Interpolation Based Transmit Beamforming for MIMO-OFDM with Partial Feedback Jihoon Choi∗ and Robert W. Heath, Jr.
The University of Texas at Austin Department of Electrical and Computer Engineering Wireless Networking and Communications Group 1 University Station C0803 Austin, TX 78712-0240 Phone: +1-512-425-1305 Fax: +1-512-471-6512 E-mail:
[email protected]
Technical Subject Areas: Communication Theory
∗
Corresponding author.
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Abstract Transmit beamforming with receive combining is a simple method for exploiting the significant diversity provided by multiple-input multiple-output (MIMO) systems, and the use of orthogonal frequency division multiplexing (OFDM) enables low complexity implementation of this scheme over frequency selective MIMO channels. Optimal beamforming requires channel state information in the form of the beamforming vectors corresponding to all the OFDM subcarriers. In an attempt to reduce the amount of feedback information, we propose a new approach to transmit beamforming that combines partial feedback and beamformer interpolation. In the proposed architecture, the receiver sends a fraction of information about optimal beamforming vectors to the transmitter, and the transmitter computes the beamforming vectors for all subcarriers through modified spherical linear interpolation of the conveyed beamforming vectors. Simulation results show that the proposed beamforming method requires much less feedback information than optimal beamforming while it exhibits slight diversity loss compared to the latter. I. Introduction In multiple-input multiple-output (MIMO) wireless systems, antenna diversity have attracted significant interest as a means to mitigate signal-level fluctuations by fading [1]–[4]. In narrowband Rayleigh matrix channels, MIMO systems can provide a diversity in proportion to the product of the number of transmit and receive antennas. When the channel is unknown at the transmitter, diversity can be obtained by using space-time codes [1]–[3]. On the other hand, the use of channel state information (CSI) at the transmitter further improves diversity gain [4]–[9]. Transmit beamforming with receive combining is a simple method for exploiting the significant diversity provided by MIMO systems [4]– [7],[10]. This method significantly improves the signal to noise ratio (SNR), however this approach requires the knowledge about the complete channel or beamforming vector at the transmitter. When the uplink and downlink channels are not reciprocal, the receiver needs
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to send back the channel state information in the form of beamforming vector through the feedback channel with limited bandwidth, and a practical solution is to use a codebook which is known to both the transmitter and receiver [11]–[13]. Transmit beamforming and receive combining proposed in narrowband MIMO channels can be easily extended to frequency selective MIMO channels by employing orthogonal frequency division multiplexing (OFDM) systems. In a MIMO-OFDM system, a wideband MIMO channel is divided into multiple narrowband MIMO channels corresponding to OFDM subcarriers [14],[15], and transmit beamforming with receive combining is independently performed for each subcarrier. In non-reciprocal channels, the receiver of MIMO-OFDM computes the optimal beamforming vectors used for all the OFDM subcarriers, and sends back the information about the beamforming vectors to transmitter. Thus the feedback information increases in proportion to the number of subcarriers. In an attempt to reduce the feedback information of MIMO-OFDM, we consider a new approach to transmit beamforming that combines partial feedback and beamformer interpolation. In MIMO-OFDM, neighboring channel matrices are significantly correlated because the number of subcarriers is generally much greater than the maximum length of channel impulse response. Since the beamforming vectors are determined by the channels corresponding to subcarriers, neighboring beamforming vectors are also correlated and this means that a beamforming vector can be estimated from neighboring beamforming vectors. To exploit the correlation between beamforming vectors, the receiver selects a fraction of subcarriers and computes the beamforming vectors only for the selected subcarriers. Then the receiver quantizes the beamforming vectors by selecting the best vectors from the codebook and sends back the indices of the beamforming vectors to the transmitter [11]–[13]. The transmitter evaluates the beamforming vectors for all the OFDM subcarriers through interpolation of the conveyed beamforming vectors. The beamforming vectors have unit norm due to transmit power constraint, and thus the transmitter requires a spherical interpolation algorithm. Some spherical averaging methods and their appli-
