Channel Estimation for MC-CDMA Uplink Transmissions With ... - Unipi
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 6, JUNE 2006
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Channel Estimation for MC-CDMA Uplink Transmissions With Combined Equalization Luca Sanguinetti, Student Member, IEEE, Ivan Cosovic, Member, IEEE, and Michele Morelli, Member, IEEE
Abstract—Combined equalization has recently been proposed to enhance the error rate performance of conventional multicarrier code-division multiple-access (MC-CDMA) systems. This technique applies pre-equalization at the transmitter in conjunction with post-equalization at the receiver, thereby splitting the overall equalization process into two separate parts. In this way, efficient power allocation over the available subcarriers is possible at the transmitter, while leaving the interference cancellation task at the receiver. In this paper, we consider the uplink of an MC-CDMA system employing combined equalization. As the users transmit from different locations, the uplink signals arrive at the base station after passing through different multipath channels and the goal is to estimate the pre-equalized channel frequency response of each user. This is pursued following two different approaches. The first operates in the frequency-domain and treats the channel gains over adjacent subcarriers as independent unknown parameters. The second operates in the time-domain and achieves better performance by reducing the number of unknown parameters. Both schemes are based on maximum-likelihood reasoning and require knowledge of the transmitted symbols. Numerical examples are given to highlight the effectiveness of the proposed methods. Index Terms—Channel estimation, combined equalization, multicarrier code-division multiple-access (MC-CDMA), prefiltering techniques.
I. INTRODUCTION
T
HE DEMAND for high data rates in wireless transmissions has increased significantly over the last years. This has led to a strong interest in multicarrier code-division multiple-access (MC-CDMA), which is an effective technique for combating multipath fading over highly dispersive wireless channels [1]. In MC-CDMA systems, the data of different users are spread in the frequency-domain using orthogonal signature sequences. The main impairment of this multiplexing technique is represented by the multiple-access interference (MAI), which occurs in the presence of multipath propagation due to loss of orthogonality among the received spreading codes. In conventional MC-CDMA systems, MAI mitigation is accomplished at
Manuscript received April 1, 2005; revised October 1, 2005. This work was supported in part by the Italian Ministry of Education under the Fondo per gli Investimenti della Ricerca di Base (FIRB) Project Piattaforme Riconfigurabili per Interoperabilità in Mobilità (PRIMO). The work of I. Cosovic was done in part while with the German Aerospace Center (DLR). L. Sanguinetti and M. Morelli are with the Department of Information Engineering, University of Pisa, 56126 Pisa, Italy (e-mail: luca.sanguinetti@ iet.unipi.it; [email protected]). I. Cosovic is with DoCoMo Communications Laboratories Europe GmbH, 80687 Munich, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/JSAC.2005.864023
the receiver using well-known single-user or multiuser detection schemes [2]. Alternatively, prefiltering techniques based on the zero-forcing (ZF) or minimum-mean-square-error (MMSE) criterion can be employed to mitigate MAI and channel distortions in uplink transmissions [3]–[5]. The idea behind prefiltering is to vary the complex gain assigned to each subcarrier so that interference is reduced and the signal at the receiver appears undistorted. In this way, channel estimation is not even necessary at the base station (BS) and low-complex detection schemes can be employed. A possible drawback of prefiltering schemes is represented by the power-boosting effect [6], which occurs in the presence of deeply faded subcarriers. This phenomenon leads to high power consumption at the transmitter side and represents a serious problem in uplink transmissions due to the limited amount of power available at the mobile terminal (MT). A different approach has recently been proposed in [7] in the form of combined equalization. This technique applies pre-equalization at the transmitter in conjunction with post-equalization at the receiver. This ensures an additional degree of freedom in the system design, since equalization at the transmitter and receiver sides can be adjusted separately. In particular, in combined schemes the prefiltering coefficients are designed so as to efficiently distribute the transmit power among the available subcarriers, whereas MAI cancellation is the main task of the receiver (post-equalization). As shown in [7], combined equalization significantly outperforms conventional prefiltering techniques, where interference cancellation is accomplished at the transmitter at the expense of significant power boosting. In order to work properly, however, combined schemes require channel knowledge at both the transmitter and receiver ends. This problem is not addressed in [7], where ideal channel state information (CSI) is assumed for pre- and post-equalization purposes. In this paper, we discuss channel estimation in the uplink of an MC-CDMA system employing combined equalization. In doing so, we assume that CSI is available at each MT. This assumption is reasonable in time-division duplex (TDD) systems (such as IEEE802.11a and HiperLAN/II) due to the channel reciprocity between alternative uplink and downlink transmissions. If the analog transmit/receive front ends are correctly calibrated [8] and channel variations are sufficiently slow (as occurs in indoor applications), the channel estimate derived at a given MT during a downlink time slot can be reused for prefiltering in the subsequent uplink time slot. The uplink signals arrive at the BS after passing through different multipath channels and the goal is to estimate the pre-equalized channel frequency responses of the active users, which are then exploited to perform post-equalization and data detection.
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So far, the problem of channel estimation in MC-CDMA uplink transmissions has received little attention and only few results are available in the technical literature [9]–[14]. The main difficulty is that the channel responses of the active users are different from one another and the BS must estimate a large number of parameters. This is expected to degrade the quality of the estimates as compared with the downlink situation where only a single channel response is present. Although designed for conventional MC-CDMA systems without any signal prefiltering, the channel estimators in [9]–[14] can also be used in conjunction with combined equalization. As shown later, however, in combined schemes the prefiltered channel responses are real-valued since compensation for the channel phase is accomplished at the transmit side. Intuitively, we expect that exploiting the above property may improve the system performance as compared with the conventional schemes in [9]–[14]. This possibility is investigated here and two different solutions are provided. The first operates in the frequency-domain and is referred to as unstructured channel estimation (UCE). The second is called structured channel estimation (SCE) and operates in the time-domain. This means that an estimate of the pre-equalized channel impulse response (CIR) is computed first, and then it is exploited to estimate the corresponding channel frequency response. In this way, the number of unknown parameters is reduced with respect to UCE, leading to a better estimation accuracy. Both schemes are based on the maximum-likelihood (ML) criterion and require knowledge of the transmitted data symbols. For this purpose, we assume that the transmission is organized in frames and some training blocks (carrying known symbols) are placed at the beginning of each frame. The contribution of this paper can be summarized as follows. First, we derive channel estimation schemes specifically designed for MC-CDMA systems employing combined equalization. This is achieved by explicitly taking into account the fact that the pre-equalized channel responses are real-valued. Second, we show how the above property can be exploited to improve the quality of the uplink channel estimates or, alternatively, to reduce the training overhead with respect to conventional channel estimation schemes. Third, we discuss the minimum number of training blocks required to perform channel estimation and provide optimal training sequences minimizing the mean-square error of the channel estimates under various operating conditions. The rest of this paper is organized as follows. Section II outlines the uplink of an MC-CDMA system with combined equalization. Channel estimation by means of UCE and SCE is addressed in Sections III and IV, respectively. Simulation results are provided in Section V, while some conclusions are drawn in Section VI. A. Notation Vectors and matrices are denoted by boldface letters, with and representing the identity and null matrices, reto indispectively. We use cate an diagonal matrix with entries . We denote and the inverse and trace of a square matrix . The convolution between two sequences is denoted by , while represents the Euclidean norm of the enclosed vector. We
for expectation and the superscript , and for comuse plex conjugation, transposition and Hermitian transposition. Fiand indicate the real and imaginary parts nally, of a complex-valued quantity, while is used for the corresponding magnitude. II. SYSTEM MODEL A. MC-CDMA Transmitter We consider the uplink of an MC-CDMA system with available subcarriers. The latter have indexes and are divided into smaller groups of elements [1, p. 72]. We the number of groups and denote the subcarrier indexes in the th group. In order to exploit the channel frequency diversity, we assume that the subcarriers of a given group are uniformly distributed over the signal bandfor . We width, i.e., we let ) denote the number of simultaneously active users ( and assume that each of them transmits one data symbol in each group. This means that a total of symbols are transmitted by a given user in each MC-CDMA block. The users within a group are separated by their specific spreading codes. The latter are usually taken from an orthogonal Walsh–Hadamard (WH) set to mitigate the MAI. Alternatively, Golay or Zadoff–Chu codes can be used in uplink transmissions to reduce the peak-to-average power ratio (PAPR) [15]. Without loss of generality, we concentrate on the th MC-CDMA block and call the symbol transmitted by the th user within the th group of subcarriers. Fig. 1(a) shows the block diagram of the th uplink transmitter. After channel coding, outer interleaving and symbol mapping, the complex valued symbols are partitioned into adjacent segments of length . The th segment is denoted and contains the symbols to be transmitted over the th MC-CDMA block. For this purpose, each ( ) is chips using a unit-energy spreading sequence spread over , where . , which are then This operation produces frequency interleaved to produce the -dimensional vector with entries (1) and is the where represents the remainder of the ratio . Next, is fed to the pre-equalization integer part of unit which delivers (2) and is the prefiltering coefficient of the th user over the th subcarrier. The entries of are finally mapped on subcarriers using an OFDM modulator which comprises an -point inverse discrete Fourier transform (IDFT) unit and the insertion of a cyclic prefix (longer than the CIR duration). In conventional MC-CDMA systems (i.e., without is the identity matrix so that any prefiltering), . Vice versa, when only signal prefiltering is utilized (without any further processing at the receiver where
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SANGUINETTI et al.: CHANNEL ESTIMATION FOR MC-CDMA UPLINK TRANSMISSIONS WITH COMBINED EQUALIZATION
