we show that the BA system can achieve diversity order MrMt for a system with Mr receive antennas and Mt transmit antennas if the codebook is prop-erly designed and has at least Mt codewords. Let the initial precoder F′ be an Mt × Mt unitary matrix (F′ = F and M = Mt). The number of bits to be transmitted in each channel use is Rb, which is distributed among M symbols (M ≤ min(Mt, Mr)). The augmented bit allocation vector b′ is of size Mt× 1.
It has at most M nonzero entries and PMt−1
i=0 b′i = Rb. Suppose the bit allocation codebook is Cb′. The minimum achievable BER is
BERmin(H) = min
b′∈Cb′
BER(b′, H), (4.44)
where BER(b′, H) is the BER in (2.3) Assume the bit allocation codebook Cb′
contains the set of codewords
Cb∗ ={Rbe0, Rbe1,· · · , RbeMt−1}, (4.45) where ei are standard vectors of size Mt× 1, i.e., [ei]i = 1 and [ei]j = 0 for j 6= i.
The following lemma shows that the BA system can achieve full diversity order
using the bit allocation vectors in Cb∗. Therefore to achieve a diversity order of MrMt we can use a codebook of size Mt, which requires only log2Mt feedback bits.
Lemma 3. For a finite-rate feedback MIMO channel with Mr receive antennas and Mt transmit antennas, the BA system with an Mt× Mt augmented unitary precoder F′ achieves diversity order MrMt if the bit allocation codebook Cb′ con-tains the Mt vectors in (4.44).
Proof. As Cb∗ is a subset of Cb′, we have BERmin(H) = min
b′∈Cb′BER(b′, H)≤ min
b′∈C∗b
BER(b′, H). (4.46) The average BER is bounded by
BER≤ E[ min
b′∈Cb∗
BER(b′, H)].
When the bit allocation b′ is chosen from Cb∗,all the Rb bits are allocated to the same symbol and this system becomes a beamforming system. For example, when b′ = [ Rb 0 · · · 0 ]T, the beamforming vector is the 0-th column of F′. When we choose b′ ∈ Cb∗ to minimize the BER, we are actually choosing the best beamforming vector from the columns of F′ to maximize the received SNR. In other words, the equivalent codebook of beamforming vectors is Cf = {f0′, f1′,· · · , fM′ t−1}, where fi′ is the i-th column of F′. From [40], we know such a beamforming system has diversity order equal to MrMt if the span ofCf is equal to CMt . Because F′ is an Mt× Mt unitary matrix, the span of Cf is the same as CMt.Therefore the BA system has diversity order MrMt when codebook Cb′
contains the vectors in (4.44).
Chapter 5 Simulations
In our simulations, the channel is of the form H =
r K
K + 1Hsp+
r 1
K + 1HwR1/2t for Ricean channel.
and
H = HwR1/2t for no line of sight.
and
H = Hw for uncorrelated channel.
Consider different channel case as following Channel I Uncorrelated channel.
Channel II No line of sight with low correlation for dt = 2, θt= 40◦. Channel III No line of sight with high correlation for dt= 2, θt = 8◦.
Channel IV Ricean channel with low correlation for dt = 2, θt= 40◦, dr = 1, θr = 20◦, K = 5.
Channel V Ricean channel with high correlation for dt= 1, θt= 20◦, dr = 1, θr = 10◦, K = 3.
We have used 106 channel realizations in the Monte Carlo simulations. The error rates are computed using (2.3) for both linear and decision feedback re-ceivers. For the decision feedback receiver, the detection order is determined
using the criterion of maximizing the rate-normalized SNRs mentioned in Sec 4.3. Antennas with spacing dt, dr and plane-wave span an angular spread of θt, θr at transmitter and receiver respectively.
Example 1. Distribution of bit allocation vectors.
