Equalization and Closed-Loop MIMO Technology
4.2 Precoder Selection Methods
We introduce several feedback-based methods for precoder selection in this section. Fig. 4.2 depicts the considered system model. Reference [14] proposed SVD-based search method and [17] proposed optimal precoder selection method. We propose max minSNR-based search method and MMSE-based exhaustive search method.
4.2.1 MaxminSNR-Based Search Method
In this method, We maximize the minimum SNR at the receiver antennas under MMSE or ZF equalization. It can be thought as minimize the maximum symbol error rate (SER) at
Æ
Figure 4.2: System model with feedback.
the receive antennas. The signal power per receive antenna can be expressed as
| gkTHFix |2 . (4.10)
The noise power is
σn2 | gTkgk | . (4.11)
The optimum precoding matrix F selected from the codebook C maximizes the minimum SNR of at the receive antennas as
F = arg max
where k is the index of the receive antennas, i is the index of the precoder, and gk is the equalizer for kth receive antenna.
4.2.2 MMSE-Based Exhaustive Search Method
In this method, we minimize the mean square error (MSE) between ˆX and X under the ZF or the MMSE equalizer. We compare all possible precoders to find the best one in the source that
min MSE = arg min
Fi k ˆX − Xk2. (4.13)
For ZF equalizer, therefore,
min MSE = arg min
Fi k(HFi)†HFiX + (HFi)†n − Xk2, (4.14)
and for the MMSE equalizer, min MSE = arg min
Fi k(HFiFHi HH + R−1n )HFiHFiX + (HFiFHi HH + R−1n )HFin − Xk2. (4.15) Then we will transmit the precoder index i back to the transmitter. The transmitter will use this precoder to allocate the power. In this method, the computation of complexity is high, because we have to evaluate all the precoders.
4.2.3 SVD-Based Search Method [14]
It was proved in [14] that for the ZF receiver, the optimum precoding matrix F selected from the codebook C maximizes the minimum singular value of HF. Reference [14] calls this the minimum singular value selection criterion (MSV-SC), i.e.,
F = arg maxFi∈Cλmin{HFi} (4.16) The optimum un-quantized precoder for MSV-SC has also been shown in [14], [18]. Let the singular value decomposition (SVD) of the channel matrix H be represented by
H = VLΣVHR. (4.17)
Then, the optimum um-quatized precoding matrix for MSV-SC is
Fopt = ¯VR, (4.18)
which is the first M columns of the right singular matrix VR. In practice, the optimum precoder shown in (4.18) is difficult to feedback. Using the criterion in (4.16), we can select a precoding matrix from a codebook, and feed back the index to the transmitter. The exhaustive search can be used to find the optimum precoding matrix in the codebook. If the codebook size is L and the number of the subcarriers is N, we then have to conduct the operation in (4.16) for LN times. The required computational complexity can be very high.
We will attempt to minimize the degradation in channel power introduced by quantizing the precoder kHFoptk2F − kHFk2F where k·kF is the Frobenius matrix norm. Note that
kHFoptk2F − kHFk2F = tr(ΣΣT) − tr(ΣVHRFFHVRΣT)
≤ tr(ΣΣT) − tr(ΣVRHFFHV¯RΣ¯T)
≤ λ2max{H}tr(IM − ¯VRHFFHV¯R)
= λ2max{H}1
2k ¯VRV¯HR − FF∗k2F
(4.19)
where the second line follows from setting ¯Σ equal to the first M columns of Σ, the third line results from substituting λmax{H} for the other nonzero singular values, and the last line cover from an alternative representation for subspace distance [14]. We will therefore design codebook F by attempting to minimize the distortion metric
EV¯R[ min
i∈{1,2,...,N }
1
2k ¯VRV¯HR − FFHk2F] (4.20) where ¯VR is isotropically distributed on U(Mt, M). Matrices in U(Mt, M) can be char-acterized as representing M-dimensional subspaces of the complex Mt-dimensional vector space. Thus [14] will adopt the common Grassmannian packing notation and define the set of all column spaces of the matrices in U(Mt, M) to be the complex Grassmannian space G(Mt, M, C). Thus if F1, F2 ∈ U(Mt, M) then column spaces of F1 and F2, P1 and P2 respectively, are contained in G(Mt, M, C). The chordal distance between two subspaces P1 and P2 is [1]
dchordal(F1, F2) = 1
√2kF1FH1 − F2FH2 k
= vu
utM −XM
i=1
λ2i{FH1 F2}.