side) must be designed so as to mitigate MAI and channel distortions. For example, the transmit zero-forcing (ZF) scheme tries to precompensate at the transmitter all the interference seen at the receiver. This is achieved by setting , where is the channel frequency response of the th user over the th subcarrier of the th MC-CDMA block. In this way, the signal appears at the receiver as undistorted and channel estimation is not even necessary at the BS. However, ZF pre-equalization leads is much to excessive power consumption when less than unity. This is a manifestation of the power-boosting phenomenon [6], which occurs in the presence of deeply faded subcarriers. Scaling the prefiltering coefficients by a fixed normalization factor is a viable method to reduce the transmit power. In this way, however, the signal-to-noise ratio (SNR) at the receive side decreases and the system performance is correspondingly degraded. A promising alternative to the transmit ZF scheme has recently been proposed in [7] in the form of combined equalization. This technique applies pre-equalization at the transmitter together with post-equalization at the receiver. In particular, pre-equalization is employed to efficiently distribute the transmit power among the available subcarriers rather than as a method to cancel the MAI. This leads to the following generalized prefiltering (GPF) coefficients [7] (3)
where the integer is a design parameter that must be chosen so as to achieve a reasonable tradeoff between efficient allocation of the transmit power and MAI mitigation capabilities. It is worth noting that GPF comprises several well-known makes pre-equalization strategies. For example, setting proportional to the complex conjugate of the corresponding channel gain, leading to the transmit equivalent of results in maximum-ratio-combining (MRC). Letting equal-gain-combining (EGC) since in this case the prefiltering coefficients have unit amplitude and only compensate for the channel phase. Finally, channel inversion is obtained with , corresponding to the transmit ZF scheme. Also, from allows one to allocate (3), we see that using GPF with more power on good subcarriers (i.e., those with higher channel gains), thereby avoiding to invest system resources on deeply faded subcarriers. This, however, is expected to enhance the loss of orthogonality among the received pre-equalized spreading codes, leading to a corresponding increase of the MAI [7]. In these circumstances, some form of multiuser detection must be employed at the receiver to counteract the detrimental effect of MAI. As a final remark, we observe that applying GPF with has the additional advantage of reducing the PAPR with respect to conventional MC-CDMA schemes [16]. This can be explained bearing in mind that the transmitted signal is given sinusoidal waveforms (one for each by the superposition of it is likely that only a subset of them subcarrier) and for have significant power (i.e., those corresponding to the best subcarriers). In the above situation, there are fewer possibilities
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for constructive addition of the sinusoidal components and the PAPR is correspondingly smaller than in conventional MC-CDMA transmissions. In the limit case where all the power is allocated on the strongest subcarrier ( approaching infinity), the transmitted signal has a constant envelope and, accordingly, will not suffer from PAPR at all. Since combined schemes provides intrinsic protection against PAPR, orthogonal WH spreading codes can reasonably be used even in uplink transmissions. Returning to (3), we see that GPF requires channel knowledge at the transmit side. This can be achieved in TDD systems by exploiting the channel reciprocity between alternative downlink and uplink transmissions provided that the channel variations are sufficiently slow and the analog front ends at the transmit and receive sides are correctly calibrated. For simplicity, in the sequel, we assume that ideal CSI is available at each MT and the channel response of each user is practically constant over a time ). The impact of channel slot (i.e., we let variations and/or estimation errors on the system performance is addressed in Section V by means of numerical simulations. B. Channel Model The signals transmitted by each user propagate through different multipath channels and undergo frequency selective fading. A complex baseband discrete-time model is assumed throughout this paper. In particular, we utilize the simplified tapped delay line multipath channel model of [17] and denote the CIR of the th user. The channel length is related to the maximum excess delay by , where is the sampling period represents the largest integer less than or equal to . and is computed as the The channel frequency response and reads discrete Fourier transform (DFT) of (4)
C. MC-CDMA Receiver The block diagram of the BS receiver is sketched in Fig. 1(b). The incoming waveforms are implicitly recombined by the receive antenna and passed to an OFDM demodulator, which eliminates the cyclic prefix and performs an -point DFT of the received samples. We denote the DFT output corresponding to the th received block. Thus, assuming ideal frequency and timing synchronization [18], we may express as (5) where is the pre-equalized channel frequency response of the th user and . The th entry of is given by
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(6)
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Fig. 1.
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 6, JUNE 2006
(a) Block diagram of the k th uplink transmitter. (b) Block diagram of the BS receiver.
whereas the quantities
are defined in (1). Also, represents thermal noise and it is modeled as a white Gaussian . Subprocess with zero mean and covariance matrix stituting (3) into (6) indicates that the entries of are real-valued and nonnegative. As shown later, this property can be exploited to ease the channel estimation task. Clearly, in conventional MC-CDMA systems without signal prefiltering, in (5) is replaced by the complex-valued vector . is exploited to detect the data symbols of all Vector users within the th MC-CDMA block. For this purpose, we employ the iterative soft parallel interference-cancellation (PIC) receiver discussed in [19], where MAI is cancelled out on the basis of reliability information about the detected symbols. In particular, at each iteration the symbols of all interfering users are estimated and demapped. Then, they are fed to a soft-input–soft-output channel decoder which computes soft estimates of the information bits. Next, the re-encoded and reinterleaved soft bits are mapped onto soft symbols and spread with the user-specific spreading code. The resulting chips are then predistorted according to the same GPF strategy employed at the transmitter. In this way, the MAI can be reconstructed . After MAI cancellation, the desired and subtracted from data symbol is finally detected using MRC, which is the best detection strategy in the absence of MAI. As is intuitively clear, the performance of PIC-based receivers depends heavily on the quality of the channel estimates. The reason is that poor channel information leads to imperfect cancellation of the interference, thereby degrading the system performance. In the following, we discuss two methods for estimating the pre-equalized channel frequency response of each user at the BS. The first scheme, called unstructured channel as unknown estimator (UCE), considers the entries of independent parameters. The second operates in the time-domain and implicitly takes advantage of the particular structure that derives from the fading correlation between adof jacent subcarriers. For this reason, it is called the structured channel estimator (SCE). As we shall see, both schemes require knowledge of the transmitted data symbols. For this purpose, we assume that the MC-CDMA blocks are organized in frames and
each frame is preceded by . dexes
training blocks with temporal in-
III. UNSTRUCTURED CHANNEL ESTIMATION A. Derivation of UCE We begin by collecting the pre-equalized channel reinto a single vector sponses . Next, we rewrite (5) in the equivalent form (7) where
is the following matrix: (8)
is perfectly known at the receiver It is worth noting that since its entries only depend on the spreading codes and training sequences of the active users as indicated in (1). Without loss of generality, we assume that the training symbols have unit amplitude, i.e., we let for . , vectors From (7), we see that, for a given are statistically independent and Gaussian distributed with mean and covariance matrix . Thus, dropping additive and multiplicative terms independent of , the logtakes the form likelihood function for
(9) is a trial value of , and we have taken where has real-valued components. The ML into account that is the location where achieves estimate of its global maximum. Setting to zero the gradient of yields
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(10)
SANGUINETTI et al.: CHANNEL ESTIMATION FOR MC-CDMA UPLINK TRANSMISSIONS WITH COMBINED EQUALIZATION
where
is the real part of the following matrix:
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is the Kronecker delta function. In these cirwhere and (10) becomes cumstances, reduces to (11)
In Appendix I, it is shown that is unbiased and its meansquare estimation error (MSEE) is given by
(16) whereas the minimum MSEE is given by (17)
(12)
B. Remarks 1) In the absence of any prefiltering operation, we have , where is a complex-valued vector. In these circumstances, UCE reduces to the channel estimation scheme discussed in [9] (13)
As discussed in [9], employing WH sequences of length is a possible method to meet the constraint (15). The into (17) corresponding MSEE is found setting and reads (18) On the other hand, substituting (14) and letting produces
into
(19)
and the corresponding MSEE is given by (14) Note that the difference between (10) and (13) relies on the fact that has real-valued components, which reduces the number of unknown parameters by a factor two with respect to a conventional system with complexvalued channel responses. As shown later, this improves the quality of the channel estimates for a given training overhead. 2) Inspection of (8) reveals that . On the other hand, using the inequality 1 2 1 2 [20, p. 61], from (11) it follows that . , we Finally, bearing in mind that . At this stage, we recall see that is a matrix of order , and thus a necessary that condition for the existence of is . so that, in a fully This amounts to saying that ), the number of training blocks loaded system ( . Comparing this result with the cannot be less than constraint that was found in [9] in the context of conventional MC-CDMA transmissions, we see that GPF allows a potential reduction of the training overhead by a factor two. 3) Using similar arguments as in [21], it can be shown that in and order to achieve the minimum MSEE for a given dedicated for training, we subject to a fixed energy is a scaled version of the identity matrix. require that Collecting (1), (8), and (11), we see that this condition occurs with training sequences satisfying the identity (15)
which represents the performance of UCE in the absence of any prefiltering (with WH codes of length employed as training sequences). Comparing (18) and (19), we see that, for the same training overhead, GPF improves the accuracy of the channel estimates by a factor two with respect to a conventional system without signal prefiltering. 4) As discussed previously, in a fully loaded system with GPF the number of training blocks cannot be less than if UCE is employed for channel estimation. At this stage, an interesting question is whether a set of sequences of minimum length and satisfying (15) can be found or not. The answer is affirmative and a possible solution is given by (20) where of length
is the set of WH codes . The corresponding MSEE is found setting into (17) and reads (21)
Comparing the above results with (19), we see that GPF allows a reduction of the training overhead by a factor two with respect to a conventional MC-CDMA system (we to ) without degrading pass from the channel estimation accuracy. In summary, from the above discussion it follows that GPF provides some potential benefits on the performance of UCE. For a given overhead, it can be employed to improve the quality of the channel estimates with respect to conventional transmissions without any prefiltering. Alternatively, for a given MSEE, it can be used to reduce the number of training blocks.
