Estimators

In this subsection, the estimator G_k is optimized for given codebook entries (precoders), partition cells, and other estimators, i.e. G_opt,k = argmin_GkMSE (see Eq. (7.7)). Due to [cf. Eq. (7.4)]

C_z,k = E z_kz^H_k

= G_k SC_h,kS^H+ C_η,k
G^H_k

we get the following alternative parameterization of the estimator Gk= C_z,k^1/2X_k^H SCh,kS^H+ Cη,k

_−1/2

(7.9) where the unknown X_k ∈ C^Ntr×N has orthonormal columns, i.e. X_k^HX_k = IN. It is very easy to see that this expression for G_k leads to C_z,k when we substitute into G_k(SCh,kS^H+ Cη,k)G^H_k. Note that the transformation of Sh_k+ ηkwith(SCh,kS^H+ C_η,k)^−1/2 leads to an uncorrelated signal with unit covariance matrix and the additional transformation with X_k^H again gives an uncorrelated signal with unit covariance matrix irrespective of the choice for C_z,k. Therefore, the optimization with respect to G_kcan be split into an optimization with respect to X_kand a subsequent optimization with respect to C_z,k.

Before the minimization of E

ku − ˆuk²2

 with respect to Xk can be performed, the MSEE

ku − ˆuk²2

must be rewritten by using the matrix A_k defined as A_k = C_h,kS^H SC_h,kS^H+ C_η,k
−1/2

∈ C^{N ×Ntr} (7.10)

and by obtaining the conditional momentsE[H|z ∈ Rⁱ] and E[H^HH|z ∈ Rⁱ]. Taking into account that h_kand z_kare jointly Gaussian, we have

hk

where C_zh,kis given by Czh,k = E

zkh^H_k

= C_z,k^1/2X_k^H SCh,kS^H+ Cη,k

_−1/2

SCh,k. (7.11) Thus, the following conditional moments read as (e.g. [121])

E[hk|z^k] = C_zh,k^H C_z,k⁻¹zk = Ch,kS^H SCh,kS^H+ Cη,k

7.3 Proposed MMSE Optimization 169

Note that µ_k,iand R_k,i only depend on the choice ofR^k,iwhich are assumed to be given in this subsection. The above results for E[H|z ∈ Rⁱ] and E[H^HH|z ∈ Rⁱ] can be substituted into Eq. (7.7). Thus, the MSE for the given codebook entries{Pⁱ, gi}^Mi=1 and partition cells{Ri}^Mi=1is expressed as

MSE= E

As mentioned above, when introducing the alternative representation of the estimator G_k in Eq. (7.9), we first find the basis X_kby minimizing the above MSE for fixed C_z,k, i.e.

X_opt,k = argmin

X_k

MSE s.t.: X_k^HX_k= IN.

The constraint ensures the sub–unitarity of X_k ∈ C^Ntr×N. The corresponding Lagrangian function reads as

L(X_k, Λ_k) = MSE + tr Λ_k X_k^HX_k− I

with the Lagrangian multiplier Λ_k ∈ C^{N ×N}, which is Hermitian by definition, as the constraint is Hermitian. A necessary condition for optimality is that

∂L(Xk, Λk)

∂X_k^T = ∂MSE

∂X_k^T + ΛkX_k^H= 0.

From this KKT condition we obtain that [cf. Eq. (7.13)]

−piµ_k,ie^T_kP_i^TgiA_k− piX_k^HA^H_kg_i²P_i^∗P_i^TA_k+ piR_k,iX_k^HA^H_kg_i²P_i^∗P_i^TA_k+ ΛkX_k^H= 0.

Since the range of the first three summands reachable for row vectors multiplied from the left is the span of the rows of A_k, the space spanned by the rows of X_k^Hmust be the same to fulfill the above condition. Thus,

range(X_k) = range A^H_k

. (7.14)

By considering the Singular Value Decomposition (SVD) of a matrix B = M DN^H, where D is a square diagonal matrix and M and N are unitary or sub–unitary, it is satisfied that the range of B is equal to the range of M [107]. With this property and the SVD of A_kgiven by

Ak= VkΦkW_k^H

with unitary V_k ∈ C^{N ×N}, diagonal Φ_k ∈ R^{N ×N} whose diagonal elements are positive, and sub–unitary W_k∈ C^Ntr×N, we have that range(A^H_k) = range(W_k). We can conclude that the optimal basis is expressed as

Xopt,k = WkU_k^H ∈ C^Ntr×N (7.15)

to fulfill the condition in Eq. (7.14). The so far undefined unitary U_k ∈ C^{N ×N} must be chosen to minimize the precoding MSE in Eq. (7.13). As Φ_kW_k^H = V_k^HAk, the optimal estimator must have the form [cf. Eq. (7.9)]

Gopt,k = C_z,k^1/2X_k^H SCh,kS^H+ Cη,k

_−1/2

= C_z,k^1/2UkΦ⁻¹_k V_k^HAk SCh,kS^H+ Cη,k

_−1/2

= C_z,k^1/2U_kΦ⁻¹_k V_k^HGMMSE-estim,k (7.16) where the conventional linear MMSE estimator is given by

GMMSE-estim,k = C_h,kS^H(SC_h,kS^H+ C_η,k)⁻¹.

