Simulation results reported in this section use the reference MIMO channel model of [52], [2]. We consider an 8×8 MIMO system. Spatial correlation matrices are generated by the power azimuth spectrum (PAS) at the BS and MS respectively according to the specific physical settings. To comply with the one-ring model, we assume that the environment surround MS is rich scattering and uncorrelated. Different physical settings at BS such as angle spread and nonzero AOD are used to examine the effect of different degrees of spatial correlation. The size of source vector, L, is 2, and the BPSK constellation is assumed.
Each element of the channel matrix is normalized to E [|Hi,j|2] = 1, for 1 ≤ i ≤ N, and 1 ≤ j ≤ M. The SNR is defined as the total transmitted power over the noise variance at each received antenna. Other assumptions used in our simulation are: (i) the antenna
∗
If λ
1≥ λ
2≥ · · · ≥ λ
Ldenote the first L eigenvalues and
λλ1L
is not significantly larger than 1, we can
roughly say that these eigenvalues are well-conditioned. Oppositely, we say that they are ill-conditioned.
spacing at transmit and receive antennas are both half wavelength, and (ii) additive noise distribution at the receiver is complex white Gaussian. The DCT bases are used in the proposed transceiver since we found that the DCT and polynomial bases give almost the same performance. For comparison purpose, we also present performance of the optimal transceiver with a full rank estimated CSI feedback (F-CSI).
0.0001 0.001 0.01 0.1 1
0 5 10 15 20 25 30
MSE
SNR (dB) Model-based / Perfect CSI - Simulation
Model-based / Reduced Rank / Perfect CSI : KT=3 Model-based / Reduced Rank / Perfect CSI : KT=4 Model-based / Reduced Rank / Estimated CSI : KT=3 Model-based / Reduced Rank / Estimated CSI : KT=4 EVD / Least Squared / Estimated CSI
EVD / Perfect CSI - Simulation EVD / Perfect CSI - Theory
Figure 4.4: MSE performance of DCT-based transceiver; angel spread = 4◦, AOD = 45◦, L = 2.
Performance curves in Fig. 4.4 represent the MSE performance for the MIMO system with AOD = 45◦ and angle spread ∆ = 4◦. The simulation results are obtained by averaging over 100 random channels. Since the channel correlation at BS is high, the corresponding correlation function lies in functional subspace of small dimension. Hence, even the number of antennas at BS increases, the amount of feedback information required by the proposed technique remains low and the performance degradation with respect to that of the optimal transceiver with F-CSI remains negligible. When the modelling order is only 3, which gives a compression rate of 0.18, the MSE performance of the proposed model-based system is still very close to that of the system with F-CSI. The MSE gap
between the performance of the optimal transceiver with perfect CSI and that of the transceivers with estimated CSI is obviously due to channel estimation error.
As the angle spread ∆ increases, correlation between the transmit antennas diminishes and a higher modelling order is necessary to describe the rapid-changing spatial correlation at the transmit site. MSE curves in Fig. 4.5 are the performance of an 8×8 MIMO system with AOD= 45◦, angle spread ∆ = 15◦, and different modelling orders. Simulation results indicate that bases of order less than 4 tend to incur larger modelling errors while those of order larger or equal to 4 provide performance which is almost the same as that of the optimal transceiver with F-CSI.
0.0001 0.001 0.01 0.1 1
0 5 10 15 20 25 30
MSE
SNR (dB) Model-based / Perfect CSI - Simulation
Model-based / Reduced Rank / Perfect CSI : KT=3 Model-based / Reduced Rank / Perfect CSI : KT=4 Model-based / Reduced Rank / Perfect CSI : KT=5 Model-based / Reduced Rank / Estimated CSI : KT=3 Model-based / Reduced Rank / Estimated CSI : KT=4 Model-based / Reduced Rank / Estimated CSI : KT=5 EVD / Least Squared / Estimated CSI
EVD / Perfect CSI - Simulation EVD / Perfect CSI - Theory
Figure 4.5: MSE performance of DCT-based transceiver; angel spread = 15◦, AOD = 45◦, L = 2.
In both figures, we notice that the proposed approach outperform the F-CSI approach at low SNRs. This observation indicates that although our approach introduces modelling error due to reduced-rank regression model it also reject the noise outside the modelling space. At higher SNRs, like other model-based methods, the modelling error dictates the performance whence this advantage gradually disappears.