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IV. STRUCTURED CHANNEL ESTIMATION
is a vector of dimension while the form
A. Problem Formulation of
We define the pre-equalized CIR of the th user as the IDFT , i.e., (22)
and observe that
that takes
(29) In the above equations, is an -dimensional column vector with unit elements and is the pre-equalized CIR matrix with entries vector. Also, is an
(23) (30) since is real-valued. In particular, from (3) and (6), is a scaled version of . Then, we see that assuming for convenience that the integer is nonnegative, it follows that is proportional to , where the convolution is applied times and is the autocorrelation of . Since has , is nonzero only for . support Accordingly, we have
-dimensional vector To proceed further, we define the . Thus, from (27), it follows that (31) where identical blocks with yields
is a block-diagonal matrix . Finally, substituting (31) into (7) (32)
(24) where (25) Although the above relation has been derived assuming that is nonnegative and even, simulations indicate that it can be rea. Note that for sonably used with any integer (transmit ZF) is independent of . This means that is proportional to the Kronecker delta function, so that for as predicted by (24) and (25). On the other hand, it is easily seen that the result (25) cannot be since in that case it would provide a negative used for , which is clearly meaningless. Fortunately, this is value of not a serious limitation since prefiltering schemes with are not of practical interest [7]. Returning to (22), we see that can be computed by . Thus, bearing in mind (23) and taking the DFT of (25), we may write
As mentioned previously, SCE operates in the time-domain and effectively exploits the structure of shown in (31). For this purpose, an estimate of the pre-equalized CIR vector is computed first and is next used to obtain the corresponding in the form estimate of (33) In this way, the number of unknowns is reduced since in practical applications is typically smaller than the total number of subcarriers. Note that a similar approach has been adopted in [22]–[24] in the context of multiple-input–multiple-output (MIMO) OFDM systems. B. Derivation of SCE The ML estimate of is derived from the linear model (32). Recalling that has real-valued components, we obtain (34) where
denotes the real part of the matrix (35)
(26) where parts of
and denote the real and imaginary , respectively. From (26), it follows that can be expressed as: (27)
where
Next, substituting (34) into (33) produces the ML estimate of in the form (36) The performance of SCE is assessed in Appendix II. It turns out that is unbiased and its MSEE is given by
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SANGUINETTI et al.: CHANNEL ESTIMATION FOR MC-CDMA UPLINK TRANSMISSIONS WITH COMBINED EQUALIZATION
C. Remarks 1) Since
, we have
. This fact, together with (35) and the property , indicates that . Thus, bearing in mind , it follows that that . We now recall that is a matrix , so that is of order a necessary condition for the existence of . This amounts to saying that or, equivalently, . Finally, in ), the constraint on a fully loaded system ( becomes (38) Comparing (38) with the result obtained with UCE, we see that SCE leads to a potential reduction . Note that only of the training overhead by a factor . one training block is required when 2) For given and , the minimum MSEE is achieved is a scaled version of the identity matrix [21]. when Recalling (8) and the structure of , from (35) we see can be partitioned into blocks ( that ), each of dimensions and expressed by
Inspection of (41) indicates that optimal training sequences must be designed at chip level rather than at symbol level. Fortunately, this fact is not restrictive since have the structure in (1) only over the the chips data blocks, while during the training period they can be selected independently for each and . 3) Optimal training sequences satisfying the set of constraints (41) can be designed using the same arguments . In these cirof [22] and [23] provided that cumstances, we see from (38) that the number of training and conditions (41) are blocks can be limited to met by
The MSEE is computed setting is expressed by
Comparing (45) with (18), we see that SCE achieves the same accuracy of UCE but reduces the training overhead from to 1. An interesting interpretation of the sequences in (44) is possible if we consider the corresponding time-domain samples, which are computed (multiplied by ) as the -point IDFT of (46) Substituting (44) into (46) and setting for otherwise
(40) or, equivalently, when the conditions shown in (41) at the bottom of the page are satisfied. In these circumstances, reduces to and (36) becomes
(42) The minimum MSEE is computed from (37) setting and reads (43)
(44) into (43) and it
(45)
(39)
Hence, the MSEE is minimum when
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, yields (47)
The above equation indicates that each user transmits a single time-domain sample during the training block. Also, samples of different users are spaced apart by . Bearing in mind that the multiples of (discrete-time) response of the pre-equalized channel to is , from (24), we see that signals transmitted by different users are received at the BS over adjacent and nonoverlap, with ping time intervals . This means that there is no interference among the active users as each received sample at the input of the DFT unit is contributed by one user only. On the other hand, transmitting a single time-domain sample
for
and
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of power as indicated in (47) is disadvantageous since it may produce an excessive PAPR at the mobile unit. 4) The set of constraints (41) is also met by the sequences defined in (20). The corresponding MSEE is cominto (43) and reads puted substituting (48) Alternatively, we may resort to WH sequences of length . In these circumstances, the MSEE is given by (49) Comparing (48) and (49) with (21) and (18), we see that . This means that, the MSEE is reduced by a factor for a given overhead, the structured approach is more accurate than UCE. 5) A possible shortcoming of SCE is that all groups of subcarriers must be assigned to the same set of users, so that the maximum number of contemporarily active users is . On the other hand, from (16), limited to it can be seen that UCE operates independently on each group and, therefore, it is suited for more versatile systems in which different groups of subcarriers may be assigned to different groups of users so as to guarantee multimedia services with variable data rates. In these circumstances, the number of contemporarily active users can be as large as the total number of subcarriers. V. PERFORMANCE EVALUATION Computer simulations have been run to assess the performance of an MC-CDMA receiver employing the proposed channel estimation schemes in conjunction with combined equalization. The system parameters are inspired by the HiperLAN/II specifications and reflect a typical indoor and low-mobility environment for wireless local area network (WLAN) applications. A. System Parameters The total number of subcarriers is and WH codes of length are used for spreading purposes. The signal MHz, so that the useful part of each bandwidth is . The MC-CDMA block has duration and a cyclic sampling period is prefix of is adopted to eliminate inter-block interference. This corresponds to an extended block (including the cyclic prefix) of 4.0 . Each frame has a duration of 256 and comprises 64 blocks. All the CIR’s have the same length , corresponding to a maximum excess delay of . Unless otherwise specified, each is kept fixed over the uplink time slot (static channel) but varies from slot to slot. The channel taps are modeled as complex-valued independent Gaussian random variables with zero-mean (Rayleigh fading) and powers (50)
Fig. 2. BER performance versus p for E =N = 6 dB and K = 4 or 8.
where is chosen such that the average energy of the channel . A rate 1/2 conis normalized to unity, i.e., volutional code is adopted with constraint length 6 and generator polynomials 131, 177 (in octals). After outer interleaving, the coded bits are mapped onto quadrature phase-shift keying (QPSK) symbols. Unless otherwise specified, ideal CSI is assumed at each MT, where the uplink signals are prefiltered according to the GPF scheme. At the BS the pre-equalized channel responses are estimated by means of UCE or SCE. To this purpose, we either exor the sequences ploit WH training sequences of length of length defined in (20). The channel estimates are kept fixed over the uplink time slot and are passed to a soft PIC-based receiver [19] which delivers the final data decisions. B. Performance Assessment Our first objective is the design of the parameter that determines the prefiltering coefficients in (3). As a design criterion, we choose the minimization of the average bit-error rate (BER) of the active users. Unfortunately, in doing so, we must resort to computer simulations since the average BER of the employed PIC-based receiver is hard to get theoretically. Fig. 2 , where shows the BER as a function of for is the transmitted energy per information bit, while is the one-sided noise power spectral density. The system is ei) or half-loaded ( ) and perfect ther fully loaded ( channel knowledge (PCK) is assumed at the receiver. As expected, the optimal depends on the number of users, but we represents a good choice with both and see that . This value is adopted in all subsequent simulations. We now address the problem of determining the (one-sided) of the pre-equalized CIR. Fig. 3 illustrates the length power delay profile of , from which it turns out that . This result validates the relation (25) for . However, since the MSEE in (48) and (49) is proportional to and becomes vanishing small for , in
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SANGUINETTI et al.: CHANNEL ESTIMATION FOR MC-CDMA UPLINK TRANSMISSIONS WITH COMBINED EQUALIZATION
Fig. 3. Power delay profile of h
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the following the SCE is designed for , which is tantamount to saying that matrix in (34) has columns. It is worth noting that is merely a design parameter that should not be confused with the true CIR . Since we choose in our simulalength tions, the SCE operates in a mismatched mode, but we expect that the effect of the mismatching is only visible at very high SNRs. Note that the idea of forcing to zero the weakest channel taps to reduce the number of unknown parameters is not novel since it has been previously considered in [25]. Recalling that , and , from (38), we see that channel estimation by means of SCE requires at least . Accordingly, the sequences of length defined in (44) cannot be used in our simulation setup. Fig. 4 shows the accuracy of UCE and SCE versus with . Marks indicate simulations, while solid lines represent theoretical analysis as given by (18), (21), (48), and (49). As expected, the best results are provided by SCE. Note that perfect agreement is observed between simulation and theory when the channel estimates are obtained with UCE. Vice versa, theoretical results for SCE are validated only for dB. As mentioned previously, the reason is that . This SCE operates in a mismatched mode with gives rise to an irreducible floor in the MSEE curves that limits the system performance at high SNRs. The above arguments are supported by other simulations (not shown for space limitations), which indicate that the floor disappears when SCE is . Unfortunately, in the latter case, designed with the estimation accuracy degrades at low/intermediate SNRs as predicted by (48) and (49). In summary, setting represents a good tradeoff that allows the system to achieve satisfactory performance over a wide range of practical SNRs. of a receiver enFig. 5 illustrates the BER versus dowed with the proposed channel estimation schemes. The ) and the number of training system is fully loaded ( or . The curves labeled GPF+UCE and blocks is either GPF+SCE are obtained employing the GPF coefficients in (3),
Fig. 4. Accuracy of UCE and SCE versus 1= with either N
= 4 or 8.
Fig. 5. BER performance over a static channel with K = 8 and N = 4 or 8.
whereas UCE refers to the uplink channel estimator discussed in [9] without any signal prefiltering (only post-equalization is present). In the latter case, we only provide simulations for since in the absence of signal prefiltering cannot of active users. Results obtained be less than the number with perfect knowledge of the pre-equalized channel responses (PCK) are also shown as a benchmark. We see that GPF significantly improves the system performance. Compared with a conventional system, a gain of nearly 3.5 dB is observed at an error rate of 10 when GPF is employed in conjunction with . The best UCE and the number of training blocks is results are obtained with GPF+SCE. For , the loss of GPF+SCE with respect to PCK is approximately 1 dB, while it becomes 2.5 dB with GPF+UCE. As expected, the system becomes smaller. In particular, performance deteriorates as
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Fig. 6. BER performance versus K for E =N = 6 dB and N = 4 or 8.
