. Most important of all, the radio channel is highly random and the statistical characteristics of the channel are environment dependent. Multipath propagation, mobility, and local scattering cause the signal to be spread in frequency, time, and angle. These spreads, which are related to the selectivity of the channel, have significant implications on the received signal. Channel estimation performance is directly related to these statistics. Different techniques are proposed to exploit these statistics for better channel estimates. There has been some studies that cover these estimation techniques, however these are limited to the comparison of few of the channel estimation techniques [2 24]. This paper focuses on an extensive overview of the channel estimation techniques commonly applied to OFDM based multi-carrier wireless systems. OFDM CHANNEL ESTIMATION Channel estimation has a long and rich history in single carrier communication systems. In these systems, the CIR is typically modeled as an unknown time-varying FIR filter, whose coefficients need to be estimated [4]. Many of the channel estimation approaches of single carrier systems can be applied to multi-carrier systems. However, the unique properties of multi-carrier transmission bring about additional perspectives that allow the development of new approaches for channel estimation of multi-carrier systems. In OFDM based systems, the data is modulated onto the orthogonal frequency carriers. For coherent detection of the transmitted data, these sub-channel frequency responses must be estimated and removed from the frequency samples. Like in single carrier systems, the time domain channel can be modelled as a FIR filter, where the delays and coefficients can be estimated from time domain received samples, which are then transformed to frequency domain for obtaining the CFR. Alternatively, radio channel can also be estimated in frequency domain using the known (or detected) data on frequency domain sub-channels. Instead of estimating FIR coefficients, one tap CFR can be estimated (Fig. ). Channel estimation techniques for OFDM based systems can be grouped into two main categories: blind and non-blind. The blind channel estimation methods exploit the statistical behavior of the received signals and require a large amount of data [25]. Hence, they suffer severe performance degradation in fast fading channels [26]. On the other hand, in the nonblind channel estimation methods, information of previous channel estimates or some portion of the transmitted signal are available to the receiver to be used for the channel estimation. In this article, only the non-blind channel estimation techniques will be investigated. The non-blind channel estimation can be studied under two main groups: data aided and DDCE. In data aided channel estimation, a complete OFDM symbol or a portion of a symbol, which is known by the receiver, is transmitted so that the receiver can easily estimate the radio channel by demodulating the received samples. Often, frequency domain pilots are employed similar to those in new generation WLAN standards (82.a and HYPERLAN2) [27]. The estimation accuracy can be improved by increasing the pilot density. However, this introduces overhead and reduces the spectral efficiency. In the limiting case, when pilot tones are assigned to all subcarriers of a particular OFDM symbol, an OFDM training symbol can be obtained (block type pilot arrangement). This type of pilot arrangement is usually considered for slow channel variation and for burst type data transmission schemes, where the channel is assumed to be constant over the burst. The training symbols are then inserted at the beginning of the bursts to estimate the CFR (e.g. WLAN and WiMAX systems) [28, 29]. When channel varies between consecutive OFDM symbols, either the training symbols should be inserted regularly within OFDM data symbols with respect to the time variation of the channel (Doppler spread), or the channel should be tracked in a decision directed mode to enhance the receiver performance. In the DDCE methods, to decode the current OFDM symbol the channel estimates for a previous OFDM symbol are used. The channel corresponding to the current symbol is then estimated by using the newly estimated symbol information. Since an outdated channel is used in the decoding process, these estimates are less reliable as the channel can vary drastically from symbol to symbol [3, 32]. Hence, additional information is usually incorporated in DDCE such as periodically sent training symbols. Channel coding, interleaving, and iterative type approaches are also commonly applied to boost the performance of DDCE~techniques. There are numerous approaches to estimate the channels for OFDM subcarriers. The direct estimation of the channel for subcarriers treats each subcarrier as if the channels are independent. However, in practice, the CFR is often oversampled via the subcarriers, and hence the estimated frequency domain channel coefficients are correlated. On the other hand, the noise in these subcarriers can be independent. By utilizing the correlation of CFR in subcarriers, the noise can be reduced significantly. Therefore, the channel estimation accuracy can be improved [28]. Several approaches have been proposed to exploit this correlation. These approaches and their relationship with each other will be discussed in the subsequent sections to provide a unified understanding. Similarly, the subcarrier correlation in time and spatial domain can be exploited since the noise can be considered to be independent in time and spatial domain as well. IEEE Communications Surveys & Tutorials 2nd Quarter 27 9