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cations to spherical interpolation have been proposed [16]–[19]. However, it is not easy to directly use the algorithms to interpolate beamforming vectors, because the optimal beamforming vector is not unique and it is difficult to determine the coefficients for higher order interpolation. In this paper, we propose a simple spherical interpolator obtained by modifying the spherical linear interpolator in [16]. The proposed algorithm uses the conveyed vectors multiplied by ejφ which is determined at the receiver to minimize the distortion induced by interpolation. Specifically, the phase ejφ is obtained in the sense of maximizing diversity gain or capacity. The quantized phase information ejφ for each beamforming vector is conveyed through the feedback channel along with the information along with the beamforming vector. In practical MIMO systems, the proposed beamforming method requires much less feedback information than the ideal transmit beamforming with feedback for all subcarriers, and it is shown through computer simulations that the proposed scheme performs close to the ideal beamforming. This paper is organized as follows. Section II reviews MIMO-OFDM communication with beamforming and combining. The proposed interpolation based beamforming scheme is presented in Section III. We compare the performance of the proposed beamformer with those of other diversity techniques in Section IV. Finally, we provide some conclusions in Section V. II. System Overview A MIMO-OFDM system with transmit beamforming and receive combining, using Mt transmit antennas and Mr receive antennas, is illustrated in Fig. 1. After the discrete Fourier transform (DFT), the received signal for each subcarrier is denoted as a narrowband MIMO signal, and the combined signal at subcarrier k, is expressed as r(k) = zH (k){H(k)w(k)s(k) + n(k)}, 1 ≤ k ≤ N
(1)
where w(k) = [w1 (k), w2 (k), · · · , wMt (k)]T and z(k) = [z1 (k), z2 (k), · · · , zMr (k)]T are beamforming and combining vectors with unit norm, respectively, and H(k) is a (Mr ×Mt ) 3
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matrix whose elements represent the channel gains of subcarrier k between transmit antennas and receive antennas. s(k) is a transmitted symbol with E[|s(k)|2 ] = E(k) and n(k) is the Mr -dimensional noise vector after DFT with i.i.d. entries distributed according to CN (0, N0 ). The signal model (1) is identical to that of narrowband MIMO systems, and thus w(k) and z(k) can be chosen to maximize the signal to noise ratio (SNR) according to the beamforming and combining scheme in a narrowband MIMO system [4],[6]. The SNR for subcarrier k is γ(k) =
E(k)|zH (k)H(k)w(k)|2 E(k)Γ(k) = . N0 N0
(2)
where Γ(k) = |zH (k)H(k)w(k)|2 is the effective channel gain. In a MIMO-OFDM system, the transmit beamforming vector and receive combining vector are designed to maximize the SNR for each subcarrier. Suppose that the receiver always use the maximum ratio combining (MRC) vector that maximizes the effective channel gain. Given w(k), it is easily shown that the MRC beamforming vector z(k) is written by z(k) =
H(k)w(k) kH(k)w(k)k
(3)
where k(·)k means 2-norm of (·). On the other hand, the selection of transmit beamforming vector w(k) requires more consideration depending on the amount of feedback information. In selection diversity transmission that selects the transmit antenna with maximum channel gain, w(k) becomes one of the columns of IMt , the Mt × Mt identity matrix. This scheme requires only log2 Mt feedback bits and achieves full diversity order, but exhibits some array gain loss. The transmitter where w(k) maximizes |zH (k)H(k)w(k)|2 given z(k) is called maximum ratio transmission (MRT). In a system with MRT and MRC, w(k) is the dominant right singular vector, the right singular vector of H(k) corresponding to the largest singular vector of H(k) [6],[9]. Note that the optimal beamforming vector for MRT/MRC is not unique, because if w(k) is optimal, then eφ w(k) with φ ∈ [0, 2π) is also optimal. This MRT/MRC scheme maximizes the SNR of each subcarrier, but it
