This chapter presents novel schemes to estimate spatial correlated MIMO fading channels based on new compact analytic models which can span the spatial and/or time correlation functions over the dominant signal subspace and provides additional directional informa-tion. Iterative algorithms are proposed for estimating spatial-correlated MIMO channels.
We then extend our work to model both spatial- and time-correlated link gains associ-ated with a MIMO channel and derive efficient estimators when the time-correlation is taken into account. We simulate the estimators’ performance in various popular industry-approved and standardized channels to validate the accuracy of our model and the useful-ness of our channel estimators. Numerical results show that in many instants the proposed
1e-05 0.0001 0.001 0.01 0.1
5 10 15 20 25 30 35 40
Normalized Mean Squared Error
Eb/N0 (dB) KT=1, KL=3, AS=8˚
KT=2, KL=3, AS=8˚
KT=3, KL=3, AS=8˚
KT=4, KL=3, AS=8˚
KT=8, KL=3, AS=8˚
Figure 3.10: MSE performance of Algorithm B in an SCM channel; AS=8◦, L = 12 and fdTs=0.02844.
algorithms give superior MSE performance. Our estimators offer tradeoffs between per-formance and complexity. They are easily extendable for use in wideband MIMO systems and are most effective when the channel’s AS is small, i.e., when the dimension of the dominant subspace is much smaller than full channel correlation rank. Not only do they offer fast and accurate estimates, give MSE performance improvement due to the noise reduction effect but, more importantly, also provide compact and useful CSI that lead to significant feedback channel bandwidth reduction and other potential post processing complexity cutbacks.
Chapter 4
Model-Based Eigen-Beamforming
In this chapter, we present a novel transceiver design based on a nonparametric MIMO channel estimator established in the previous chapter. Optimal MIMO system perfor-mance is achieved when CSI is available at both sites of the communication link. This is usually accomplished by deriving the CSI at the receiving site and feeding it back to the transmitting site. To maintain the promised system performance, large amount of CSI must be regularly updated at the transmit side through a feedback channel. Providing channel tracking information to the transmitter either consumes feedback bandwidth or increases the feedback delay. By using a reduced-order nonparametric MIMO channel model that characterizes the channel spatial correlations, we are able to reduce the feed-back requirement while compromising no system performance. We obtain bounds of the reception mean squared error and feedback information loss that can be used to assess the system performance. Numerical and simulation results based on several environment settings are given to validate the proposed method.
4.1 Modelling of Correlated MIMO Channels
4.1.1 Notations
Boldface upper-case and boldface lower-case letters denote matrices and column vectors, respectively. Italics denote scalars. Ri×j and Ci×j denotes the set of i × j real and complex matrices, respectively. The super-scripts (·)T, (·)∗ and (·)H denote transpose, complex conjugate, and Hermitian operations, respectively. Tr(·) denotes the trace of a matrix and det(·) denotes the determine of a square matrix. kXkF is the Frobenius norm
of a matrix X, and kYk2 is the 2-norm of a matrix Y. R(A) denotes the range (column space) of the matrix A. diag(x) is a diagonal matrix with its diagonal entries given by the elements of vector x, while diag(Y) denotes the column vector whose entries are the diagonal elements of matrix Y. vec(X) denotes a column vector obtained by stacking the columns of matrix X into a single vector. Operator ⊗ denotes the Kronecker product. IK denotes a K × K identity matrix. [X]i,j denotes the (i, j)th element of X while [X]L×M
signifies that X is an L × M matrix. Operator (x)+ is defined as max(x, 0).
4.1.2 System Setup
Following the same system configuration used in Chapter 2 and Chapter 3, the base station (BS) and mobile station (MS) are equipped with linear arrays of M and N antennas, re-spectively. Independent data streams x(t) = [x1(t), x2(t), x3(t), · · · , xM(t)]T are transmit-ted at BS at time t, where xm(t) denotes the source signal at the mth transmit antenna. At the MS, the received baseband waveform is given by y(t) = [y1(t), y2(t), y3(t), · · · , yN(t)]T, where yn(t) is the received signal at the nth receive antenna at time t. For notational simplicity, we define the two M-dimensional vectors xi = x(i△t) and yi = y(i△t), where
△t is the sampling interval.