In this example, the Channel I is considered. The number of receive antennas Mr is 5, and the number of transmit antennas Mtis 4. we compute the empirical distribution of bit allocation vectors. For a given channel realization, the best bit allocation vector in the codebook is chosen using the BER criterion. The number of bits transmitted per channel use is Rb = 12 and the number of substreams that the transmitter and receiver can process is M = 4. The corresponding opti-mal precoder F is the identity matrix and the receiver is linear. The number of possible integer bit allocation vectors is 455. We include in the codebook all 455 integer bit allocation vectors. Fig. 5.1(a) shows the distribution of the bit alloca-tion vectors, where the indexes of the vectors are ordered so that the probabilities are in decreasing order. The cdf (cumulative distribution function) is shown in Fig 5.1(b). We can see that some bit allocation vectors are far more probable than others. The probability of the 53 most probable bit allocation vectors is more than 99%. The distribution of the bit allocation vectors is highly skewed, rather than uniform. In all following examples with quantize bit allocation, we will choose the most probable 2B bit allocation vectors obtained in experiments like this example and use them as codewords when the number of feedback bits is B.
0 100 200 300 400 0
0.02 0.04 0.06 0.08 0.1 0.12
k
(a)
0 100 200 300 400
0 0.2 0.4 0.6 0.8 1
k
(b)
Figure 5.1: (a) Probability mass function of the bit allocation vectors for Channel I, Mr = 5, Mt = 4, M = 4, and Rb = 12; (b) corresponding cumulative distribution function.
Example 2. Precoder and distribution of bit allocation.
The correlated Channel II with zero mean is considered for Mr = 4, Mt= 5, M = 4. The number of bits transmitted per channel use is Rb = 8. We condider two type of the precoder F = Ut,M and F =
I 0
used, the receiver is linear. The number of possible integer bit allocation vector is 460. The codebook contains all 460 integer bit allocation vectors. Fig. 5.2(a) shows the distribution of the bit allocation vectors. The cumulative distribution function (cdf) is shown in Fig.
0 50 100 150 200 250 300 350 400 450 0
0.02 0.04 0.06 0.08 0.1 0.12
k
F=Ut,m F=[IM 0]T
(a)
0 50 100 150 200 250 300 350 400 450
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
k
F=Ut,m F=[IM 0]T
(b)
Figure 5.2: (a) Probability mass function of bit allocation vectors for Channel II, Mr = 4, Mt = 5, M = 4 andRb = 8; (b) Corresponding CDF.
5.2(b). From Fig. 5.2(a) we can see when F = Ut,M is used, the distribution of bit allocation vectors is more concentrated.
Example 3. BER bound.
In Fig. 5.3(a), Channel II is used. Mr = 4, Mt = 5, M = 3, Rb = 10, the precoder is F′ = Ut. We show the BER bounds BERbd,lin andBERbd,df. We have also computed BER0 in (4.8) over 106 channels for a linear receiver and for a decision feedback receiver. The results are called, respectively, BER0,lin and BER0,df. The gap between BERbd,lin and BERbd,df is around 3.5dB. We can see that the curve BERbd,df is an upper bound for BER0,df in low SNR and a lower bound for BER0,df in high SNR, consistent with what we have shown in Sec. 4.2.
The same can be observed for the case of linear receiver. In Fig. 5.3(b) Channel IV with both mean and covariance information is used. Mr = 5, Mt= 4, M = 4, Rb = 12. We use the approximation in (4.18) and choose F′ = bUt. In. 5.4 shows the same set of curves. We can have conclusions similar to those for correlated Channel II with zero mean.
5 10 15 20 25 30
Example 4. BER for different feedback bits.
In Fig. 5.4(a), Mr = 4, Mt = 5, M = 4, Rb = 8, Channel II is considered, the precoder is F′ = Ut. We shows the BER performance of the BA system for different number of feedback bits. The codewords are selected to minimize BER. The performance is shown for both linear and decision feedback receivers for different number of feedback bits. The BER is improved when the number of feedback bits B increases. We can see that BER of B = 5 is close to that of B = 9, in which case all the integer bit allocation codewords are used. Observe that the curves correspond to B = 7 and B = 9, are indistinguishable in the figure. We can understand this by examining the distribution plot in Fig. 5.2 The cdf is very close to one for k ≥ 150. When we increase B from 7 to 8 to 9, the added codewords are almost never chosen so the performance has no improvement. Fig. 5.4(b) also shows BER of the BA system when Channel V is considered with Mr = 5, Mt= 4, M = 4 and Rb = 12. The precoder is chosen as F′ = bUt. For the case B = 9 which considers all integer bit allocation codewords, the gain of the decision feedback receiver over the linear receiver is around 3.5dB, similar to the gap between BERbd,df and BERbd,lin observed in Fig. 5.3(a).
