Bit Allocation

When using scalar quantization (transform coding, [108]) instead of vector quantization, the available bits have to be allocated to the different scalar coefficients. Since in real systems the bandwidth of feedback channels is very limited, the total number of bitsNbit

should be very small and, therefore, strategies such as optimum bit allocation can greatly improve the performance with a negligible increase in computational complexity.

The average distortion or MSE is given by Eq. (6.73). Let ˘h^Re_k,i[q] and ˘h^Im_k,i[q] be the real or imaginary part of thei–th element of ˘h_k[q]. With Φk = diag(ϕk,1, . . . , ϕk,d) and using scalar quantizers, Eq. (6.73) can be expressed as

εk = ck+ E

" _d X

i=1

ϕ²_k,i

h˘^Re_k,i[q]− Q^′k,i

˘h^Re_k,i[q]2

+

˘h^Im_k,i[q]− Q^′k,i

˘h^Im_k,i[q]2#

= ck+ Xd

i=1

ϕ²_k,iε˘hk,i (6.76)

whereε˘hk,i = E[|˘hk,i[q]− Q^′k,i(˘h^Re_k,i[q])− j Q^′k,i(˘h^Im_k,i[q])|²] is the MSE between ˘h_k,i[q] and its quantized version. Remember thatd is the number of coefficients resulting from the rank reduction process to be sent from each user to the transmitter. Note thatε˘hk,i is fixed

for a given number of bits b_k,i since ˘h_k[q] ∼ N^C(0, I). Therefore, for a given number of bits, we can calculateε˘hk,i off-line to be stored at the users prior to transmission. Let ε˘hk,i = 2MSE(b_k,i) be the function that determines the mean squared error in terms of the number of bits used to quantize the real or imaginary part of each channel coefficient [see Eq. (6.75)]. Then, the bit allocation problem can be solved by means of the optimization problem

{bopt,k,1, . . . , b_opt,k,d} = argmin

{bk,1,...,bk,d}

Xd i=1

ϕ²_k,i2MSE (b_k,i) s.t.:

Xd i=1

2b_k,i= Nbit (6.77)

where Nbit is the number of bits per user sent through the feedback channel. It should be mentioned that we provide the same number of bits to quantize both the real and the imaginary part of each coefficient and, therefore, it is obvious that each quantized coefficient uses an even number of bits.

In principle, we would have to test all the possible bit allocations whose total number of bitsNbit is fixed, which can make the search difficult when the number of bits to be allocated is high. However, the MSE of each quantizer decreases with a higher number of bits and, due to ϕk,1 ≥ . . . ≥ ϕ^k,d, the total MSE is always smaller when more bits are allocated to quantize the coefficients with lower indices. Thus, we only have to test bit allocations whose number of bits decreases or stays constant with the coefficient index (see Table 6.2). In the sequel, we refer to this bit allocation algorithm as optimum bit allocation.

6.3.5 MSE Error Matrix for Robust Multi-User Precoder Design

For the robust precoder design, we must find the conditional moments E[h_k[q]|˜hQ,k[q]]

andE[h_k[q]h^H_k[q]|˜hQ,k[q]] of the probability density function f_h

k[q]|˜h_Q,k[q](h_k[q]|˜hQ,k[q]), since the transmitter only knows ˜hQ,k[q], but the cost function depends on hk[q] ³. The closed-form expressions will be obtained for the special case thatQ^′_k(•) performs separate scalar quantization as assumed in the previous two subsections. Remember that the transmission over the feedback channel introduces a delay ofD = q− ν slots, i.e. the precoder is designed during the time slot q and the channel estimate is obtained during the time slotν = q− D. Remember also that ˘h^k[q]∼ N^C(0, Id) is the input vector to the quantizer given by ˘hk[q] = G^′_opt,kyk[q], where G^′_opt,kis the estimator that results from the quantizer redefinition and y_k[q] is the received pilot signal (see Subsection 6.3.3).

3For example, the precoder in Eq. (5.17) depends on ˆH and T . The row of ˆH corresponding to userk is E[h^T_k[q]|˜h^Q,k[q]] and CΘin T [see Eq. (5.14)] containsE[hk[q]h^H_k[q]|˜h^Q,k[q]] in the Bayesian framework employed in this section.

6.3 Bayesian Error Modeling based on Joint CSI MSE 145

Bits per user d = 2 d = 3 d = N = 4

Nbit= 8 [4, 0]^T, [3, 1]^T [4, 0, 0]^T, [3, 1, 0]^T [4, 0, 0, 0]^T, [3, 1, 0, 0]^T 4 for real part [2, 2]^T [2, 2, 0]^T, [2, 1, 1]^T [2, 2, 0, 0]^T, [2, 1, 1, 0]^T

4 for imaginary part [1, 1, 1, 1]^T

Nbit= 12 [6, 0]^T, [5, 1]^T [6, 0, 0]^T, [5, 1, 0]^T [6, 0, 0, 0]^T, [5, 1, 0, 0]^T 6 for real part [4, 2]^T, [3, 3]^T [4, 2, 0]^T, [4, 1, 1]^T [4, 2, 0, 0]^T, [4, 1, 1, 0]^T 6 for imaginary part [3, 3, 0]^T, [3, 2, 1]^T [3, 3, 0, 0]^T, [3, 2, 1, 0]^T [2, 2, 2]^T [3, 1, 1, 1]^T, [2, 2, 2, 0]^T