Fig. 7. Effect of imperfect CSI at all MTs over the BER performance for K = 8 and N = 8.
a loss of nearly 1 dB is observed with both GPF+UCE and is reduced from 8 to 4. GPF+SCE when Fig. 6 compares the performance of the investigated schemes as a function of . The number of training blocks is either or , while is fixed to 6 dB. Again, results . We see obtained without GPF are only provided for that the BER degrades as the number of users grows large and, in any case, GPF+SCE has the best performance. Fig. 7 illustrates the BER of GPF+UCE and GPF+SCE versus when signal prefiltering is accomplished at each MT using imperfect CSI (ICSI). This means that in computing the GPF coefficients the true channel gains in (3) are replaced by some suitable estimates. The latter are derived at each MT using the downlink channel estimator discussed in [9], which exploits a single training block placed at the end of the downlink time
Fig. 8. BER performance versus the mobile speed v for E =N = 6 dB and N = 8.
slot. The system is fully loaded and the length of the uplink . The curves labeled PCSI corretraining sequences is spond to perfect CSI at all MT’s and are shown for comparison. We see that ICSI entails a loss of approximately 1 dB on the performance of both schemes. Similar results (not shown for space . limitations) are obtained with The impact of channel variations on the system performance is addressed in Fig. 8, where the BER is shown as a function dB. Channel estimates of the mobile speed for are computed at the beginning of each uplink slot and are kept fixed over the entire slot, while the CIRs vary continuously from block to block due to Doppler effects. In particular, the channel taps are generated by filtering statistically independent white Gaussian processes in a third-order low-pass Butterworth filter. The 3-dB bandwidth of the filter is taken as a measure of the , where GHz is the employed Doppler rate 10 ms is the speed-of-light. carrier frequency, while and the number The training sequences have length or . We see that the BER deof active users is either grades with , but the performance loss is negligible for mobile speeds up to 10 m/s. This means that the considered schemes are well suited for indoor applications or outdoor systems with low/medium mobility. VI. CONCLUSION We have discussed two channel estimation schemes for TDD MC-CDMA uplink transmissions employing combined equalization. They are based on the ML criterion and exploit a sequence of training blocks placed at the beginning of each uplink time slot. The first scheme, UCE, ignores any correlation among adjacent subcarriers, while the second, SCE, uses a structured approach that improves the quality of the channel estimates. Both methods exploit the fact that the pre-equalized channel frequency response of each user has real-valued components. The performance of a soft PIC-based receiver endowed with the proposed channel estimators has been assessed by
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SANGUINETTI et al.: CHANNEL ESTIMATION FOR MC-CDMA UPLINK TRANSMISSIONS WITH COMBINED EQUALIZATION
simulation. It is shown that signal pre-equalization based on the GPF criterion can be used to improve the quality of the channel estimates or, alternatively, to reduce the training overhead with respect to conventional transmissions without any signal prefiltering. Optimal training sequences minimizing the mean square channel estimation error are designed under different operating conditions.
mean. On the other hand, collecting (56) and (58) we see that is given by the covariance matrix of
(59) or, bearing in mind (35)
APPENDIX I
(60)
In this appendix, we compute the covariance matrix of . Substituting (7) into (10) and bearing in mind (11) produces (51) where
At this stage, we recall that . Thus, from (60), it follows that (61)
denotes the real part of (52)
Thus, collecting (51), (52), and recalling that mean, it follows that is unbiased. From (51), we see that the covariance matrix of the form
, we may express
Finally, using the identity and side of (61) with
in the right-hand produces
has zero (62) takes (53)
where
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. Using the identity as (54)
or, alternatively (55) At this stage, we recall that vectors are statistically independent for different values of , with and . Thus, substituting (52) into (55) yields (56) from which it follows that (57) where we have born in mind (11) and (53). Finally, recalling that , we obtain the result (12). APPENDIX II In this appendix, we compute the expectation and the covariance matrix of . Substituting (32) into (36) and bearing in mind (31) and (35) produces (58) denotes the real part of the vector defined in (52). where From (58), it follows that is unbiased since has zero
which is equivalent to the result (37) since
.
REFERENCES [1] K. Fazel and S. Kaiser, Multi-Carrier and Spread Spectrum Systems. New York: Wiley, 2003. [2] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [3] D. Mottier and D. Castelain, “SINR-based channel pre-equalization for uplink multi-carrier CDMA systems,” in Proc. IEEE Int. Symp. Pers., Indoor, Mobile Radio Commun., Sep. 2002, pp. 1488–1492. [4] I. Cosovic, M. Schnell, and A. Springer, “On the performance of different channel pre-compensation techniques for uplink time division duplex MC-CDMA,” in Proc. IEEE Veh. Technol. Conf. (VTC-Fall), vol. 2, Oct. 2003, pp. 857–861. [5] P. Bisaglia, N. Benvenuto, and S. Quitadamo, “Performance comparison of single-user pre-equalization techniques for uplink MC-CDMA systems,” in Proc. IEEE Global Telecommun. Conf., vol. 6, Dec. 2003, pp. 3402–3406. [6] M. Brandt-Pearce and A. Dharap, “Transmitter-based multiuser interference rejection for the down-link of a wireless CDMA system in a multipath environment,” IEEE Trans. Signal Processing, vol. 18, pp. 607–617, Mar. 2000. [7] I. Cosovic, M. Schnell, and A. Springer, “Combined-equalization for uplink MC-CDMA in Rayleigh fading channels,” IEEE Trans. Commun., vol. 53, pp. 1609–1614, Oct. 2005. [8] W. Keusgen, C. Walke, and B. Rembold, “A system model considering the influence of front-end imperfections on the reciprocity of up- and downlink system impulse responses,” in Proc. Aachen Symp. Signal Theory, Aachen, Germany, Sep. 2001, pp. 243–248. [9] L. Sanguinetti, M. Morelli, and U. Mengali, “Channel estimation and tracking for MC-CDMA signals,” Eur. Trans. Telecommun., vol. 15, pp. 249–258, Mar. 2004. [10] U. Tureli, D. Kivanc, and H. Liu, “Channel estimation for multi-carrier CDMA,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, vol. 5, Instanbul, Turkey, Jun. 5–9, 2000, pp. 2909–2912. [11] K. Deng, Q. Yin, M. Luo, and Y. Zeng, “Blind uplink channel and DOA estimator for space-time block coded MC-CDMA systems with uniform linear array,” in Proc. IEEE Veh. Technol. Conf., vol. 1, May 17–19, 2004, pp. 69–73. [12] K. Zheng, G. Zeng, and W. Wang, “DFT-based uplink channel estimation for uplink MC-CDMA systems,” in Proc. IEEE Int. Symp. Spread Spectrum Tech. Appl., Sydney, Australia, Aug.-Sep. 2004, pp. 570–574. [13] P. Marques, A. Gameiro, and J. Fernandes, “Vectorial channel estimation for uplink MC-CDMA in beyond 3G wireless systems,” in Proc. IEEE Int. Symp. Pers., Indoor, Mobile Radio Commun., vol. 1, Sep. 2003, pp. 984–988.
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[14] Z. Li and M. Latva-Aho, “Analysis of MRC receivers for asynchronous MC-CDMA with channel estimation errors,” in Proc. IEEE Int. Symp. Spread Spectrum Tech. Appl., Sep. 2002, pp. 343–347. [15] B. V. Popovic, “Spreading sequences for multicarrier CDMA systems,” IEEE Trans. Commun., vol. 47, pp. 918–926, Jun. 1999. [16] I. Cosovic and L. Sanguinetti, “On the peak-to-average power ratio of pre-equalized multi-carrier code-division multiple-access transmissions,” in Proc. IEEE Veh. Technol. Conf.-Spring, Stockholm, Sweden, May-Jun. 28–1, 2005. [17] J. Proakis, Digital Communications. New York: McGraw-Hill, 1995. [18] A. C. McCormick, P. M. Grant, and J. S. Thompson, “A hybrid uplink multi-carrier CDMA interference cancellation receiver,” Proc. Inst. Elect. Eng. Commun., vol. 148, no. 2, pp. 119–124, Dec. 2001. [19] S. Kaiser and J. Hagenauer, “Multi-carrier CDMA with iterative decoding and soft-interference cancellation,” in Proc. IEEE Global Telecommun. Conf., Nov. 1997, pp. 6–10. [20] H. Lutkepohl, Handbook of Matrices. New York: Wiley, 1996. [21] T. L. Tung, K. Yao, and R. E. Hudson, “Channel estimation and adaptive power allocation for perfomance and capacity improvement of multiple-antenna OFDM systems,” in Proc. IEEE 3rd Workshop Signal Processing Advances Wireless Commun., vol. 5, Mar. 2001, pp. 82–85. [22] Y. G. Li, “Simplified channel estimation for OFDM systems with multiple transmit antennas,” IEEE Trans. Wireless Commun., vol. 1, pp. 67–75, Jan. 2002. [23] I. Barhumi, G. Leus, and M. Moonen, “Optimal training design for MIMO OFDM systems in mobile wireless channels,” IEEE Trans. Signal Processing, vol. 51, pp. 1615–1624, Jun. 2003. [24] F. Horlin and L. V. d. Perre, “Optimal training sequences for low-complexity ML multi-channel estimation in multi-user MIMO OFDM-based communications,” in Proc. IEEE Global Telecommun. Conf., vol. 4, Dallas, TX, Jun. 2004, pp. 2427–2431. [25] Y. Li, N. Seshadri, and S. Ariyavisitakul, “Channel estimation for OFDM systems with transmitter diversity in mobile wireless channels,” IEEE J. Sel. Areas Commun., vol. 17, pp. 461–471, Mar. 1999.
Luca Sanguinetti (S’04) received the Laurea degree (cum laude) in information engineering from the University of Pisa, Pisa, Italy, in 2002. He is currently working towards the Ph.D. degree in information engineering at the Department of Information Engineering, University of Pisa. In 2004, he was a visiting Ph.D. student at the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. His research interests span the areas of communications and signal processing, estimation, and detection theory. Current research topics focus on transmitter and receiver diversity techniques for single- and multiuser fading communication channels, antenna array processing, channel estimation and equalization, MIMO systems, multicarrier systems, linear and nonlinear prefiltering for interference mitigation in multiuser environment.
Ivan Cosovic (S’03–M’06) was born in Foca, Yugoslavia, on July 21, 1977. He received the Dipl.-Ing. degree in electrical engineering from the University of Belgrade, Serbia and Montenegro, in 2001, and the Ph.D. degree (Highest Hon.) in electrical engineering from the Johannes Kepler University Linz, Austria, in 2005. From February 2002 to April 2006, he was with the Institute of Communications and Navigation, German Aerospace Center (DLR), Wessling, Germany. He joined DoCoMo Communications Laboratories Europe GmbH, Munich, Germany, in April 2006, as a Researcher. His research interests include post-equalization and pre-equalization, transmitter and receiver diversity techniques, channel estimation, and spectrum sharing techniques for multicarrier, CDMA, and hybrid multicarrier CDMA wireless communications systems. Dr. Cosovic was awarded the Best Student Paper Award for the paper “Physical Layer Design for a Broadband Overlay System in the VHF Band” at the Digital Avionics Systems Conference (DASC’05), Washington, DC, in 2005.
Michele Morelli (M’04) received the Laurea (cum laude) degree in electrical engineering and the Premio di Laurea SIP from the University of Pisa, Pisa, Italy, in 1991 and 1992, respectively, and the Ph.D. degree in electrical engineering from the Department of Information Engineering, University of Pisa, in 1995. In September 1996, he joined the Centro Studi Metodi e Dispositivi per Radiotrasmissioni (CSMDR) of the Italian National Research Council (CNR), Pisa, where he held the position of Research Assistant. Since 2001, he has been with the Department of Information Engineering, University of Pisa, where he is currently an Associate Professor of Telecommunications. His research interests are in wireless communication theory, with emphasis on equalization, synchronization and channel estimation in multiple-access communication systems.
