Interpolation Based Transmit Beamforming for MIMO-OFDM with Limited Feedback Jihoon Choi and Robert W. Heath, Jr. Dept. of Electrical and Computer Engineering, The University of Texas at Austin Austin, TX 78712–0240, email:
[email protected],
[email protected]
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. This paper proposes a beamforming technique that reduces feedback requirements for optimal beamforming. In the proposed architecture, the receiver sends back quantized versions of select beamforming vectors and the transmitter reconstructs the missing beamforming vectors through interpolation. Since a beamforming vector is phase invariant and has unit norm, a new spherical linear interpolator is proposed that exploits additional parameters for phase rotation. These parameters are determined at the receiver in the sense of maximizing the minimum channel gain or capacity, and sent back to the transmitter along with the quantized beamforming vectors. Simulation results show that the proposed beamforming method requires much less feedback information than optimal beamforming with only slight diversity loss.
I. I NTRODUCTION In multiple-input multiple-output (MIMO) wireless systems, antenna diversity has 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. Diversity can be obtained by using space-time codes [1]–[3] or through the use of channel state information (CSI) at the transmitter [4]–[9]. Transmit beamforming and receiver combining is a simple approach to achieve the full diversity order as well as additional array gain [4]–[6], thus it significantly improves system performance. This approach, however, requires the knowledge about the transmit beamforming vector at the transmitter. In nonreciprocal channels, this necessitates that the receiver sends back the CSI in the form of the beamforming vector through a feedback channel with limited rate. Beamforming and combining techniques proposed for narrowband channels can be easily extended to frequency selective MIMO channels by employing orthogonal frequency division multiplexing (OFDM). In a MIMO-OFDM system, a broadband MIMO channel is converted into multiple narrowband MIMO channels corresponding to OFDM subcarriers This material is based in part upon work supported by the Texas Advanced Technology Program under Grant No. 003658-0614-2001 and 003658-03802003, the National Instruments Foundation, the Samsung Advanced Institute of Technology, and the Post-Doctoral Fellowship Program of Korea Science and Engineering Foundation (KOSEF).
IEEE Communications Society
[10], [11]. Transmit beamforming with receive combining can be performed independently for each subcarrier of MIMOOFDM [12]. In non-reciprocal channels, the MIMO-OFDM receiver computes the optimal beamforming vector for each OFDM subcarrier and sends them back to the transmitter. To transmit beamforming vectors through a feedback channel with limited data rate, we can quantize each beamforming vector using a codebook designed for narrowband MIMO channels (see e.g., [13]–[16]). The feedback requirements, though, still increase in proportion to the number of subcarriers. To reduce the feedback requirements of MIMO-OFDM, this paper proposes a new beamforming technique that sends back only a fraction of beamforming vectors combined with intelligent interpolation of beamforming vectors to the transmitter. In OFDM, the number of subcarriers is generally much greater than the maximum length of time domain channel impulse response, and thus neighboring subchannels are significantly correlated. Since a beamforming vector is determined by the corresponding subchannel matrix, the beamforming vectors for neighboring subchannels are also substantially correlated. Using the correlation between beamforming vectors, the receiver selects a fraction of subcarriers and finds the optimal quantized beamforming vectors for the selected subcarriers using a codebook. The receiver sends back the indices of the beamforming vectors to the transmitter, and the transmitter evaluates the beamforming vectors for all OFDM subcarriers through interpolation. Since the beamforming vectors have unit norm by the transmit power constraint, it is natural to introduce spherical interpolators in [17]–[19]. Unfortunately, it is not easy to directly use the spherical interpolation algorithms to interpolate beamforming vectors, because the optimal beamforming vector is not unique. In this paper, we propose a new spherical linear interpolator through some modification of the interpolator in [19]. Considering the nonuniqueness of the beamforming vectors, the proposed interpolator employs a new parameter for phase rotation. The phase parameter denoted as ejθ is optimized at the receiver in the sense of maximizing the minimum channel gain or capacity. We define a cost function based on the channel gain or capacity, however it is not trivial to fine the optimal phase from the cost function. Since we require only the quantized value of θ, we determine the best θ through a numerical grid search. The phase θ is sent back to the transmitter along with the information about the beamforming
249
0-7803-8533-0/04/$20.00 (c) 2004 IEEE
vectors. It is shown through computer simulations that the proposed beamforming method requires much less feedback information than ideal beamforming with feedback for all the OFDM subcarriers at the cost of slight diversity gain loss. Compared with prior work that deals with unquantized partial channel information [12], our approach focuses on methods for reducing feedback requirements by subsampling in frequency and quantizing separately the optimal beamforming vectors computed from the instantaneous channel state. This paper is organized as follows. Section II reviews MIMO-OFDM communication with beamforming and combining. The proposed beamforming scheme is presented in Section III. We compare the proposed method with existing diversity techniques in terms of bit error rate (BER) and capacity in Section IV. Finally, we provide some conclusions in Section V.