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requires the knowledge about the complete channel impulse responses or beamforming vectors {w(k)|k = 1, 2, · · · , N }. In this paper, we consider a communication link where channel state information is not available to the transmitter, but there exists a low-rate, error-free, zero-delay feedback link which conveys channel state information to the transmitter. Due to the limited bandwidth of the feedback channel, it is essential to quantize the channel state information before transmission. When we sends back the channel impulse responses, the amount of feedback information increases in proportion to Mt Mr , and it is difficult to estimate performance degradation by quantization error because the channel is used after inverse DFT at the transmitter. Alternatively, if we conveys the beamforming vectors to the transmitter, the feedback data increases in proportion to Mt , and we can use an efficient quantization method using a codebook which is known to both transmitter and receiver [11]–[13]. For this reason, we consider a MIMO-OFDM system where the receiver sends back the channel state information in the form of beamforming vectors to the transmitter. In this system, the receiver selects the best beamforming vectors for all subcarriers from the codebook and conveys the indices of the beamforming vectors to the transmitter. Using the criteria about the codebook design in [13], we can easily choose the number of bits required for transmission of a beamforming vector, however the feedback information increases in proportion to the number of subcarriers. III. Interpolation Based Beamforming for MIMO-OFDM In this section, we propose a new approach to transmit beamforming that combines partial feedback and beamformer interpolation to reduce the feedback information for MIMO-OFDM. In the proposed scheme, the receiver sends back only a fraction of beamforming vectors to the transmitter, and the transmitter computes the beamforming vectors for all the OFDM subcarriers through proposed spherical linear interpolation. We will obtain the proposed interpolator by modifying spherical linear interpolation and use it to maximize the diversity gain or capacity. 5
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A. Proposed Interpolation Method In OFDM, the length of cyclic prefix (CP) is designed to be much less than the number of subcarriers to increase spectral efficiency (see e.g., [20],[21]), and thus the channels which correspond to neighboring subcarriers are significantly correlated. Since the beamforming vectors for MIMO-OFDM are determined by the channels, they are also correlated with each other, and this correlation can be used to reduce the feedback information. A simple beamforming method with less feedback, which will be referred to as clustering, is to combine the neighboring subcarriers into a cluster and use the beamforming vector corresponding to the center subcarrier for all the subcarriers in the cluster. In this method, the feedback information is decreased as the cluster size increases. However, the beamforming performance is significantly degraded in the subcarriers near cluster boundary. As an alternative, we consider an interpolation method that computes the beamforming vectors for all subcarriers from a part of beamforming vectors. The receiver downsamples the subcarriers, i.e. selects a fraction of OFDM subcarriers, and finds the best beamforming vectors for the selected subcarriers from the codebook under the assumption that MRC is used at the receiver. Suppose that N is divided by the downsampling rate K. Then the receiver sends back the indices of {w(1), w(K + 1), w(2K + 1), · · · , w(N − K + 1)} to the transmitter where w(k) is the best beamforming vector for subcarrier k selected from the codebook. The transmitter reconstructs the beamforming vectors for all subcarriers through interpolation of {w(1), w(K + 1), w(2K + 1), · · · , w(N − K + 1)}. This requires a spherical interpolator different from a standard interpolator because the beamforming vectors have unit norm, and a solution is to exploit the spherical interpolation algorithms in [16] and [17]. However, it is not proper to directly apply the algorithms to interpolation of beamforming vectors because the optimal beamforming vector is not unique as mentioned in Section II. For this reason, we propose a new spherical linear interpolater given by w(lK ˆ + k) =