3 Cyclic prefix Remove Cyclic prefix X S / P IFFT - point P S /S /P Ant # Ant # Wireless channel IFFT - point P /S Y Data bits Coding, modulation, interleaving X Ntx S / P Cyclic prefix IFFT - point P /S Ant #Ntx Ant #Nrx S / P Remove Cyclic prefix IFFT - point P /S Y Nrx Deinterleaving, demodulation, decoding CSI Output bits nfigure 2. MIMO-OFDM transceiver model. Although it is a common approach to assume the channel to be constant over an OFDM symbol duration [9, 27], for fast fading channels the same assumption leads to ICI [33], which degrades the channel estimation performance. Hence, the methods employed in data-aided and decision directed channel estimation need to be modified so that the variation of the channel over the OFDM symbol is taken into account for better estimates. External interfering sources also affect the performance of channel estimation. The effect of interfering sources can be mitigated by exploiting their statistical properties. Although most systems treat ICI and external interference as part of noise, better channel estimation performance can be obtained by more accurate modeling [34]. There are basically three basic blocks affecting the performance of the non-blind channel estimation techniques. These are the pilot patterns, the estimation method, and the signal detection part. Each method covered in this article either tackles one of the above basic block or several at a time. The specific choice depends on the wireless system specifications and the channel condition. The aspects of each method are presented such that a suitable method can easily be selected for a given wireless system and channel conditions. It can be observed that each method can be approximated to the other methods by using the same set of variables. For example, in this paper it is shown that each estimation method is indeed a subset of LMMSE technique. In the literature, initial channel estimation methods have been mostly developed for SISO-OFDM systems, that is, single antenna systems. With the emergence of MIMO-OFDM, these methods need some modifications as the received signal in MIMO-OFDM is the superposition of all the transmitted signals of a given user. In many cases, the methods of SISO- OFDM are easily adopted for MIMO-OFDM but novel methods exploiting space-time codes or other MIMO specific elements are also introduced. In the rest of the article, starting from a generic system model, the channel estimation techniques will be presented starting from the less complicated techniques. More emphasis will be given on data aided channel estimation as it provides some unique approaches for OFDM systems. Discussions on ICI, external interferers, and MIMO systems as well as related issues will also be given. Finally, some concluding remarks and potential research areas will be given at the end of the article. NOTATION Matrices and the vectors are denoted with boldface letters, where the upper/lower letters will be used for frequency/time domain variables; (.) H denotes conjugate-transpose; E{.} denotes expected value; diag(x) stands for diagonal matrix with the column vector x on its diagonal; a b denotes a matrix of a b with zero entries; IN denotes N N identity matrix; and j=. SYSTEM MODEL A generic block diagram of a basic baseband-equivalent MIMO-OFDM system is given in Fig. 2. A MIMO-OFDM system with N tx transmit and N rx receive antennas is assumed. The information bits can be coded and interleaved. The coded bits are then mapped into data symbols depending on the modulation type. Another stage of interleaving and coding can be performed for the modulated symbols. Although the symbols are in time domain, the data up to this point is considered to be in the frequency domain. The data is then demultiplexed for different transmitter antennas. The serial data symbols are then converted to parallel blocks, and an IFFT is applied to these parallel blocks to obtain the time domain OFDM symbols. For the transmit antenna, tx, time domain samples of an OFDM symbol can be obtained from frequency domain symbols as x [ n, m] = IFFT { X [ n, k]} () () tx = Xtx[ n, ke ] j2πmk/ km, k= where X tx [n, k] is the data at the kth subcarrier of the nth OFDM symbol, is the number of subcarriers, and m is the time domain sampling index. After the addition of CP, which is larger than the expected maximum excess delay of the channel, and D/A conversion, the signals from different transmit antennas are sent through the radio channel. The channel between each transmitter/receiver link is modelled as a multi-tap channel with the same statistics [3]. The typical channel at time t is expressed as, L ht (, τ) = α () tδ( τ τ ), l= l tx where L is the number of taps, α l is the lth complex path gain, and τ l is the corresponding path delay. The path gains are WSS complex Gaussian processes. The individual paths can be l (2) (3) 2 IEEE Communications Surveys & Tutorials 2nd Quarter 27

4 correlated, and the channel can be sparse. At time t, the CFR of the CIR is given by, + j2π fτ Htf (, ) = ht (, τ) e dτ. With proper CP and timing, the CFR can be written as [3], where h[n, l] = h(nt f, kt s ), F = e j2π/, T f is the symbol length including CP, f is the subcarrier spacing, and t s = /Df is the sample interval. In matrix notations, for the nth OFDM symbol, Eq. 5 can be rewritten as H = Fh (6) where H is the column vector containing the channel at each subcarrier, F is the unitary FFT matrix, and h is the column vector containing the CIR taps. At the receiver, the signal from different transmit antennas are received along with noise and interference. After perfect synchronization, down sampling, and the removal of the CP, the simplified received baseband model of the samples for a given receive antenna, rx, can be formulated as where rx =,, N rx, the time domain effective CIR, h m rxtx[n, l], over an OFDM symbol is given as time-variant linear filter