Twireless_first_pre_DFT[1]

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004117Symbol-Based Space Diversity for Coded OFDM SystemsDefeng (David) Huang, Student Member, IEEE and Khaled Ben Letaief, Fellow, IEEEAbstract—The authors present a coded orthogonal frequency division multiplexing (COFDM) system with multiple-input/multiple-output (multiple transmit and/or multiple receive) antennas for high-rate wireless data transmission. A symbol-based space diversity technique, which can take advantage of the inherent space diversity, is proposed. In contrast to conventional subcarrier-based space diversity, it is shown that the proposed technique can be implemented using only one discrete Fourier transform block and the same weighting coefficients for the whole OFDM symbol. This significantly reduces the system complexity while achieving almost the same diversity order as that of the traditional space diversity approach. They also propose an iterative algorithm to obtain the antenna weighting coefficients. Simulation results show that the proposed algorithm converges fast and approaches the global optimal solution for most channel realizations. It is also shown that, when the proposed technique is employed in a time division duplex scenario, where the uplink and downlink channels are reciprocal, the system complexity can be further reduced. Index Terms—Broad-band wireless communications, multipleinput/multiple-output (MIMO) systems, orthogonal frequency division multiplexing (OFDM), space diversity.I. INTRODUCTIONORTHOGONAL frequency division multiplexing (OFDM) [1]–[3] is an effective technology for robust and reliable high-rate and high-speed data transmission in the next-generation wireless communication systems because of its spectrum efficiency and ability to mitigate the effects of delay spread and intersymbol interference. As a result, OFDM has been receiving remarkable attention and it now forms the basis of key wireless local area network (LAN) standards such as ETSI-BRAN HIPERLAN/2 and IEEE 802.11a [4], [5]. Multiple transmit and/or multiple receive [multiple-input/multiple-output (MIMO)] [6]–[8] antennas have also recently attracted much attention because they have the potential to provide an enormous increase in the capacity of wireless communication systems. In particular, MIMO antennas can be used with OFDM to improve system capacity and quality. Conventionally, subcarrier-based space diversity [9]–[11] isManuscript received June 22, 2002; revised October 9, 2002 and December 16, 2002; accepted December 18, 2002. The editor coordinating the review of this paper and approving it for publication is I. Collings. This work was supported in part by the Hong Kong Telecom Institute of Information Technology and the Hong Kong Research Grant Council. This paper was presented in part at the IEEE Global Telecommunications Conference, Taipei, Taiwan, November 2002. The authors are with the Center for Wireless Information Technology, Electrical and Electronic Engineering Department, Hong Kong University of Science and Technology, Kowloon, Hong Kong (e-mail: huangdf@; eekhaled@t.hk). Digital Object Identifier 10.1109/TWC.2003.821133employed since space diversity achieved by MIMO antennas is effective over a flat fading channel. However, the subcarrier-based space diversity induces great complexity because multiple discrete Fourier transform (DFT) blocks, one for each antenna, are required and the complexity of DFT blocks is often a major concern in system implementation [12], [13]. Recently, some techniques [14], [15], [17], [18], which reduce the OFDM system implementation complexity by reducing the number of required DFT blocks, have been proposed. In [14], it was shown that the number of DFT blocks could be reduced by 50% at the expense of a 3-dB performance degradation. Motivated by [14], [15] put forward a general architecture for an OFDM receiver, which can further reduce the number of DFT blocks based on the theory of orthogonal designs [16]. For coded OFDM (COFDM) systems with multiple receive antennas, a scheme, which can be seen as a special case of the system proposed in this paper, was put forward in [17]. It only uses one DFT block at the expense of a little performance degradation. Because the antenna weighting and combining is done before the DFT processing, the technique is referred to as pre-DFT combining space diversity. Reference [18] extended the analysis of the system to an environment with cochannel interference (CCI). All the techniques mentioned above are about complexity reduction for OFDM systems with multiple receive antennas and only one transmit antenna. To the best of our knowledge, however, few studies have been devoted to the complexity reduction in OFDM systems with MIMO antennas. In this paper, a symbol-based space diversity technique, which can reduce the complexity of MIMO/OFDM systems, is presented