0 5 10 15 20 25
Example 5. BER for different Precoders.
In Fig. 5.5(a), Mr = 4, Mt = 5, M = 4, Rb = 8, B = 9 and channel III be considered. The BER plots are given for four different types of Mt × Mt
precoders and decision feedback at receiver. (1) the identity matrix, (2) the normalized DFT matrix , (3) the DCT matrix and (4) F = Ut . We can see that Uthas the best performance among. Fig. 5.5(b) shows the same set of curves for four precoders with linear receiver. Channel IV be considered. It has the same result as covariance feedback case that optimal precoder is F = bUt.
6 8 10 12 14 16 18 20 22 10−5
10−4 10−3 10−2 10−1 100
Po/No(dB)
BER
F=identity F=DFT F=DCT F=Ut
(a)
5 10 15 20 25
10−5 10−4 10−3 10−2 10−1 100
Po/No(dB)
BER
F=identity F=DFT F=DCT F=Ut
(b)
Figure 5.5: (a) BER for different precoder Mr = 4, Mt = 5, M = 4, Rb = 8 for Channel III. (b) BER for different precoder Mr = 5, Mt = 4, M = 4, Rb = 12 for Channel IV.
Example 6. BER for different Cb.
In Fig. 5.6, Mr = 5, Mt = 4, M = 4, Rb = 12 and F = Ut, Channel II and linear receiver are considered. In this case we show BER for two codebook, one trained using H and one trained using HF. Even though the precoder is chosen as F = Ut, the performance of the codebook trained using HF is better than the other for about 1dB for the same feedback rate. So we can conclude codebook training is important for system performance.
12 14 16 18 20 22 24
10−5 10−4 10−3 10−2 10−1 100
Pt/N 0
BER
trained using H, B=3 trained using H, B=5 trained using HF, B=3 trained using HF, B=5
5.6 Figure 5.6: BER with different Cb, Mr = 5, Mt = 4, M = 4, Rb = 12 for Channel II
Example 7. BER for MMSE and ZF receivers.
In this case, Mr = 5, Mt = 4, M = 4, Rb = 12 and F = IMt, Channel I is considered. We show the BER performance of MMSE and ZF receivers with linear and decision feedback receivers. Fig. 5.7(a) is linear receiver. In each case, the codebook is trained based the channel at receiver. From Fig. 5.7(a) we can see the ZF receiver is close to that of MMSE receiver. Fig. 5.7(b) show the two curves again when the receiver has decision feedback. We can draw conclusions similar to that for the linear receiver case.
8 10 12 14 16 18 20 22 24 26 Channel I (b) BER for decision feedback receiver for Channel I.
Example 8. Codeword selection criterion.
In this example, Mr = 3, Mt= 4, M = 3, Rb = 10, Channel II is considered and linear receiver is used. We compare the results using the BER criterion and the maximin criterion. In the first case, the codeword in the codebook that leads to the minimum BER is chosen. In the second case, the suboptimal codeword is chosen by quantizing the optimal bit allocation vector using the maximin criterion b = arg maxbb∈C
bmin(b∗k − bbk) described in Sec. 4.3. The results for B = 8 are shown in Fig. 5.8. The BER using the suboptimal maximin criterion is close to that using the minimum BER criterion.
8 10 12 14 16 18 20 22 24
10−6 10−5 10−4 10−3 10−2 10−1 100
Pt/N0
BER
suboptimal optimal
5.8 Figure 5.8: BER of BA system for Mr = 3, Mt= 4, M = 3, Rb = 10, Channel II is considered using the optimal BER criterion and suboptimal maximin criterion.
Example 9. BER for different K and precoder.
