synchronization

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 5, SEPTEMBER 2008 2991 A Space–Time Receiver With Joint Synchronization and Interference Cancellation in Asynchronous MIMO-OFDM Systems Taiwen Tang, Student Member, IEEE, and Robert W. Heath, Jr., Senior Member, IEEE Abstract—We consider the scenario in which multiple transmitters send signals to a receiver in the asynchronous multiple-input– multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) mode. A receiver structure that can perform joint synchronization and space–time equalization for the asynchronous MIMO-OFDM systems is proposed. We first introduce a novel space–time equalization algorithm based on the channel state information. This algorithm aims to null the cochannel interference and combat asynchronism, as well as minimize a lower bound to the symbol error rate in the asynchronous MIMOOFDM system. In the second part of this paper, we propose a training-based minimizing mean-square-error (MMSE) algorithm to jointly obtain frequency offsets and equalization delays. We compensate for the effects of the frequency offsets and the propagation delays with these estimated parameters and then use the proposed direct training-based equalizer design algorithm to obtain the equalizer coefficients. The simulation section illustrates the desirable bit-error-rate (BER) performance of our proposed algorithms. Index Terms—Asynchronous, frequency offset, interference, multiple-input–multiple-output (MIMO) systems, orthogonal frequency-division multiplexing (OFDM). I. I NTRODUCTION HE COMBINATION of multiple-input–multiple-output (MIMO) communication and orthogonal frequencydivision multiplexing (OFDM) modulation is attractive for high spectral efficiency wireless communication systems [1]–[4]. Multiple receive antennas can be used for interference cancellation in multiuser MIMO-OFDM systems and, thus, support the reception of multiple data streams from different users on the same channel. We consider the scenario that multiple transmitters send signals to a receiver in the asynchronous MIMO-OFDM mode. Under this scenario, data streams from T Manuscript received June 25, 2007; revised November 16, 2007 and December 6, 2007. This material is based in part upon the work supported by Rockwell-Collins Inc. and by the Office of Naval Research under Grant N00014-05-1-0169. This material was presented in part at the IEEE 6th Workshop on Signal Processing Advances in Wireless Communications, June 2005, and in part at the Military Communications Conference, October 2005. The review of this paper was coordinated by Dr. J. W. Lee. T. Tang is with the Chengdu Goldtel Electronic Technology Company, Ltd., Chengdu, Sichuan, China (e-mail: ttang@ece.utexas.edu). R. W. Heath, Jr. is with the Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712 USA (e-mail: rheath@ece.utexas.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2008.915522 different users are not synchronized to the receiver because of the effects of propagation delays and frequency offsets. For asynchronous systems, the receiver must be able to synchronize the transmit and receive links for frequency offsets and propagation delays. This can usually be achieved via feedback or feedforward mechanisms. The feedback-based approach generally relies on feedback loops at the receiver to track and compensate the phase evolution that is induced by the frequency offset and the timing offset [5]–[7]. The principle of the feedforward operation, however, eliminates the use of such a tracking loop but directly compensates the phase with the estimates to the frequency offset and the timing offset [8]–[12]. These synchronization methods in the literature fail to operate in the presence of strong interference. In this paper, we adopt the feedforward architecture for the receiver, which is suitable for packet-oriented communications [8]. For asynchronous MIMO-OFDM multiuser systems, the receiver must also be able to deal with various sources of interference, including multiple-access interference, coantenna interference, and, more importantly, intercarrier interference (ICI). ICI in OFDM systems is mainly caused by the time variation of the channel and synchronization errors [13]. The residual frequency offset also causes ICI. Such effects are pronounced in multiuser OFDM systems, where the ICI caused by frequency offsets of each of the individual users interferes with each other. Furthermore, the asynchronism between different users, i.e., the difference of propagation delays between users, may also introduce ICI and undermine system performance [14]–[17]. As for the transmit signaling strategies, in this paper, we consider spatial multiplexing instead of the diversity-based transmit strategy. The reason is that the diversity mode is usually used in the data transmission phase and not necessarily in the initial synchronization phase [18], [19]. Dealing with different signal models for the training phase, the data phase under the diversity and spatial multiplexing modes complicates this paper’s presentation. We consider only the spatial multiplexing mode in this paper for simplicity. Our first contribution is to propose an equalizer design principle based on channel information that achieves interference cancellation and deals with the effects of residual frequency offsets of various users. A similar idea has been used for single user systems in [20] and [21]. In this paper, we design equalizers to suppress various sources of interference and consider the symbol error rate (SER) performance. 0018-9545/$25.00 © 2008 IEEE Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. 2992 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 5, SEPTEMBER 2008 Fig. 1. MIMO-OFDM multiuser system with a spatial multiplexing scheme (Mt = 2 and Mr ≥ 2). To design the equalizer, we need to have knowledge about the channel and the synchronization parameters for all transmitted data streams. In particular, obtaining accurate estimates of the frequency offsets is an important issue. Related results on frequency offset estimation have been presented in [22]–[25]. For MIMO systems, the problem of frequency offset estimation for MIMO flat-fading channels has been addressed in [26] and [27]. In multiuser MIMO-OFDM systems with a frequency-selective channel, this problem, however, is relatively unaddressed. For asynchronous MIMO-OFDM systems, estimating all the parameters, including wireless channels, timing, and frequency offsets, jointly involves computationally intensive operations since all the interference components are mixed together. In this paper, we propose a simpler solution to directly obtain all these parameters from the training sequences based on the idea of joint parameter estimation and interference cancellation. By nulling the interference, this enables separate estimation of the parameters for each user. Our proposed direct-training-based algorithm is suitable for asynchronous packet-oriented wireless systems. In conventional multiuser cellular systems, base stations coordinate the initial acquisition for all users to achieve synchronization. Careful frequency planning allows the acquisition to operate nearly without the presence of interference. In mobile packet communication systems, due to the lack of central coordination of communication, simultaneous packet transmissions may interfere with each other. Thus, synchronization must be performed in the presence of interference. It is very difficult to estimate the synchronization parameters without properly dealing with multiple-access interference. To obtain the proper equalizers to deal with interference, though, it is a prerequisite to properly compensate the synchronization errors. These issues make joint synchronization and interference suppression an attractive physical layer technology to support packet communications [16], [28]. The second contribution of this paper is that we propose a two-stage feedforward receiver architecture to perform joint synchronization and equalization. We propose a direct-trainingbased minimizing mean-square-error (MMSE) algorithm to estimate the frequency offsets and the equalization delays in MIMO-OFDM systems. The main advantage of our scheme is that we avoid the complexity of jointly estimating all the frequency offsets and the propagation delay parameters. We propose a novel direct-training-based method to finetune the space–time equalizer based on the design principles with channel information after obtaining the synchronization parameters. This paper is organized as follows. In Section II, we propose the space–time/space–frequency