depending on the time selectivity of the channel. Please note that n represents OFDM symbol number, while m denotes the sampling index in time domain so that h m rxtx[n, l] is the CIR at the sampling time index m for the symbol n. When the CIR is constant over an OFDM symbol duration, then h m rxtx[n, l] will be the same for all m values, and hence the superscript m can be dropped. Moreover, i rx [n, m] is the term representing external interference, w rx [n, m] is the AWGN sample with zero mean and variance of σ w 2. After taking FFT of the time domain samples of Eq. 7, the received samples in frequency domain can be expressed as, Y rx kl Hnk [, ] HnT (, k f) = hnlf [, ], N t x L m y [ n, m] = x [ n, m l] h [ n, l] rx tx rxtx tx= l= + i [ n, m] + w [ n, m], rx [ n, k] = yrx[ n, m] e + i [ n, m] + w [ n, m] ] e rx m= 2πkm j = N tx L m xtx[ n, m l] hrxtx[ n, l] m= f tx= l= N tx L = xtx[ n, k ] e tx= m= l= k = j2π ( m l) k / ( () j km rx rx L l= 2π j km m hrxtx[ n, l] 2π e + Irx[ n, k] + Wrx[ n, k] where I rx [n, k] and W rx [n, k] are the corresponding frequency domain components calculated from i rx [n, m] s and w rx [n, m] s, respectively. After arranging the terms, and representing the (4) (5) (7) (8) (9) variables in matrix notation, for rxth receive antenna and nth OFDM symbol, we get Ntx H Yrx = FΞrxtxF X tx + Irx + Wrx, tx= N tx () = ΨXtx + Irx + Wrx. (2) tx= Here, Y rx is column vector storing the received signal at each subcarrier, F is the unitary FFT matrix with entries e j2πmk/ with m and k being the row and column index and Ψ = FΞ rxtx F H, which can be considered as the equivalent channel between each received and all the transmitted subcarriers. Moreover X tx denotes the column vector for transmitted symbols from txth transmit antenna, I rx is the column vector for interferers, W rx is the column vector for noise, and Ξ rxtx is the matrix containing the channel taps at each m index. The entries of Ξ are given by hrxtx[ n, ] hrxtx[ n, ] hrxtx[ n, ] Ξ rxtx = L L hrxtx [ n, L ] hrxtx [ n, L 2] hrxtx[ n, 2] hrxtx[ n, 3] hrxtx[ n, ] hrxtx[ n, 2]. hrxtx [ n, L ] hrxtx [ n, ] (3) When the channel is assumed to be constant over one OFDM symbol and the CP is larger than the CIR length, then h m rxtx[n, l] is the same for all m s, making Ξ rxtx a circulant matrix [35]. The multiplication of FΞ rxtx F H then results in a diagonal matrix, and hence no cross-terms between subcarriers exist, that is, no ICI occurs. In this case, h is equivalent to the first column of Ξ. However, when the channel varies over an OFDM symbol, then ICI occurs, and for the equalization the channel at each time sample of OFDM symbol is needed, that is, at each m samples. For the frequency domain estimation, this requirement translates into the knowledge of the channel coefficients at each carrier frequency as well as their cross-terms. The number of unknowns in time domain estimation are L, whereas the number of unknowns in frequency domain (the entries of Ψ) are 2. In either case, the number of unknowns will be higher than the number of equations, and hence a system of under-determined equations will result in. Simplifications are needed so that the unknowns in the system of equations are reduced. Different approaches will be described in detail in the subsequent sections. Once the received signals for each transmit antennas are detected with the help of channel estimation, the reverse operation at the receiver is performed, that is, they are demodulated, de-interleaved, and decoded. As it will be seen later, the information at different stages of decoding process can be exploited to enhance the performance of channel estimation methods. IEEE Communications Surveys & Tutorials 2nd Quarter 27 2

5 Frequency Frequency Training symbols Time Data symbols Pilot subcarriers Time Data subcarriers (a) (b) nfigure 3. Typical training symbols and pilot subcarriers arrangement. OFDM CHANNEL ESTIMATION TECHNIQUES There are several basic techniques to estimate the radio channel in OFDM systems. The estimation techniques can be performed using time or frequency domain samples. These estimators differ in terms of their complexity, performance, practicality in applications to a given standard, and the a priori information they use. The a priori information can be subcarriers correlation in frequency [36], time [3], and spatial domains [37]. Moreover, the transmitted signals being constant modulus [38], CIR length [39], and using a known alphabet for the modulation can also be a priori information [4, 4]. The more the a priori information is exploited, in general the better the estimates are [42]. For frequency domain channel estimates, MSE is usually considered as the performance measure of channel estimates, and it is defined by MSE = E{ H[n, k] H^[n, k] 2 }, (4) where H^[n, k] is the estimate of equivalent channel at kth subcarrier of nth OFDM symbol. Although MSE is used extensively, sometimes, other measures like BER performance are also used [43, 44]. BER performance is mainly used when the performance of OFDM system with the channel estimation error is to be evaluated [45, 46]. Before introducing the estimation techniques, it is worthwhile to look at the data aided channel estimation in general and the pilot allocation mechanisms. DATA AIDED CHANNEL ESTIMATION In this subsection, we will review commonly used methods in the data aided channel estimation. Initially, we will consider the methods developed for SISO-OFDM. ICI is assumed not to exist and the CIR is assumed to be constant for at least one OFDM symbol. Hence, Ψ is a diagonal matrix, where each diagonal element represents the channel between the corresponding received and the transmitted subcarriers. In this case, for the nth OFDM symbol, the channel given in Eq. 5 at each subcarrier can be