(1 − ck )w(lK + 1) + ck {ejθl w((l + 1)K + 1)} k(1 − ck )w(lK + 1) + ck {ejθl w((l + 1)K + 1)}k 6
(4)
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where the linear weighting ck = (k − 1)/K, w(N + 1) = w(1), 1 ≤ k ≤ K, and 0 ≤ l ≤ N/K − 1. {θl |0 ≤ l ≤ N/K − 1} are parameters for phase rotation. Note that kw(k)k ˆ = 1 by normalization. Also, notice that the proposed interpolator uses w(lK + 1) and ejθl w((l + 1)K + 1) to obtain {w(k), ˆ lK + 1 ≤ k ≤ (l + 1)K}, while conventional spherical linear interpolators utilize w(lK + 1) and w((l + 1)K + 1). It means that the proposed interpolator fixes the relative phase between w(lK + 1) and w((l + 1)K + 1) to minimize the interpolation error induced by the nonuniqueness of the optimal beamforming vector. For the proposed interpolation, the receiver evaluates the optimal {θl } in the sense of maximizing diversity gain or capacity depending on the constraint, and conveys the phase information along with the information of the beamforming vectors. For transmission through the feedback channel, it is assumed that θl and w((l + 1)K + 1) are separately quantized, i.e. θl is uniformly quantized and w((l + 1)K + 1) is quantized using the codebook in [13]. In fact, the whole vector ejθl w((l + 1)K + 1) can be quantized using spherical quantization methods based on the Euclidean distance [22],[23]. Through simulations, however, this approach requires almost the same feedback information as the separate quantization, yet makes it more complicated to determine {θl }. Therefore, we develop two criteria to determine the optimal {θl } when θl and w((l + 1)K + 1) are separately quantized. B. Maximization of Diversity Gain or Capacity Suppose that {H(k), 1 ≤ k ≤ N } are known at the receiver. Given a codebook, we can select the best beamforming vectors {w(1), w(K + 1), w(2K + 1), · · · , w(N − K + 1)} using the channel matrices. To maximize the diversity gain in the proposed beamforming method, we find {θl |0 ≤ l ≤ N/K − 1} by maximizing the minimum effective channel gain 2 kH(k)w(k)k ˆ for all subcarriers. In (4), θl is only used for computing {w(k), ˆ lK + 1 ≤
k ≤ (l + 1)K}, thus the θl maximizing the minimum effective channel gain is separately
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found by θl = arg max min{kH(lK + k)w(lK ˆ + k)k2 , 1 ≤ k ≤ K} θl ∈[0,2π)
(5)
where 0 ≤ l ≤ N/K − 1. It is difficult to get a closed-form solution for (5). Fortunately, we only need a quantized value of θl instead of the exact value. Therefore, the best θl can be determined using the grid search that repeatedly calculates the effective channel gain kH(lK + k)w(lK ˆ + k)k2 for all possible quantized phases and that finds the value optimizing (5). In the grid search, finding the minimum effective channel gain requires the computation of {kH(lK + k)w(lK ˆ + k)k2 for 1 ≤ k ≤ K given a quantized phase. In (5), the subcarrier (lK + K/2 + 1) has the minimum mean effective channel gain because w(lK ˆ + K/2 + 1) suffers from the largest average distortion by interpolation. So, to reduce the complexity of the grid search, θl can be approximately obtained by θl = arg max kH(lK + K/2 + 1)w(lK ˆ + K/2 + 1)k2
(6)
θl ∈P
, 4π , · · · , 2π(Q−1) } and Q is the number of possible quantized phases used where P = {0, 2π Q Q Q in the grid search. On the other hand, θl can be determined to maximize the sum rate of all the OFDM subcarriers when the proposed transmit beamforming and MRC are used. To compute the sum capacity of the MIMO-OFDM with transmit beamforming and receive combining, it is assumed that the transmit power is identically assigned into all subcarriers. In a similar manner to maximizing the diversity gain, {θl |0 ≤ l ≤ N/K − 1} maximizing the sum rate of all subcarriers are separately found by the following equation. θl = arg max
θl ∈[0,2π)
K X k=1
½ log2
kH(lK + k)w(lK ˆ + k)k2 1+ N0
¾ .