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Channel Estimation for MC-CDMA Uplink Transmissions With Combined Equalization Luca Sanguinetti, Student Member, IEEE, Ivan Cosovic, Member, IEEE, and Michele Morelli, Member, IEEE
Abstract—Combined equalization has recently been proposed to enhance the error rate performance of conventional multicarrier code-division multiple-access (MC-CDMA) systems. This technique applies pre-equalization at the transmitter in conjunction with post-equalization at the receiver, thereby splitting the overall equalization process into two separate parts. In this way, efficient power allocation over the available subcarriers is possible at the transmitter, while leaving the interference cancellation task at the receiver. In this paper, we consider the uplink of an MC-CDMA system employing combined equalization. As the users transmit from different locations, the uplink signals arrive at the base station after passing through different multipath channels and the goal is to estimate the pre-equalized channel frequency response of each user. This is pursued following two different approaches. The first operates in the frequency-domain and treats the channel gains over adjacent subcarriers as independent unknown parameters. The second operates in the time-domain and achieves better performance by reducing the number of unknown parameters. Both schemes are based on maximum-likelihood reasoning and require knowledge of the transmitted symbols. Numerical examples are given to highlight the effectiveness of the proposed methods. Index Terms—Channel estimation, combined equalization, multicarrier code-division multiple-access (MC-CDMA), prefiltering techniques.
I. INTRODUCTION
T
HE DEMAND for high data rates in wireless transmissions has increased significantly over the last years. This has led to a strong interest in multicarrier code-division multiple-access (MC-CDMA), which is an effective technique for combating multipath fading over highly dispersive wireless channels [1]. In MC-CDMA systems, the data of different users are spread in the frequency-domain using orthogonal signature sequences. The main impairment of this multiplexing technique is represented by the multiple-access interference (MAI), which occurs in the presence of multipath propagation due to loss of orthogonality among the received spreading codes. In conventional MC-CDMA systems, MAI mitigation is accomplished at
Manuscript received April 1, 2005; revised October 1, 2005. This work was supported in part by the Italian Ministry of Education under the Fondo per gli Investimenti della Ricerca di Base (FIRB) Project Piattaforme Riconfigurabili per Interoperabilità in Mobilità (PRIMO). The work of I. Cosovic was done in part while with the German Aerospace Center (DLR). L. Sanguinetti and M. Morelli are with the Department of Information Engineering, University of Pisa, 56126 Pisa, Italy (e-mail: luca.sanguinetti@ iet.unipi.it; [email protected]). I. Cosovic is with DoCoMo Communications Laboratories Europe GmbH, 80687 Munich, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/JSAC.2005.864023
the receiver using well-known single-user or multiuser detection schemes [2]. Alternatively, prefiltering techniques based on the zero-forcing (ZF) or minimum-mean-square-error (MMSE) criterion can be employed to mitigate MAI and channel distortions in uplink transmissions [3]–[5]. The idea behind prefiltering is to vary the complex gain assigned to each subcarrier so that interference is reduced and the signal at the receiver appears undistorted. In this way, channel estimation is not even necessary at the base station (BS) and low-complex detection schemes can be employed. A possible drawback of prefiltering schemes is represented by the power-boosting effect [6], which occurs in the presence of deeply faded subcarriers. This phenomenon leads to high power consumption at the transmitter side and represents a serious problem in uplink transmissions due to the limited amount of power available at the mobile terminal (MT). A different approach has recently been proposed in [7] in the form of combined equalization. This technique applies pre-equalization at the transmitter in conjunction with post-equalization at the receiver. This ensures an additional degree of freedom in the system design, since equalization at the transmitter and receiver sides can be adjusted separately. In particular, in combined schemes the prefiltering coefficients are designed so as to efficiently distribute the transmit power among the available subcarriers, whereas MAI cancellation is the main task of the receiver (post-equalization). As shown in [7], combined equalization significantly outperforms conventional prefiltering techniques, where interference cancellation is accomplished at the transmitter at the expense of significant power boosting. In order to work properly, however, combined schemes require channel knowledge at both the transmitter and receiver ends. This problem is not addressed in [7], where ideal channel state information (CSI) is assumed for pre- and post-equalization purposes. In this paper, we discuss channel estimation in the uplink of an MC-CDMA system employing combined equalization. In doing so, we assume that CSI is available at each MT. This assumption is reasonable in time-division duplex (TDD) systems (such as IEEE802.11a and HiperLAN/II) due to the channel reciprocity between alternative uplink and downlink transmissions. If the analog transmit/receive front ends are correctly calibrated [8] and channel variations are sufficiently slow (as occurs in indoor applications), the channel estimate derived at a given MT during a downlink time slot can be reused for prefiltering in the subsequent uplink time slot. The uplink signals arrive at the BS after passing through different multipath channels and the goal is to estimate the pre-equalized channel frequency responses of the active users, which are then exploited to perform post-equalization and data detection.
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So far, the problem of channel estimation in MC-CDMA uplink transmissions has received little attention and only few results are available in the technical literature [9]–[14]. The main difficulty is that the channel responses of the active users are different from one another and the BS must estimate a large number of parameters. This is expected to degrade the quality of the estimates as compared with the downlink situation where only a single channel response is present. Although designed for conventional MC-CDMA systems without any signal prefiltering, the channel estimators in [9]–[14] can also be used in conjunction with combined equalization. As shown later, however, in combined schemes the prefiltered channel responses are real-valued since compensation for the channel phase is accomplished at the transmit side. Intuitively, we expect that exploiting the above property may improve the system performance as compared with the conventional schemes in [9]–[14]. This possibility is investigated here and two different solutions are provided. The first operates in the frequency-domain and is referred to as unstructured channel estimation (UCE). The second is called structured channel estimation (SCE) and operates in the time-domain. This means that an estimate of the pre-equalized channel impulse response (CIR) is computed first, and then it is exploited to estimate the corresponding channel frequency response. In this way, the number of unknown parameters is reduced with respect to UCE, leading to a better estimation accuracy. Both schemes are based on the maximum-likelihood (ML) criterion and require knowledge of the transmitted data symbols. For this purpose, we assume that the transmission is organized in frames and some training blocks (carrying known symbols) are placed at the beginning of each frame. The contribution of this paper can be summarized as follows. First, we derive channel estimation schemes specifically designed for MC-CDMA systems employing combined equalization. This is achieved by explicitly taking into account the fact that the pre-equalized channel responses are real-valued. Second, we show how the above property can be exploited to improve the quality of the uplink channel estimates or, alternatively, to reduce the training overhead with respect to conventional channel estimation schemes. Third, we discuss the minimum number of training blocks required to perform channel estimation and provide optimal training sequences minimizing the mean-square error of the channel estimates under various operating conditions. The rest of this paper is organized as follows. Section II outlines the uplink of an MC-CDMA system with combined equalization. Channel estimation by means of UCE and SCE is addressed in Sections III and IV, respectively. Simulation results are provided in Section V, while some conclusions are drawn in Section VI. A. Notation Vectors and matrices are denoted by boldface letters, with and representing the identity and null matrices, reto indispectively. We use cate an diagonal matrix with entries . We denote and the inverse and trace of a square matrix . The convolution between two sequences is denoted by , while represents the Euclidean norm of the enclosed vector. We
for expectation and the superscript , and for comuse plex conjugation, transposition and Hermitian transposition. Fiand indicate the real and imaginary parts nally, of a complex-valued quantity, while is used for the corresponding magnitude. II. SYSTEM MODEL A. MC-CDMA Transmitter We consider the uplink of an MC-CDMA system with available subcarriers. The latter have indexes and are divided into smaller groups of elements [1, p. 72]. We the number of groups and denote the subcarrier indexes in the th group. In order to exploit the channel frequency diversity, we assume that the subcarriers of a given group are uniformly distributed over the signal bandfor . We width, i.e., we let ) denote the number of simultaneously active users ( and assume that each of them transmits one data symbol in each group. This means that a total of symbols are transmitted by a given user in each MC-CDMA block. The users within a group are separated by their specific spreading codes. The latter are usually taken from an orthogonal Walsh–Hadamard (WH) set to mitigate the MAI. Alternatively, Golay or Zadoff–Chu codes can be used in uplink transmissions to reduce the peak-to-average power ratio (PAPR) [15]. Without loss of generality, we concentrate on the th MC-CDMA block and call the symbol transmitted by the th user within the th group of subcarriers. Fig. 1(a) shows the block diagram of the th uplink transmitter. After channel coding, outer interleaving and symbol mapping, the complex valued symbols are partitioned into adjacent segments of length . The th segment is denoted and contains the symbols to be transmitted over the th MC-CDMA block. For this purpose, each ( ) is chips using a unit-energy spreading sequence spread over , where . , which are then This operation produces frequency interleaved to produce the -dimensional vector with entries (1) and is the where represents the remainder of the ratio . Next, is fed to the pre-equalization integer part of unit which delivers (2) and is the prefiltering coefficient of the th user over the th subcarrier. The entries of are finally mapped on subcarriers using an OFDM modulator which comprises an -point inverse discrete Fourier transform (IDFT) unit and the insertion of a cyclic prefix (longer than the CIR duration). In conventional MC-CDMA systems (i.e., without is the identity matrix so that any prefiltering), . Vice versa, when only signal prefiltering is utilized (without any further processing at the receiver where
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SANGUINETTI et al.: CHANNEL ESTIMATION FOR MC-CDMA UPLINK TRANSMISSIONS WITH COMBINED EQUALIZATION