signal model for the multiuser MIMO-OFDM systems with synchronization parameters. In Section III, we discuss the equalizer design issues for such a system based on channel information and the synchronization parameters. In Section IV, we propose a direct-training-based synchronization architecture and a direct-training-based equalizer design algorithm. We set up the optimization problems and propose optimization algorithms. In Section V, we present simulation results for our proposed receiver. We use the following notations throughout this paper: Boldface letters denote matrices and vectors, the superscript T and ∗ denote the transpose and conjugate operations, respectively, and the superscript H stands for conjugate transpose. II. S YSTEM M ODEL In this section, we describe the asynchronous MIMO-OFDM multiuser system and the proposed receiver architecture to combat asynchronism in time and frequency. We discuss our assumption on channel models and study the signal model in the time and frequency domains. A. System Overview We consider a MIMO multiple-access channel where the receiver that has Mr antennas receives signals from multiple active mobile transmitters. Each mobile transmitter employs spatial multiplexing [29] and sends independent data streams from its transmit antennas. The total number of active transmit antennas in the system is assumed to be Mt , creating an Mt × Mr MIMO communication system, as illustrated in Fig. 1. We assume that one local oscillator is present at each transmitter Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. TANG AND HEATH: SPACE–TIME RECEIVER WITH JOINT SYNCHRONIZATION AND INTERFERENCE CANCELLATION 2993 Fig. 2. Block diagram of the receiver. The space–time equalization is followed by frequency offset compensation. and at the receiver. Therefore, there is only one frequency offset between each transmitter and the receiver. Each OFDM symbol that is sent by active transmit antennas has N tones and a length LCP cyclic prefix. Each data stream may only use some portion of the OFDM tones for data transmission. We denote the index set of the tones that are used for the data transmission of the ith stream as Si . We can also use a diagonal matrix T i consisting of zeros and ones in its diagonal axis to select these tones. We illustrate the proposed receiver structure in Fig. 2. A space–time equalizer bank is deployed at the front-end to suppress the cochannel interference and shorten the channel response in the case in which the channel response is longer than the cyclic prefix. By performing interference cancellation at the equalizer bank, different transmitted data streams are separated from each other. After space–time equalization, we use a two-step feedforward scheme to achieve the frequency offset and the timing synchronization for each transmitted data stream. The first step is to independently estimate the frequency offset and the timing based on the training sequences for each stream. The second step is to correct the phase rotation and the timing for this particular data stream after equalization. The space–time equalizer at the jth receive antenna for the ith output data stream of the space–time equalizers is represented by a vector wi,j = [wi,j (0), . . . , wi,j (L)]T of L+1 taps, where (·)T denotes the transpose. Each finite impulse response equalizer wi,j may have a different equalization decision delay Δi,j . The equalizer set {wi,1 , . . . , wi,Mr } forms a space–time filter bank for the ith output data stream. The frequencydomain signal of the bth OFDM symbol of the mth data stream is denoted by vm (b) = [vm (0, b), . . . , vm (N −1, b)]T . We consider the case where the OFDM training symbols are sent in the frequency domain. The time-domain signal is transformed by a discrete Fourier transform (DFT) basis matrix Q of size N × N as sm (b) = QH vm (b), where Q is the DFT basis matrix, and sm (b) = [sm (0, b), . . . , sm (N − 1, b)]T . For the sample sm (p, b), the relation between b, p, and k is that k = b(N + LCP ) + p + LCP , which is the universal time in the system. B. Channel Model The channel is assumed to be frequency-selective and blockwise time invariant over the duration of multiple OFDM symbols. We denote the channel response by hj,m (l) for the lth tap of the channel between the mth transmit antenna and the jth receive antenna, where l = 0, 1, 2, . . . , ν, m = 1, 2, . . . , Mt , and j = 1, 2, . . . , Mr . Note that ν + 1 is considered as the length of the channel response between the mth transmit antenna and the jth receive antenna. We assume that ν is known at the receiver, and it takes into account the maximal span of the channels in the time domain between all active Mt transmit antennas and the receive antennas over different channel realizations. The absolute propagation delay of the link between the mth transmit antenna and the jth receive antenna is denoted by dj,m . We model the effects of propagation delays as part of the channel response. The first dj,m taps of the channel response are considered to be zeros. Assume that the number of nonzero taps of the channel response between the jth transmit antenna and the mth receive antenna is νj,m + 1. We have νj,m + dj,m + 1 ≤ ν + 1. We assume that there is only one local oscillator that drives the receive RF chains on different receive antennas. Thus, the carrier frequency offset between the mth transmit RF chain and the receiver is denoted by m , which is normalized by the sampling interval Ts . We make the assumption that ν is less than the number of OFDM tones N , but ν may be less than or greater than LCP . C. MIMO Signal Model 1) Received Signal Model: Let sm (k) be the signal of the data stream transmitted by the mth antenna at discrete time k. The basic discrete time system equation, which is extended from the single-input–single-output case in [22] to the multiuser MIMO scenario, can be written as Mt ν rj (k) = m=1 l=0 hj,m (l)sm (k − l)ej2π mk + nj (k) (1) where nj (k) is the noise for receive antenna j at time k. Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. 2994 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 5, SEPTEMBER 2008 We construct a Toeplitz channel matrix Hj,m of size (L + 1) × (L + ν + 1) for the mth transmit antenna and the jth receive antenna constructed from hj,m (l)’s, where Hj,m is shown in (2) at the bottom of the page. Notice that the first dj,m taps [hj,m (l), l = 0, 1, . . . , dj,m − 1] are zeros in this Toeplitz matrix. We also define a diagonal matrix Γ( m , L) to represent the phase rotation that is introduced by the frequency offset, which has the following form: Γ( m , L) = diag{e−j2π mL , e−j2π m (L−1) , . . . , 1}. (3) We stack L + ν + 1 (the length of the effective channel response after equalization) samples of the transmitted signal into a vector sm (k − L − ν : k) = [sm (k − L − ν), . . . , sm (k)]T , where p : q denotes the pth to the qth sample of the signal in MATLAB notation. The signal vector at the jth receive antenna, which is denoted by rj (k − L : k), is given as Mt rj (k − L : k) = m=1 Γ( m , L)Hj,m sm (k − L − ν : k) × ej2π mk + nj (k − L : k) (4) the signal vectors sm (k − L − ν : k) are coded in space and time when transmitters send data or training sequences. Also, the diversity mode is usually used in the data transmission phase and not necessarily in the initial synchronization phase. Dealing with different signal models for the training phase, the data phase under the diversity and spatial multiplexing modes complicates this paper’s presentation. In this paper, we focus on the spatial multiplexing case and do not assume any transmit diversity scheme for simplicity. Based on these observations, we propose a receiver to support the joint space–time interference cancellation and the correction to the frequency offsets and the propagation delays on each stream independently. The receiver needs to obtain an estimate for the frequency offset m , which we denote by ˆm . The phase rotation that is caused by the effects of frequency offsets can, thus, be corrected with the frequency offset estimate after space–time equalization. Also, propagation delays are independently adjusted with equalization delays for each stream. 