related to Ψ as H[n, k] = Ψ[k, k]. (5) Furthermore, the external interference is folded into the noise with noise statistics being unchanged. With the above assumption, the expression in (2) can be expressed as Y = diag(x) H + W, (6) or Y[n, k] = H[n, k] X[n, k] + W[n, k]. (7) Here H and W are the column vectors representing the channel and the noise at each subcarrier for the nth OFDM symbol, respectively. In data aided channel estimation, known information to the receiver is inserted in OFDM symbols so that the current channel can be estimated. Two techniques are commonly used: sending known information over one or more OFDM symbols with no data being sent, or sending known information together with the data. The previous arrangement is usually called channel estimation with training symbols while the latter is called pilots aided channel estimation (Fig. 3). Channel estimation employing training symbols periodically sends training symbols so that the channel estimates are updated [29]. In some cases training symbols can be sent once, and the channel estimation can then be followed by decision directed type channel estimation. The details of the decision directed will be given later in the article. In the pilots aided channel estimation, the pilots are multiplexed with the data. For time domain estimation, the CIR is estimated first. The estimate of the CIR are then passed through a FFT operation to get the channel at each subcarrier for the equalization in frequency domain. For frequency domain estimation, the channel at each pilot is estimated, and then these estimates are interpolated via different methods. Pilots Allocation for Data Aided Channel Estimation For the pilot aided channel estimation, the pilot spacing needs to be determined carefully. The spacing of pilot tones in frequency domain depends on the coherence frequency (channel frequency variation) of the radio channel, which is related to the delay spread. According to the Nyquist sampling theorem, 22 IEEE Communications Surveys & Tutorials 2nd Quarter 27

6 Correlation coefficient (abs) Subcarrier index nfigure 4. Periodic behavior of subcarriers cross-correlation for = 64. the number of subcarrier spacing between the pilots in frequency domain, D p, must be small enough so that the variations of the channel in frequency can be all captured, that is, Dp (8) τ max df where τ max is the maximum excess delay of channel. When the above is not satisfied, then the channel available at the pilot tones does not sample the actual channel accurately. In this case, an irreducible error floor in the estimation technique exists since this causes aliasing of the CIR taps in the time domain [47]. When the channel is varying across OFDM symbols, in order to be able to track the variation of channel in time domain, the pilot tones need to be inserted at some ratio that is a function of coherence time (time variation of channel), which is related to Doppler spread. The maximum spacing of pilot tones across time is given by Dt (9) 2 fd max Tf where f dmax is the maximum Doppler spread and T f is the OFDM symbol duration. For comb-type pilot arrangements, the pilot tones are often inserted for every OFDM symbols. When the spacing between the pilot tones does not satisfy the Nyquist criteria, then the pilots can still be exploited in a combined pilot-plus DDCE [48]. The pilots can be sent continuously for each OFDM symbol. Since the channel might be varying both in time and frequency domains, for the reconstruction of the channel, this 2-D function needs to be sampled at least a Nyquist rate. Hence, the rate of insertion of pilots in frequency domain and from one OFDM symbol to another cannot be set arbitrarily. The spacing of pilots should be according to Eq. 8 and Eq. 9. In general, within an OFDM symbol the number of pilots in frequency domain should be greater than the CIR length (maximum excess delay), which is related to the channel delay spread. Over the time, the Doppler spread is the main criteria for the pilot placement. Many studies are performed in order to get the optimum pilot locations in time-frequency grid given a minimum number of pilots that sample the channel in 2-D at least Nyquist rate. This optimality is in general based on the MSE of the LS estimates [6, 49]. It should be noted that an optimum pilot allocation is a trade-off between wasted energy in unnecessary pilot symbols, the fading process not being sampled sufficiently, the channel estimation accuracy, and the spectral efficiency of the system [5]. Hence, an optimum pilot allocation for a given channel might not be optimal for another channel as the fading process will be different. In addition to minimizing MSE of the channel estimates, pilots also need to simplify the channel estimation algorithms so that the system resources are not wasted. For example, it is noted that the use of constant modulus pilots simplify the channel estimation algorithms as matrix operations become less complex [38, 5]. Some other important elements for pilot arrangements are the allocation of power to the pilots with respect to the data symbols, the modulation for the pilot tones etc. In many cases, the power for pilot tones and data symbols are equally distributed. The channel estimation accuracy can be improved by transmitting more power at the pilot tones compared to the data symbols [52]. For a given total power, this reduces the SNR over the data transmission. As for the pilot power at different subcarriers, studies show that based on the MSE of the LS estimates pilots should be equipowered [6, 53]. Moreover, due to the lack of the pilot subcarriers at the edge of OFDM symbols, the estimation via the extrapolation for the edge subcarriers results in a higher error [54, 55]. Simulations also reveal that the channel