Optimal precoders design with Ricean channel

Suppose A is a Mr× Mt matrix each row of which is independently drawn from a Mt-variate normal distribution with zero mean each row of A is independently and let the ith column of A^†be gi , then the autocorrelation matrix of gi is equal to R_t. It is known that A^†A = PMr−1

i=0 g_ig_i^† has a complex Wishart distribution with M_rdegrees of freedom, denoted asWMt(R_t, M_r) [38]. When B has a Wishart distribution, we say B⁻¹ has inverse Wishart distribution. For Ricean channel model, the channel be considered as

H =

K+1Hsp, is called non-centrality parameter matrix means the expectation of H. Mr is degree of freedom and Rt is the autocorrelation matrix of HwR^1/2_t . This non-central Wishart distribution can be approximated by a

Wishart distribution [39]

N WMt(Rt, M, Mr)∼ WMt( bRt, Mr), (4.18) where bRt= Rt+ M^†M/Mr.

Linear receiver We can obtain σ²_ek by averaging the error covariance matrix R_e= N₀(F^†H^†HF)⁻¹ over the channel. If F is nonsingular matrix then the ma-trix F^†H^†HF is W^M(F^†RbtF, Mr) and so the matrix R⁻¹_e = 1/N0F^†H^†HF has a Wishart distribution W^M(N₀⁻¹F^†RbtF, Mr). Then Re has an inverse Wishart distribution. It has been shown in [43] that when a matrix B is Wishart distri-butionW^p(A, r) with r > p , then E[B⁻¹] = 1/(r− p)A⁻¹. Using this result and assuming Mr> M, Re= E[Re] is given by

Re= N₀

Mr− M(F^†RbtF)⁻¹. (4.19) Let the eigen decomposition of bRt be bRt = bUtΛbtUb^†_t, where bΛt is a diagonal matrix and the diagonal elementsλ_t,i are the eigen value of bR_t. Let the diagonal elements of bΛt be ordered such that λt,0 ≥ λ^t,1 ≥ . . . ≥ λ^t,Mt−1 and assume λt,Mt−1 > 0

Theorem 1. For the linear receiver, the BER bound BERbd in (4.14) satisfies

BERbd ≥ BERbd,lin, whereBERbd,lin = 4M The inequality becomes an equality when F = bUt,M, where bUt,M is the submatrix of bUt that consists of the first M column vectors of bUt.

Proof. Majorization theorem [36] will be used to prove the theorem. For com-pleteness, some related definitions are given below.

(1)Given a sequence a[0], a[1], . . . , a_{[M −1]},the notation a[k] refers to the permuted majorizes b if the following two conditions are satisfied: P_{M −1}

k=0 a_k = P_{M −1}

k=0 b_k

and Pn

k=0a[k] ≥Pn

k=0b[k] , 0≤ n ≤ M − 2 . Let g(y) be a real-valued function of a real vector y. We say that g(y) is Schur-concave if g(a)≤ g(b) whenever a majorizes b. is a diagonal matrix. The matrix R⁻¹_e is the inverse of Re, their eigen val-ues are related by λi(Re) = 1/λ_{M −1−i}(R⁻¹_e ). As R⁻¹_e = ^Mr_N^−M

0 F^†RbtF and F is unitary, we can apply the Poincare separation theorem to bound the eigen values of R⁻¹_e using the eigen values of bRt. Poincare separation theorem says λi(B) ≥ λi(C^†BC),i = 0, 1, . . . , r− 1, for any n × n Hermitian matrix B and In (4.20),The lower bound Q_{M −1}

i=0 N0

Mr−M 1

λi(Rt) can be achieved by choosing F = bUt,M. Using the above inequality and the monotone increasing property of f (·), we can establish the inequality in (4.20). Therefore, to minimize the BER bound BER_bd the optimal precoder is F = bU_t,M. 

detector

Mr

r

G

B

Figure 4.1: Block diagram of the decision feedback receiver.