[2, 2, 1, 1]^T Nbit= 16 [8, 0]^T, [7, 1]^T [8, 0, 0]^T, [7, 1, 0]^T [8, 0, 0, 0]^T, [7, 1, 0, 0]^T 8 for real part [6, 2]^T, [5, 3]^T [6, 2, 0]^T, [6, 1, 1]^T [6, 2, 0, 0]^T, [6, 1, 1, 0]^T 8 for imaginary part [4, 4]^T [5, 3, 0]^T, [5, 2, 1]^T [5, 3, 0, 0]^T, [5, 2, 1, 0]^T [4, 4, 0]^T, [4, 3, 1]^T [5, 1, 1, 1]^T, [4, 4, 0, 0]^T [4, 2, 2]^T, [3, 3, 2]^T [4, 3, 1, 0]^T, [4, 2, 2, 0]^T [4, 2, 1, 1]^T, [3, 3, 2, 0]^T [3, 3, 1, 1]^T, [3, 2, 2, 1]^T

[2, 2, 2, 2]^T Table 6.2: Number of Bits Assigned per User Coefficient for CSI–MSE Metric.

Taking into account that the conditional moments needed for the robust design and for the conditional covariance matrix

C_h

k[q]|˘hk[q] = Ch,k− Vopt,kΦ²_kV_opt,k^H . (6.81) Therefore, the conditional momentE[hk[q]h^H_k[q]|˘h^k[q]] is given by

Eh

h_k[q]h^H_k[q]|˘hk[q]i

= C_{hk[q]|˘hk[q]}+ µ_{hk[q]|˘hk[q]}µ^H_h

k[q]|˘h_k[q]

= C_h,k − Vopt,kΦ²_kV_opt,k^H + V_opt,kΦ_kh˘_k[q]˘h^H_k[q]Φ_kV_opt,k^H . (6.82) Thus, both the conditional mean and the conditional correlation matrix in Eq. (6.78), henceforth denoted respectively by µ_h

k[q]|˜h_Q,k[q]and R_h

6.3 Bayesian Error Modeling based on Joint CSI MSE 147 cells of the quantizerQ^′_k,i(•) applied to the real part of ˜h^k,i[q] corresponding to the fed–

back indexℓk of user k. Similarly, α^Im_k,i(ℓk) and β_k,i^Im(ℓk) are the lower and upper limits, respectively, of the partition cells of the quantizerQ^′_k,i(•) applied to the imaginary part of ˜hk,i[q] when the index ℓk is fed back to the transmitter by the user k. Note that w∼ N^C(0, I) is used instead of ˘h_k[q] for brevity.

Taking into account that C_˘h,k = Id, the above expressions can be written as

mk= µ^Re_k + j µ^Im_k (6.86) The second term of M_kin Eq. (6.87) is diagonal, i.e.

Σ_k = diag (σ_k,1, . . . , σ_k,d)

whosei-th diagonal element can be expressed as (see Appendix F.3)

σk,i= τ_k,i^Re+ τ_k,i^Im (6.90)

−10 −5 0 5 10 15 20 25 30 10⁻²

10⁻¹

SNR in dB

uncoded BER

Ntr=4, channel 2, WF−THP Ntr=6, channel 2, WF−THP Ntr=8, channel 2, WF−THP Ntr=10, channel 2, WF−THP Ntr=12, channel 2, WF−THP Ntr=14, channel 2, WF−THP Ntr=16, channel 2, WF−THP perf. CSI, channel 2, WF−THP

Figure 6.7: Effect of Estimation Error on the Proposed Robust WF–THP with Approach III from Section 6.3 as a Function of Different Training Lengths in an Urban Macrocell Environment.

and, correspondingly,

τ_k,i^Im = 1

2 − µ^Im,2k,i + 1 2√ π

α^Im_k,i(ℓk) exp

−α^Im,2k,i (ℓk)

− βk,i^Im(ℓk) exp

−βk,i^Im,2(ℓk) Φ √

2α^Im_k,i(ℓk)

− Φ √

2β_k,i^Im(ℓk)
. The above results enable us to compute the conditional covariance matrix

C_{hk[q]|˜hQ,k[q]} = R_{hk[q]|˜hQ,k[q]}− µhk[q]|˜h_Q,k[q]µ^H_h

k[q]|˜h_Q,k[q]

= Ch,k− V^opt,kΦ²_kV_opt,k^H + Vopt,kΦ²_kΣkV_opt,k^H

= Ch,k+ Vopt,kΦ²_kΥ_kV_opt,k^H (6.91)

where Υ_k = Σk− Id. Note that the non-zero elements of the diagonal matrix Υ_k ∈ R^d×d0,+

only depend on the properties of Q^′_k(•). They can therefore be computed in advance and stored as parameters ofQ^′_k(•). The first and the second term in the second line of Eq. (6.91) come from the erroneous knowledge about h_k, if we had ˜h_k. But since we only have ˜h_Q,k available, the variance of the error is increased by the third term in Eq. (6.91).

As seen in this section, the uncertain knowledge about the channel at the transmitter is modeled by the conditional probability density functionf_h

k[q]|˜h_Q,k[q](h_k[q]|˜hQ,k[q]) whose