side) must be designed so as to mitigate MAI and channel distortions. For example, the transmit zero-forcing (ZF) scheme tries to precompensate at the transmitter all the interference seen at the receiver. This is achieved by setting , where is the channel frequency response of the th user over the th subcarrier of the th MC-CDMA block. In this way, the signal appears at the receiver as undistorted and channel estimation is not even necessary at the BS. However, ZF pre-equalization leads is much to excessive power consumption when less than unity. This is a manifestation of the power-boosting phenomenon [6], which occurs in the presence of deeply faded subcarriers. Scaling the prefiltering coefficients by a fixed normalization factor is a viable method to reduce the transmit power. In this way, however, the signal-to-noise ratio (SNR) at the receive side decreases and the system performance is correspondingly degraded. A promising alternative to the transmit ZF scheme has recently been proposed in [7] in the form of combined equalization. This technique applies pre-equalization at the transmitter together with post-equalization at the receiver. In particular, pre-equalization is employed to efficiently distribute the transmit power among the available subcarriers rather than as a method to cancel the MAI. This leads to the following generalized prefiltering (GPF) coefficients [7] (3)
where the integer is a design parameter that must be chosen so as to achieve a reasonable tradeoff between efficient allocation of the transmit power and MAI mitigation capabilities. It is worth noting that GPF comprises several well-known makes pre-equalization strategies. For example, setting proportional to the complex conjugate of the corresponding channel gain, leading to the transmit equivalent of results in maximum-ratio-combining (MRC). Letting equal-gain-combining (EGC) since in this case the prefiltering coefficients have unit amplitude and only compensate for the channel phase. Finally, channel inversion is obtained with , corresponding to the transmit ZF scheme. Also, from allows one to allocate (3), we see that using GPF with more power on good subcarriers (i.e., those with higher channel gains), thereby avoiding to invest system resources on deeply faded subcarriers. This, however, is expected to enhance the loss of orthogonality among the received pre-equalized spreading codes, leading to a corresponding increase of the MAI [7]. In these circumstances, some form of multiuser detection must be employed at the receiver to counteract the detrimental effect of MAI. As a final remark, we observe that applying GPF with has the additional advantage of reducing the PAPR with respect to conventional MC-CDMA schemes [16]. This can be explained bearing in mind that the transmitted signal is given sinusoidal waveforms (one for each by the superposition of it is likely that only a subset of them subcarrier) and for have significant power (i.e., those corresponding to the best subcarriers). In the above situation, there are fewer possibilities
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for constructive addition of the sinusoidal components and the PAPR is correspondingly smaller than in conventional MC-CDMA transmissions. In the limit case where all the power is allocated on the strongest subcarrier ( approaching infinity), the transmitted signal has a constant envelope and, accordingly, will not suffer from PAPR at all. Since combined schemes provides intrinsic protection against PAPR, orthogonal WH spreading codes can reasonably be used even in uplink transmissions. Returning to (3), we see that GPF requires channel knowledge at the transmit side. This can be achieved in TDD systems by exploiting the channel reciprocity between alternative downlink and uplink transmissions provided that the channel variations are sufficiently slow and the analog front ends at the transmit and receive sides are correctly calibrated. For simplicity, in the sequel, we assume that ideal CSI is available at each MT and the channel response of each user is practically constant over a time ). The impact of channel slot (i.e., we let variations and/or estimation errors on the system performance is addressed in Section V by means of numerical simulations. B. Channel Model The signals transmitted by each user propagate through different multipath channels and undergo frequency selective fading. A complex baseband discrete-time model is assumed throughout this paper. In particular, we utilize the simplified tapped delay line multipath channel model of [17] and denote the CIR of the th user. The channel length is related to the maximum excess delay by , where is the sampling period represents the largest integer less than or equal to . and is computed as the The channel frequency response and reads discrete Fourier transform (DFT) of (4)
C. MC-CDMA Receiver The block diagram of the BS receiver is sketched in Fig. 1(b). The incoming waveforms are implicitly recombined by the receive antenna and passed to an OFDM demodulator, which eliminates the cyclic prefix and performs an -point DFT of the received samples. We denote the DFT output corresponding to the th received block. Thus, assuming ideal frequency and timing synchronization [18], we may express as (5) where is the pre-equalized channel frequency response of the th user and . The th entry of is given by
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(6)
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 6, JUNE 2006
(a) Block diagram of the k th uplink transmitter. (b) Block diagram of the BS receiver.
whereas the quantities
are defined in (1). Also, represents thermal noise and it is modeled as a white Gaussian . Subprocess with zero mean and covariance matrix stituting (3) into (6) indicates that the entries of are real-valued and nonnegative. As shown later, this property can be exploited to ease the channel estimation task. Clearly, in conventional MC-CDMA systems without signal prefiltering, in (5) is replaced by the complex-valued vector . is exploited to detect the data symbols of all Vector users within the th MC-CDMA block. For this purpose, we employ the iterative soft parallel interference-cancellation (PIC) receiver discussed in [19], where MAI is cancelled out on the basis of reliability information about the detected symbols. In particular, at each iteration the symbols of all interfering users are estimated and demapped. Then, they are fed to a soft-input–soft-output channel decoder which computes soft estimates of the information bits. Next, the re-encoded and reinterleaved soft bits are mapped onto soft symbols and spread with the user-specific spreading code. The resulting chips are then predistorted according to the same GPF strategy employed at the transmitter. In this way, the MAI can be reconstructed . After MAI cancellation, the desired and subtracted from data symbol is finally detected using MRC, which is the best detection strategy in the absence of MAI. As is intuitively clear, the performance of PIC-based receivers depends heavily on the quality of the channel estimates. The reason is that poor channel information leads to imperfect cancellation of the interference, thereby degrading the system performance. In the following, we discuss two methods for estimating the pre-equalized channel frequency response of each user at the BS. The first scheme, called unstructured channel as unknown estimator (UCE), considers the entries of independent parameters. The second operates in the time-domain and implicitly takes advantage of the particular structure that derives from the fading correlation between adof jacent subcarriers. For this reason, it is called the structured channel estimator (SCE). As we shall see, both schemes require knowledge of the transmitted data symbols. For this purpose, we assume that the MC-CDMA blocks are organized in frames and
each frame is preceded by . dexes
training blocks with temporal in-
III. UNSTRUCTURED CHANNEL ESTIMATION A. Derivation of UCE We begin by collecting the pre-equalized channel reinto a single vector sponses . Next, we rewrite (5) in the equivalent form (7) where
is the following matrix: (8)
is perfectly known at the receiver It is worth noting that since its entries only depend on the spreading codes and training sequences of the active users as indicated in (1). Without loss of generality, we assume that the training symbols have unit amplitude, i.e., we let for . , vectors From (7), we see that, for a given are statistically independent and Gaussian distributed with mean and covariance matrix . Thus, dropping additive and multiplicative terms independent of , the logtakes the form likelihood function for
(9) is a trial value of , and we have taken where has real-valued components. The ML into account that is the location where achieves estimate of its global maximum. Setting to zero the gradient of yields
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where
is the real part of the following matrix:
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is the Kronecker delta function. In these cirwhere and (10) becomes cumstances, reduces to (11)
In Appendix I, it is shown that is unbiased and its meansquare estimation error (MSEE) is given by
(16) whereas the minimum MSEE is given by (17)
(12)
B. Remarks 1) In the absence of any prefiltering operation, we have , where is a complex-valued vector. In these circumstances, UCE reduces to the channel estimation scheme discussed in [9] (13)
As discussed in [9], employing WH sequences of length is a possible method to meet the constraint (15). The into (17) corresponding MSEE is found setting and reads (18) On the other hand, substituting (14) and letting produces
into
(19)
and the corresponding MSEE is given by (14) Note that the difference between (10) and (13) relies on the fact that has real-valued components, which reduces the number of unknown parameters by a factor two with respect to a conventional system with complexvalued channel responses. As shown later, this improves the quality of the channel estimates for a given training overhead. 2) Inspection of (8) reveals that . On the other hand, using the inequality 1 2 1 2 [20, p. 61], from (11) it follows that . , we Finally, bearing in mind that . At this stage, we recall see that is a matrix of order , and thus a necessary that condition for the existence of is . so that, in a fully This amounts to saying that ), the number of training blocks loaded system ( . Comparing this result with the cannot be less than constraint that was found in [9] in the context of conventional MC-CDMA transmissions, we see that GPF allows a potential reduction of the training overhead by a factor two. 3) Using similar arguments as in [21], it can be shown that in and order to achieve the minimum MSEE for a given dedicated for training, we subject to a fixed energy is a scaled version of the identity matrix. require that Collecting (1), (8), and (11), we see that this condition occurs with training sequences satisfying the identity (15)
which represents the performance of UCE in the absence of any prefiltering (with WH codes of length employed as training sequences). Comparing (18) and (19), we see that, for the same training overhead, GPF improves the accuracy of the channel estimates by a factor two with respect to a conventional system without signal prefiltering. 4) As discussed previously, in a fully loaded system with GPF the number of training blocks cannot be less than if UCE is employed for channel estimation. At this stage, an interesting question is whether a set of sequences of minimum length and satisfying (15) can be found or not. The answer is affirmative and a possible solution is given by (20) where of length
is the set of WH codes . The corresponding MSEE is found setting into (17) and reads (21)
Comparing the above results with (19), we see that GPF allows a reduction of the training overhead by a factor two with respect to a conventional MC-CDMA system (we to ) without degrading pass from the channel estimation accuracy. In summary, from the above discussion it follows that GPF provides some potential benefits on the performance of UCE. For a given overhead, it can be employed to improve the quality of the channel estimates with respect to conventional transmissions without any prefiltering. Alternatively, for a given MSEE, it can be used to reduce the number of training blocks.