estimation error at the edge subcarriers are higher than those at the mid-bands due to this extrapolation [56 58]. One quick solution would be to increase the number of pilot subcarriers at the edge subcarriers [58], however this would decrease the spectral efficiency of the system [57]. Due to the periodic behavior of the Fourier Transform, the subcarriers at the beginning and the end of the OFDM symbol are correlated, and this can be used to improve the channel estimates at the edge subcarriers (Fig. 4). Simulations exploiting this property are reported to enhance the estimation accuracy of the edge subcarriers [57]. Another issue related to pilot arrangement is the pattern of the pilots, that is, how to insert the pilots to efficiently track the channel variation both in time and frequency domains. The selection of a pilot pattern may affect the channel estimation performance, and hence the BER performance of the system. Equation 8 states that the pilot spacing in frequency domain needs to satisfy the Nyquist criteria. More insight into Eq. 8 reveals that the number of required pilots in frequency domain can be taken as the CIR length. At a first glance, this does not pose any restriction on the pilot spacing that a sufficient number of pilots can be inserted in adjacent subcarriers. However, when the MSE of the time domain LS estimation, which is covered in the next subsection, is analyzed, it is observed that the minimum MSE is obtained when the pilots are equispaced with maximum distance [6, 3, 39]. This is due to the reason that when the pilots are inserted in adjacent subcarriers, then the FFT matrix used in the time domain LS estimation approaches to an ill conditioned matrix, making the system performance vulnerable to the noise effect [39]. Hence, from the MSE of LS estimation, the pilots in frequency domain need to be equipowered, equispaced, and their number should not be less than the CIR length. Since the use of pilots is a trade-off between extra overhead and the accuracy of the estimation, adaptive allocation of pilots based on the channel length estimation can offer a better trade-off [52, 56, 59]. As will be seen later in the article, with MIMO and ICI additional requirements will be observed on the pilot subcarriers spacing and properties. When it comes to the pilot allocation for subsequent OFDM symbols, either the set of subcarriers chosen in a previous OFDM symbol or a different set of pilots can be used (Fig. 3). The use of the same subcarriers as the pilots is a IEEE Communications Surveys & Tutorials 2nd Quarter 27 23

7 widely used pilot arrangement. In such a pilot arrangement, first the channel between subcarriers is estimated via interpolation in frequency domain. This is followed by interpolation over OFDM symbols in time domain. In some cases, interpolation can be first performed in time domain, followed by the frequency domain interpolation. The details of different interpolation techniques will be given later in this section. The analysis of MSE of time domain LS estimation over several OFDM symbol indicates that for a lower MSE, the pilots should be cyclically shifted for the next OFDM symbol [6, 6]. This pilot allocation is similar to those used in DTV applications, and is similar to the pilot scheme given in Fig. 3. In this pilot allocation scheme, the interpolation is first performed in frequency domain, followed by the interpolation in time domain. Similar to the pilot scheme used in DTV, a hexagonal type pilot scheme is also proposed [6 63]. In both schemes, different subcarriers are utilized for each OFDM symbol, and hence the possibility of sticking into terribly fading subcarriers is eliminated, that is, diversity is exploited. In addition to the above pilot schemes, different types of pilot schemes are tested through simulations [56]. The pilots having more density than the others, those utilizing different subcarriers over time and at the edge subcarriers are expected to perform better for channels varying both in time and frequency domains. The previous pilot allocation schemes were solely based on the MSE analysis of the channel estimation. In some cases, other system parameters can also be considered for the pilots to be used. For example, due to the IFFT block at the transmitter side, PAPR of OFDM systems can be very high. It is observed that different training symbols (not scattered pilots) results in different PAPR [64]. Moreover, different scattered pilot allocation schemes can result in different PAPR when multiplexed with data. Since the data is random, the optimum allocation for minimum PAPR will be different for each transmission. However, pre-defined pilot allocation schemes can be tested for the best PAPR [65]. With such a scheme however, the information about the pilot scheme needs to be conveyed to the receiver side, and this reduces the spectral efficiency of the system. It is clear from the discussion about the pilot allocation that a better system performance can be obtained when the system is adaptive [52, 59, 6, 66]. In this case, the information about the channel statistics becomes very critical. The pilot allocation in the frequency domain requires the delay spread estimation, whereas the one in over OFDM symbols (over time evolution) requires Doppler spread estimation. If these estimates are available, then a pilot scheme using just the right amount of pilots can yield an acceptable performance. If this information is not available, then the pilot scheme can be designed based on the worst channel condition, that is, the maximum expected delay and Doppler spreads. In addition to unknown channel statistics, randomly generated pilots can be utilized for the reduction of interference from adjacent cells. However, it is shown via simulations that such