The output of V_k^H in Eq. (7.16) is uncorrelated and with Φ⁻¹_k , the estimate is white, i.e.

with unit variance. Again, as in Chapter 6, some rotation with U_kis applied that does not change the property of unit covariance. Finally, the estimate is colored with C_z,k^1/2. This result is quite surprising, since we do not optimize the mean squared error between the true channel and the channel recovered at the transmitter. Instead, the precoding MSE E

ku − ˆuk²₂

is minimized [see Eq. (7.8)]. Additionally, the notation introduced in the previous chapter, where the MSE between the true channel and the CSI at the transmitter was minimized, explicitly included a rank reduction. We choose a different formulation now, since a rank reduction can be included by an appropriate restriction of the partition cellsRk,i(i.e. by bit allocation). Therefore, V_kin Eq. (7.16) is square and is not used for rank reduction.

We also see from Eq. (7.16) that the optimal estimator G_opt,kcan be obtained in closed form except for the covariance matrix C_z,k and the unitary matrix U_k. The optimization of these parts of the estimator is difficult and cannot be found analytically. However, the last stages of the estimator G_opt,k can be moved into the quantizerQ_k(•) as we have

7.3 Proposed MMSE Optimization 171

already done in Chapter 6. Thanks to this step, the partition cellsRk,i are just redefined and optimality is not spoilt. Therefore, we can set without loss of optimality that

G_opt,k = Φ⁻¹_k V_k^HGMMSE-estim,k ∈ C^{N ×Ntr}. (7.17)

Note that the optimal estimator is independent of any properties of the codebook and the other estimators. Additionally, note that the output z_k of the estimator G_opt,k is white Gaussian. Then, we rename the output of the estimator as w_k ∼ N^C(0, I). Due to the relationship of X_opt,kand A_k[see Eq. (7.15)], we have

Ch,k− A^kXkX_k^HA^H_k = Ch,k− A^kWkU_k^HUkW_k^HA^H_k

= C_h,k− VkΦ_kW_k^HW_kW_k^HW_kΦ_kV_k^H = C_h,k− VkΦ²_kV_k^H

and

AkWkU_k^HRk,iUkW_k^HA^H_k = AkWk

| {z }

VkΦk

Rk,iW_k^HA^H_k

| {z }

ΦkV_k^H

because Gaussian distributions are invariant to unitary rotations (see Appendix D.2).

Bearing in mind the above results, the conditional moments from Eq. (7.12) can be rewritten as

E [H|z ∈ Rⁱ] = [µ1,i, . . . , µK,i]^T E

H^HH|z ∈ Rⁱ

= XK k=1



Ch,k− V^kΦ²_kV_k^H
T

+ R^T_k,i

= XK k=1

C_h,k − VkΦ²_kV_k^H+ R_k,i
T

where µ_k,iand R_k,iare redefined as

µ_k,i = V_kΦ_kE [w_k| wk∈ Rk,i] Rk,i = VkΦkE

wkw_k^H| wk ∈ Rk,i

ΦkV_k^H. (7.18)

Now,E[H^HH|z ∈ Ri] can be further written as applied. C_estim,kis the MSE error matrix due to the estimation error and Cquantize,k,i is the error covariance matrix due to the quantization error. The matrix Γ_k,i= I−C^Q,k,i ∈ R^0,+

depends only on the quantizer parameters. Note that when we assume perfect channel knowledge at the receiver, i.e. when there are no errors caused by estimation, C_estim,k = 0, and when there is no limited rate for the feedback, i.e. no quantization errors, we have that Cquantize,k,i = 0. Therefore, the regularization that is introduced due to imperfect CSI at the transmitter is given by C_estim,k + Cquantize,k,i. Remember that the effect of feedback delay is omitted in Eq. (7.19). In the event that we assume a simple Jakes model, we would have that [cf. Eq. (6.5)]

E

hk[q]h^H_k[ν]

= J0(2πfD,max,kD/fslot)Ch,k = rkCh,k

with the slot indexq, the delay in slots D = q−ν, the maximum Doppler frequency of the k-th user fD,max,k, the slot ratefslot, and the zero-th order Bessel function of the first kind J0(•) [34]. The factor rkin the last equality is implicitly defined. Note that the delay can be neglected by considering a speed value ofv = 0 km/h (rk = 1). As done in Chapter 6, the only impact is that this term rk must be included into the expression of A_k in Eq. (7.10) since the input of the quantizer z_kgiven by Eq. (7.4) is obtained from outdated channel vectors and therefore C_zh,k = r_kC_z,k^1/2X_k^H SC_h,kS^H+ C_η,k
−1/2

SC_h,k [cf.

Eq. (7.11)].