Fig. 1. Block diagram of a MIMO-OFDM system with Mt transmit antennas, Mr receive antennas, and N subcarriers.
(MRC) with z(k) =
II. S YSTEM OVERVIEW A MIMO-OFDM system with transmit beamforming and receive combining, using Mt transmit antennas, Mr receive antennas, and N subcarriers, is illustrated in Fig. 1. At the transmitter, the k-th subcarrier modulates the symbol s(k) using the beamforming vector w(k) = [w1 (k), w2 (k), · · · , wMt (k)]T . Assuming the sampled impulse response of the channel is shorter than the cyclic prefix (CP), the channel for the kth subcarrier after the DFT can be described by a Mr -byMt channel matrix H(k) whose entries represent the channel gains experienced by subcarrier k. After processing with the combining vector z(k) = [z1 (k), z2 (k), · · · , zMr (k)]T , the combined signal at subcarrier k can be expressed as r(k) = zH (k){H(k)w(k)s(k) + n(k)}, 1 ≤ k ≤ N
γ(k) =
H
2
Es Es |z (k)H(k)w(k)| = Γ(k). N0 N0
(2)
where Γ(k) = |zH (k)H(k)w(k)|2 is the effective channel gain. In the MIMO-OFDM system under consideration, w(k) and z(k) are designed based on H(k) to maximize the SNR for each subcarrier k. Given w(k), it is possible to show that the SNR maximizing solution uses maximum ratio combining
IEEE Communications Society
(3)
At the transmitter, we can use the maximum ratio transmission (MRT) that uses the unconstrained beamforming vector w(k) that maximizes the effective channel gain Γ(k) given z(k). With MRT and MRC, w(k) is simply the right singular vector of H(k) corresponding to the largest singular vector of H(k) [6], [9]. The MRT/MRC scheme obtains full diversity order in Rayleigh fading channels as well as the full array gain available [4]–[6]. Unfortunately, this improved performance comes at the expense of knowledge of the beamforming vectors {w(k), k = 1, 2, · · · , N }. The extensive feedback required for the optimal MRT/MRC solution motivates developing new methods that achieve near maximum SNR performance but with more realistic feedback requirements.