pilots cause severe degradation in the channel estimation MSE [67]. So far the pilots in the frequency domain are discussed. In some cases, the estimation can be performed using the data in time domain, that is, data before the FFT block at the receivers. Training symbols for this case can be set to all s in frequency domain that result in an impulse in the time domain. When this impulse is passed through the channel, then CIR can be obtained. By careful arrangement of s in frequency domain, the multiple replicas of the CIR can be obtained, and these can be improved through noise averaging. In a similar way, PN sequences superimposed with the data can be utilized for the channel estimation. In such a case, correlators at the receiver can be used for the expected samples of the OFDM symbols [68 7]. However, it is shown that superimposing training with data is not optimal for channel estimation [7]. Having reviewed the pilot schemes employed in OFDM systems, it is time to look at the channel estimation techniques. Starting from the methods using the least a priori information, in this article we will review channel estimation methods such as LS estimation, ML, transform domain techniques, and LMMSE. Simple interpolation techniques will be covered along with LS estimation technique. LS ESTIMATION Before going into the details of the estimation techniques, it is necessary to give the LS estimation technique as it is needed by many estimation techniques as an initial estimation. Starting from system model of SISO-OFDM given in Eq. 7 as[72] Y[n, k] = X[n, k]h[n, k] + W[n, k], (2) the LS estimation of H[n, k] is Ĥ LS [n,k]= Y[n,k] W[n,k] = H [n,k]+ (2) X[n,k] X[n, k]. In matrix notations, H^LS = diag(x) Y + diag(x) W. (22) Note that this simple LS estimate for H^LS does not exploit the correlation of channel across frequency carriers and across OFDM symbols. The MSE of LS estimation of Eq. 22 is given by [73] MSELS = E (23) H SNR where E H = E{H[n, k]}. LS method, in general, is utilized to get initial channel estimates at the pilot subcarriers [72], which are then further improved via different methods. It is also common to introduce CIR to Eq. 6 to exploit CIR length for a better performance [2, 74]. In this case, Eq. 6 can be modified as [74] Y = diag(x)fh + W where H = Fh. The LS estimation of Eq. 24 is then H^ = Q LS F H diag(x) H Y (25) where Q LS = (F H diag(x) H diag(x)f). (26) The above LS estimation will be referred as time domain LS. When no assumptions on the number of the CIR taps or length are made, then the time domain LS reduces to that of frequency domain, and it does not offer any advantages. However, with the assumption that there are only L number of channel taps, which then reduces the dimension of the matrices F and hence Q, an improved performance due to the noise reduction can be obtained [75, 76]. The resultant LS estimation has higher computational complexity than the frequency domain LS but the performance increase is the plus side of the approach. The increase in the performance can be considered as the exploitation of subcarrier correlation. A comparison study showed that when the frequency domain LS also exploits the correlation of the subcarriers, then its performance can be that of time domain LS (2). Further compari- 24 IEEE Communications Surveys & Tutorials 2nd Quarter 27

8 son studies showed that based on the SNR information, either method can be used [74]. For example if the SNR is low then the time domain LS can be less accurate as additional filtering in time domain is based on less accurate CIR length. In this case, the probability of not accounting for all the taps and discarding some of them are high. However, for other SNR regions, the time domain LS gives better results as it utilizes a more accurate CIR length. The use of time domain LS becomes inevitable when OFDM is combined with MIMO systems [77]. This will be explored more when channel estimation techniques for MIMO systems are presented. Similar to the time domain LS, the ML estimate of the CIR taps for the same system model given in Eq. 24 can be derived. With the assumption of L channel taps and N p number of pilot subcarriers, the ML estimate of the channel coefficients is shown to be [58, 78], ^ML H = (F H p F p ) F H p diag(x) H Y (27) where F p is N p L truncated unitary Fourier matrix. In the above formulation, for the sake of simplicity, it is assumed that pilots symbols are from PS constellation and hence diag(x) H diag(x) = I, and they do not appear in the parenthesis for the inverse operation. It can be observed that when the number of pilots is greater than the channel length and the noise is AWGN, the time domain LS estimate in Eq. 25 is equivalent to the ML estimate given in Eq. 27 [58, 79]. Furthermore, it should be noted that the ML estimate given in (27) makes the assumption about the CIR length, which improves the performance of the estimation accuracy [8]. Unlike LMMSE channel estimation, both LS and ML are based on the assumption that the CIR is a deterministic quantity with unknown parameters. This implies that LS and ML techniques do not utilize the long term channel statistics and hence are expected to perform worse than the LMMSE channel estimation method [58]. However, the computational complexity is the main trade-off factor between the two groups of the channel estimation techniques. Before introducing the other channel estimation techniques, it is worthwhile to review the methods used for the training sequences as well as the pilot subcarriers. The corresponding implications on the channel estimation techniques will also be covered briefly. CHANNEL ESTIMATION TECHNIQUES IN TRAINING MODE As mentioned before, in the training mode, all the subcarriers of an OFDM symbol are dedicated to the known