2) Postequalization Space–Time Model: The kth sample of the ith output data stream after the space–time equalization and the correction to the frequency offset, which is denoted by xi (k), can be written as Mr where nj (k − L : k) = [nj (k − L), . . . , nj (k)]T is the noise vector at the jth receive antenna. Remark 1: Based on (4), the optimal receiver jointly detects all the users in the time domain [30]. One of the main issues associated with this approach, however, is the computational complexity of joint multiuser detection. Furthermore, due to the asynchronism of the system, we cannot leverage the OFDM structure to lower the detection complexity if such joint timedomain detection is applied. Remark 2: As addressed earlier, with additional spatial dimensions leveraged by multiple receive antennas, time-domain equalization with multiple antennas can be applied to cancel the cochannel interference that is caused by interfering data streams, and we may further perform joint channel shortening to constrain the postequalization channel response to be less than or equal to LCP + 1. All these allow us to perform perstream and per-tone detection for the desired data stream. This solution has the supreme advantages of low complexity over the joint detection schemes. Remark 3: Another important observation is that the frequency offset and timing effects (equivalently, equalization delay) can be adjusted after the space–time equalization for each stream since the interference from other streams can be nulled by the equalizers. If transmit diversity as space–time coding [29] is employed, the overall signal model remains the same as (4), except that xi (k) = j=1 Mr T wi,j rj (k + Δi,j −L : k + Δi,j )e−j2πˆi k = j=1 T wi,j Γ( i , L)Hj,i ej2π i Δi,j × si (k + Δi,j −L−ν : k + Δi,j )ej2π( Mt Mr T wi,j Γ( m=1,m=i j=1 j2π m , L)Hj,m e i−ˆi )k + m Δi,j × sm (k + Δi,j −L−ν : k + Δi,j )ej2π( Mr m−ˆi )k + j=1 T wi,j nj (k + Δi,j −L : k + Δi,j )e−j2πˆi k . (5) We write the effective channel between the mth transmit antenna and the jth receive antenna as follows: He = Γ( j,m j2π m , L)Hj,m e m Δi,j . (6) Remark 4: From (5), we observe that the effective channels are time invariant; thus, it is possible to use time-invariant space–time equalizers (wi,j ) to null the other users’ signal. This is equivalent to having Mr T wi,j He = 0T . j,m j=1 (7) ⎡h ⎢ Hj,m = ⎢ ⎣ j,m (ν) 0 . . . 0 hj,m (ν − 1) hj,m (ν) . . . 0 ... hj,m (ν − 1) .. . 0 hj,m (0) ... .. . ... 0 hj,m (0) .. . hj,m (ν) 0 0 .. . hj,m (ν − 1) ... ... . . . ... 0 0 . . . hj,m (0) ⎤ ⎥ ⎥ ⎦ (2) Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. TANG AND HEATH: SPACE–TIME RECEIVER WITH JOINT SYNCHRONIZATION AND INTERFERENCE CANCELLATION 2995 Remark 5: The equivalent channel response after the space– time equalization has a length of L + ν + 1, which might be larger than LCP + 1. In this case, the signal from adjacent OFDM symbols might propagate and, thus, cause inter-OFDMsymbol interference (IOSI). If we can constrain the number of the nonzero taps of the effective channel out of the total L + ν + 1 taps to be LCP + 1, such a problem can be avoided. After the frequency and timing adjustment, the receiver discards the cyclic prefix and performs the DFT. The detection happens in parallel for all tones in the frequency after the DFT. 3) Postequalization Space–Frequency Model: From (4), we stack the received signal vectors at the jth receive antenna into a matrix Rj (b, Δi,j ) of size N × (L + 1), shown in (8) at the bottom of the page, with delay parameter Δi,j for the bth OFDM symbol. The matrix Sm (b, Δi,j ) is generated by stacking the transmitted data vector sm with the delay parameter Δi,j . The matrix Nj (Δi,j ) denotes the stacked noise matrix at the jth receive antenna. The matrix Θ( i , N, b) is an N × N diagonal matrix, which is defined as Θ( i , N, b) = ej2π i (N +LCP )b (L + ν + 1) × 1, is the postequalization channel response for the mth transmit antenna and the jth receive antenna. The quantity ei (b) includes the colored noise, and the ICI due to the frequency offset estimation error, the possible IOSI due to the long channel response after equalization, and the residual cochannel interference due to imperfect interference cancellation. The training-based algorithm, as we will address later, however, does not require exploitation of its complicated structure but simply treats it as an aggregate error term. III. S PACE –T IME E QUALIZER D ESIGN In this section, we study the methods of the space–time equalizer design for an asynchronous multiuser MIMO-OFDM system, given the ideal channel information and synchronization parameters such as frequency offsets and propagation delays. This section is organized as follows. The Min-SER-based design method is elaborated upon in Section III-A. The design method that is based on maximizing the signal-to-interferencepower ratio is discussed in Section III-B. For these two design methods, we address the design objectives and formulate optimization problems to meet the objectives. The equalizer design methods based on channel information, as we propose here, are the rationales of the training-based algorithms addressed in Section IV. They also provide baseline performance as compared to the training-based algorithms in Section V. A. Min-SER Equalizer Design diag{1, ej2π i , . . . , ej2π i (N−1) }. (9) The space–frequency model can then be derived as follows: Mr yi (b) = j=1 QΘ(ˆi , N, b)∗ Rj (b, Δi,j )wi,j (10) ¯ = Vi (b) Qbi,i + ICI + IOSI + CCI + Noise fi,i ei (b) where yi (b) is an N × 1 vector, which has the samples of the bth OFDM symbol of the ith output data stream. The matrix Rj (b, Δi,j ) is the stacked version of the data matrix at the jth receive antenna with delay parameter Δi,j of size N × (L + 1). We define Vm (b) = diag{vm (0, b), . . . , vm (N − 1, b)}, which has the bth frequency-domain samples of the data stream transmitted by the mth active transmit antenna. Notice that only those tones whose indexes belong to the set Sm have nonzero values. The vector fi,i is the frequency-domain effective channel response for the ith output stream, the ¯ partial DFT basis is denoted by Q, which has a size of N × (LCP + 1), and bi,i is the shortened channel response of size (LCP + 1) × 1 for the shortened effective channel between the ith transmit antenna and the ith output data stream. Note that Mr HT Γ( i , L)wi,j ej2π m Δi,j , which is of size j,m j=1 In general, it is very costly to optimize the equalizer delays Δi,j for all receive antennas. We make the following assumption that Δi,j = Δi for all j = 1, . . . , Mr to simplify the Mr variable optimization to a single variable optimization. Let us define a vector channel response after the space–time T ¯ equalization as bT = Mr wi,j Γ( m , L)Hj,m ej2π m Δi . m,i j=1 1) Objective 1—Nulling Cochannel Interference: To guarantee that there is no cochannel interference on the selected tones for the ith output data stream, we must have the following: ˜¯ T i Qbm,i = 0N ×1 , m=i (11) ˜ where Q is the partial DFT matrix to transform the time-domain channel response into the frequency-domain channel response. This basically means that the equivalent channel response of the interferers in the frequency domain after the space–time equalization should be null. We assume that rank{T i } ≥ L + ν + 1, which is equivalent to using more than L + ν + 1 tones ˜ for the ith data stream. Since the matrix T i Q is a tall matrix, ⎡ ⎢ Rj (b, Δi,j ) = ⎣ Mt rj (b(LCP + N ) + LCP − L + Δi,j : b(LCP + N ) + LCP + Δi,j )T . . . rj ((b + 1)(LCP + N ) − L + Δi,j − 1 : (b + 1)(LCP + N ) + Δi,j − 1)T ⎤ ⎥ ⎦ = m=1 Θ( T j2π m , N, b)Sm (b, Δi,j )Hj,m Γ( m , L)e m Δi,j + Nj (Δi,j ) (8) Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. 2996 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 5, SEPTEMBER 2008 then, the condition of canceling the cochannel interference is equivalent to Mr T wi,j Γ( j=1 j2π m , L)Hj,m e m Δi = 0T L+ν+1 m = 1, . . . , Mt . (12) ¯ Mr (L + 1) > rank(HCi ). Then, to find a feasible wi and make (14) hold for all H, it is equivalent to having Mr (L + 1) > ¯ maxH {rank(HCi )}. ¯ Clearly, rank(HCi ) ≤ Td , where Mt ∀m = i, Td = Mt (L + 1) + m=1 νm − LCP − 1 (17) Notice that there are νj,m + 1 nonzero taps for the channel Hj,m . The rank of the effective channel matrix Γ( m , L)Hj,m is min{Mr (L + 1), L + νj,m + 1}. 2) Objective 2—Shortening the Channel to Mitigate ICI: Another design objective is to make the nonzero taps of ¯ the effective channel response after equalization bm,i = ]. This guarantees that no ICI arises [0T i , bT , 0T i,i Δ L+ν−LCP −Δi after the DFT operation following the space–time equalization. 