Decision feedback receiver To consider the precoder design for a decision feedback receiver, we can use the receiver structure in Fig .4.1 based on the

QR decomposition of HF [21] [28]. This corresponds to the case of a reverse detection ordering. Let the QR decomposition of HF be QR, where Q is an Mr×M unitary matrix and R is an M ×M upper triangular matrix with diagonal element [R]ii= rii. The feedforward matrix G and feedback matrix B are given, respectively, by [32]

G = (r⁻¹₀₀ r⁻¹₁₁ · · · r_{M −1,M−1}⁻¹ )Q^† (4.22) B = (r₀₀⁻¹ r⁻¹₁₁ · · · r⁻¹_{M −1,M−1})R− IM (4.23) Assuming there is no error propagation, the kth subchannel error ek = bs^k − sk

has variance σ²_ek = N0r_kk⁻², k = 0, 1, . . . , M − 1. The average error variance is σ²_ek = N0E[r⁻²_kk]. The value E[r_kk⁻²] as been shown to be related to the Cholesky decomposition of F^†Rb_tF in [28]. The result is summarized in the following lemma.

Lemma 2. [28] When K = 0, H = HwR^1/2_t , the following result was derived.

Let the Cholesky decomposition of F^†RtF be LDL^† where L is a lower triangular matrix with unity diagonal elements and D is diagonal. For Mr > M, E[r⁻²_kk] = d⁻¹_kk/(Mr− k − 1), for k = 0, 1, . . . , M − 1 where dkk is the kth diagonal element of D.

Using lemma 2 and the approximation in (4.18), the results in Lemma 2 allows us to establish the following bound for BER_bd

Theorem 2. For the decision feedback receiver with Mr > M, the BER bound BERbd in (4.14) satisfies

BERbd ≥ BER^bd,df, where bUt,M is the submatrix of bUt that consists of the first M column vectors of Ubt.

Proof. Using Lemma 2, we can obtain σ²_ek = N0d⁻¹_kk/(Mr − k − 1) . Thus

The BERbd in(4.14) can be expressed as

BERbd = 4M

Applying the Poincare separation theorem (also stated in the proof of Theorem 1), we have the inequality

BERbd = 4M

In this special case, we assume no correlation at transmitter and receiver. Channel is considered as H = q

K

K+1Hsp+q

1

K+1Hw, where Hsp , ara^T_t, ar and at are LOS array response at transmitter and receiver described in chapter 2. Let H^†_spHsp = VΛV^†, we have nonzero eigenvalue, the other eigenvalues are 0 and the eigenspaces of λ0 and 0 are othogonal. Because Hsp = ara^T_t = [ar0at ar1at. . . ar_Nrat]^T, we can see that when we take the hermitian of the first row and normalize, it is equal to ea^∗_t. When Rt= I, the first column of optimal precoder F is the hermitian of the first

row of Hsp with normalization, the other columns of F can be arbitrarily chosen, except for the restriction that the columns of F are orthonormal. That is

F = h

at

katk f2 · · · f^Mi

, (4.30)

where f2,· · · fM are arbitrarily vectors such that F is unitary. Such a precoder has been shown in [11] to maximize capacity of a beamforming system.

No line of sight (K = 0) case

In this case, we consider the special case that K = 0 such that H in (4.17) becomes to

H = HwR^1/2_t

it is known H^†H has a complex Wishart distribution with M_r degree of freedom, denoted as W^Mt(Rt, Mr) instead of complex non-central Wishart distribution so we don’t need take approximate. Let the eigen decomposition of Rt be Rt = UtΛtUt, where Λt is a diagonal matrix and the diagonal elementsλt,i are the eigen value of Rt. Let the diagonal elements of Λt be ordered such that λt,0 ≥ λt,1 ≥ . . . ≥ λt,Mt−1 and assume λt,Mt−1 > 0.

Linear receiver Using the property of Wishart distribution , the matrix F^†H^†HF is WM(F^†RtF, Mr) and we have

R_e = N0

Mr− M(F^†R_tF)⁻¹ (4.31) Using the proof of theorem 1 , we have

M −1Y

thus

Decision feedback receiver We can use lemma 2 and theorem 2. Let the Cholesky decomposition of F^†RtF be LDL^†, thus we have

With the same result as mean feedback, then F = Ut,M the bound BERbd can be minimized.