pilots. In some systems like WLAN or WiMAX, two of the symbols are reserved for the training. If the training symbols are employed over two OFDM symbols, for very slowly varying channels, the channels at two OFDM symbols for the same subcarriers can be assumed to be the same. In this case, the estimates can be averaged for further noise reduction [72]. If the noise variances of the OFDM symbols are different, then alman filtering can be used such that noise variances are exploited as weighting parameters [8]. Once the channel is estimated over the training OFDM symbols, it can be exploited for the estimation of the channels of the OFDM symbols sent in between the training symbols. Depending on the variation of the channel along time, different techniques can be utilized. A very common method is to assume the channel being unchanged between OFDM training symbols [23, 28 3, 69]. In this method, the channel that is estimated at training symbols is used for the subsequent symbols until a new training sequence is received. The channel is then updated by using the new training sequence, and the process continues. In fact, this is one of the algorithms employed for IEEE 82.a/b/g and fixed WiMAX systems. However, these approaches introduce an error floor for non-constant channels, that is, outdoor channels. The highest performance degradation occurs at the symbols farthest from the training symbols. For video transmission systems, the critical information can be sent over the symbols closer to the training symbols, while non-critical information can be sent over those farther from the training symbols [29, 3]. It is observed that such an arrangement can improve the performance without increasing the number of training blocks. However, for systems requiring equal priority packets like data networks, such an approach cannot be taken. In this case satisfactory results can be obtained by increasing the rate at which the training symbols are sent at the expense of system efficiency. For the fast varying channels, interpolation methods can be utilized in time domain. Interpolating the channel linearly between the training symbols is one simple solution [59, 72, 82]. The disadvantage with such an approach is the latency introduced in the system [83]. Indeed, if the system can tolerate more latency, then the channel estimation for non-training OFDM symbols can be improved by higher order polynomials [66, 84, 85]. CHANNEL ESTIMATION TECHNIQUES IN PILOT MODE In the pilot mode, only few subcarriers are used for the initial estimation process. Depending on the stage where the estimation is performed, estimation techniques will be considered under time and frequency domains techniques. In frequency domain estimation techniques, as a first step, CFR for the known pilot subcarriers is estimated via (22). These LS estimates are then interpolated/extrapolated to get the channel at the non-pilot subcarriers. The process of the interpolation/extrapolation can be denoted as ^ H = QH^LS (28) where Q is the interpolation/extrapolation matrix. The goal of the estimation technique is to obtain Q with lower computational complexity but at the same time is to achieve higher accuracy for a given system. In this subsection, the calculation of matrix Q for simple interpolation techniques will be discussed. Piecewise Linear Interpolation Two of the simplest ways of interpolation are the use of piecewise constant [86] and linear interpolation [22, 84, 87, 88]. In the piecewise constant interpolation, the CFR between pilot subcarriers is assumed to be constant, while in piecewise linear interpolation the channel for non-pilot subcarriers is estimated from a straight line between two adjacent pilot subcarriers. Mathematically, for piecewise constant interpolation, Q is a matrix consisting of columns made up from shifted versions of the column vector T c = [,,,,,, ], D p where D p is the spacing of the pilots. For the the piecewise linear interpolation, Q consists of coefficients that are a function of the slope of the line connecting two pilot subcarriers and the distance of the pilots to the subcarrier for which the channel is to be estimated. In the first method, acceptable results can be obtained if the CFR is less frequency selective or the CIR maximum excess delay is very small. Such a constraint makes the CFR at the subcarriers very correlated that CFR at a group of subcarriers can be assumed to be the same. IEEE Communications Surveys & Tutorials 2nd Quarter 27 25

9 In piecewise linear interpolation some variation is allowed between the pilot subcarriers. Such an approach can result in a lower MSE since noise averaging is performed. Moreover, when the channel becomes more frequency selective, the piecewise linear interpolation results in a better performance compared to the piecewise constant [86 89]. For a better insight into the performance of the piecewise linear interpolation, its MSE is derived and is expressed in terms of the channel statistics and the pilot spacing as [87] MSE = + + R f + + w ( ζ)( ζ) [ ] ( ζ ) σ Dp 4ζ ( ζ 3 2 ) R { Rf [ l]} + ( ζ ) R{ Rf[ Dp]} l= (29) where /ζ = D p, R f [l] is the frequency domain correlation of CFR, R denotes the real part of a complex number, and σ w 2 is the noise variance. When the piecewise linear interpolation is to be performed between OFDM symbols over time, then the parameters above need to be replaced with their time domain equivalence. As can be seen from the expression, lower MSE results in: When many pilots are used When the noise is low When the channel is very correlated Higher Order Polynomial Fitting Piecewise linear interpolation requires more pilot subcarriers for an acceptable performance in highly frequency selective channels [52, 86, 89]. However, by using a priori information about the