3) Matrix Formulation for Objectives 1 and 2: Define an aggregated effective channel matrix (13), shown at the bottom of the page. We aggregate the equalizers (wi,j ) on all receive T T antennas into a vector wi = [wi,1 , . . . , wi,Mr ]T . We define a matrix Ci to be a binary diagonal matrix, which selects the samples in a certain window of length LCP + 1 of the effective channel response H after space–time filtering for the ith data ¯ stream. The matrix Ci selects the samples that are outside the window. Both selection matrices are determined by the equalization delay Δi . Then, we have the following constraint: T ¯ wi HCi = 0T t (L+ν+1)−LCP −1)×1 . (M and the equality holds when the channel vectors hj,m (hj,m = [hj,m (0), . . . , hj,m (ν)]T ) are linearly independent from each other. Thus, we conclude that the order of the equalizer should be selected to satisfy (16). We denote the extra degrees of freedom by Mt De = Mr (L + 1) − Mt (L + 1) + m=1 νm − LCP . (18) 5) Objective 3—Minimize an SER Bound: Let us define a Toeplitz matrix Wi,j of size N × (N + L), which is generated by the equalizer wi,j as ⎤ ⎡ 0 wi,j (L) . . . wi,j (0) . . . ⎥ ⎢ . . . .. .. . . . Wi,j = ⎣ ⎦ . (19) . . . . . 0 ... wi,j (L) ... wi,j (0) Assume that the noise on each receive antenna follows an independent identically distributed (i.i.d.) Gaussian distribution 2 with zero mean and σn . The noise covariance matrix after the space–time equalization and DFT operation can be obtained as Mr 2 Gi = σ n j=1 H QΘ( i , N, 0)∗ Wi,j Wi,j Θ( i , N, 0)QH . (20) (14) Also, the equivalent channel response for the ith stream after space–time equalization can be written as T bT = wi HCi . i,i (15) 4) Parameter Selection Condition: To avoid the degenerate solution that the equalizer coefficients are all zeros, we further introduce a total power constraint on the equalizer response Mr 2 = 1. Notice that the equalizer order L and the j=1 wi,j number of receive antennas Mr are fixed parameters before system deployment; thus, we must select them offline for various channel conditions. We present a simple condition for selecting these parameters. Proposition 1: A necessary and sufficient condition of the existence of equalizer wi to satisfy (14) for all H and the constraint on the equalizer power is that Mt Mr (L + 1) ≥ Mt (L + 1) + m=1 νm − LCP . (16) Hence, the noise power on tone p for the ith output data stream can be represented as Gi (p, p), i.e., the pth diagonal element of the matrix Gi . If the equalizer satisfies condition (14), another design objective is to optimize the average uncoded SER using the extra degrees of freedom De in the system. Given a quadraticamplitude modulation (QAM) scheme, the average symbol ¯ error probability of the OFDM system Pe can be approximated by the following expression [31]: ⎞ ⎛ N constant1 constant2 ⎠ ¯ Pe ≈ (21) Q⎝ N SNR−1 p=1 p Proof: From linear algebra, we know that there exists a nonzero solution of wi to satisfy (14) if and only if where constant1 and constant2 only depend on the chosen QAM constellation, and Q(·) is the error probability function. In [32], Γ( 1 , L)H1,1 ej2π 1 Δi ⎢ . . H=⎣ . Γ( 1 , L)HMr ,1 ej2π 1 Δi ⎡ ... .. . ... Γ( Γ( j2π Mt , L)H1,Mt e M t Δi ⎤ ⎥ ⎦ (13) . . . Mt , L)HMr ,Mt e j2π M t Δi Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. TANG AND HEATH: SPACE–TIME RECEIVER WITH JOINT SYNCHRONIZATION AND INTERFERENCE CANCELLATION 2997 it has been shown that the complementary error function is convex with respect to SNR−1 in the moderate-to-high SNR region. It is also shown in [32] that the average SER can be tightly lower bounded by the following at moderate-to-high SNRs for binary phase-shift keying modulation: ⎞ ⎛ constant2 ¯ ⎠. Pe ≥ constant1 · Q ⎝ (22) N 1 SNR−1 p p=1 N This SER lower bound can be extended to any general QAM with the approximate SER expression in (21) using the convexity result established in [32]. Our rationale of designing the equalizer is to minimize this tight lower bound of the ¯ average error probability Pe . This gives the following objective function: gi (wi , Δi ) = 1 |Si | Gi (p, p) |fi,i (p)|2 (23) (27) by its norm. Thus, the optimal equalizer vector is given as follows: wi → Us 1, tT i i . 1 + ti 2 T (28) The Min-SER method is robust to the variation of the interference power level since the interferers’ channels are cancelled perfectly. Given the constraint of perfectly nulling the interference, the more extra degrees of freedom De the receiver has, the better the SER performance may be achieved. B. Maximizing Power Ratio Design (MaxPwr) The MaxPwr algorithm proposed in [15] uses the power ratio of the desired signal to the interference and noise as the cost function for optimization. Here, we generalize this approach to deal with different frequency offsets. Given the channel estimates in (13), we state the problem of maximizing the signal-to-interference ratio (henceforth, the MaxPwr method introduced in [15]) as max H T wi H∗ Ci Ci HT wi , H ¯T ¯ wi H∗ Ci Ci HT wi p∈Si where |Si | denotes the cardinality of the set Si . ¯ 6) Linear Constraint: Define the matrix Ki = Ci HT . We apply the singular value decomposition to this matrix and obtain the following: Ki = Zi Σi UH . i Clearly, we can write Ui as follows: Ui = [Un , Us ] i i (25) (24) wi ,Ci s.t. wi 2 = 1. (29) where Un and Us are of sizes Mr (L + 1) × Td and Mr (L + i i 1) × [Mr (L + 1) − Td ], respectively [Td is defined in (17)]. The matrix Un generates the subspace corresponding to all i singular values in Σi , and the matrix Us generates the null i space to the subspace that is generated by Un . Any equalizer i vector wi lying in the subspace that is generated by Us satisfies i the constraint Ki wi = 0. To avoid the degenerate solution wi = 0, we further restrict wi to be of the following form: wi = Us i 1 ti = U (:, 1) + s Thus, by performing the optimization over wi and Ci , the energy of the postequalization channel response will be mainly concentrated in the selected window, and the energy outside the selected window is minimized. When the length of the window is set to be equal to LCP + 1, joint channel shortening and cochannel interference suppression is performed through the optimization. As we can see, under condition (16), the interference term can be nulled. Thus, the MaxPwr method is equivalent to maximizing the energy of the postequalization channel response under the constraint of nulling the cochannel interference, i.e., wi ,Ci T max wi HCi 2 Us i T ¯ s.t. wi HCi = 0T t (L+ν+1)−LCP −1)×1 , (M wi 2 = 1. (30) (:, 2 : (De + 1)) ti (26) where ti is of length De = Mr (L + 1)−Td −1, and Us (:, p : q) i denotes the pth to the qth column of the matrix Us in standard i MATLAB notation. 7) Min-SER Problem Formulation and the Proposed Optimization Algorithm: We phrase the optimization problem of designing the equalizer as given by Ci , i.e., min gi (ti , Ci ). ti As addressed earlier, we may only use the tones that belong to the set Si for the ith data stream; therefore, the cost function is modified as wi ,Ci max p∈Si |fi,i (p)|2 wi 2 T ¯ s.t. wi HCi = 0T t (L+ν+1)−LCP −1)×1 , (M =1 (31) (27) We can solve this unconstrained problem by applying the steepest descent method [33]. The algorithm is described in Appendix I, where we also present the expression of the derivative and analyze the computational complexity of each iteration of the algorithm. Since any nonzero scalar multiplication to the optimal equalizer vector is still optimal, we normalize the optimal solution in ¯ where fi,i = QCi HT wi . We perform an optimization for every Ci . When Ci is given, by the use of change of variables, we define wi = Us di . The optimization problem can be i restated as max di p∈Si |fi,i (p)|2 , s.t. di 2 = 1. (32) This method of designing equalizer does not give optimal uncoded performance; however, the optimization is much simpler Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. 