The usefulness of the upper bounds derived in Theorems 4.1 and 4.2 are demonstrated in Fig. 4.6. It is shown that both bounds predict correct trend of the MSE performance of the system. These bounds becomes tighter as the modelling order increases. Although the bound of (4.29) is tighter than (4.28), (4.28) needs only the knowledge of the smallest eigenvalue, ǫL.
We use the two perturbation upper bounds given in Lemma 4.2 to review the effect of CSI error from a geometric perspective in Fig. 4.7(a), assuming AS = 2◦. The distance between the subspace associated with perturbed CSI using a rank 1 (KT = 1) approxima-tion and that associated with the perfect CSI increases as the quality of channel estimaapproxima-tion deteriorates at lower SNRs, and remains steady for the case of good channel estimation (high SNR).
In Fig. 4.7(b), we plot the distance between the above-mentioned two subspaces for the cases of AS equals to 4◦. These curves show that in higher correlation case, a rank-1 model is sufficient to describe the spatial correlation and thus the corresponding distance is small. For the larger AS case, since the channel correlation decreases the subspace distance increases for the rank 1 system and a larger modelling order is needed.
4.6 Summary
This chapter presents a novel regression model-based transceiver design for spatial corre-lated MIMO fading channels. Orthogonal bases and an additional AOD information are used to model the spatial correlation functions associated with MEAs in BS and MS so that compact CSI representation can be obtained. Optimum precoding strategies are pro-vided based on the proposed channel representation. Computer simulation results show that excellent performance is attainable if proper modelling basis and order are used.
The modelling order provides trade-off between reception performance and feedback com-plexity. Significant feedback compression is achieved if the channel spatial correlation is high. To analyze the performance loss, we derive perturbation bounds for the reception MMSE caused by CSI modelling errors. We also provide bounds for the distance between the signal subspace associated with perfect CSI and that associated with the proposed
0.002 0.0025 0.003 0.0035 0.004
20.8 21 21.2 21.4 21.6 21.8
MSE
SNR (dB) Perfect CSI - Simulation
Model-based / Reduced Rank : KT=2 Model-based / Reduced Rank : KT=3 Model-based / Reduced Rank : KT=4
Upper Bound 1 / Model-based / Reduced Rank : KT=2 Upper Bound 2 / Model-based / Reduced Rank : KT=2 Upper Bound 1 / Model-based / Reduced Rank : KT=3 Upper Bound 2 / Model-based / Reduced Rank : KT=3 Upper Bound 1 / Model-based / Reduced Rank : KT=4 Upper Bound 2 / Model-based / Reduced Rank : KT=4
Figure 4.6: MSE upper bounds of DCT-based transceiver; angel spread = 10◦, AOD
= 45◦, L = 2.
approach for which only imperfect CSI is available. Numerical results for these bounds are given to show that performance trends can be accurate predicted.
4.7 Acknowledgement
We would like to thank the European Union IST project ISI-2000-30148 I-METRA Deliv-erable D2 for providing a MIMO simulation package as the reference space-time channels.
0
Upper Bound : Lemma 2 - First Inequality Upper Bound : Lemma 2 - Second Inequality
0
Upper Bound : Lemma 2 - First Inequality Upper Bound : Lemma 2 - Second Inequality
0.1
Upper Bound : Lemma 2 - First Inequality Upper Bound : Lemma 2 - Second Inequality
0.1
Upper Bound : Lemma 2 - First Inequality Upper Bound : Lemma 2 - Second Inequality
Figure 4.7: (a): angle spread = 2◦, AOD = 45◦, and (b): angle spread = 4◦, AOD = 45◦.
: distance between signal subspaces of perfect CSI and model-based CSI using a rank-1 approximation; △ : perturbation bound of (4.36). ◦ : perturbation bound of (4.37).
Chapter 5
Conclusion and Future Work
This thesis presents a framework of nonparametric model-based MIMO systems which are basically based on the proposed compact analytic models for correlated MIMO fad-ing channels. The proposed work models both spatial- and time-correlated link gains associated with a MIMO channel and derive efficient estimators when the spatial and/or time-correlation is taken into account. For correlated MIMO channels, by spanning the spatial and/or time correlation functions over the dominant signal subspace using a set of orthogonal modelling bases, we obtain an efficient channel representation that can al-leviate the processing complexity for many post-channel-estimation processes and reduce the feedback bandwidth requirement for MIMO precoding systems as well. Tremendous computation saving and large reduction of feedback data rate are accessible especially for large MIMO systems and highly correlated environments. Based on the proposed models, we develop channel estimation schemes against several typical channel situations. Iter-ative batch algorithms are proposed to accomplish the task of channel estimation, and the sequential adaptive algorithms are also available for channel tracking [17]. Various popular industry-approved and standardized channels are simulated to validate the accu-racy of our model and the usefulness of our channel estimators. Numerical results show that the proposed algorithms can provide tradeoffs between performance and complexity.