In this example Mr = 3, Mt= 4, M = 3, Rb = 8 and Channel IV with mean and covariance information is considered. The feedback bits is 8 and receiver is linear for Fig. 5.9. The BER plots are given for three types of Mt× Mt augmented precoders, (1) F′ = eig(Rt) which is the best precoder when there is no mean information. (2) The precoder is chosen as in (4.30), the optimal precoder when there is no correlation at transmitter, i,e. Rt = IM case in section 4.2.1. (3) F′ = eig( bRt). When the Ricean factor K is small, precoder 1 is better than precoder 2 and precoder is not as good for o large K. We also show the decision feedback receiver case in Fig. 5.10. The result is similar to Fig. 5.9. The BER performance is close for precoder 1 and precoder 2 when the Ricean factor K is small and precoder 2 is better than precoder 1 when the Ricean factor K is large.
We can see the precoder 3 is better then the other two for small or large K.
5 10 15 20 25 30
Example 10. Comparisons of BER for Mt = M case.
Mr = 5, Mt= 4, M = 4, Rb = 12 and Channel II is considered. In Fig. 5.11(a) we compare the BA system with the precoder system [4], in which the feedback is the index of the optimal precoder in the codebook and bits are uniformly loaded on all M symbols transmitted. In addition, we compare with the QR based system with bit allocation (VBLASTba) [21], the VBLAST system with feedback of ordering(VBLASTordering) [19]. The VBLASTordering system in [19]
feedback detection ordering for a fixed bit allocation. This is equivalent to having a codebook of all permutation of a single bit allocation vector. We also compare with VBLAST system with optimal precoder design (VBLASTprecoder) in [28].
The VBLASTprecoder in [28] requires no instant feedback. It designs for precoder based on statistics of the channelfor minimizing MSE. We can see if system has no bit allocation i.e. VBLASTprecoder and the precoder system, the BER performance is not as good. For VBLASTordering, the required number of feedback bits is log2(4!) ≈ 5. The number of feedback bits is made as close to 5 as possible except VBLASTprecoder system. For VBLASTba, the original codebook containing all integer vectors satisfying the sum rate constraint is trimmed by setting b1 ≥ 2 and b2, b3, b4 ≥ 0 as in [21], which results in a codebook of 35 codewords. For the precoder and BA systems, the codebook size is 32. The BER performance of the BA system with linear receiver is much better than of the the precoder system VBLASTprecoder and is comparable to VBLASTordering with decision feedback receiver. The VBLASTba system has BER similar to the BA system with a decision feedback receiver in low SNR. In Fig. 5.11(b) we show the result for Channel V, high correlation case with mean information. We see that the BA system achieve a good performance due to statistical precoder design.
5 10 15 20 25 30
Example 11. Comparisons of BER for Mt > M case.
Mr = 4, Mt = 4, M = 3, Rb = 8 and Channel I is considered. In Fig.
5.12(a) we compare with VBLASTordering, VBLASTba and precoder system. The VBLASTprecoder system is not compared in this example because it can be used only when M = Mt. In the case, we use augmented precoder for BA system.
For VBLASTordering, the required number of feedback bits is log2(3!) ≈ 3. The number of feedback bits is made as close to 3 as possible for all other cases. For VBLASTba, the original codebook containing all integer vectors satisfying the sum rate constraint is trimmed by setting b1 ≥ 2 and b2, b3 ≥ 0, which results in a codebook of 10 codewords. The BER performance of BA system with linear receiver is better than VBLASTba due to the flexible codebook design and aug-mented precoder is used. We can also see the BA system with linear receiver is very close to VBLASTordering which uses more complex decision feedback receiver in this case. Fig. 5.12(b) also show the high correlated case of Channel III. We can have conclusions similar to In Fig. 5.12(a).
10 15 20 25 30
Chapter 6 Conclusion
In this paper we considered the feedback of bit allocation for MIMO systems with limited feedback and the system is called a BA system. We first introduced system and channel model. Secondly, we derived the optimal unconstrained bit allocation for a given precoder. The optimal bit allocation is treated as a vector signal.