2998 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 5, SEPTEMBER 2008 to solve than the Min-SER approach. The optimal solution to the problem in (32) is the eigenvector corresponding to the max¯ ¯ imal eigenvalue of the matrix (Us )H H∗ Ci QH T i QCi HT Us . i i The computational complexity for this algorithm is shown in Appendix II. IV. D IRECT -T RAINING -B ASED S YNCHRONIZATION AND E QUALIZER D ESIGN The methods proposed in the previous section are all based on the utilization of channel knowledge. In real systems, the equalizer can be designed based on the estimated channel responses and frequency offsets. Complexity, however, may be a main concern for this approach because the complexity of jointly obtaining channel responses and frequency offsets is high in MIMO systems. Also, such a design strategy leads to the propagation of the estimation noise to the equalizers. Furthermore, this approach is not possible when the training sequences are not all known to the receiver, i.e., the case of having certain interferers or jammers whose transmitted data sequences are unknown to the receiver. To deal with these problems, we propose a direct-trainingbased two-stage architecture for synchronization and equalizer design. The first stage utilizes an MMSE joint synchronization and interference cancellation scheme (based on the rationale of MaxPwr) to obtain the synchronization parameters—frequency offsets and equalization delays, coarse equalizer coefficients, and the postequalization channel response. In the second stage, we refine the design to the equalizer and the postequalization channel response by a direct-training-based approach based on the rationale of Min-SER. The organization of this section is given as follows. The first stage of the two-stage training-based method is discussed in Section IV-A. The second design stage is discussed in Section IV-B. The complexity issues are addressed in Section IV-C. A. MMSE Joint Equalization Delay and Frequency Offset Estimation Here, we discuss the MMSE joint synchronization and interference cancellation scheme to estimate the equalization delays and the frequency offsets. Define two matrices Pj,i ( i , Δi , b) of size N × (L + 1) as Pj,i ( i , Δi , b) = QΘ(− i , N, b)Rj (b, Δi ) and Di (b) of size N × (LCP + 1) as ¯ Di (b) = Vi (b)Q. (34) (33) trix P( i , Δi ), which is defined as ⎛ ... P1,i ( i , Δi , 0) ⎜ . .. . Pi ( i , Δi ) =⎝ . . ⎞ PMr ,i ( i , Δi , 0) ⎟ . . ⎠ . P1,i ( i , Δi , K −1) . . . PMr ,i ( i , Δi , K −1) (36) and a matrix Di defined as follows: Di = Di (0)T , . . . , Di (K − 1)T T . (37) The equalizer vectors are stacked into a vector wi = T T [wi,1 , . . . , wi,Mr ]T . The objective of the joint synchronization and interference cancellation is to fine-tune all parameters to minimize the mean square error of the residual error term after equalization. Thus, we have wi ,bi,i ,Δi , min Pi ( i , Δi )wi − Di bi,i i 2 , s.t. bi,i 2 =1 (38) where we use the constraint bi,i = 1 to avoid degenerate solutions to the optimization. Let us define a matrix Mi ( , Δ) as follows: Mi ( , Δ) = DH Di − DH Pi ( , Δ) Pi ( , Δ)H Pi ( , Δ) i i −1 2 × Pi ( , Δ)H Di . (39) Applying the technique of separation of variables, the offset and equalizer delay estimator can be derived as follows: opt opt i , Δi = arg min (λ(Mi )) ,Δ (40) where λ(·) denotes any eigenvalue of a matrix. Given any integer delay, we use the steepest descent method [33] to find a locally optimal estimate to the frequency offset. Also, we apply a grid search at each integer frequency q in the range of [q − 0.5, q + 0.5]/N to find a locally optimal solution. The number of grids over which to search is denoted by Ng . Thus, the globally optimal solution to the frequency offset corresponds to the minimum cost over all ranges. The derivative of the eigenvalue of the matrix function Mi ( , Δ) is also derived in Appendix III. For each iteration of the algorithm, we obtain the computational complexity result. B. Direct-Training-Based Min-SER Space–Time Equalizer Design After obtaining the equalization delay Δi and the frequency offset i based on the MMSE method addressed earlier, we can compensate for the effects of asynchronism with these estimated parameters. To further improve SER performance, we refine the space–time equalizer coefficients based on the Min-SER principle in Section III. By (10), the residual error term has the following form: Mr Based on the space–frequency model (10), the residual error term ei ( i , Δi , b) has the following form: Mr ei ( i , Δi , b) = j=1 Pj,i ( i , Δi , b)wi,j − Di (b)bi,i . (35) The mean square error over K training symbol duration amounts to (1/K) K ei ( i , Δi , b) 2 . We introduce a mab=1 ˜ ei ( i , Δi , b) = j=1 T i [Pj,i ( i , Δi , b)wi,j −Di (b)bi,i ] (41) Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. TANG AND HEATH: SPACE–TIME RECEIVER WITH JOINT SYNCHRONIZATION AND INTERFERENCE CANCELLATION 2999 where T i is a diagonal tone selection matrix that is used to select the tones loaded with data. We also define a matrix Λi as Λi (bi,i ) = diag |fi,i (0)|−2 , . . . , |fi,i (N − 1)|−2 (42) TABLE I SUMMARY OF THE COMPUTATIONAL COMPLEXITY OF THE PROPOSED ALGORITHMS. IZFIC , IOffset , AND ITraining DENOTE THE NUMBER OF I TERATIONS T HAT V ARY THE S TEP S IZES OF THE S TEEPEST DESCENT ALGORITHM, NΔ REPRESENTS THE NUMBER OF DELAYS SEARCHED OVER, AND Ng REPRESENTS THE NUMBER OF FREQUENCY GRIDS SEARCHED OVER ¯ where fi,i = Qbi,i . The cost function takes the form of the summation of the inverse of the per-tone signal-to-noise-andinterference ratio, i.e., K−1 φi (wi , bi,i ) = b=0 ˜ ˜ ei ( i , Δi , b)H Λi ei ( i , Δi , b). (43) The design objective is to minimize the cost function with a certain constraint on bi,i to avoid degenerate solution, i.e., wi ,bi,i min φi (wi , bi,i ), s.t. bi,i 2 = 0. (44) We can apply the technique of separation of variables to reduce the number of variables since the optimal wi can be expressed as follows: ¯ wi = Pi ( i , Δi )H Λi Pi ( i , Δi ) −1 an asymptotic upper bound, and lg to denote the base-two logarithmic. We make a note that the Min-SER equalizer design in Section III-A is also called the Zero Forcing Interference Cancellation method (ZFIC) since we null cochannel interference based on the zero-forcing approach. We observe that the ZFIC method is much more computationally costly than the MaxPwr approach. When the channels and the frequency offsets are perfectly known, we can first apply the MaxPwr method to find the equalization delay and then use the ZFIC method to fine-tune the equalizers, given the equalization delay. The combined method can balance the computational time and the performance. V. S IMULATION R ESULTS A. Simulation Setup We assume that there are two data streams that are transmitted by two active antennas. Throughout the simulation, we assume that the channel is fixed over a frame that contains a certain number of OFDM symbols. For each frame, we observe an independently faded channel. The time-domain channel for both streams has six taps, and on each tap, there is an equivalent Mt × Mr channel matrix. The elements of the channel matrices are all independently generated according to a complex Gaussian distribution with zero mean and unit variance. Both streams use equal transmit power. For an intended stream, the carrier-to-interference-power ratio (CIR) at each receive antenna is 0 dB throughout the simulation section. The SNR corresponds to the ratio of the transmit power to the noise power. We use QPSK modulation and N = 64 tones. The cyclic prefix length is L = 16. In the simulation, we assume that each data stream uses all the tones (T i = I). We select the number of receive antennas Mr and the equalizer order L based on (16). For each channel realization, we use one OFDM symbol for parameter estimation and equalizer training (K = 1). The training sequence is an i.i.d. constant modulus with a random phase. In the training stage, the synchronization parameters are estimated via the MMSE method. Then, the equalizer coefficients and the postequalization channel response are obtained from the Min-SER method. After the training stage, we apply the equalizer that is obtained at the training stage to equalize the received signal. We also note here that for all search results, we restrict the maximum number of iterations to 500. We study the uncoded performance. We assume that there is no interleaving in space for the transmitted data streams. At the receiver, we apply the equalization schemes discussed in the previous sections to obtain the equalizer coefficients and the postequalization channel response. For uncoded streams, we ¯ Pi ( i , Δi )H Λi Di bi,i (45) ¯ where Λi = IK×K ⊗ Λi T i , and ⊗ denotes the Kronecker product. Also, notice that we assume that K|Si | > Mr (L + 1) to guarantee the invertibility of the matrix ¯ Pi ( i , Δi )H Λi Pi ( i , Δi ). Let us also define a matrix Ki to be ¯ ¯ Ki ( i , Δi , bi,i ) = DH Λi − Λi Pi ( i , Δi ) i ¯ × Pi ( i , Δi )H Λi Pi ( i , Δi ) −1 ¯ Pi ( i , Δi )H Λi Di . (46) To avoid the degenerate solutions, we constrain the first tap of the bi,i to be 1 to satisfy that bi,i 2 = 0. Thus, we can substitute bi,i with the vector [1, pi ]T , where pi is of length LCP . The optimization problem can be stated with respect to the variable pi as follows: min 1, pH Ki ( i , Δi , pi ) 1, pT i i pi T . (47) Again, we solve this optimization by the steepest descent algorithm [33]. The derivative to the cost function in (47) is given in Appendix IV, where we also discuss the complexity of the algorithm at each iteration. We normalize the optimal solution pi from (47) and obtain the postequalization response as bi,i → [1, pT ]/ [1, pT ] . i i Then, the optimal solution to the equalizer is provided in (45). C. Computational Complexity of the Proposed Algorithms Last, we summarize the computational complexity of the proposed algorithms in Table I (the brief derivations are given in the appendices). We follow the notation in [34] and use Θ(·) to denote an asymptotically tight bound, O(·) to denote Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. 