frequency or time selectivity of the channel, the use of higher order polynomial can result in better performance. Higher order polynomials indeed can approximate the wireless channels accurately, since the channel itself is smooth in both time and frequency domains [66]. The degree of this smoothness depends on the selectivity of the channel. For highly time and frequency selective channels, the higher the polynomial order, the better the estimation at the expense of higher computational complexity [23]. However, when the channel is changing very slowly both in frequency and time, then the use of very high order polynomials can degrade the performance, as the modelling uses noise as a means to represent the channel [66]. This behavior also suggests dynamic polynomial fitting based on the channel statistics [23]. Simulations show that adaptive polynomial fitting performs better than the static polynomial fitting when the channels become more selective [23]. In a move towards reducing the computational complexity of such an adaptation, instead of estimating the true channel statistics, variation of channel between two adjacent subcarriers can be monitored, and an idea of how fast the channel is changing can be obtained [9]. Further computational complexity can be achieved if the coefficients of Q are made power of 2 to eliminate the multiplication/division via bit shifting. It is observed that such an approach can yield accurate channel estimates [9]. In the higher order polynomial approaches, the entries of the Q are calculated by using more information about the channel. Higher order polynomial fitting uses more than two pilot subcarriers for the CFR estimation. While some of the polynomial fitting methods utilize no channel statistics [52, 9], others assume to have some information about the statistics [66, 85]. The most common higher order interpolation methods are spline interpolation [22, 6, 89], Gaussian interpolation [22], and polynomial fitting [66, 85, 9, 93, 94]. In the spline interpolation, basis function of some orders or Beizer curve are defined over a group of subcarriers [6, 84]. These basis functions are determined such that they are unity at the pilot locations at which they are defined for, and vanishes at the other pilot locations. The channel at non-pilot subcarriers can then be found as N p Hnk [, ] = Bp[ nkhnp, ] [, ], (3) p= where N p is the number of pilots over a range, B p [n, k] is the basis function at subcarrier k, and H[n, p] is the CFR at the pilot location p. The rows of the interpolation matrix, Q, are then formed using B p [n, k] s. For more frequency selective channels the order of the basis functions, B p [n, k] s, can be increased for a better performance. This corresponds to having more columns in Q, and implies the use of more pilot subcarriers for the estimation of a single subcarrier. Gaussian interpolation is another interpolation technique, where the coefficients of Q are obtained from a Gaussian function [95]. The Gaussian function resembles the sinc function, the ultimate function for ideal low pass filtering. The Gaussian function can be considered as an approximation to the sinc function. The width of the Gaussian function or equivalently the coefficients used in the interpolation are dependent on the frequency selectivity of the channel. Hence, as with many approaches, the knowledge of the channel statistics can improve the performance of the Gaussian interpolation. Similar to the Gaussian interpolation, radial basis functions utilizing Gaussian function are also used for the interpolation purpose [96]. The coefficients of the radial basis functions are determined through some non-linear training mechanism similar to those used in neural networks. Overall, the goal is to find the coefficients of the interpolation using the Gaussian function as a basis, and the training process indeed reflects the information about the channel statistics to the coefficients to be used in the interpolation. Hence, the approach of the radial basis function interpolation can be considered as an adaptive low-pass filtering. The improved performance due to this adaptation comes at the cost of training process using pilot subcarriers. 2-D regression models for the pilot subcarriers scattered in frequency and time domains are also studied [85, 94]. In these models, a 2-D polynomial whose coefficients are obtained using the channel correlation and the initial LS estimates at the pilot subcarriers is developed. Although higher order polynomials can be used, second order approximation is found to yield close to ideal BER performance for certain channels [85]. All of the above interpolators can be seen as a simple lowpass filter. This is due to the fact that CIR has a finite length that is in general much smaller than the number of subcarriers. The above interpolation methods are not ideal low-pass filters, and hence they introduce an error floor due to either the suppression of some of the channel taps or the inclusion of noise whose effect becomes effective at high SNR regions. A low-pass filtering can eliminate the noise in non-tap locations, which in turn means the elimination of most of the noise in the estimated subcarriers. For example, it is shown that the use of raised cosine filter as a low-pass filter provides accurate channel estimates for WLAN systems [28]. The sharper the low-pass filtering the better the estimates are. Since the Fourier Transform of a rectangular function (or a window) is the sinc function, the sinc interpolator with the known CIR length provides ideal low-pass filtering. However, sinc interpolator is not realizable in practical implementations. Moreover, it is computationally heavy as it requires more CFR samples. 26 IEEE Communications Surveys & Tutorials 2nd Quarter 27