Based on the results of optimal bit allocation and statistical of the channel, we can use a approximation distribution of statistical to design the statistical precoder for Ricean channel. For line of sight case, a non-approximate distribution of statistical to design the optimal precoder. Furthermore when the number of transmit antenna is larger than the number of symbols transmitted, augmented precoding improve the performance and the use of augmented precoding does not require additional feedback. We have also shown that the proposed BA system can achieve full diversity order. Simulations have demonstrated the proposed BA system achieves a nice good trade-off between performance and feedback rate.
Appendix
Bibliography
[1] D. Love, and R. W. Heath, Jr., V. K. N. Lau, D. Gesbert, B. D. Rao, and M. Andrews, “An overview of limited feedback in wireless communication systems,”IEEE J. Sel. Areas Commun, vol. 26, No. 8, pp. 1341-1365, Oct.
2008.
[2] D. J. Love, and R. W. Heath, Jr.,“ Limited feedback unitary precoding for spatial multiplexing systems,” IEEE Trans. Information Theory, vol. 51, no.
8, Aug. 2005.
[3] W. Santipach, and M. Honig, “Asymptotic performance of MIMO wireless channels with limited feedback,” Proc. Military Communications Confer-ence, pp. 141-146, 2003.
[4] S. Zhou, and B. Li, “BER criterion and codebook construction for finite-rate precoded spatial multiplexing with linear receivers,” IEEE Transaction on Signal Processing, vol. 54, no. 5, pp. 1653-1665, May 2006.
[5] J. C. Roh, and B. D. Rao, “Design and analysis of MIMO spatial multiplex-ing systems with quantized feedback,” IEEE Trans. Signal Processmultiplex-ing, vol.
54, no. 8, Aug. 2006.
[6] S. Thoen, L. Van der Perre, B. Gyselinckx, and M. Engels, “Performance analysis of combined transmit-SC/receive-MRC,” IEEE Trans. Communi-cations, vol. 49, no. 1, pp. 5-8, Jan. 2001.
[7] R. W. Heath, Jr., and D. J. Love, “Multimode antenna selection for spatial multiplexing systems with linear receivers,” IEEE Transaction on Signal Processing, vol. 53, no. 8, pp. 3042- 3056, Aug. 2005.
[8] D. J. Love, and R. W. Heath, Jr., “Multimode precoding for MIMO wireless systems,,” IEEE Transaction on Signal Processing, vol. 53, no. 10, pp. 3674-3687, Oct. 2005.
[9] V. Raghavan, V. V. Veeravalli, and A. M. Sayeed, “Quantized multimode precoding in spatially correlated multiantenna channels,” IEEE Trans. Sig-nal Processing, vol. 56, no. 12, Dec. 2008.
[10] X. Song, and H.-N. Lee, “Multimode precoding for MIMO systems: perfor-mance bounds and limited feedback codebook design,” IEEE Trans. Signal Processing, vol. 56, no. 10, Oct. 2008.
[11] Syed Ali Jafar and Andrea Goldsmith “Transmitter Optimization and Op-timality of Beamforming for Multiple Antenna System,” IEEE Transaction on wireless Communications , vol. 3, no. 4, Oct. 2004.
[12] Eugene Visotsky and Upamanyu Madhow, “Space-Time Transmit Precoding With Imperfect Feedback,” IEEE Transaction on information theory , vol.
47, no. 6, 2001.
[13] W. Wang, G. Shen, S. Jin, D. Wang, and J. Chen “Approximate Minimum SER Preconding for Spatial Multiplexing Over Ricean channels,” in IEEE 20th International Symposium onPersonal, Indoor and Mobile Radio Com-munications, (PIMRC09), 2009, pp. 2360-2364.
[14] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela,
“V-BLAST : An architecture for realizing very high data rates over the righ-scatterning wireless channels,” Proc. International Symposium on Signals, Systems, and Electronics,, pp. 295-300, 1998.
[15] N. Wang and S. D. Blostein, “Approximate minimum BER power allocation for MIMO spatial multiplexing systems,” IEEE Trans. Communi., vol. 55, no. 1, Jan. 2007.