and discussed. This technique only requires one DFT block at the receiver and one inverse DFT (IDFT) block at the transmitter with all antennas weighting coefficients kept constant during one OFDM frame. In contrast to the subcarrier-based space diversity, where each subcarrier requires an eigenvalue decomposition (EVD) as well as a matrix multiplication to obtain the weighting coefficients [9], the proposed method only requires that the weighting coefficients be calculated only once for the whole OFDM symbol. As a result, significant complexity reduction is further obtained. Compared with the study in [17], where only one transmit antenna is considered and the justification is only given for repetition codes, we consider multiple transmit antennas and prove that the proposed technique can be applied for any coding schemes as long as an interleaver is employed. It is shown that, by using more transmit antennas, system performance can be significantly improved. We also propose an iterative algorithm to optimize the antenna weighting coefficients, which is shown1536-1276/04$20.00 © 2004 IEEE118IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004sl (1)Intput Code cn and Interleave IDFT#1 #2 ... #FOutputsl Add Guard Intervalsl (2) s (F)lψ1ψ 2 ... ψ FDiversity Weights(a) #1 #2...r (1)lrl ( 2 ) rl( M )ω1 ω 2 ... ω MDiversity Weightsrl '#MRemove Guard IntervalDFTvnDeinterleave and Decode(b)Fig. 1. Block diagram of the coded OFDM system with MIMO antennas: (a) transmitter and (b) receiver.to be of low complexity and converges fast. By extensive simulations, it is also observed that the proposed algorithm gives a global optimal solution for most channel realizations. In addition the above contributions, this paper investigates the case when the proposed technique is employed in a time-division duplex (TDD) environment. Generally, the weighting coefficients can be obtained by running an iterative algorithm at the receiver, where the channel state information can be available by channel estimation. The transmit weighting coefficients are then sent back to the transmitter for further processing. When the proposed system is employed in a TDD environment, where the uplink and downlink channels are reciprocal, the transmit and receive weighting coefficients can be achieved using the signal covariance matrix (instead of the channel state information) and optimized at the base station and mobile station independently. As a result, the system complexity can be further drastically reduced while achieving reliable performance. The rest of this paper is organized as follows. In Section II, the base-band model of the COFDM system with MIMO antennas is described. The iterative algorithm, which optimizes the antenna weighting coefficients, is discussed in Section III. Simulation results and discussions are provided in Section IV. Finally, our conclusion is given in Section V. II. SYSTEM MODEL We consider a MIMO/OFDM system as shown in Fig. 1, where a coding scheme and an interleaver are employed to achieve frequency domain diversity. Assume that the number of OFDM subcarriers is . Then, the signal after the IDFT processing at the transmitter is given by (1) is the number of samples in the guard interval, is the associated coded quadrature amplitude mod. We ulation (QAM) symbol of the th subcarrier, and to keep high transmission efficiency. assume that whereAssume that there are transmit antennas at the base station receive antennas at the mobile station. After weighting, and the signal at each transmit antenna is given by (2) where is the antenna weighting coefficient at the th transmit antenna. The received signal at the th receive antenna is given by (3) where denotes the convolution product, denotes the channel impulse response (CIR) between the th transmit antenna and the th receive antenna, and denotes the additive white Gaussian noise (AWGN) component at the th receive antenna. Here, we assume that has a nonzero value , where is the maximum lag only for the duration of the CIR. After weighting and combining, the received signal is given by (4) where is the weighting coefficient for the th receive antenna. We assume that the weighting coefficients and are normalized. That is (5) and (6) Note that is normalized to guarantee that the total transmitted power remains unchanged.HUANG AND LETAIEF: SYMBOL-BASED SPACE DIVERSITY FOR CODED OFDM SYSTEMS119We further assume that the CIRs decay to zero during the . After guard interval cyclic extension, or removal, the weighted and combined signal is then applied to a DFT processor. The output of the DFT processor is given byAfter weighting and combining, the received codeword is vector (in the frequency domain) as follows: given by an(14) For convenience, we let (15) (7) and where (8) is the channel frequency response for the th subchannel between the th transmit antenna and the th receive antenna and (9) is the