(7)
The optimal θl is also obtained by the grid search. This approach can be applied to a MIMO-OFDM system with transmit beamforming and adaptive modulation. For compu-
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tational simplicity, equation (7) can be approximated by ½ ¾ kH(lK + K/2 + 1)w(lK ˆ + K/2 + 1)k2 θl = arg max log2 1 + θl ∈P N0 = arg max kH(lK + K/2 + 1)w(lK ˆ + K/2 + 1)k2 . θl ∈P
(8)
Note that (8) is identical with (6). After approximation, we find the optimal θl considering only one subcarrier (lK+K/2+1) so that maximizing the diversity gain becomes equivalent to maximizing the capacity. IV. Simulation Results Through computer simulations, we compared the proposed beamforming scheme with antenna selection diversity [8], the theoretical bound of space-time block codes (STBC)1 [2], and the simple clustering method. In the simulation, the following parameters were assumed: Mt = 4, Mr = 2, N = 64, K = 8, each channel impulse response had 6 resolvable multipaths with uniform profile, and the channels between different transmit and receiver antenna pairs were independent. All of the simulations assume an MRC receiver. In the selection diversity, the transmit antenna with maximum channel gain was selected independently for each subcarrier, so 128 (= 2N ) feedback bits per OFDM symbol were used. For quantization of beamforming vectors, the codebook in [13] that sends back 6 bits per beamforming vector was used, and θl for the proposed interpolator was quantized in 2 bits. Thus, we used 64 (=8K) feedback bits per symbol for the proposed method, 48 (=6K) feedback bits per symbol for the clustering, and 384 (= 6N ) feedback bits for the ideal beamforming which conveys the indices of all beamforming vectors. To compare the diversity gain, θl was determined by (5). Fig. 2 compares the average bit error rate (BER) performance of different diversity techniques when quadrature phase shift keying (QPSK) modulation was used for all subcarriers. The average BER was estimated by averaging the BER values over more than 5000 independent generation of 1
In [2], it was shown that there are no orthogonal STBC for complex constellations when Mt ≥ 3, but the
theoretical performance bound can be obtained for any Mt . 9
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frequency selective MIMO channels. The proposed method performed comparable to selection diversity, and outperformed the clustering scheme and the theoretical lower bound of STBC. The proposed scheme exhibited slight loss in diversity order which determines the slope of BER curves compared with the ideal beamforming, because of the interpolation error. Selection diversity achieves full diversity but have some loss in array gain which shifts BER curves. For this reason, the BER curves of the proposed and the selection diversity cross at around Eb /N0 = 8 dB. To compare the effective channel gains and capacity, θl was determined by (7). Figs. 3 and 4 show the average effective channel gains and the average capacity of different schemes, respectively. For evaluation of capacity, it was assumed that the transmit power was identically assigned to all subcarriers. The average channel gain and capacity were obtained by averaging the values over more than 3000 independent iterations. In both figures, the proposed algorithm performed very close to the ideal beamforming with much more feedback bits, and outperformed the selection diversity and the bound of STBC. Also, the proposed method exhibited better performance than the clustering, and in Fig. 4, it showed 0.5 dB gain compared with the clustering. These results demonstrate that the proposed beamforming scheme with partial feedback and beamformer interpolation is advantageous in the sense of diversity gain and capacity compared with existing techniques . V. Conclusion In an attempt to reduce the feedback information for MIMO-OFDM, a new transmit beamforming scheme with partial feedback and beamformer interpolation was proposed. In the proposed scheme, the receiver sends back the information about only a fraction of beamforming vectors and the information for the phase rotation of the selected vectors to the transmitter through the feedback channel, and the transmitter computes the beamforming vectors for all subcarriers through proposed spherical linear interpolation. In the proposed interpolator, the information for the phase rotation can be determined to 10
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maximize the diversity gain or capacity. Through computer simulations, it was shown that the proposed beamformer has benefits compared with existing diversity techniques. We only considered the spherical linear interpolation, and the use of higher order interpolation remains further research area. References [1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1456–1467, July 1999. [3] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [4] J. B. Andersen, “Antenna arrays in mobile communications: gain, diversity, and channel capacity,” IEEE Antennas and Propagation Magazine, vol. 42, pp. 12–16, Apr. 2000. [5] T. K. Y. Lo, “Maximum ratio transmission,” IEEE Trans. Commun., vol. 47, pp. 1458–1461, Oct. 1999. [6] P. A. Dighe, R. K. Mallik, and S. S. Jamuar, “Analysis of transmit-receive diversity in Rayleigh fading,” IEEE Trans. Commun., vol. 51, no. 4, pp. 694–703, Apr. 2003. [7] S. Thoen, L. V. der Perre, B. Gyselinckx, and M. Engels, “Performance analysis of combined transmitSC/receive-MRC,” IEEE Trans. Commun., vol. 49, pp. 5–8, Jan. 2001. [8] R. W. Heath, Jr. and A. Paulraj, “A simple scheme for transmit diversity using partial channel feedback,” in Proc. IEEE Asilomar Conference on Signals, Systems and Computers, Nov. 1998, vol. 2, pp. 1073–1078. [9] C.-H. Tse, K.-W. Yip, and T.-S. Ng, “Performance tradeoffs between maximum ratio transmission and switched-transmit diversity,” in Proc. IEEE PIMRC, Sept. 2000, vol. 2, pp. 1485–1489. [10] M. Kang and M.-S. Alouini, “Largest eigenvalue of complex wishart matrices and performance analysis of MIMO MRC systems,” IEEE J. Select. Areas Commun., vol. 21, pp. 418–426, Apr. 2003. [11] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use of side information in multiple-antenna data transmission over fading channels,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1423–1436, Oct. 1998. [12] D. J. Love and R. W. Heath, Jr., “Equal gain transmission in multi-input multi-output wireless systems,” IEEE Trans. Commun., vol. 51, pp. 1102–1110, July 2003.