[16] H. Zhuang, L. Dai, S. Zhou, and Y. Yao, “A Low complexity per-antenna rate and power control approach for closed-loop V-BLAST,” IEEE Trans.
Communi., vol. 51, no. 11, Nov. 2003.
[17] A. Kannan, B. Varadarajan, J. R. Barry, “Joint optimization of rate alloca-tion and BLAST ordering to minimize outage probability,” IEEE Wireless Communications Networking Conference, 2005.
[18] S. T. Chung, A. Lozano, H. C. Huang, A. Sutivong, and J. M. Cioffi,
“Approaching the MIMO capacity with a low-rate feedback channel in V-BLAST,” EURASIP journal on Applied Signal Processing, no. 5, pp. 762-771, 2004.
[19] Y. Jiang and M. K. Varanasi, “Spatial multiplexing architectures with jointly designed ratetailoring and ordered BLAST decoding part-II: a practical method for rate and power allocation,” IEEE Trans. Wireless Communi.
vol. 7, no. 8, Aug. 2008.
[20] N. Prasad and M.K. Varanasi, “Analysis of decision feedback detection for MIMO Rayleighfading channels and the optimization of power and rate al-locations,” IEEE Trans. Information Theory. vol. 50, no. 6, June 2004.
[21] C. C. Weng, C. Y. Chen, and P. P. Vaidyanathan, “MIMO transceivers with decision feedback and bit loading: theory and optimization,” IEEE Trans.
Signal Processing. vol. 58, no. 3, March 2010.
[22] J. Du, Y. Li, D. Gu, A. F. Molisch, and J. Zhang, “Statistical Rate Allocation for Layered Space-Time Structure,” IEEE Trans. Communications, vol. 55, no. 3, March 2007.
[23] E. Visotsky and U. Madhow, “MSpace-time transmit precoding with im-perfect feedback,” IEEE Trans. Information Theory, vol. 47, no. 6, Sept.
2001.
[24] S. A. Jafar and A. Goldsmith, “MTransmitter optimization and optimal-ity of beamforming for multiple antenna systems,” IEEE Trans. Wireless Communi. vol. 3, no. 4, July 2004.
[25] M. Kiessling and J. Speidel, “Statistical prefilter design for MIMO ZF and MMSE receivers based on majorization theory,” IEEE International Conf.
Acoust., Speech, and Signal Processing. 2004.
[26] S. H. Moon, J. S. Kim, and I. Lee, “Statistical precoder design for spatial multiplexing systems in correlated MIMO fading channels,” IEEE Vehicular Technology Conference, 2010
[27] Y. Li, H. Liu, and L. Jiang, “Statistical precoder design for spatial correlated MIMO channels in V-BLAST systems,” International Conference on Neural Networks and Signal Processing, 2008
[28] T. Liu, J.-K. Zhang, K. M. Wong, “Optimal precoder design for correlated MIMO communication systems using zero-forcing decision feedback equal-ization,” IEEE Trans. Signal Processing, vol. 57, no. 9, Sept. 2009.
[29] R. Narasimhan, “Spatial multiplexing with transmit antenna and constel-lation selection for correlated MIMO fading channels,” IEEE Trans. Signal Processing, vol. 51, no. 11, Nov. 2003.
[30] S. Jarmyr, B. Ottersten, and E. A. Jorswieck , “SStatistical precoding with decision feedback equalization over a correlated MIMO channels,” IEEE Trans. Signal Processing, vol. 58, no. 12, Dec. 2010.
[31] X. Zhang, D. P. Palomar, and B. Ottersten, “SStatistically robust design of linear MIMO transceivers,” IEEE Trans. Signal Processing, vol. 56, no. 8, Aug. 2008.
[32] P. P. Vaidyanathan, See-May Phoong, and Yuan-Pei Lin, “Signal process-ing and optimization for transceiver systems,” Cambridge University Press, April 2010.
[33] D. S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correla-tion and its effects on the capacity of multielement antenna systems,” IEEE
[33] D. S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correla-tion and its effects on the capacity of multielement antenna systems,” IEEE