3000 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 5, SEPTEMBER 2008 Fig. 3. Plot of BER versus SNR for the proposed schemes with two transmit data streams. We select the equalizer memory length L = 12 and use QPSK modulation. The numbers of channel taps from the two transmit streams to the receive antennas are both six. Fig. 4. Plot of NMSE versus SNR for the proposed frequency offset estimator for the case of two transmit data streams. The numbers of channel taps from the two transmit streams to both receive antennas are three and four, respectively. perform per-stream and per-tone detection in the frequency domain and apply the maximum-likelihood-type scalar detection. Throughout the simulations, we assume that there is a common time reference in the system, namely, time 0. The propagation delays for the two users that are considered in the rest of this paper are the absolute time lags with respect to the common time reference. B. BER Comparison of Different Design Methods In Fig. 3, we compare the uncoded bit-error-rate (BER) performance of the proposed schemes. In this comparison, we average the BER over 400 channel realizations. For each channel realization, we randomly generate a normalized frequency offset (normalized by the number of OFDM tones) within the range of −0.5 to 0.5. The propagation delay difference between both transmitted streams is set to be 0. With ideal channel information and all synchronization parameters, at a 20-dB SNR, the Min-SER algorithm remarkably outperforms the MaxPwr method by 10 dB. At a 20-dB SNR, the BER of the Min-SER algorithm is below 10−3 . We also show the BER performance of the proposed space–time receiver with synchronization and equalization. Notice that with only one training OFDM symbol, this algorithm outperforms the MaxPwr method in a high SNR region and can achieve an uncoded BER on the order of 10−3 when the SNR is higher than 20 dB. C. NMSE of the Frequency Offset Estimator In Fig. 4, we illustrate the normalized mean square error (NMSE) versus the SNR for the proposed MMSE training algorithm. The simulation results are averaged over 1000 channel realizations. The normalized frequency offsets are set to be 0.45 and 0.36 in this experiment. The NMSE is defined as the mean square error of the estimated frequency offset that is normalized by the square of the true frequency offset. Let i denote the true frequency offset, and let ˆi stand for the estimate to the NMSE is given by N Ei = E|ˆi − i |2 2 i i. Then, (48) where E denotes the expectation with respect to ˆi . Notice that the CIR is 0 dB in our simulation. Conventional frequency offset estimation algorithms [8]–[12], [22], [23] do not successfully operate at this CIR level. We also plot a lower bound to the NMSE as a baseline for comparison. The bound is obtained in [28], and it is basically the Cramer–Rao lower bound that is normalized by the true frequency offset for each individual stream using one training OFDM symbol (LCRB), assuming that there are no other interfering data streams. We do not show the actual Cramer–Rao bound (CRB) here since it is difficult to characterize in the presence of interference. The NMSE is monotonically decreasing and is on the order of 10−4 for SNRs that are above 20 dB. Note that the LCRB is not the actual Cramer–Rao lower bound for our system since it neglects the presence of interference. The gap between our method and the actual CRB should be smaller than the gap between our method and the LCRB. Furthermore, note that the linear interference canceller that we use treats the interference component as colored noise. This is not optimal compared with nonlinear cancellation (e.g., successive interference cancellation). This also explains a part of the gap. However, the complexity of our approach is much smaller than using nonlinear cancellers. In Fig. 5, we compare the result of the Schmidl and Cox method [9] with our proposed method for different CIRs. Our method’s performance is robust to different interference levels; however, the Schmidl and Cox method is significantly affected by the interference. Our method outperforms the Schmidl and Cox method at a high SNR region for different CIR levels (0–20 dB). When CIR = 20 dB, the Schmidl and Cox method performs better than our method only when the SNR is less than 17 dB. Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. TANG AND HEATH: SPACE–TIME RECEIVER WITH JOINT SYNCHRONIZATION AND INTERFERENCE CANCELLATION 3001 Fig. 5. Plot of NMSE versus SNR for different CIRs when there are two streams (a desired stream and an interferer). We compare the performance of the Schmidl and Cox (S&C) method with our proposed method. The numbers of channel taps from the two transmit streams to both receive antennas are three and four, respectively. Fig. 7. Plot of BER versus SNR for the proposed space–time receiver with two transmit data streams with different propagation delays. We select the equalizer memory length L = 12 and use QPSK modulation. The numbers of channel taps from the two transmit streams to the receive antennas are both six. E. BER Comparison for Different Propagation Delays The performance of the proposed space–time receiver is also studied for different propagation delays between transmitted data streams. We average the BER over 400 channel realizations. For each channel realization, we randomly generate a normalized frequency offset (normalized by the number of OFDM tones) within the range of −0.5 to 0.5. The propagation delays between the two transmitted data streams vary from 0 to 12 samples; however, the BER curves under different propagation delays are close to each other. We then conclude that the space–time receiver is robust to different ranges of propagation delays between users (Fig. 7). VI. C ONCLUSION Fig. 6. Plot of BER versus SNR for the proposed space–time receiver with two transmit data streams with different frequency offsets. We select the equalizer memory length L = 12 and use QPSK modulation. The numbers of channel taps from the two transmit streams to the receive antennas are both six. D. BER Comparison for Different Frequency Offsets We study the performance of the proposed space–time receiver with different frequency offsets in Fig. 6. The propagation delay difference between transmitted data streams is set to be 0, and we average the simulation results over 400 channel realizations. We use three sets of frequency offsets in the simulation. The first pair of frequency offsets for the first and second transmitted streams is [0.045, −0.036], respectively, and the other two pairs are set to be [0.45, −0.36] and [5.45, 4.64], respectively. The BER performance for the three pairs of frequency offsets is very similar. Thus, we can conclude that the space–time receiver is robust to the different range of frequency offsets. In this paper, we have studied the receiver design for asynchronous MIMO-OFDM systems. We have proposed an equalizer design algorithm when both channel information and synchronization parameters are available to the receiver. We have also proposed an MMSE joint frequency and equalization delay estimation algorithm using training sequences. A training-based equalizer design algorithm has been studied to refine the design to the space–time equalizers after MMSE synchronization. Simulation results under various asynchronous scenarios have shown good performance for our algorithms. A PPENDIX I ZFIC A LGORITHM A. Derivatives of the Cost Function of Min-SER We denote the real part of any quantity as (·)r or Re{·} and the imaginary part as (·)i or Im{·}. From (23), given a selection matrix Ci , we can write the derivative of the cost Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. 3002 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 5, SEPTEMBER 2008 function with respect to tr as those given in (A.1)–(A.4), shown i at the bottom of the page. We also have the derivatives of the matrix Wi,j with respect to the real and imaginary parts of the lth component wi (l) of the vector wi , i.e., ⎡ 0 ∂Wi,j ⎣. = . . ∂wi (l)r 0 ⎡ 0 ∂Wi,j . =⎣. . ∂wi (l)i 0 ... . . . ... ... . . . ... 1 .. . ... i .. . ... ... .. . 