Effect of Scalar Feedback Quantization

6 8 10 12 14 16 18 20

SNR

Sum Rate [bits/s/Hz]

Perfect CSIT Metric I Metric II Metric III Metric IV

Random Beamforming

Figure 4.8: Sum rate achieved by different feedback approaches versus aver-age SNR, forB = 9 bits, M = 3 transmit antennas andK = 10 users.

quantizations, as discussed in [28].

Figure 4.8 depicts the performances of different schemes in the low-mid SNR region, in a setting with K = 10 users. As the average SNR in the system increases, the sum rate of schemes using metrics I and III for feedback converges to the same value. They exhibit linear increase in the high SNR region as expected, which corresponds to a TDMA solution. The scheme that uses Metric IV for scheduling also benefits from a variable number of active beams, although providing worse performance than the systems using metrics I and III. Since in the simulated system the number of codebook bits Bis not increased proportionally to the average SNR, as discussed in [51], the scheme using Metric II (M_o =M) exhibits an interference-limited behavior, flattening out at high SNR.

4.8.1 Effect of Scalar Feedback Quantization

In the reminder of this section, we provide simulation results for the simplified approach described in Section 4.7, considering scalar feedback quantization

4.8 Numerical Results 131

and ZFBF at the transmitter side. We evaluate through simulations the sum rate performance in a system with M = 2 transmit antennas and = 0.4.

The total number of available feedback bits isB_tot= 7 bits. CQI quantization is performed through Lloyd’s algorithm. Once both the input quantization levels q_i and output representative levels ξ_qi are found, the quantizer sets ξ_qi = q_i, 0 ≤ i ≤ 2^B2 −1 in order to avoid outage events as discussed in Section 4.7.1.

Figure 4.9 and 4.10 show the sum rate as a function of the number of users for SNR = 10 dB and SNR = 20 dB respectively for different CDI and CQI feedback bit split. As expected, it is more beneficial to allocate more bits on channel direction quantization in a system with low number of active users. On the other hand, as the number of users increases, it becomes more beneficial to allocate bits on CQI quantization instead. The black curve B₁ = 1 bit corresponds to the random unitary beamforming for M = 2 transmit antennas proposed in [2]. In a system with optimal quantization, i.e. matched to the pdf of the maximum CQI value among K users, the amount of necessary quantization levels is reduced as the number of users in the cell increases. Thus, less amount of feedback bits is needed for CQI quantization in order to capture the multiuser diversity.

0 100 200 300 400 500 600 700 800 900 1000

In Figure 4.11, the envelope of the curves in the two previous figures is shown, which corresponds to a system that chooses the best B₁/B₂ balance for each average SNR and K pair. In this figure, we compare how this best pair of (B₁, B₂) changes as the system average SNR increases. Both curves are divided in different regions, according to the optimal (B₁, B₂) pair in each region. It can be seen that the optimal threshold for switching from B₁ →B₁−1 bits (and thus B₂ → B₂ + 1) is shifted to the right for higher average SNR values (upper curve). This means that as the average SNR increases, more bits should be allocated on channel direction information.

Summarizing, given a pair of average SNR and K values, there exists an optimal compromise of B₁ and B₂, given thatB_tot = B₁ + B₂.

4.9 Conclusions

A design framework for scalar feedback in MIMO broadcast channels with limited feedback has been presented. In order to perform user scheduling, these metrics may contain information such as channel power gain, quantiza-tion error, orthogonality factor between beamforming vectors and/or number

4.9 Conclusions 133

0 100 200 300 400 500 600 700 800 900 1000

5 6 7 8 9 10 11 12 13 14

Number of users

Sum Rate [bits/s/Hz]

Optimal Tradeoff − SNR = 10 dB Optimal Tradeoff − SNR = 20 dB B1 = 6 B1 = 5

B1 = 4 B1 = 3 B1 = 1

B1 = 6

B1 = 5 B1 = 4 B1 = 3 B1 = 1

SNR = 20 dB

SNR = 10 dB

Figure 4.11: Sum rate vs. number of users in a system with optimal B₁/B₂ balancing for different SNR values.

of active beams. An approximation on the sum-rate has been provided for the proposed family of metrics, which has been validated through simulations.