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IV. STRUCTURED CHANNEL ESTIMATION
is a vector of dimension while the form
A. Problem Formulation of
We define the pre-equalized CIR of the th user as the IDFT , i.e., (22)
and observe that
that takes
(29) In the above equations, is an -dimensional column vector with unit elements and is the pre-equalized CIR matrix with entries vector. Also, is an
(23) (30) since is real-valued. In particular, from (3) and (6), is a scaled version of . Then, we see that assuming for convenience that the integer is nonnegative, it follows that is proportional to , where the convolution is applied times and is the autocorrelation of . Since has , is nonzero only for . support Accordingly, we have
-dimensional vector To proceed further, we define the . Thus, from (27), it follows that (31) where identical blocks with yields
is a block-diagonal matrix . Finally, substituting (31) into (7) (32)
(24) where (25) Although the above relation has been derived assuming that is nonnegative and even, simulations indicate that it can be rea. Note that for sonably used with any integer (transmit ZF) is independent of . This means that is proportional to the Kronecker delta function, so that for as predicted by (24) and (25). On the other hand, it is easily seen that the result (25) cannot be since in that case it would provide a negative used for , which is clearly meaningless. Fortunately, this is value of not a serious limitation since prefiltering schemes with are not of practical interest [7]. Returning to (22), we see that can be computed by . Thus, bearing in mind (23) and taking the DFT of (25), we may write
As mentioned previously, SCE operates in the time-domain and effectively exploits the structure of shown in (31). For this purpose, an estimate of the pre-equalized CIR vector is computed first and is next used to obtain the corresponding in the form estimate of (33) In this way, the number of unknowns is reduced since in practical applications is typically smaller than the total number of subcarriers. Note that a similar approach has been adopted in [22]–[24] in the context of multiple-input–multiple-output (MIMO) OFDM systems. B. Derivation of SCE The ML estimate of is derived from the linear model (32). Recalling that has real-valued components, we obtain (34) where
denotes the real part of the matrix (35)
(26) where parts of
and denote the real and imaginary , respectively. From (26), it follows that can be expressed as: (27)
where
Next, substituting (34) into (33) produces the ML estimate of in the form (36) The performance of SCE is assessed in Appendix II. It turns out that is unbiased and its MSEE is given by
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SANGUINETTI et al.: CHANNEL ESTIMATION FOR MC-CDMA UPLINK TRANSMISSIONS WITH COMBINED EQUALIZATION
C. Remarks 1) Since
, we have
. This fact, together with (35) and the property , indicates that . Thus, bearing in mind , it follows that that . We now recall that is a matrix , so that is of order a necessary condition for the existence of . This amounts to saying that or, equivalently, . Finally, in ), the constraint on a fully loaded system ( becomes (38) Comparing (38) with the result obtained with UCE, we see that SCE leads to a potential reduction . Note that only of the training overhead by a factor . one training block is required when 2) For given and , the minimum MSEE is achieved is a scaled version of the identity matrix [21]. when Recalling (8) and the structure of , from (35) we see can be partitioned into blocks ( that ), each of dimensions and expressed by
Inspection of (41) indicates that optimal training sequences must be designed at chip level rather than at symbol level. Fortunately, this fact is not restrictive since have the structure in (1) only over the the chips data blocks, while during the training period they can be selected independently for each and . 3) Optimal training sequences satisfying the set of constraints (41) can be designed using the same arguments . In these cirof [22] and [23] provided that cumstances, we see from (38) that the number of training and conditions (41) are blocks can be limited to met by
The MSEE is computed setting is expressed by
Comparing (45) with (18), we see that SCE achieves the same accuracy of UCE but reduces the training overhead from to 1. An interesting interpretation of the sequences in (44) is possible if we consider the corresponding time-domain samples, which are computed (multiplied by ) as the -point IDFT of (46) Substituting (44) into (46) and setting for otherwise
(40) or, equivalently, when the conditions shown in (41) at the bottom of the page are satisfied. In these circumstances, reduces to and (36) becomes
(42) The minimum MSEE is computed from (37) setting and reads (43)
(44) into (43) and it
(45)
(39)
Hence, the MSEE is minimum when
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, yields (47)
The above equation indicates that each user transmits a single time-domain sample during the training block. Also, samples of different users are spaced apart by . Bearing in mind that the multiples of (discrete-time) response of the pre-equalized channel to is , from (24), we see that signals transmitted by different users are received at the BS over adjacent and nonoverlap, with ping time intervals . This means that there is no interference among the active users as each received sample at the input of the DFT unit is contributed by one user only. On the other hand, transmitting a single time-domain sample
for
and
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of power as indicated in (47) is disadvantageous since it may produce an excessive PAPR at the mobile unit. 4) The set of constraints (41) is also met by the sequences defined in (20). The corresponding MSEE is cominto (43) and reads puted substituting (48) Alternatively, we may resort to WH sequences of length . In these circumstances, the MSEE is given by (49) Comparing (48) and (49) with (21) and (18), we see that . This means that, the MSEE is reduced by a factor for a given overhead, the structured approach is more accurate than UCE. 5) A possible shortcoming of SCE is that all groups of subcarriers must be assigned to the same set of users, so that the maximum number of contemporarily active users is . On the other hand, from (16), limited to it can be seen that UCE operates independently on each group and, therefore, it is suited for more versatile systems in which different groups of subcarriers may be assigned to different groups of users so as to guarantee multimedia services with variable data rates. In these circumstances, the number of contemporarily active users can be as large as the total number of subcarriers. V. PERFORMANCE EVALUATION Computer simulations have been run to assess the performance of an MC-CDMA receiver employing the proposed channel estimation schemes in conjunction with combined equalization. The system parameters are inspired by the HiperLAN/II specifications and reflect a typical indoor and low-mobility environment for wireless local area network (WLAN) applications. A. System Parameters The total number of subcarriers is and WH codes of length are used for spreading purposes. The signal MHz, so that the useful part of each bandwidth is . The MC-CDMA block has duration and a cyclic sampling period is prefix of is adopted to eliminate inter-block interference. This corresponds to an extended block (including the cyclic prefix) of 4.0 . Each frame has a duration of 256 and comprises 64 blocks. All the CIR’s have the same length , corresponding to a maximum excess delay of . Unless otherwise specified, each is kept fixed over the uplink time slot (static channel) but varies from slot to slot. The channel taps are modeled as complex-valued independent Gaussian random variables with zero-mean (Rayleigh fading) and powers (50)
Fig. 2. BER performance versus p for E =N = 6 dB and K = 4 or 8.
where is chosen such that the average energy of the channel . A rate 1/2 conis normalized to unity, i.e., volutional code is adopted with constraint length 6 and generator polynomials 131, 177 (in octals). After outer interleaving, the coded bits are mapped onto quadrature phase-shift keying (QPSK) symbols. Unless otherwise specified, ideal CSI is assumed at each MT, where the uplink signals are prefiltered according to the GPF scheme. At the BS the pre-equalized channel responses are estimated by means of UCE or SCE. To this purpose, we either exor the sequences ploit WH training sequences of length of length defined in (20). The channel estimates are kept fixed over the uplink time slot and are passed to a soft PIC-based receiver [19] which delivers the final data decisions. B. Performance Assessment Our first objective is the design of the parameter that determines the prefiltering coefficients in (3). As a design criterion, we choose the minimization of the average bit-error rate (BER) of the active users. Unfortunately, in doing so, we must resort to computer simulations since the average BER of the employed PIC-based receiver is hard to get theoretically. Fig. 2 , where shows the BER as a function of for is the transmitted energy per information bit, while is the one-sided noise power spectral density. The system is ei) or half-loaded ( ) and perfect ther fully loaded ( channel knowledge (PCK) is assumed at the receiver. As expected, the optimal depends on the number of users, but we represents a good choice with both and see that . This value is adopted in all subsequent simulations. We now address the problem of determining the (one-sided) of the pre-equalized CIR. Fig. 3 illustrates the length power delay profile of , from which it turns out that . This result validates the relation (25) for . However, since the MSEE in (48) and (49) is proportional to and becomes vanishing small for , in
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SANGUINETTI et al.: CHANNEL ESTIMATION FOR MC-CDMA UPLINK TRANSMISSIONS WITH COMBINED EQUALIZATION
Fig. 3. Power delay profile of h
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.
the following the SCE is designed for , which is tantamount to saying that matrix in (34) has columns. It is worth noting that is merely a design parameter that should not be confused with the true CIR . Since we choose in our simulalength tions, the SCE operates in a mismatched mode, but we expect that the effect of the mismatching is only visible at very high SNRs. Note that the idea of forcing to zero the weakest channel taps to reduce the number of unknown parameters is not novel since it has been previously considered in [25]. Recalling that , and , from (38), we see that channel estimation by means of SCE requires at least . Accordingly, the sequences of length defined in (44) cannot be used in our simulation setup. Fig. 4 shows the accuracy of UCE and SCE versus with . Marks indicate simulations, while solid lines represent theoretical analysis as given by (18), (21), (48), and (49). As expected, the best results are provided by SCE. Note that perfect agreement is observed between simulation and theory when the channel estimates are obtained with UCE. Vice versa, theoretical results for SCE are validated only for dB. As mentioned previously, the reason is that . This SCE operates in a mismatched mode with gives rise to an irreducible floor in the MSEE curves that limits the system performance at high SNRs. The above arguments are supported by other simulations (not shown for space limitations), which indicate that the floor disappears when SCE is . Unfortunately, in the latter case, designed with the estimation accuracy degrades at low/intermediate SNRs as predicted by (48) and (49). In summary, setting represents a good tradeoff that allows the system to achieve satisfactory performance over a wide range of practical SNRs. of a receiver enFig. 5 illustrates the BER versus dowed with the proposed channel estimation schemes. The ) and the number of training system is fully loaded ( or . The curves labeled GPF+UCE and blocks is either GPF+SCE are obtained employing the GPF coefficients in (3),
Fig. 4. Accuracy of UCE and SCE versus 1= with either N
= 4 or 8.
Fig. 5. BER performance over a static channel with K = 8 and N = 4 or 8.
whereas UCE refers to the uplink channel estimator discussed in [9] without any signal prefiltering (only post-equalization is present). In the latter case, we only provide simulations for since in the absence of signal prefiltering cannot of active users. Results obtained be less than the number with perfect knowledge of the pre-equalized channel responses (PCK) are also shown as a benchmark. We see that GPF significantly improves the system performance. Compared with a conventional system, a gain of nearly 3.5 dB is observed at an error rate of 10 when GPF is employed in conjunction with . The best UCE and the number of training blocks is results are obtained with GPF+SCE. For , the loss of GPF+SCE with respect to PCK is approximately 1 dB, while it becomes 2.5 dB with GPF+UCE. As expected, the system becomes smaller. In particular, performance deteriorates as
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Fig. 6. BER performance versus K for E =N = 6 dB and N = 4 or 8.