1 ... .. . i ⎤ 0 . ⎦ . . ... ⎤ 0 . ⎦ . . . ... B. Main Iteration of the Steepest Descent With the Armijo Rule With Polynomial Line Search Define the gradients of the cost function gi (ti , Ci ) with respect to the real and imaginary parts of ti as ∂gi (ti , Ci )/∂tr i and ∂gi (ti , Ci )/∂ti . At the mth iteration, we update a new i value for ti according to tr [m + 1] = tr [m] − δm i i ∂gi (ti , Ci ) ∂tr i ∂gi (ti , Ci ) ∂ti i (A.13) (ti [m]) (A.5) (A.6) ti [m + 1] = ti [m] − δm i i . (ti [m]) (A.14) ¯ Let (q(p) )T be the pth row of the matrix Q. Thus ∂ |fi,i (p)|2 = 2fi,i (p)r Re r ∂wi + 2fi,i (p)i Im ∂ |fi,i (p)|2 = −2fi,i (p)r Im i ∂wi + 2fi,i (p)i Re T The step size δm is selected based on the Armijo rule with polynomial line search [33]. We count iterations when the step size δm varies according to the Armijo rule. C. Computational Complexity of Each Iteration q(p) Ci HT T q(p) T Ci HT (A.7) q(p) q(p) Ci HT T Ci HT (A.8) r ∂wi s r = Re {Ui (:, 2 : (De + 1))} ∂ti i ∂wi s r = Im {Ui (:, 2 : (De + 1))} ∂ti r ∂wi = −Im {Us (:, 2 : (De + 1))} i ∂ti i i ∂wi = Re {Us (:, 2 : (De + 1))} . i i ∂ti (A.9) (A.10) To simplify the analysis, we assume that all tones are used; thus, |Si | = N . The terms Gi (p, p) can be evaluated based on fast Fourier transform (FFT). By [34], we can compute the FFT in time Θ(N lgN ), where Θ(·) denotes an asymptotically tight bound, and lg denotes base-two logarithmic. Computing Θ( i , N, 0)∗ Wi,j requires time O(N (N + L)), where O(·) denotes an asymptotic upper bound. The computational cost to compute all Gi (p, p) is O(Mr N (N + L)) + Θ(Mr (N + L)N lgN ) ≈ Θ(Mr N 2 lgN ). We can also obtain that it takes time O(Mr (L + 1)(L + ν + 1) + N lgN ) to compute fi,i (p). Assuming that N > L + ν + 1, the cost to evaluate gi (ti , Ci ) is O(Mr N 2 lgN ). Similarly, we can find that the cost for 2 evaluating the gradient is O(Mr (L + 1)N 2 lgN ). Therefore, the computational cost of each iteration is on the order of 2 O(Mr (L + 1)N 2 lgN ). A PPENDIX II C OMPUTATIONAL C OMPLEXITY OF THE M AX P WR M ETHOD ¯ ¯ The target matrix (Us )H H∗ Ci QH T i QCi HT Us is of dii i mension (Mr (L + 1) − Td ) × Mr (L + 1). Assuming that all (A.11) (A.12) ∂gi (ti , Ci ) 1 = r ∂ti |Si | p∈Si 1 |fi,i (p)|2 1 |fi,i (p)|2 r i ∂Gi (p, p) ∂wi ∂Gi (p, p) ∂wi + r i ∂wi ∂tr ∂tr ∂wi i i − Gi (p, p) |fi,i (p)|4 Gi (p, p) |fi,i (p)|4 r i ∂ |fi,i (p)|2 ∂wi ∂ |fi,i (p)|2 ∂wi + r i ∂wi ∂tr ∂tr ∂wi i i (A.1) 1 ∂gi (ti , Ci ) = i |Si | ∂ti p∈S Mr i ∂Gi (p, p) r ∂wi r ∂wi ∂ti i + ∂Gi (p, p) i ∂wi i ∂wi ∂ti i − ∂ |fi,i (p)| r ∂wi 2 r ∂wi ∂ti i + ∂ |fi,i (p)| i ∂wi 2 i ∂wi ∂ti i (A.2) ∂Gi 2 = 2σn QΘ( i , N, 0)∗ Re ∂wi (l)r j=1 r ∂Gi 2 = 2σn QΘ( i , N, 0)∗ Re ∂wi (l)i j=1 ∂Wi,j WH ∂wi (l)r i,j ∂Wi,j WH ∂wi (l)i i,j Θ( i , N, 0)QH Θ( i , N, 0)QH (A.3) M (A.4) Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. TANG AND HEATH: SPACE–TIME RECEIVER WITH JOINT SYNCHRONIZATION AND INTERFERENCE CANCELLATION 3003 tones are used, evaluating (Us )H H∗ Ci Ci HT Us requires time i i Θ((Mr (L + 1) − Td )Mr (L + 1)(LCP + 1)). We need time Θ((Mr (L + 1) − Td )3 ) to compute the eigenvector of the target matrix. Overall, the computational cost is on the order 3 of O(Mr (L + 1)3 ). Compared with the ZFIC method, the MaxPwr method can yield much lower complexity. A PPENDIX III F REQUENCY O FFSET E STIMATION A LGORITHM A. Derivative of the MMSE Cost Function With Respect to the Frequency Offset For a given Δ, the characteristic polynomial of the matrix M( , Δ) is given as φ(λ, ) = det (λI − M( , Δ)) . (A.15) Notice that ∂M( , Δ)/∂ can be derived from the derivative of Pj,i ( , Δ, b) with respect to as ∂Pj,i ( , Δ, b) = QΘ(− , Δ, b)J(b)Rj (b, Δ) ∂ (A.23) where J = diag{−j2π(N +LCP )b, . . . , −j2π[(N +LCP )b+ N −1]}. B. Computational Complexity of the Proposed Algorithm The steepest descent algorithm that is used here is the same as the one described in Appendix I-B. We briefly describe the computational complexity result of the proposed algorithm at each iteration. Computing the matrix Mi ( , Δ) requires time Θ(N KMr (L + 1)(LCP + 1)). We also obtain that performing the eigendecomposition of the matrix Mi ( , Δ) takes time Θ((LCP + 1)3 ). Overall, for each evaluation of the cost function, we need a computational time of Θ(N KMr (L + 1)(LCP + 1)). Similarly, we need time Θ(N KMr (L + 1)(LCP + 1)) to compute ∂M( , Δ)/∂ . Therefore, at each iteration, the computational cost is on the order of Θ(N KMr (L + 1)(LCP + 1)). A PPENDIX IV T RAINING -B ASED E QUALIZER D ESIGN A. Derivative of the Training-Based Equalizer Design Let us define the real and imaginary parts of the vector pi as pr and pi , and gi = [1, pT ]T . We assume T i = I here for i i i simplicity. The result for T i = I follows similarly. We denote H γ(pi ) = gi Ki gi ˆ ˆ Let λ be the smallest eigenvalue of the matrix M( , Δ) and z be the corresponding eigenvector, which must satisfy the following characteristic equation: ˆ φ(λ, ) = 0. (A.16) Taking the derivative to the implicit function with respect to , we have ∂φ ∂λ ∂φ + = 0. ∂λ ∂ ∂ Thus, we have ˆ ∂φ ∂λ =− ∂ ∂λ −1 ˆ λ=λ (A.17) (A.24) ∂φ ∂ . ˆ λ=λ (A.18) Notice that we assume that ∂φ/∂λ|λ=λ = 0 here, and it is ˆ equivalent to having the smallest eigenvalue without multiplicity. In particular ∂φ ∂λ ∂φ ∂ ˆ = Tr adj λI − M( , Δ) ∂M( , Δ) ˆ = −Tr adj λI − M( , Δ) ∂ (A.19) (A.20) where we drop the parameters to simplify the notation. From (47), the derivative of the cost function with respect to pr and i pi can be derived as i r i ∂γ(pi ) ∂γ(pi ) ∂fi,i ∂γ(pi ) ∂fi,i = r r r + i ∂pi ∂fi,i ∂pi ∂fi,i ∂pr i r i ∂γ(pi ) ∂γ(pi ) ∂fi,i ∂γ(pi ) ∂fi,i = + . r i ∂fi,i ∂pi ∂pi ∂fi,i ∂pi i i i (A.25) ˆ λ=λ (A.26) ˆ λ=λ ¯ ¯ Ki ( i , Δi , bi,i ) = DH Λi − Λi Pi ( i , Δi ) i ¯ × Pi ( i , Δi )H Λi Pi ( i , Δi ) ¯ · Pi ( i , Δi )H Λi Di . −1 where Tr(·) denotes the trace of any matrix, and adj(·) stands for the classical adjoint of any matrix. We can obtain that ˆ ˆˆ adj(λI − M( , Δ)) = zzH since ˆ ˆ ˆˆ zzH λI − M( , Δ) = O = det λI − M( , Δ) · I Simply, we have ˆ ∂M( , Δ) ∂λ ˆ ˆ = zH z. ∂ ∂ (A.22) (A.21) (A.27) Let ki (b) = [(vi (0, b)∗/fi,i (0)), . . . , (vi (N −1, b)∗/fi,i (N − 1))]H and vi (b) = [vi (0, b), . . . , vi (N − 1, b)]T . Thus K−1 γ(pi ) = b=0 vi (b) −1 2 − ki (0)H , . . . , ki (K −1)H H ¯ × Pi PH Λi Pi i PH ki (0)H , . . . , ki (K −1)H i (A.28) Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. 3004 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 5, SEPTEMBER 2008 ∂γ(pi ) = −2Re ∂fi,i (l)r ∂ki (0)H ∂ki (K − 1)H ¯ ,..., Pi PH Λi Pi i ∂fi,i (l)r ∂fi,i (l)r −1 −1 PH ki (0)H , . . . , ki (K − 1)H i −1 H ¯ − ki (0)H , . . . , ki (K − 1)H Pi PH Λi Pi i PH i ¯ ∂ Λi ¯ Pi PH Λi Pi i ∂fi,i (l)r −1 PH ki (0)H , . . . , ki (K − 1)H i H (A.29) ∂γ(pi ) = −2Re ∂fi,i (l)i ∂ki (0)H ∂ki (K − 1)H ¯ ,..., Pi PH Λi Pi i i ∂fi,i (l) ∂fi,i (l)i −1 PH ki (0)H , . . . , ki (K − 1)H i −1 H ¯ − ki (0)H , . . . , ki (K − 1)H Pi PH Λi Pi i PH i ¯ ∂ Λi ¯ Pi PH Λi Pi i ∂fi,i (l)i PH ki (0)H , . . . , ki (K − 1)H i H (A.30) where vi = [vi (0), . . . , vi (N − 1)]T are the transmitted data of the ith stream. We can derive (A.29) and (A.30), shown at the top of the page, where ∂ki (b) vi (l, b)∗ = 0, . . . , − ,...,0 ∂fi,i (l)r fi,i (l)2 H R EFERENCES [1] D. Gesbert, M. Shafi, D.-S. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: An overview of MIMO space–time coded wireless systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 281–302, Apr. 2003. [2] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, “Capacity limits of MIMO channels,” IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 684– 702, Jun. 2003. [3] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [4] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003. [5] F. Daffara and A. Chouly, “Maximum likelihood frequency detectors for orthogonal multicarrier systems,” in Proc. Int. Conf. Commun., Jun. 1993, pp. 766–771. [6] M. Morelli, A. N. D’Andrea, and U. Mengali, “Feedback frequency synchronization for OFDM applications,” IEEE Commun. Lett., vol. 5, no. 1, pp. 28–30, Jan. 2001. [7] E. Chiavaccini and G. M. Vitetta, “Maximum-likelihood frequency recovery for OFDM signals transmitted over multipath fading channels,” IEEE Trans. Commun., vol. 52, no. 2, pp. 244–251, Feb. 2004. [8] R. Mehlan, Y.-E. Chen, and H. Meyr, “A fully digital feedforward MSK demodulator with joint frequency offset and timing estimation for burst mode mobile radio,” IEEE Trans. Veh. Technol., vol. 42, no. 4, pp. 434– 443, Nov. 1993. [9] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613–1621, Dec. 1997. [10] J. J. van de Beek, P. O. Borjesson, M.