As it has been shown, the proposed sum-rate function is a powerful design tool and enables simple analysis. A sum-rate comparison between SDMA and TDMA has been provided in several extreme regimes. Particularly, SDMA outperforms TDMA as the number of users becomes large. TDMA provides better rates than SDMA in the high SNR regime (interference-limited region).

Moreover, the importance of optimizing the orthogonality factor in the low SNR regime has been highlighted. Several metrics have been presented based on the proposed design framework, illustrating their performances through numerical simulations. The system sum-rate can be drastically improved by considering a variable number of active beams adapted to each scenario. In addition, scalar metrics based on SINR lower bounds can provide benefits from a point of view of QoS and feedback reduction.

We have also formulated the problem of optimal feedback balancing in order to exploit spatial multiplexing gain and multiuser diversity gain under

a sum feedback rate constraint. A low complexity approach has been intro-duced to illustrate the performance improvement of systems with optimally balanced feedback. The scaling of CDI feedback load in order to achieve full multiplexing gain is also provided, revealing an interesting interplay between the number of users, the average SNR and the number of feedback bits.

4.A Proof of Proposition 4.1 135

APPENDIX 4.A Proof of Proposition 4.1

Define the following changes of variables

ψ := sin²θ_k x:= ¹_δφ(1−ψ)

φ :=h_k² y:= ¹_δφψ. (4.45)

Then, the metric in (4.9) can be expressed as

ξ = x ψ are independent random variables for i.i.d. channels, the joint pdf ofxand y is obtained from f_xy(x, y) = _J(x,y)¹ f_φ[δ(x+y)]f_ψ obtained from the cdf of ψ given in (4.6). Hence, we get the joint density

f_xy(x, y) = δ

Γ(M −1)e^−δ(x+y)y^M−2. (4.49) The cdf of the proposed SINR metric is found by solving the integral

F_ξ(s) =

x,y∈D_s

f_xy(x, y) dx dy. (4.50)

The bounded region D_s in the xy-plane represents the region where the inequality ^x

αx+βy+2γ√

xy+λ ≤ s holds. Isolating x on the left side of the in-equality, D_s can be equivalently described as x≤g(y), with g(y) given by

g(y) =(2γ²s²+βs(1−αs))y+2γs(

(γ²s²+βs(1−αs))y²+λs(1−αs)y

(1−αs)² +ϕ(s) (4.51) whereϕ(s) = ₁₋^λs_αs. Since using g(y) in the integration limits yields difficult integrals, we use the following linear approximation

g(y)≈m(s)y+ϕ(s) (4.52) where the slopem(s) corresponds to the oblique asymptote of g(y)

m(s) = lim

Depending on the slopes of these linear boundaries, the integral in (4.50) is carried out over different regions

F_ξ(s)≈ corresponds to the value ofy in which the linear boundaries intersect

y_c= λs(1−αs)δ

(1−αs)²(1−δ)−βs(1−αs)δ−2γs(γs+(

βs(1−αs) +γ²s²)δ. (4.55) Expressing the regions of the domain of F_ξ(s) as function of s_c, defined as the crossing point between m(s) and ¹⁻_δ^δ, and substituting (4.49) into (4.54), the cdf of ξ is found from the following integrals

F_ξ(s)≈

4.B Proof of Theorem 4.1 137

where s_c is given by

s_c = α(1−δ)²+β(1−δ)δ−2(

γ²(1−δ)³δ

α²(1−δ)²+ 2αβ(1−δ)δ+δ(β²δ−4γ²(1−δ)). (4.57) Solving the integrals in (4.56), the resulting cdf becomes

F_ξ (x) =

x t^a−1e^−tdt is the (upper) incomplete gamma function.

Note that this is a generalization of previous results in the literature. In the particular case of B = 0, then δ= 1 and thus s_c becomes 0, yielding the cdf derived in [2] for random beamforming. If the metric refers to an upper bound on the SINR, with = 0, then s_c = ¹⁻_δ^δ. If in addition M_o = M is considered as in Metric II, the cdf of (4.58) becomes the one provided in [28].

In order to obtain a tractable expression for F_ξ(s), we assume that s_c is small so that F_ξ(s) can be approximated as described in (4.10). Note that a small s_c value corresponds to a low value of B and thus the obtained cdf approximates better the low resolution regime.