(1)
where n(k) is the Mr -dimensional noise vector whose entries have the independent and identically distributed (i.i.d.) complex Gaussian distribution with zero mean and variance N0 . We assume that the power is allocated equally across all subcarriers thus E[|s(k)|2 ] = Es is a constant and w(k) = 1 (where (·) means 2-norm of (·)) to maintain the overall power constraint. The signal model in (1) is identical to that of a narrowband MIMO system, thus w(k) and z(k) can be chosen to maximize the signal to noise ratio (SNR) using similar results on beamforming and combining for narrowband MIMO systems [4], [6]. Without loss of generality, we can fix z(k) = 1. Then the signal to noise ratio (SNR) for subcarrier k can be written as
H(k)w(k) . H(k)w(k)
III. I NTERPOLATION BASED B EAMFORMING FOR MIMO-OFDM To reduce the feedback information for MIMO-OFDM, we propose a new approach to transmit beamforming that combines subsampling of the beamforming vectors at the receiver and beamformer interpolation at the transmitter. We introduce a new beamformer interpolator based on a spherical linear interpolator but with an additional parameters for phase rotation, which takes the invariance of the optimal beamforming vectors into account. We optimize the phase rotation parameters in the sense of maximizing the minimum effective channel gain or mutual information. Finally, we describe how to combine this approach with quantization to limit the feedback required. A. Proposed Interpolation Based Beamforming Method In OFDM, the CP length is designed to be much less than the number of subcarriers to increase spectral efficiency (see e.g., [20], [21]), and thus the neighboring subchannels are significantly correlated. This results in high correlation between neighboring beamforming vectors because a beamforming vector is determined by the corresponding subchannel matrix. A simple beamforming method for using the correlation of the beamforming vectors is to combine the neighboring
250
0-7803-8533-0/04/$20.00 (c) 2004 IEEE
subcarriers into a cluster and use the beamforming vector corresponding to the center subcarrier in the cluster. This method will be referred to as clustering 1 . Suppose that N is divided by the subsampling rate K. If we combines K subcarriers into one cluster, the amount of feedback information is reduced to 1/K. As the cluster size K increases, however, the beamforming performance is significantly degraded because of the distortion experienced by subcarriers near the cluster boundary. This limits the cluster size K and consequently the reduction of feedback information is restricted. As an alternative, we consider an interpolation based beamforming method that sends back a part of beamforming vectors and computes the beamforming vectors for all subcarriers through interpolation. The receiver subsamples the subcarriers, i.e. selects a fraction of OFDM subcarriers, and finds the beamforming vectors for the selected subcarriers under the assumption that MRC is used at the receiver. Unfortunately, conventional interpolation is not suitable for beamformer interpolation because it does not preserve the unit norm constraint on w(k). An alternative approach is to exploit the spherical interpolation algorithms in [17]–[19] that will naturally enforce the unit norm constraint. Unfortunately, these algorithms cannot be directly applied to our approach for the following reason. Recall the effective channel gain Γ(k) = H(k)w(k)2 in (2). When w(k) is the optimal beamforming vector maximizing the effective channel gain, ejφ w(k) also maximizes the effective channel gain. In other words, the optimal beamforming vector is not a unique point on the unit sphere but represents a complex line. As a consequence, algorithms that are used to compute the optimal beamforming vector, e.g. the singular value decomposition, typically choose the phase to force the first coefficient of each w(k) to be real. The phase, however, has a dramatic impact on the resulting interpolated vectors. This makes it difficult to directly apply the spherical interpolation algorithms to the interpolation of beamforming vectors. To solve this problem, we propose a new algorithm through modification of the interpolater in [19]. This interpolator performs a weighted average of the beamforming vectors and renormalizes the result to place it on the unit sphere. Given {w(lK + 1), 0 ≤ l ≤ N/K − 1}, the proposed interpolator is expressed as w(lK ˆ + k; θl ) = (1 − ck )w(lK + 1) + ck {ejθl w((l + 1)K + 1)} (1 − ck )w(lK + 1) + ck {ejθl w((l + 1)K + 1)}
(4)
where ck = (k −1)/K is the linear weight value, w(N +1) = w(1), 1 ≤ k ≤ K, θl is a parameter for phase rotation with 0 ≤ l ≤ N/K −1. Note that {w(k), N −K +1 ≤ k ≤ N } are obtained by w(N − K + 1) and ejθN/K−1 w(N + 1) (equivalently ejθN/K−1 w(1)). While the spherical linear interpolator in [19] only utilizes w(lK + 1) and w((l + 1)K + 1), the 1 In [22], a clustering approach based on the channel correlation was proposed to reduce the feedback information for an OFDM system with adaptive modulation.