Moreover, we also show that under different channel conditions, the modelling order that leads to dimension reduction may also achieve the best MSE performance due to the noise reduction effect discussed in Chapter 3. In this situation, we can provide compact and useful CSI that lead to significant feedback channel bandwidth reduction and other
po-tential post processing complexity cutbacks while retaining good reception performance at the same time.
Based on the proposed channel representation, optimum precoding/eigen-beamforming strategies are provided for feedback MIMO systems. Computer simulation results show that excellent performance is attainable provided proper modelling basis and order are used. Over a wide range of interested SNR, the reception performance of the model-based eigen-beamforming system using optimal modelling order is shown to be better than that of the conventional eigen-beamforming systems, which use full dimensional LS channel estimation results as feedback CSI. Significant feedback compression is achieved if the channel spatial correlation is high. We derive perturbation bounds for the reception MMSE to analyze the performance loss caused by CSI modelling errors. We also provide bounds for the distance between the signal subspace associated with perfect CSI and that associated with the proposed approach for which only imperfect CSI is available. Numer-ical results for these bounds are given to show that performance trends can be accurate predicted. For limited feedback MIMO systems, we prove that the proposed model will lead to fewer distortion or compact compression if a conventional quantization scheme, such as Grassmannian packing, is used.
The framework of the proposed model-based MIMO system leads to a new class of model-based MIMO processing techniques. Similar to the proposed precoding/eigen-beamforming systems, MIMO detection schemes that incorporate the MIMO channel matrices, such as LLL aglorithm which is based on the theory of lattice reduction [53], and the sphere decoding algorithm that performs the QR factorization of the channel matrix [54],[55], can be derived accordingly. Besides the potential saving of computational complexity, the proposed channel representation may lead to essential improvement of the lattice structure embedded in the underlying channel matrix. Such improvement is suspected to bring additional reception benefits. Hence, experimental algorithms and theoretical studies are going to be conducted to reveal the potential usages of the proposed model-based schemes.
The performance improvement made by the proposed model-based systems somehow
depends on the accuracy of the selected modelling order. Although, in general, we can use a modelling order larger than is needed to guarantee a negligible modelling error, repre-sentation efficiency and processing advantages will diminish for an over-modelling system.
The order determination scheme provided in this thesis calculates the optimal order based on the long-term channel statistics. In practice, for a more non-stationary channel, the modelling order should be update in a short period. Hence, a order determination scheme, which is computational more efficient and works reliably under non-stationary environ-ment is what we try to explore next.
Appendix A
AoD Information Extraction
For small ∆, the correlation between two transmit antennas i, j can be approximated by [6]
In addition, correlation between two receive antennas p, q can be approximated by E
hpih∗qi
which can be further decomposed by using the W defined in (2.17)
ΦT = W·
The correlation matrix at the receive site can also be decomposed as
ΦR =
where ˜ddef= 2πdλ . The above two equations immediately lead to (A.4) and (A.5). Here, the separable model (2.2) is equivalent to
H= ¯Φ
and the canonical model (2.4) is equivalent to
H= ¯ΦR12HindΦ¯T12 TW, (A.5)
where ¯ΦT and ¯ΦR denote the power correlation matrices at the transmit and the receive sites, respectively. Using Φ1/2T = W ¯Φ1/2T and following a procedure similar to (2.12)–
(2.15), we obtain (2.16) of the main text.
The separable W can also be obtained directly from the physical model [56],[57]. In [56], the directional term exp
−j2π(i−j)d sin φ λ
is also shown to be separable in the expres-sion of spatial correlation (i.e., Eq. (9) in [56]), and has the similar form of (A.1). Note that Forenza et al. [58] have recently showed that, for a clustered MIMO channel with uniform linear or circular array, the cross-correlation coefficients also have a regression form similar to (A.1). Hence, if we assume a similar environment, we will obtain an analytical model of the same form as (2.16).