discrete Fourier transformed AWGN component at the th receive antenna. We note here that the Fourier transform of an AWGN process is also an AWGN process. It is assumed that the coding and interleaving are across one whole OFDM symbol. Then, the codeword can be represented vector , where in the by an superscript stands for transpose. For the convenience of mathematical development, we use the following notation: Then (17) We note here that the covariances of the noise term remain the since . same as those of the noise term When the maximum likelihood decoding criterion is employed, the pair-wise error probability (PEP) can be used to denote the system performance [19], which is further determined by the pair-wise codeword distance. The PEP of deciding erroneously instead in favor of a coded sequence of the transmitted coded sequence , conditioned on the channel , is given by (16)(18) where is the average energy of the coded symbol and the is given by pair-wise codeword distance (19)(10) where for (11) and the vector (12) The received OFDM symbol for the th receive antenna can vector then be represented in the frequency domain by an as follows:where in the superscript denotes conjugate transpose. When an ideal interleaver is employed, we have the following: (20) where is a constant that relates to the specific codeword pair is the statistical average of . As a result, at the reand ceiver, the average pair-wise codeword distance is given by(13) is an vector that stands for the additive inwhere denoting a dependent white Gaussian noise term with diagonal matrix with the main diagonal elements given by .(21)120IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004where in the superscript stands for complex conjugate. From (10)–(12), we haveWe note, here, that is an Hermitian matrix. Using these definitions, (25) can be rewritten as (30)(22) By substituting (22) into (21), the average pair-wise codeword distance iswhere is the Kronecker product and matrix. Let us further defineis anidentity(31) is the covariance matrix of the CIRs when We note here that multiple transmit antennas are equivalent to one transmit anis also an Hermitian matrix. Then, (30) becomes tenna, and (32)(23) When the codeword is across multiple OFDM symbols, say OFDM symbols, by replacing in (19) with , where are the components of the codeword at the th OFDM symbol, and using the following (assume that an ideal interleaver is used): (24) we can still get (23) following similar steps as those from (19)–(22). Next, recall that the larger the pairwise codeword distance, the better the FER performance. Hence, to achieve a better perand to maximize formance, we should select appropriate the average pair-wise codeword distance. As a result, we need to maximizeSimilarly, (25) can be rewritten as follows: (33) where is an identity matrix, and (34) The entries of the matrix are given by (35) Let us now further define (36) Then, (33) can be rewritten as (37) Next, we will develop an algorithm to obtain the antenna weighting coefficients both at the transmitter and receiver. III. TRANSMIT AND RECEIVE ANTENNA WEIGHTING COEFFICIENTS In this section, we derive and which maximize in (25), (32), or (37). That is, we want to find(25) For the convenience of mathematical development, we rewrite (25) into a vector form. Let us define the weighting coefficients vectors as (26) and (27) The covariance matrix of the CIRs of all the antennas is defined as (28) where is an matrix with entries (29)(38) represents the squared Euclidean norm of . where in (32) is known. Then, it is well Let us first assume that known that the optimal is the dominant eigenvector of (This fact is used in both [9] and [17]. One can also refer [24] to in (37) is known, then the optimal prove it.) Similarly, if is the dominant eigenvector of . Based upon this, we present a weighting coefficients estimation algorithm to obtain the weighting coefficients as follows. Weighting Coefficients Estimation Algorithm: , and iniInitialization: Set ( can be any tialize other normalized vector).HUANG AND LETAIEF: SYMBOL-BASED SPACE DIVERSITY FOR CODED OFDM SYSTEMS121Transmit#1 #2...ψ1 ψ2 ... ψFCovariance MatrixDiversity Weightsψ1 ψ2... ψFAdd/Remove Guard IntervalIDFT/DFTEncode/ DecodeInput/Output Data#F Duplexer ReceiveFig. 