IEEE Communications Society
proposed interpolator evaluates the beamforming vector from w(lK + 1) and ejθl w((l + 1)K + 1). Essentially the role of θl is to remove the distortion caused by the arbitrary phase rotation of the optimal beamforming vectors. To maximize the performance of the interpolator, the receiver evaluates the optimal phase {θl , 0 ≤ l ≤ N/K − 1} based on a performance metric, and conveys {θl } along with the indices of the selected beamforming vectors {w(lK + 1), 0 ≤ l ≤ N/K − 1} to the transmitter. At the transmitter, the beamforming vectors for all the subcarriers are computed by (4). The feedback of {θl } slightly increases the amount of feedback information compared to the clustering, yet the proposed interpolation can significantly improve performance as shown in Section IV. B. Phase Optimization for the Proposed Interpolator Suppose that the channel matrices {H(k), 1 ≤ k ≤ N } are known to the receiver. Given the subsampling rate K and the codebook W, the receiver calculates the quantized beamforming vectors {w(lK + 1), 0 ≤ l ≤ N/K − 1}. In this subsection, we define a cost function to find the optimal {θl } in the sense of maximizing the minimum effective channel gain or capacity. Ideally, {θl } would be determined to minimize the average BER performance. This criterion, however, makes it complicated to define the cost function with respect to θl especially when modulation, coding, and interleaving are taken into account. Instead, we propose to find {θl } by maximizing the minimum of the effective channel gains for all subcarriers. We find through simulations that this criterion tends to decrease the average BER. ˆ + k), 1 ≤ In (4), θl is only used for computing {w(lK k ≤ K}, and thus the optimal θl maximizing the minimum effective channel gain can be found by θl = arg max min{H(lK + k)w(lK ˆ + k; θ)2 , 1 ≤ k ≤ K} θ (5) where 0 ≤ l ≤ N/K − 1. Due to the normalization factor in w(lK ˆ + k; θ), it is not easy to get a closed-form solution for (5). Since we are quantizing θl , it is natural to use a numerical grid search to find the best θl . By modifying (5), we get θl = arg max min{H(lK + k)w(lK ˆ + k; θ)2 , 1 ≤ k ≤ K} θ∈Θ (6) 2(P −1)π 4π }, and P is the number where Θ = {0, 2π P , P ,··· , P of quantized levels which determines the performance and complexity of the search. Given θ, finding the minimum effective channel gain requires the computation of {H(lK + ˆ k)w(lK+k; ˆ θ)2 for 1 ≤ k ≤ K. In general, w(lK+K/2+1) suffers from the largest distortion induced by interpolation, because the subcarrier (lK + K/2 + 1) is the furthest from the subcarriers (lK + 1) and ((l + 1)K + 1) with feedback of the beamforming vectors. Using this fact, (6) can be approximated to maximize the effective channel gain of the subcarrier (lK + K/2 + 1), and then θl is obtained by
251
θl = arg max H(lK + K/2 + 1)w(lK ˆ + K/2 + 1; θ)2 . (7) θ∈Θ
0-7803-8533-0/04/$20.00 (c) 2004 IEEE
θl = arg max log2 {1+
Theoretical lower bound of OSTBC Clustering w/ 48 bit feedback Selection diversity w/ 128 bit feedback Proposed w/ 64 bit feedback Ideal beamforming w/ 384 bit feedback
−1
10
−2
10
BER
Alternatively, θl can be determined to maximize the sum rate of all the OFDM subcarriers, assuming that the transmit power is identically assigned to all subcarriers. In a similar approach to maximizing the minimum effective channel gain, the optimal phase maximizing the sum rate is determined by K H(lK + k)w(lK ˆ + k; θ)2 θl = arg max log2 1 + , θ N0 k=1 (8) and the optimal solution is obtained by using the grid search as follows. K H(lK + k)w(lK ˆ + k; θ)2 log2 1 + . θl = arg max θ∈Θ N0 k=1 (9) Using the fact that the subcarrier (lK +K/2+1) has the worst average effective channel gain, (9) can be approximated by
−3
10
−4
10
−5
10
−6
10
−4
−2
0
2
4
6
8
10
12
14
Eb/N0 (dB)
Fig. 2. Comparison of BER performance when Mt = 4, Mr = 2, N = 64, and K = 8.