In the above single-directional model, the AoD from the transmitting antennas at the transmitter can be captured by a mean AoD. In contrast, the principle of maximum entropy [59] assumes i.i.d. uniformly distributed AoA angles over [0, 2π] and leaves no mean arriving direction being modelled at the mobile side. It models the separate power azimuthal spectra (PAS) of AoA and AoD, with a common direction being described by the mean AoD at the base station [57].
Appendix B
Proof of Lemma 3.1
According to Lemma 5.1.3 of [18], the ith entry of the vector
(1E ⊗ A) ⊙ (BT ⊗ 1N) cis identical to the (i, i)th diagonal entry of the square matrix
(1E ⊗ A)diag(c)(B ⊗ 1TN) , for i = 1, 2, . . . , NE. Define ˜A= [˜am,n]def= (1E⊗ A) and ˜B= [˜bm,n]def= (BT ⊗ 1N). Then, for i = N(p − 1) + q, p ∈ {1, . . . , E} and q ∈ {1, . . . , N}, we have
(1E ⊗ A)diag(c)(B ⊗ 1TN)
i,i= XM
j=1
˜ai,jcj˜bi,j = XM
j=1
aq,jcjbj,p = [vec (Adiag(c)B)]i, (B.1)
where aq,j is the (q, j)th entry of A and bj,p is the (j, p)th entry of B. [D]i,i denotes the (i, i)th entry of the matrix D, while [e]i denotes the ith entry of the vector e. Hence we conclude that
(1E ⊗ A) ⊙ (BT ⊗ 1N) c
i = [vec (Adiag(c)B)]i, ∀ i = 1, . . . , NE, (B.2) which proves the Lemma 3.1.
Appendix C
Proof of Theorem 4.1
Proof. Since ΦΛ
LΦ is a diagonal matrix, its eigenvalues are the permutation of the di-agonal elements {λiφ2i}. Let {λπ,iφ2π,i} be the permutation of {λiφ2i} such that λπ,1φ2π,1≥ λπ,2φ2π,2 ≥ . . . ≥ λπ,Lφ2π,L are in descending order. Due to the fact that ΦUHEUΦ is Hermitian, we have the following perturbation bound from Weyl theorem [18],λπ,iφ2π,i+ ǫL≤ γi. (C.1)
Since the function f (z)def= 1+z1 is monotonically decreasing for z > −1, we have X
i
f (ai) ≥X
i
f (bi), if − 1 ≤ ai ≤ bi, ∀ i. (C.2)
Combining (4.27), (C.1) and (C.2), we complete the proof by recognizing that XL
i=1
1
1 + λπ,iφ2π,i+ c = XL
i=1
1
1 + λiφ2i + c, ∀ c /∈ {−1 − λiφ2i}. (C.3) In order to keep f (·) being monotonically decreasing, we must have mini(λiφ2i) + ǫL> −1, which implicitly satisfies the constraint in (C.3).
Appendix D
Proof of Theorem 4.2
Proof. Notice that ΦΛΦ is a diagonal matrix, its descending ordered eigenvalues are
exactly the permuted diagonal elements {λπ,iφ2π,i}Li=1. From theorem (9.G.1) of [39], we have(γ1, γ2, . . . , γL)
= µ1(ΦUHRHUΦ), µ2(ΦUHRHUΦ), . . . , µL(ΦUHRHUΦ)
≺ µ1(ΦΛΦ) + µ1(ΦUHEUΦ), . . . , µL(ΦΛΦ) + µL(ΦUHEUΦ)
= λπ,1φ2π,1+ ǫ1, λπ,2φ2π,2+ ǫ2, . . . , λπ,Lφ2π,L+ ǫL
, (D.1)
where µi(G) is the ith largest eigenvalues of the matrix [G]L×L and µ1 ≥ µ2 ≥ . . . ≥ µL. Let h({zi}) = P
if (zi) = P
i 1
1+zi, where zi ≥ zi+1. Since f′(z) = (1+z)−1 2 is negative, continuous and monotonically increasing for z > −1, f(z) is convex for z > −1. From (3.C.1) of [39], h({zi}) is thus Schur-convex, hence, h({xi}) ≤ h({yi}), if {xi} ≺ {yi}.
Combining the above results with (D.1), we complete the proof by restricting the domain to min {γi}, {λπ,iφ2π,i+ ǫi}
= λπ,Lφ2π,L+ ǫL = mini{λiφ2i} + ǫL > −1.
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