2. Proposed TDD-OFDM system structure at either end of the communication link.Loop: Step 1: Set obtained by eigenvector Step 2: Set is obtained eigenvector. is finding the dominant . of . by finding the dominant . ofGo back to Loop until the difference between the dominant eigenvalues in adjacent steps is below a predefined value. We note that the proposed algorithm converges because the dominant eigenvalue is nondecreasing along with the progress of the algorithm (based upon the Nondecreasing Theorem [23]). A rigorous proof is given in Appendix A. In the following, we discuss the asymptotic optimality of the , (25) is reduced to proposed algorithm. When (39) As a result, the global optimal solution to is the dominant . In the first iteration of the proposed eigenvector of . Therefore, our proposed algorithm can algorithm , the give a global optimal solution. Similarly, when global optimal solution for is the dominant eigenvector . Since in the first iteration of of the proposed algorithm, our algorithm also gives a global optimal solution. When both and are not equal to one, it is very difficult to analyze the asymptotic optimality of the proposed algorithm. Instead, we employ extensive simulations to investigate it (see Appendix B). For each channel realization, we randomly (i.e., ). set the value of the initial normalized vector We then run our iterative algorithm and see if the dominant converge to the same value. If they do, eigenvalues of we conclude that the proposed algorithm does converge to the global optimum as long as there are enough tests. From our observations, we find that our algorithm does not always converge to the global optimum solution (i.e., for some , the algorithm only converges to a local minimum). Nonetheless,based upon extensive simulations, we found that for most channel realizations, the dominant eigenvalue does converge to the global optimal solution for any . This observation can be further justified by the bit-error rate (BER) performance that is shown and discussed in Section IV. Next note that when the uplink CIRs and downlink CIRs only deviate by a common factor (a complex attenuation), the weighting coefficients in the proposed algorithm do not change (because the eigenvector is normalized). As a result, the workload of the radio frequency (RF) chains calibration [20] is reduced. In general, the weighting coefficients estimation algorithm requires channel estimation to obtain the covariance matrix and . Channel estimation is normally done at the receiver, and both the transmit and receive antenna weighting coefficients can also be derived there. The transmit antenna weighting coefficients can then be sent to the transmitter. By observing (25), it can be found that the transmit antenna weighting coefficients and the receive antenna weighting coefficients are exchangeable. Therefore, when the uplink channel and the downlink channel are the same, the weighting coefficients are the same for both transmitting and receiving at either end of the link. In a TDD-OFDM system, the uplink and downlink use the same physical channel and can be assumed to be identical [21]. In this case, the same antenna weighting coefficients can be applied for both transmitting and receiving (as shown in Fig. 2). Furthermore, the dominant eigenvector and can be obtained independently at either end of (base station or mobile station) of the communication link. and In fact, we do not need to calculate the matrix directly. When the spectrum of the transmitted signal is white, the dominant eigenvectors of and are equal to the dominant eigenvectors of the signal covariance matrix [17] at the base station and mobile station, respectively. The signal covariance matrix is defined by (40)(or ) is the received signal vector at either end of the communication link and is the number of training sampleswhere122IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004Base Station Mobile StationFig. 3.Preamble DownlinkReceive...Downlink Data100F=1 F=2 F=4 F=8ReceiveCumulative ProbabilityPreamble Uplink...Receive10-1Proposed system frame format for the downlink data transmission.to calculate the signal covariance matrix during each training. Because of the use of the signal covariance matrix instead of and , channel estimation will not be required. Thus, the system complexity can be drastically reduced. Fig. 3 shows the frame format of a TDD-OFDM system. For every burst of data, some training sequences (the preamble) are first transmitted interactively in order to estimate the domand by calculating the domiinant eigenvectors of nant eigenvector of the signal covariance matrix . It is worth noting that the request-to-send (RTS) and clear-to-send (CTS) mechanism of 802.11 can be regarded as a special case of the above TDD-OFDM system. Since the only requirement is that the signal spectrum is white, some system configuration information can still be transmitted during the training period. We note, here, that because iterative training is required, the TDD-OFDM system is more appropriate for use in indoor environments where the link transmission delay is relatively short. We also note that such a system structure guarantees graceful evolution. That is, the design of mobile stations will not depend on the number of transmit antennas at the base station, and vice versa. IV. SIMULATION RESULTS AND DISCUSSION The OFDM system simulated is a TDD-OFDM system with the frame structure as shown in Fig. 3. The TDD-OFDM system has 64 subcarriers, and each subcarrier is