θ
H(lK + K/2 + 1)w(lK ˆ + K/2 + 1; θ)2 / N0 . (10)
By the concavity of log function, this equation is equivalent to (7). After the approximation, θl depends only on the subcarrier (lK + K/2 + 1) so that maximizing capacity becomes equivalent to maximizing the minimum effective channel gain. C. Quantization of the Beamforming Vectors To accommodate the limited bandwidth of the feedback channel, we propose to quantize the transmit beamforming vectors using the codebooks designed for narrowband MIMO systems [14]–[16]. These codebooks are specially designed to take into account the phase invariance of the optimal beamforming vector. Given H(k), the quantized beamforming vectors are determined by selecting the element of the codebook that maximizes the effective channel gain as follows. w(lK + 1) = arg max H(lK + 1)x x∈W
(11)
where W is the codebook including all possible beamforming vectors and 0 ≤ l ≤ N/K − 1. The quantized beamforming vectors are used to find the optimal θl and θl is uniformly quantized on [0, 2π). Then the receiver sends back the indices from the codebook corresponding to each of {w(lK+1), 0 ≤ l ≤ N/K−1} to the transmitter, as N/K−1 well as the quantized phase values {θl }l=0 . The transmitter reconstructs the beamforming vectors corresponding to all the OFDM subcarriers through interpolation of the quantized beamforming vectors {w(lK + 1), 0 ≤ l ≤ N/K − 1} with the help of the phase coefficients. As an aside, instead of separately quantizing w((l+1)K+1) and θl , the whole vector θl w((l + 1)K + 1) can be quantized by spherical quantization based on the Euclidean distance [23], [24]. This approach, however, requires almost the same feedback information as the separate quantization, yet makes it more complicated to determine θl . Therefore, we find it natural to consider only the separate quantization of w((l + 1)K + 1) and θl .
IEEE Communications Society
IV. S IMULATION R ESULTS Through computer simulations, we compare the proposed beamforming scheme with antenna selection diversity [7], the theoretical bound of orthogonal space-time block codes (OSTBC)2 [2], and the simple clustering method. In the simulation, we consider a MIMO-OFDM system with parameters: Mt = 4, Mr = 2, N = 64, K = 8. We assumed that the discrete-time channel impulse responses had 6 taps with uniform profile and i.i.d. complex Gaussian distribution; the channels between different transmit and receiver antenna pairs were independent; n(k) was i.i.d. complex Gaussian with zero mean; the feedback channel had no delay and no transmission error; the receiver used MRC with perfect channel knowledge; quadrature phase shift keying (QPSK) modulation was used for BER simulations; θl for the proposed scheme was obtained by the grid search in (7). Every point of the simulation results was obtained by averaging over more than 5000 independent realizations of channel and noise. To quantize the beamforming vectors, a codebook was designed according to equation (22) in [15]. We used the codebook with |W| = 64 that requires 6 bits of feedback for each beamforming vector. θl was uniformly quantized by the grid search with P = 4 which requires 2 bits of feedback. Consequently, we used 64 (=8N/K) feedback bits per OFDM symbol for the proposed beamforming method. In 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. We used 48 (=6N/K) feedback bits per symbol for the clustering approach, and 384 (= 6N ) feedback bits for the ideal beamforming which conveys the indices of all beamforming vectors. Fig. 2 shows the average BER performance of various diversity techniques. The proposed method outperformed clustering 2 In [2], it was shown that there are no OSTBC for complex constellations when Mt ≥ 3, but the theoretical performance bound can be obtained for any Mt .