Fig. 7. Effect of imperfect CSI at all MTs over the BER performance for K = 8 and N = 8.
a loss of nearly 1 dB is observed with both GPF+UCE and is reduced from 8 to 4. GPF+SCE when Fig. 6 compares the performance of the investigated schemes as a function of . The number of training blocks is either or , while is fixed to 6 dB. Again, results . We see obtained without GPF are only provided for that the BER degrades as the number of users grows large and, in any case, GPF+SCE has the best performance. Fig. 7 illustrates the BER of GPF+UCE and GPF+SCE versus when signal prefiltering is accomplished at each MT using imperfect CSI (ICSI). This means that in computing the GPF coefficients the true channel gains in (3) are replaced by some suitable estimates. The latter are derived at each MT using the downlink channel estimator discussed in [9], which exploits a single training block placed at the end of the downlink time
Fig. 8. BER performance versus the mobile speed v for E =N = 6 dB and N = 8.
slot. The system is fully loaded and the length of the uplink . The curves labeled PCSI corretraining sequences is spond to perfect CSI at all MT’s and are shown for comparison. We see that ICSI entails a loss of approximately 1 dB on the performance of both schemes. Similar results (not shown for space . limitations) are obtained with The impact of channel variations on the system performance is addressed in Fig. 8, where the BER is shown as a function dB. Channel estimates of the mobile speed for are computed at the beginning of each uplink slot and are kept fixed over the entire slot, while the CIRs vary continuously from block to block due to Doppler effects. In particular, the channel taps are generated by filtering statistically independent white Gaussian processes in a third-order low-pass Butterworth filter. The 3-dB bandwidth of the filter is taken as a measure of the , where GHz is the employed Doppler rate 10 ms is the speed-of-light. carrier frequency, while and the number The training sequences have length or . We see that the BER deof active users is either grades with , but the performance loss is negligible for mobile speeds up to 10 m/s. This means that the considered schemes are well suited for indoor applications or outdoor systems with low/medium mobility. VI. CONCLUSION We have discussed two channel estimation schemes for TDD MC-CDMA uplink transmissions employing combined equalization. They are based on the ML criterion and exploit a sequence of training blocks placed at the beginning of each uplink time slot. The first scheme, UCE, ignores any correlation among adjacent subcarriers, while the second, SCE, uses a structured approach that improves the quality of the channel estimates. Both methods exploit the fact that the pre-equalized channel frequency response of each user has real-valued components. The performance of a soft PIC-based receiver endowed with the proposed channel estimators has been assessed by
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SANGUINETTI et al.: CHANNEL ESTIMATION FOR MC-CDMA UPLINK TRANSMISSIONS WITH COMBINED EQUALIZATION
simulation. It is shown that signal pre-equalization based on the GPF criterion can be used to improve the quality of the channel estimates or, alternatively, to reduce the training overhead with respect to conventional transmissions without any signal prefiltering. Optimal training sequences minimizing the mean square channel estimation error are designed under different operating conditions.
mean. On the other hand, collecting (56) and (58) we see that is given by the covariance matrix of
(59) or, bearing in mind (35)
APPENDIX I
(60)
In this appendix, we compute the covariance matrix of . Substituting (7) into (10) and bearing in mind (11) produces (51) where
At this stage, we recall that . Thus, from (60), it follows that (61)
denotes the real part of (52)
Thus, collecting (51), (52), and recalling that mean, it follows that is unbiased. From (51), we see that the covariance matrix of the form
, we may express
Finally, using the identity and side of (61) with
in the right-hand produces
has zero (62) takes (53)
where
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. Using the identity as (54)
or, alternatively (55) At this stage, we recall that vectors are statistically independent for different values of , with and . Thus, substituting (52) into (55) yields (56) from which it follows that (57) where we have born in mind (11) and (53). Finally, recalling that , we obtain the result (12). APPENDIX II In this appendix, we compute the expectation and the covariance matrix of . Substituting (32) into (36) and bearing in mind (31) and (35) produces (58) denotes the real part of the vector defined in (52). where From (58), it follows that is unbiased since has zero
which is equivalent to the result (37) since
.
REFERENCES [1] K. Fazel and S. Kaiser, Multi-Carrier and Spread Spectrum Systems. New York: Wiley, 2003. [2] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [3] D. Mottier and D. Castelain, “SINR-based channel pre-equalization for uplink multi-carrier CDMA systems,” in Proc. IEEE Int. Symp. Pers., Indoor, Mobile Radio Commun., Sep. 2002, pp. 1488–1492. [4] I. Cosovic, M. Schnell, and A. Springer, “On the performance of different channel pre-compensation techniques for uplink time division duplex MC-CDMA,” in Proc. IEEE Veh. Technol. Conf. (VTC-Fall), vol. 2, Oct. 2003, pp. 857–861. [5] P. Bisaglia, N. Benvenuto, and S. Quitadamo, “Performance comparison of single-user pre-equalization techniques for uplink MC-CDMA systems,” in Proc. IEEE Global Telecommun. Conf., vol. 6, Dec. 2003, pp. 3402–3406. [6] M. Brandt-Pearce and A. Dharap, “Transmitter-based multiuser interference rejection for the down-link of a wireless CDMA system in a multipath environment,” IEEE Trans. Signal Processing, vol. 18, pp. 607–617, Mar. 2000. [7] I. Cosovic, M. Schnell, and A. Springer, “Combined-equalization for uplink MC-CDMA in Rayleigh fading channels,” IEEE Trans. Commun., vol. 53, pp. 1609–1614, Oct. 2005. [8] W. Keusgen, C. Walke, and B. Rembold, “A system model considering the influence of front-end imperfections on the reciprocity of up- and downlink system impulse responses,” in Proc. Aachen Symp. Signal Theory, Aachen, Germany, Sep. 2001, pp. 243–248. [9] L. Sanguinetti, M. Morelli, and U. Mengali, “Channel estimation and tracking for MC-CDMA signals,” Eur. Trans. Telecommun., vol. 15, pp. 249–258, Mar. 2004. [10] U. Tureli, D. Kivanc, and H. Liu, “Channel estimation for multi-carrier CDMA,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, vol. 5, Instanbul, Turkey, Jun. 5–9, 2000, pp. 2909–2912. [11] K. Deng, Q. Yin, M. Luo, and Y. Zeng, “Blind uplink channel and DOA estimator for space-time block coded MC-CDMA systems with uniform linear array,” in Proc. IEEE Veh. Technol. Conf., vol. 1, May 17–19, 2004, pp. 69–73. [12] K. Zheng, G. Zeng, and W. Wang, “DFT-based uplink channel estimation for uplink MC-CDMA systems,” in Proc. IEEE Int. Symp. Spread Spectrum Tech. Appl., Sydney, Australia, Aug.-Sep. 2004, pp. 570–574. [13] P. Marques, A. Gameiro, and J. Fernandes, “Vectorial channel estimation for uplink MC-CDMA in beyond 3G wireless systems,” in Proc. IEEE Int. Symp. Pers., Indoor, Mobile Radio Commun., vol. 1, Sep. 2003, pp. 984–988.
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[14] Z. Li and M. Latva-Aho, “Analysis of MRC receivers for asynchronous MC-CDMA with channel estimation errors,” in Proc. IEEE Int. Symp. Spread Spectrum Tech. Appl., Sep. 2002, pp. 343–347. [15] B. V. Popovic, “Spreading sequences for multicarrier CDMA systems,” IEEE Trans. Commun., vol. 47, pp. 918–926, Jun. 1999. [16] I. Cosovic and L. Sanguinetti, “On the peak-to-average power ratio of pre-equalized multi-carrier code-division multiple-access transmissions,” in Proc. IEEE Veh. Technol. Conf.-Spring, Stockholm, Sweden, May-Jun. 28–1, 2005. [17] J. Proakis, Digital Communications. New York: McGraw-Hill, 1995. [18] A. C. McCormick, P. M. Grant, and J. S. Thompson, “A hybrid uplink multi-carrier CDMA interference cancellation receiver,” Proc. Inst. Elect. Eng. Commun., vol. 148, no. 2, pp. 119–124, Dec. 2001. [19] S. Kaiser and J. Hagenauer, “Multi-carrier CDMA with iterative decoding and soft-interference cancellation,” in Proc. IEEE Global Telecommun. Conf., Nov. 1997, pp. 6–10. [20] H. Lutkepohl, Handbook of Matrices. New York: Wiley, 1996. [21] T. L. Tung, K. Yao, and R. E. Hudson, “Channel estimation and adaptive power allocation for perfomance and capacity improvement of multiple-antenna OFDM systems,” in Proc. IEEE 3rd Workshop Signal Processing Advances Wireless Commun., vol. 5, Mar. 2001, pp. 82–85. [22] Y. G. Li, “Simplified channel estimation for OFDM systems with multiple transmit antennas,” IEEE Trans. Wireless Commun., vol. 1, pp. 67–75, Jan. 2002. [23] I. Barhumi, G. Leus, and M. Moonen, “Optimal training design for MIMO OFDM systems in mobile wireless channels,” IEEE Trans. Signal Processing, vol. 51, pp. 1615–1624, Jun. 2003. [24] F. Horlin and L. V. d. Perre, “Optimal training sequences for low-complexity ML multi-channel estimation in multi-user MIMO OFDM-based communications,” in Proc. IEEE Global Telecommun. Conf., vol. 4, Dallas, TX, Jun. 2004, pp. 2427–2431. [25] Y. Li, N. Seshadri, and S. Ariyavisitakul, “Channel estimation for OFDM systems with transmitter diversity in mobile wireless channels,” IEEE J. Sel. Areas Commun., vol. 17, pp. 461–471, Mar. 1999.
Luca Sanguinetti (S’04) received the Laurea degree (cum laude) in information engineering from the University of Pisa, Pisa, Italy, in 2002. He is currently working towards the Ph.D. degree in information engineering at the Department of Information Engineering, University of Pisa. In 2004, he was a visiting Ph.D. student at the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. His research interests span the areas of communications and signal processing, estimation, and detection theory. Current research topics focus on transmitter and receiver diversity techniques for single- and multiuser fading communication channels, antenna array processing, channel estimation and equalization, MIMO systems, multicarrier systems, linear and nonlinear prefiltering for interference mitigation in multiuser environment.
Ivan Cosovic (S’03–M’06) was born in Foca, Yugoslavia, on July 21, 1977. He received the Dipl.-Ing. degree in electrical engineering from the University of Belgrade, Serbia and Montenegro, in 2001, and the Ph.D. degree (Highest Hon.) in electrical engineering from the Johannes Kepler University Linz, Austria, in 2005. From February 2002 to April 2006, he was with the Institute of Communications and Navigation, German Aerospace Center (DLR), Wessling, Germany. He joined DoCoMo Communications Laboratories Europe GmbH, Munich, Germany, in April 2006, as a Researcher. His research interests include post-equalization and pre-equalization, transmitter and receiver diversity techniques, channel estimation, and spectrum sharing techniques for multicarrier, CDMA, and hybrid multicarrier CDMA wireless communications systems. Dr. Cosovic was awarded the Best Student Paper Award for the paper “Physical Layer Design for a Broadband Overlay System in the VHF Band” at the Digital Avionics Systems Conference (DASC’05), Washington, DC, in 2005.
Michele Morelli (M’04) received the Laurea (cum laude) degree in electrical engineering and the Premio di Laurea SIP from the University of Pisa, Pisa, Italy, in 1991 and 1992, respectively, and the Ph.D. degree in electrical engineering from the Department of Information Engineering, University of Pisa, in 1995. In September 1996, he joined the Centro Studi Metodi e Dispositivi per Radiotrasmissioni (CSMDR) of the Italian National Research Council (CNR), Pisa, where he held the position of Research Assistant. Since 2001, he has been with the Department of Information Engineering, University of Pisa, where he is currently an Associate Professor of Telecommunications. His research interests are in wireless communication theory, with emphasis on equalization, synchronization and channel estimation in multiple-access communication systems.
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