differential QPSK (DQPSK) modulated. DQPSK instead of QPSK is employed because DQPSK does not require channel estimation by using the previous OFDM symbol as a reference. We also assume that the guard interval is longer than the maximum channel delay spread. Throughout this section, and unless otherwise mentioned, a quasistatic equal gain two-ray Rayleigh-fading channel model is assumed. This implies that the CIRs do not change during the iterative calculation of the antenna weighting coefficients. For each bit-error probability calculation, we use over 3000 channel realizations. We take an OFDM system with the subcarrier-based space diversity as a baseline system. In the baseline system, the coding scheme and the modulation scheme are the same as those in the proposed OFDM system. Perfect channel information is always assumed to be known in the baseline system. As in [9], for subcarrier , the transmit antenna weighting coefficients vector, , is equal to the dominant eigenvector of the denoted by , where the matrix is used to matrix denote the channel matrix for the th subcarrier. The receive antenna weighting coefficients vector for the th subcarrier is . For the symbol-based space diversity, the equal to weighting coefficients are obtained by calculating the dominant eigenvector of the signal covariance matrix.10-210 –20-3–15–10–505Dominant eigenvalue of the matrix G' [dB]Fig. 4. Cumulative probability density of the dominant eigenvalue of the after antenna weighting and combining over a 12-ray exponentially matrix = 2. decaying Rayleigh-fading channel whenGMA. Symbol-Based Space Diversity for OFDM System With Convolutional Coding From (23), (25), and (32), it follows that the dominant eigenvalue of the matrix is proportional to the average pair-wise codeword distance (after weighting and combining). Thus, in Fig. 4, we plot the cumulative probability distribution function (after ten iterations in (CDF) of the dominant eigenvalue of the proposed iterative algorithm) to denote the system performance. The channel model used is a 12-ray exponentially decaying Rayleigh-fading channel. The root mean square delay spread of the channel is set to one time-domain OFDM sample. When the cumulative probability is equal to 10 , it can be seen that doubling the number of transmit antennas achieves 3 dB in signal-to-noise ratio (SNR) performance. The proposed OFDM system with a convolutional code is as shown in Fig. 1, and the main parameters in our simulations are listed in Table I. The interleaver is an 8 16 block interleaver. Thus, after coding and interleaving, each bit in the codeword is scattered across the whole OFDM symbol. Fig. 5 shows the BER performance of the proposed OFDM equal gain system over a two OFDM sample spaced two-ray Rayleigh-fading channel. It can be seen that the performance between the subcarrier-based space diversity scheme and the symbol-based space diversity scheme is quite limited. For example, for two transmit antennas and two receive antennas, the performance degradation is only about 1.8 dB when . It can also be observed from the slopes of the curves in Fig. 5 that the space diversity order achieved by the symbol-based space diversity is the same as that of the subcarrier-based space diversity. By comparing Fig. 5(a) and (b), it can be also observed that the performance gap between the symbol-based space diversity scheme and the subcarrier-based . Note that when space diversity scheme is larger when , the proposed scheme is exactly the same scheme as proposed in [17]. By increasing from one to two or four, Fig. 5 shows that system performance is significantly increased.HUANG AND LETAIEF: SYMBOL-BASED SPACE DIVERSITY FOR CODED OFDM SYSTEMS123TABLE I MAIN PARAMETERSIN THE PROPOSED OFDM CONVOLUTIONAL CODE100SYSTEM WITH64AEb/N0=6dB Eb/N0=9dB10-1 F=1Number of sub-carriers Guard interval Modulation mode Number of training sequences for each frame Length of each training sequence Interleaving depth Code rate Constraint length Decoding algorithm Generators in octal100 Single antenna 10-1Bit Error Rate12 DQPSK 310-210-3 F=264 samples 81/210-410-57 Soft Viterbi (133,171)10-6 123456789101112Lag between two rays [sample]Fig. 6. BER performance of the proposed system versus the lag between the direct and delayed propagation ray when = 2. Subcarrier-based space diversity and the symbol-based space diversity schemes are represented by the dashed-dotted line and the solid line, respectively.MF=1,M=2 F=1,M=4 F=2,M=2 F=2,M=4100Single antennaBit Error RateF=2,M=1 F=4,M=1 F=2,M=2 F=4,M=210-210-1Bit Error Rate0 2 4 6 8 10 1210-310-210-410-310-4Eb/N0 per receive antenna branch [dB](a)100 Single antenna 10-1F=2,M=1 F=4,M=1 F=4,M=2024681012Eb/N0 per receive antenna branch [dB]Fig. 7. BER performance of the proposed OFDM system over a 12-ray exponentially decaying Rayleigh-fading channel. Subcarrier-based space diversity and the symbol-based