252
0-7803-8533-0/04/$20.00 (c) 2004 IEEE
7
R EFERENCES
6
Capacity (bps/Hz)
5
4
3
2
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 OSTBC
1
0 −5
0
5
10
15
SNR (dB)
Fig. 3. Comparison of Capacity when Mt = 4, Mr = 2, N = 64, and K = 8.
and the theoretical lower bound of OSTBC. The proposed interpolator was designed to maximize the minimum effective channel gain, however this criterion does not guarantee the full diversity order. Thus, the proposed technique exhibited slight loss in diversity order compared to the ideal beamforming. Selection diversity achieves full diversity order but has some array gain loss. For this reason, the proposed method performed better than selection diversity in low Eb /N0 region, and the BER curves crossed at about Eb /N0 = 8 dB as the performance gap decreased with Eb /N0 . Note that the proposed scheme requires only 64 feedback bits per OFDM symbol, which is a half of the feedback for selection diversity. Fig. 3 compares the average channel capacity. In the figure, the proposed method outperformed clustering, selection diversity, and the theoretical upper bound of OSTBC. In addition, the proposed scheme performed very close to the ideal beamforming and the SNR loss was less than 0.2 dB. Overall, these results demonstrate that the proposed beamforming scheme is advantageous over existing diversity techniques in practical systems. V. C ONCLUSION To reduce the feedback information for a closed-loop MIMO-OFDM system, a new transmit beamforming scheme with limited feedback and beamformer interpolation was proposed. The proposed scheme can be optimized by adjusting the phase parameters for the proposed spherical interpolator in terms of BER or capacity. Through computer simulations, it was shown that the proposed beamformer has considerable benefits compared to existing diversity techniques. For the beamformer interpolation, we only considered the spherical linear interpolator. In fact, it is apparent that higher order interpolation can decrease the distortion of the beamforming vectors. However, this makes it more complicated to determine θl , thus requires a practical method for finding the optimal phases {θl } with reasonable complexity.
IEEE Communications Society
[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] ——, “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 transmitreceive 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 transmit-SC/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, vol. 2, Nov. 1998, 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, vol. 2, Sept. 2000, pp. 1485–1489. [10] 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. [11] H. B¨olcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDMbased spatial multplexing systems,” IEEE Trans. Commun., vol. 50, no. 2, pp. 225–234, Feb. 2002. [12] P. Xia, S. Zhou, and G. B. Giannakis, “Adaptive MIMO-OFDM based on partial channel state information,” IEEE Trans. Signal Processing, vol. 52, no. 1, pp. 202–213, Jan. 2004. [13] 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. [14] D. J. Love and R. W. Heath, Jr., “Equal gain transmission in multi-input multi-output wireless systems,” IEEE Trans. Commun., vol. 51, no. 7, pp. 1102–1110, July 2003. [15] D. J. Love, R. W. Heath, Jr., and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2735–2747, Oct. 2003. [16] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On beamforming with finite rate feedback in multiple-antenna systems,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2562–2579, Oct. 2003. [17] 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. [18] K. Shoemake, “Animating rotation with quaternion curves,” in Proc. SIGGRAPH’85, ACM, San Franscisco, vol. 19, July 1985, pp. 245–254. [19] G. S. Watson, Statistics on spheres. New York: Wiley, 1983. [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] I. S. 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] R. Gr¨unheid, E. Bolinth, and H. Rohling, “A blockwise loading algorithm for the adaptive modulation technique in OFDM systems,” in Proc. IEEE VTC Fall, vol. 2, Oct. 2001, pp. 948–951. [23] 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. [24] ——, “Asymptotically dense spherical codes – Part II: laminated spherical codes,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1786–1798, 1997.
253
0-7803-8533-0/04/$20.00 (c) 2004 IEEE