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[13] D. J. Love, R. W. Heath, Jr., and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless systems,” to appear in IEEE Trans. Inform. Theory, vol. 49, Oct. 2003. [14] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE J. Select. Areas Commun., vol. 46, no. 3, pp. 357–366, Mar. 1998. [15] H. Bolcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDM-based spatial multplexing systems,” IEEE Trans. Commun., vol. 50, no. 2, pp. 225–234, Feb. 2002. [16] S. R. Buss and J. P. Fillmore, “Spherical averages and applications to spherical splines and interpolation,” ACM Trans. Graphics, vol. 20, no. 2, pp. 95–126, Apr. 2001. [17] K. Shoemake, “Animating rotation with quaternion curves,” in Proc. SIGGRAPH’85, ACM, San Franscisco, July 1985, vol. 19, pp. 245–254. [18] G. Wagner, “On means of distances on the surface of a sphere (lower bounds),” Pacific J. of Mathematics, vol. 144, no. 2, pp. 389–398, 1990. [19] G. Wagner, “On means of distances on the surface of a sphere. ii (upper bounds),” Pacific J. of Mathematics, vol. 154, no. 2, pp. 381–396, 1992. [20] Digital Broadcasting Systems for Television; Framing Structure, Channel Coding, and Modulation for Digital Terrestrial Services, ETSI DVB Standard ETS 300 7XX, Geneva, Switzerland, 1996. [21] IEEE Std 802.11a, Part 11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications–High-speed physical layer in the 5 GHz band, LAN/MAN Standards Committee, 1999. [22] J. Hamkins and K. Zeger, “Asymptotically dense spherical codes – Part I: wrapped spherical codes,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1774–1785, 1997. [23] J. Hamkins and K. Zeger, “Asymptotically dense spherical codes – Part II: laminated spherical codes,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1786–1798, 1997.
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Fig. 1.
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Block diagram of MIMO-OFDM with Mt transmit antennas, Mr receive antennas, and N
subcarriers.
Theoretical lower bound of STBC Clustering w/ 48 bit feedback Selection diversity w/ 128 bit feedback Proposed w/ 64 bit feedback Ideal beamforming w/ 384 bit feedback
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BER
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4 6 Eb/No (dB)
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Fig. 2. Comparison of BER performance when Mt = 4, Mr = 2, N = 64, and K = 8.
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Effective Channel Gains
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Ideal beamforming w/ 384 bit feedback Proposed w/ 64 bit feedback Clustering w/ 48 bit feedback Selection diversity w/ 128 bit feedback Upper bound of STBC 2
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Fig. 3. Comparison of effective channel gains when Mt = 4, Mr = 2, N = 64, and K = 8. 8
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Capacity (bps/Hz)
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Ideal beamforming w/ 384 bit feedback Proposed w/ 64 bit feedback Clustering w/ 48 bit feedback Selection diversity w/ 128 bit feedback Theoretical upper bound of STBC
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Fig. 4. Comparison of Capacity when Mt = 4, Mr = 2, N = 64, and K = 8.
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