-L. Boucheret, D. Landstrom, J. M. Arenas, P. Odling, C. Ostberg, M. Wahlqvist, and S. K. Wilson, “A time and frequency synchronization scheme for multiuser OFDM,” IEEE J. Sel. Areas Commun., vol. 17, no. 11, pp. 1900–1914, Nov. 1999. [11] H. Liu and U. Tureli, “A high-efficiency carrier estimator for OFDM communications,” IEEE Commun. Lett., vol. 2, no. 4, pp. 104–106, Apr. 1998. [12] M. Morelli and U. Mengali, “Feedforward carrier frequency estimation with MSK-type signals,” IEEE Commun. Lett., vol. 2, no. 8, pp. 235–237, Aug. 1998. [13] A. Stamoulis, S. N. Diggavi, and N. Al-Dhahir, “Intercarrier interference in MIMO OFDM,” IEEE Trans. Signal Process., vol. 50, no. 10, pp. 2451–2464, Oct. 2002. [14] T. A. Thomas and F. W. Vook, “Asynchronous interference suppression in broadband cyclic-prefix communications,” in Proc. IEEE Wireless Commun. Netw. Conf., Mar. 2003, pp. 568–572. [15] M. B. Breinholt, M. D. Zoltowski, and T. A. Thomas, “Space– time equalization and interference cancellation for MIMO-OFDM,” in Proc. Asilomar Conf. Signals, Syst. Comput., Nov. 2002, vol. 2, pp. 1688–1693. [16] T. Tang and R. W. Heath, Jr., “Space–time interference cancellation in MIMO-OFDM systems,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1802–1816, Sep. 2005. (A.31) H ∂ki (b) vi (l, b)∗ = 0, . . . , −i ,...,0 i ∂fi,i (l) fi,i (l)2 ¯ ∂ Λi = IK×K ⊗ diag {0, . . . , 2fi,i (l)r , . . . , 0} ∂fi,i (l)r ¯ ∂ Λi = IK×K ⊗ diag 0, . . . , 2fi,i (l)i , . . . , 0 . ∂fi,i (l)i Also, we have r ∂fi,i ¯ = Re Q (:, 2 : (LCP + 1)) ∂pr i i ∂fi,i ¯ = Im Q (:, 2 : (LCP + 1)) ∂pr i r ∂fi,i ¯ = −Im Q (:, 2 : (LCP + 1)) ∂pi i i ∂fi,i ¯ = Re Q (:, 2 : (LCP + 1)) . ∂pi i (A.32) (A.33) (A.34) (A.35) (A.36) (A.37) (A.38) B. Computational Cost of the Proposed Algorithm 2 Notice that we need time Θ(N KMr (L + 1)2 ) to evaluate Ki ( i , Δi , pi ). Therefore, computing the cost function 2 γ(pi ) requires time Θ(N KMr (L + 1)2 ). We comment that the steepest descent algorithm that is used for training-based equalizer design is the same as the one described in Appendix I-B. It requires an additional time of Θ(N KMr (L + 1)) to compute the derivatives at each iteration. We neglect the details to derive this result; however, we point out that the values such as ¯ PH Λi Pi are available from the procedure of computing γ(pi ). i We can use those stored values when computing the derivatives. Last, we conclude that the computational time at each iteration 2 is on the order of Θ(N KMr (L + 1)2 ). Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply. TANG AND HEATH: SPACE–TIME RECEIVER WITH JOINT SYNCHRONIZATION AND INTERFERENCE CANCELLATION 3005 [17] Y. Fu, C. Tellambura, and W. A. Krzymien, “Transmitter precoding for ICI reduction in closed-loop MIMO OFDM systems,” IEEE Trans. Veh. Technol., vol. 56, no. 1, pp. 115–125, Jan. 2007. [18] IEEE 802.11n. [Online]. Available: http://grouper.ieee.org/groups/ 802/11 [19] IEEE 802.16 Task Group 3. [Online]. Available: http://grouper. ieee.org/groups/802/16/tg3/ [20] C. Tellambura, I. R. Johnson, Y. J. Guo, and S. K. Barton, “Equalization and frequency offset correction for HIPERLAN,” in Proc. Int. Symp. Pers., Indoor Mobile Radio Commun., Helsinki, Finland, Sep. 1997, pp. 796–800. [21] L. Tong, “Joint blind signal detection and carrier recovery over fading channels,” in Proc. Int. Conf. Acoust., Speech Signal Process., May 1995, vol. 2, pp. 1205–1208. [22] M. Morelli and U. Mengali, “Carrier-frequency estimation for transmissions over selective channels,” IEEE Trans. Commun., vol. 48, no. 9, pp. 1580–1589, Sep. 2000. [23] M. Luise, M. Marselli, and R. Reggiannini, “Low-complexity blind carrier frequency recovery for OFDM signals over frequency-selective radio channels,” IEEE Trans. Commun., vol. 50, no. 7, pp. 1182–1188, Jul. 2002. [24] Y. Yao and G. B. Giannakis, “Blind carrier frequency offset estimation of SISO, MIMO and multi-user OFDM,” IEEE Trans. Commun., vol. 53, no. 1, pp. 173–183, Jan. 2005. [25] A. Saemi, V. Meghdadi, J.-P. Cances, M. Reza Zahabi, and J.-M. Dumas, “ML time and frequency synchronization for MIMO-OFDM systems in unknown frequency selective fading channels,” in Proc. Int. Symp. Pers., Indoor Mobile Radio Commun., Helsinki, Finland, Sep. 2006, pp. 1–5. [26] O. Besson and P. Stoica, “On parameter estimation of MIMO flat-fading channels with frequency offsets,” IEEE Trans. Signal Process., vol. 51, no. 3, pp. 602–613, Mar. 2003. [27] T. Cui and C. Tellambura, “Joint channel and frequency offset estimation and training sequence design for MIMO systems over frequency selective channels,” in Proc. IEEE Global Telecommun. Conf., Dallas, TX, Nov. 29–Dec. 3, 2004, pp. 2344–2348. [28] T. Tang and R. W. Heath, Jr., “Joint frequency offset estimation and interference cancellation in MIMO-OFDM systems,” in Proc. IEEE Veh. Technol. Conf., Los Angeles, CA, Sep. 2004, pp. 1553–1557. [29] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space–Time Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2003. [30] S. Verdu, Multiuser Detection, 1st ed. Cambridge, U.K.: Cambridge Univ. Press, 1998. [31] K. Cho and D. Yoon, “On the general expression of one- and twodimensional amplitude modulations,” IEEE Trans. Commun., vol. 50, no. 7, pp. 1074–1080, Jul. 2002. [32] Y. Ding, T. N. Davidson, Z.-Q. Luo, and K. M. Wong, “Minimum BER block precoder for zero-forcing equalization,” IEEE Trans. Signal Process., vol. 51, no. 9, pp. 2410–2423, Sep. 2003. [33] C. T. Kelley, Iterative Methods for Optimization. Philadelphia, PA: SIAM, 1999. [34] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms. Cambridge, MA: MIT Press, 2001. Taiwen Tang (S’04) received the B.E. degree in electronic engineering from Beijing University of Posts and Telecommunications, Beijing, China, in 2001, the M.S. degree in electrical and computer engineering from the Colorado State University, Fort Collins, in 2003, and the Ph.D. degree in electrical and computer engineering from The University of Texas at Austin in 2006. From August 2001 to July 2003, he was a member of the Radar and Communications Group, Colorado State University, working on remote sensing with dual polarized meteorological radar systems. From August 2003 to May 2006, he was a Graduate Research Assistant with the Wireless Networking and Communications Group, The University of Texas at Austin. He conducted an internship with Beceem Communications Inc., Santa Clara, CA, participating in IEEE 802.16 d/e system development and standard contributions in the summer of 2004. From September 2007 to January 2008, he was a Firmware Engineer in the Microsoft Asia Center for Hardware, Shenzhen, China. Since May 2008, he has been a Digital Design Engineer in the Chengdu Goldtel Electronic Technology Company, Ltd., Chengdu, China. His research interests are in wireless digital communications and wireless networks with emphasis on MIMO-OFDM communications, scheduling algorithms, and ad hoc networks. Robert W. Heath, Jr. (S’96–M’01–SM’06) received the B.S. and M.S. degrees from the University of Virginia, Charlottesville, in 1996 and 1997, respectively, and the Ph.D. degree from Stanford University, Stanford, CA, in 2002, all in electrical engineering. From 1998 to 2001, he was a Senior Member of the Technical Staff, and then became a Senior Consultant, with Iospan Wireless Inc., San Jose, CA, where he worked on the design and the implementation of the physical and link layers of the first commercial MIMO-OFDM communication system. In 2003, he founded MIMO Wireless Inc.: a consulting company that is dedicated to the advancement of MIMO technology. Since January 2002, he has been with the Department of Electrical and Computer Engineering, The University of Texas at Austin, where he is currently an Associate Professor and a member of the Wireless Networking and Communications Group. His research interests cover a broad range of MIMO communication, including limited feedback techniques, multihop networking, multiuser MIMO, antenna designs, and scheduling algorithms, as well as 60-GHz communication techniques. Dr. Heath is a member of the Signal Processing for Communications Technical Committee in the IEEE Signal Processing Society. He has been an Editor for the IEEE TRANSACTIONS ON COMMUNICATION and an Associate Editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. He was a Technical Cochair for the Fall 2007 Vehicular Technology Conference. He is the General Chair of the 2008 Communication Theory Workshop and is a Coorganizer of the 2009 Signal Processing for Wireless Communications Workshop. He was the recipient of the David and Doris Lybarger Endowed Faculty Fellowship in Engineering. He is a registered Professional Engineer in Texas. Authorized licensed use limited to: HANGZHOU DIANZI UNIVERSITY. Downloaded on January 18, 2009 at 01:37 from IEEE Xplore. Restrictions apply.