4.1.3 Nonparametric Channel Modelling
For the convenience of reference, we summarize the channel representation proposed in Chapter 2 as following,
H≃ QN,KRCQTM,KTW, (4.1)
where W is the matrix bearing the directional information, QM,KT and QN,KR are M ×KT and N × KR matrices whose column vectors are the orthonormal bases used to describe the discrete root power correlations, KT and KR being the associated modelling orders.
Reminding that when the angle spread ∆ at BS is zero and antennas at BS are fully correlated, the waveform transmitted from the BS MEA can be regarded as a plane wave with a fixed AOD φ and W is therefore equivalent to a diagonal steering matrix,
W= diag ([w1, w2, · · · , wM]) (4.2)
where wi = exph
−j2π(i−1)dλ sin(φ)i
, d is the inter-element distance. On the other hand, if the angle spread is large enough and the MEA at the BS tends to be fully uncorrelated, the resulting modelling order KT used in QM,KT equals M.
There are several classes of basis functions to choose from. The Taylor and Weier-strass theorem arguments and the results of [27] suggest the use of polynomial regression estimators. If we use polynomials of degree K as basis functions for estimating spatial correlation functions of length L, the corresponding basis matrix PL,K has entries
[P]l,k = (l − 1)k−1, l = 1, 2, . . . , L, and k = 1, 2, . . . , K, (4.3) where the modelling order is K ≤ L. Although the column vectors in (4.3) can be used as bases, they are not orthogonal. Furthermore, these vectors have different norms, which might result in numerical instability. By applying the QR decomposition to PL,K [28], we obtain the orthonormalized basis matrices for QM,KT and QN,KR.
Another class of candidate basis matrices is the discrete cosine transform (DCT) ma-trices. The reasons for using DCT are twofold. Firstly, DCT is very good at energy compaction for most correlated sources, especially for Markov sources with high correla-tion coefficient. Furthermore, the channel correlacorrela-tion matrix RH defined in (4.11) below tends to be a toeplitz matrix, which can be approximately diagonalized by DCT. Sec-ondly, DCT has several well established computing structures that are both efficient and robust. A typical L × (K + 1) DCT matrix is defined as
[Q]l,k = q(k) cosπ(2l − 1)(k − 1)
2L , l = 1, 2, . . . , L, and k = 1, 2, . . . , K, (4.4) where q(k) = q
1
L for k = 1 and q(k) = q
2
L for 2 ≤ k ≤ K. If the modelling order K equals to L, both the orthonormalized polynomial basis matrix and the DCT matrix become full-ranked.
4.1.4 Nonparametric Space-Time Channel Estimation
For a receiver to extract both the coefficient matrix C and the directional information W in (4.1), in Chapter 3, we develop iterative schemes which consist of two processing phases to estimate several typical channels, such as block fading, time-variant frequency non-selective, and time-variant frequency-selective channels. The proposed channel estimators
incorporate the following two steps : (i) at the ith iteration, estimate the coefficient matrix c
Wi based on the estimate bCi−1 obtained from the previous iteration, and (ii) estimate the directional matrix bCi based on the tentative estimate cWi. Both estimators improve as one proceeds with more iterations. The proposed nonparametric estimators can be summarized as follows.
• Given a data block of B length-M input symbol vectors, X = [x1, x2, · · · , xB], and the channel output block, Y = [y1, y2, · · · , yB], the proposed channel estimator outputs the optimal solution { ˆφ, bC} for the least squared (LS) problem,
nφ, bˆ Co
= arg min
φ,C ||Y − QN,KRCQTM,K
TW(φ)X||2F, (4.5) where bCis a KR× KT complex matrix.
• Let ˆwi = exph
−j2π(i−1)dλ sin( ˆφ)i
and define cWdef= W( ˆφ) = diag ([ ˆw1, ˆw2, · · · , ˆwM]), a diagonal steering matrix associated with the mean AOD estimate ˆφ. The LS channel estimate is obtained by bH= QN,KRCQb TM,KTW.c
It is clear that once the low dimensional channel representationn φ, bˆ Co
becomes available, we can use them to synthesize the required CSI for feedback. In other words, these two matrices, bCand cW, serve as an alternative CSI that provide potential saving of feedback information.