Optimal Design of the Precoding Sequence

In Section 2.2, we see that in order to identify the channel, the precoding sequence must be selected so that the resulting matrix M_j is full column rank such that F_j can be exactly solved as (2.18). However, when noise is present, the covariance matrix R_¯x contains the contribution of noise and numerical error is present in the estimation of R_¯x in practice. This implies that (2.16) usually has no solution and (2.18) becomes a least squares approximate solution. The choice of M_j will affect error in the computation of F_j since different M^T_jM_j in (2.18) usually have different condition numbers. In this section, we discuss the optimal design of the precoding sequence, which takes into account the effect

of noise and numerical error in estimating ˆR_¯x, so as to increase the accuracy of F_j and thus reduce the channel estimation error.

2.3.1 Optimality Criterion

Now we consider the general case that noise is present and discuss the design of the precoding sequence p(n). From (2.4) and assumption (A1), the covariance matrix of the received signal is

R_¯x= H₀G²H^∗₀+ H₁G²H^∗₁+ σ_w²IJ ⊗ IP. (2.22) From (2.22) and (2.6), we see that noise has only contribution to the diagonal entries of R_¯x. Therefore the (L + 1) decoupled groups of equations in (2.16) remain unchanged, except for the j = 0 group, which becomes

Υ₀(R_¯x) = Υ₀

H₀G²H^∗₀+ H₁G²H^∗₁

+ σ²_wΥ₀(I_J ⊗ IP) = M₀F₀+ Y, (2.23) where Y = σ_w²[IJ IJ · · · IJ]^T ∈ R^{J P×J}. Thus from (2.18), ˆF₀, the least squares approxi-mation of F₀, can be written by

Fˆ₀ = (M^T₀M₀)⁻¹M^T₀ (M₀F₀+ Y)

 

Υ0(Rx¯)

= F₀+ (M^T₀M₀)⁻¹M^T₀Y = F₀+ Z, (2.24)

which is F₀ plus a perturbation term due to noise. The perturbation term Z is the least squares solution of the equation M₀Z = Y. We note that if every column of Y is orthogonal to every column of M₀, then Z = 0, which implies ˆF₀ = F₀. But that is impossible since the entries of M₀ are positive and those of Y are nonnegative. Therefore, we seek to appropriately choose the precoding sequence p(n) such that every column of Y is as close to being orthogonal to that of M₀ as possible. To this end, we first define q_ki and y_i shown below as the columns of M₀ and Y, respectively:

M₀ =



q₀₁ q₀₂ · · · q_0J

 

M0(:,1:J)

q₁₁ q₁₂ · · · q_1J

 

M0(:,J+1:2J)

· · · q L1 q_L2· · · qLJ

M0(:,LJ+1:(L+1)J)



, (2.25)

Y = σ_w²[I_J I_J · · · IJ]^T = [y₁ y₂ · · · yJ]. (2.26)

Then, due to the special structure of the block matrix M₀ and Y, it is easy to check that

Thus we only need to consider the relation between columns of q₀₁ and y₁ (the case of k = 0 and i = 1). Define the correlation coefficient

γ = q^T₀₁y₁

q₀₁₂y₁₂. (2.27)

Since γ is nonnegative and by Cauchy-Schwarz inequality, 0≤ γ ≤ 1. In order to make the perturbation term Z small, we choose q₀₁ so that the correlation coefficient γ is as small as possible. Based on this point of view, we formulate the optimal selection problem as minimizing γ subject to

Roughly, constraint (2.28) normalizes the power gain of the precoding sequence of each transmitter to 1; constraint (2.29) requires that at each instant, the power gain is no less than τ . Note that the problem of selecting the precoding sequence is identical to the SISO case considered in [16]. Thus the optimal precoding sequence p(n) is a two-level sequence with a single peak in one period [16]. More specifically, for each m, 0≤ m ≤ P − 1,

is an optimal precoding sequence. Because the precoding sequence is periodic with period P , the single peak can be placed at any one of the P positions which yield the same γ =

√ 1

P(1−τ)²+τ(2−τ). Note that γ decreases as τ decreases, which implies that the noise effect in the estimation of covariance matrix R_¯x is minimized and thus identification performance improves. However the peak location m does significantly affect the numerical condition of the linear equation (2.16). We discuss the selection of m next.

2.3.2 On Selection of m

We now consider the selection of m. We know that different choices of m result in dif-ferent matrix M_j and affect the numerical computation of F_j, j = 1, 2,· · · , L, in (2.18) and Fˆ₀ in (2.24), since different M^T_jM_j may have different condition number. If the condition number is large, then the matrix M^T_jM_j is ill-conditioned and the computations in (2.18) and (2.24) are sensitive to data error. Let

µ = max

0≤j≤Lκ(M^T_jM_j), (2.31)

where κ(A) is the condition number of A. Our goal is to choose m so as to minimize the largest condition number of the corresponding matrices M^T_jM_j, j = 0, 1,· · · , L. Since the peak appears at one of the P possible positions in the periodic precoding sequence, there are P precoding sequences which may result in P different µ. The following result shows that some choices of m are to be avoided since they result in some M_j being rank deficient and thus µ =∞.

Proposition 2.2 : At least one M_j, 0 ≤ j ≤ L, is not full column rank if and only if P − L + 1 ≤ m ≤ P − 2.

Proof : See Appendix A.

Hence if we choose, either 0≤ m ≤ P − L or m = P − 1, then each Mj is full column rank and the channel is identifiable. The following result shows that we can classify the remaining choices into 2 groups that are relevant to the optimal choice of m.

Proposition 2.3 :

(a) Each of the (P − L) choices, m = 0, m = 1, · · · , m = P − L − 1, results in the same µ denoted by µ₁.

(b) The two choices m = P − L and m = P − 1 result in the same µ denoted by µ₂. Also

µ₂ ≥ µ1.

Proof : See Appendix A.

From Proposition 2.3, we know if µ₂ > µ₁, then we choose case (a); if µ₂ = µ₁, we proceed to compare the second largest condition numbers of the set of matrices{M^T_jM_j}^L_j=0 for these two cases and choose the case whose value is smaller. If they are again equal, the same procedure can be done by comparing the third largest condition numbers and so on. Moreover, for 0 ≤ m ≤ P − L − 1 (case (a)), since the condition numbers of M^TjM_j are the same for each fixed j, j = 0, 1,· · · , L, (see Appendix A), we can use m = 0 to represent case (a). Similarly, m = P − 1 can be used to represent case (b). Hence the optimal selection of m reduces to one of two cases: m = 0 or m = P − 1. In other words, the optimal precoding sequence has a peak either at the beginning or at the end.