space diversity schemes are represented by the dotted line and the solid line, respectively.Bit Error Rate10-210-310-4024681012Eb/N0 per receive antenna branch [dB](b)Fig. 5. BER performance of the proposed OFDM system over an equal = 3. Subcarrier-based space gain two-ray Rayleigh-fading channel with diversity and the symbol-based space diversity schemes are represented by the . (b) . dashed-dotted line and the solid line, respectively. (a)KF MF >MNext, we consider Fig. 6, which shows that the lags between the two CIR rays do not impact the BER performance significantly. , the performance degrades only because the channel For becomes a flat fading channel and no frequency diversity can be achieved. In Fig. 7, we show the BER performance of our proposed system over a 12-ray exponentially decaying Rayleigh-fading channel. The root mean square delay spread of the channel is set to one time-domain OFDM sample. From Fig. 7, it can be clearly observed that the system performance of the proposed scheme is improved significantly by employing more transmit antennas. By comparing Fig. 5 and Fig. 7, it can be seen that for the subcarrier-based space diversity, the performance over a 12-ray exponentially decaying channel is better than that over124IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004100Eb/N0=6dB Eb/N0=9dB100CNR=2dB CNR=6dB CNR=12dB CNR=20dB CNR=Inf10-1Bit Error Rate10-1Bit Error Rate10-3 10-2 10-110-210-210-310-3 10-410-4 012345678910Normalized doppler frequencyFig. 8. BER performance of the proposed system versus Doppler frequency normalized by an OFDM symbol duration when F = M = 2 and K = 3.Eb/N0 per receive antenna branch [dB]Fig. 9. BER performance of the proposed system over asymmetric channels when F = M = 2 and K = 3.1001 iteration 2 iterations 3 iterations 5 iterations 10 iterationsBit Error Ratean equal gain two-ray channel. In contrast, the performance of the proposed scheme remains relatively the same over the two channels. As a result, the performance gap of the proposed system compared with the subcarrier-based space diversity is larger over a 12-ray exponentially decaying channel than that over an equal gain channel. B. Channel Variations The simulation results presented so far assume that the channel is quasistatic. That is, the CIRs are time invariant during one transmission frame. In this part, we consider the channel variations caused by the movement of the communication objects and/or environment and other factors such as the ill-calibrated RF chains [20]. Fig. 8 shows the BER performance of the TDD-OFDM system with a convolutional code over fast Rayleigh-fading channels [22]. During the transmission of each frame, we use three training sequences to estimate the weighting coefficients and the payload in each frame is set to 15 OFDM symbols. The transmission time delay is assumed to be one OFDM symbol. It is shown in Fig. 8 that only when the normalized Doppler frequency is larger than 6 10 , the performance begins to degrade significantly. To study the sensitivity of the TDD-OFDM system with convolutional coding to other factors such as those induced by RF components, we model the uplink channel as for (41)10-110-210-310-4012345678910Eb/N0 per receive antenna branch [dB]Fig. 10. BER performance of the proposed system with different number of training iterations when F = M = 2 and K = 3.C. Weighting Coefficients Estimation Algorithm The convergence property of the weighting coefficients estimation algorithm and the number of samples required during each training period for our proposed OFDM system with a convolutional code are as shown in Figs. 10 and 11, respectively. A close observation of Fig. 10 shows that, for a system with two transmit antennas and two receive antennas, the performance is good enough after only two iterations. Fig. 11 depicts the BER performance versus the number of samples required during each training period [i.e., in (40)]. It is shown that about 15 samples are enough for an effective training. Therefore, the overheads required by training are very limited. There are many options to initialize the weighting coefficients (i.e., ) in the weighting coefficients estimation algorithm. The power transmitted can initially be equally distributed, randomly distributed, or put to one specific antenna. However, from our observation, the initial weighting coefficients do not impact the ultimate BER performance. This fact can be used to further jus-where and are the channel impulse responses for the downlink and the uplink, respectively. The downlink channel is modeled as an equal gain two OFDM samples spaced is an indepentwo-ray slow Rayleigh-fading channel, and dent complex Gaussian random variable. To characterize the link asymmetry, we define the term channel nonideal recip. Fig. 9 rocal ratio (CNR), which is given by demonstrates that asymmetric channels cause limited performance degradation.。