(4.21)
Reference [14] use this sub-optimum codeword selection criterion which minimizes the chordal distance between the chosen codeword and the ideal (un-quantized) optimum
pre-coder, i.e.,
F = arg min
Fi∈Cdchordal(Fi, Fopt). (4.22) Simulation results show that this criterion can perform comparably to the MSV-SC criterion.
4.2.4 Optimum Precoder Selection [17]
To obtain the set of transmit matrices {Fk}, reference [17] now consider the minimization of an arbitrary objective function of the diagonal elements of MSE. The MSE can be expressed as
MSEk,i = [(I + FHkHHkR−1nkHkFk)−1]ii
= 1
1 + fkHHHkR−1k,iHkfk,i
. (4.23)
The minimization of arithmetic mean of the MSEs was considered in few papers. The objective function is
f0({MSEk,i}) =X
k,i
(MSEk,i). (4.24)
When f0 is Schur-concave, then precoder F is
F = UH,1ΣF,1 (4.25)
where UH,1∈ CnT×L has as columns the eigenvectors of RH corresponding to the L largest eigenvalues in increasing order, and ΣF,1 = [0diag({σF,i})] ∈ CL×L has zero elements, except along the rightmost main diagonal. This objective function is Schur-concave on each carrier k. Therefore, the diagonal structure is optimal. The problem in convex form is
{zmink,i}
X
k,i
1 1 + λk,izk,i
s.t.X
k,i
zk,i ≤ PT and zk,i ≥ 0, 1 ≤ k ≤ N, 1 ≤ i ≤ Lk. (4.26)
For simplicity of notation, reference [17] define zk,i , σFk,i2 and λk,i , λHk,i This partic-ular problem can be solved very efficient by because the solution has a nice water-filling
interpretation (from the KKT optimality conditions):
zk,i = (µ−1/2λ−1/2k,i − λ−1k,i)+ (4.27)
where µ−1/2 is the waterlevel chosen to satisfy the power constraint with equality.
4.2.5 Matrix Computation Complexity
In the design of signal processing algorithms, an analysis of the required number of flops is often derivable. Where a flop means a floating point operation. In this regard, a dot product of length n vectors is considered to require 2n flops because there are n multiplications and n adds. We derive the expressions for the number of needed flops that the equalization in the CL MIMO requires. Fig. 4.3 gives the flop counts of some common matrix operations [16].
In an order of magnitude study, we often throw away the lower order terms since their inclusion does not contribute to the overall flop count. But if the n is small, we have to keep the all the terms. This happens to be our case. We list the total flop counts per subcarrier in Fig. 4.4.
We can see that optimum precoder performs well in complexity because it does not have a complex computation flow. Also we can see that ZF equalizer is much easier than
å
Figure 4.3: Number of flops of different matrix operations [16].
MMSE equalizer. SVD-based and max minSNR-based search method have less complexity in high order matrix. We can have a conclusion that if the matrix size is low, we can use the MMSE-based search method. When the matrix size is high, we use SVD-based and max minSNR-based search method.
,-./012 3412 5 6712 489312
. /E 628812 98..12 F663.12
645312 924412 39 7.812
/E 23412 7.3412 77F7812
GG;H
6...12 6738.12 9437412
GG;H:N O
9 /E 58.12 232712 7F27.12
Figure 4.4: Number of flops per subcarrier of each method.