4.B Proof of Theorem 4.1

Given M_o beams active for transmission, using (4.13) we approximate the rate as

Expanding the binomial in the integral, we get A closed-form solution for the integral in the above equation can not be found, and thus we use the Bernouilli inequality to obtain an approximation

_1/α

Note that the integral above is also difficult to solve, since m is a nonlinear function of s, as shown in Proposition 4.1. In order to provide good sum rates,will take in general small values. Under this assumption, the following approximation can be made

By substituting the approximated value of the integral found above into (4.61), and using the definitions of Bn, Ki,n and Pn given in Theorem 4.1, we obtain the desired approximation for the sum rate.

Chapter 5

Optimization of Channel Quantization Codebooks

The design of channel quantization codebooks for MIMO broadcast channels with limited feedback is addressed. Rather than separating CQI and CDI feedback, we consider a simple scenario in which each user quantizes di-rectly its vector channel. Our goal consists of finding simple quantization codebooks which, in scenarios with spatial or temporal correlation, provide performance gains over classical quantization techniques. In addition, as we show, our optimized codebooks outperform existing techniques that rely on separate CDI and CQI feedback, in systems with a sum-rate feedback con-straint. A design criterion that effectively exploits the spatial correlations in the cell is proposed, based on minimizing the average sum-rate distortion in a system with joint linear beamforming and multiuser scheduling. Moreover, we show how to apply Predictive Vector Quantization (PVQ) to quantize time-correlated broadcast channels. PVQ exploits the temporal correlation to reduce the quantization error, and thus to improve the sum rate of the system. In this chapter we show how the corresponding codebooks can be de-signed, and we present a prediction strategy. Numerical results are provided, which illustrate the benefits of effectively designing quantization codebooks in MIMO broadcast channels with spatial or temporal correlations.

139

5.1 Introduction

MIMO systems can significantly increase the spectral efficiency by exploit-ing the spatial degrees of freedom created by multiple antennas [74]. As discussed in previous chapters, due to the complexity and need for accurate CSIT, linear beamforming techniques relying on limited feedback are a good alternative to DPC, the capacity achieving technique. In this chapter, we consider a limited feedback framework different from the one discussed in Chapter 3. Rather than separating CQI and CDI feedback, we consider a simpler system in which each user quantizes directly its vector channel. Our goal is to find simple quantization codebooks which, in scenarios with spatial or temporal correlation, provide performance gains over classical quantiza-tion techniques.

Codebook designs for MIMO broadcast channels with limited feedback follow in simple design criteria, that aim at simplifying codebook generation and system analysis. Random beamforming has been proposed in [2] as an SDMA extension of opportunistic beamforming [63], in which feedback from the users to the base station is conveyed in the form of a beamforming vector index and an individual SINR value. An extension of RBF is proposed in [3], coined as opportunistic SDMA with limited feedback (LF-OSDMA), in which the transmitter has a codebook containing an arbitrary number of unitary bases. In that approach, the users quantize the channel direction (channel shape) to the closest codeword in the codebook, feeding back the quantiza-tion index and the expected SINR. Multiuser scheduling is performed based on the available feedback, using the unitary basis in the codebook that max-imizes the system sum rate as a beamforming matrix. Other schemes for MIMO broadcast channels, like the approach described in [51], propose to use simple Random Vector Quantization (RVQ) [75] for quantizing the user vector channels. A simple geometrical framework for codebook design is pro-posed in [49], which divides the unit sphere in quantization cells with equal surface area. This framework is also used for channel direction quantization in [28], where feedback to the base station consists of a quantization index along with a channel quality indicator for user selection. These codebook designs take neither spatial nor temporal correlations present in the system into account. Taking them into account could yield better quantization code-books and in turn better sum-rate performance.

The gains of adaptive cell sectorization have been studied in [76] in the context of CDMA networks and single antenna communications, with the

5.1 Introduction 141 aim of minimizing the total transmit power in the uplink of a system with non-uniform user distribution over the cell. This situation is analogous to a system with multiple transmit antennas in which beamforming is performed, adapting its beams to uneven user distributions. In a scenario with limited feedback available, adaptation of quantization codebooks can be performed instead, in order to improve the system performance. In [77], an approach for exploiting long term channel state information in the downlink of mul-tiuser MIMO systems is proposed. A flat-fading multipath channel model is assumed, with no line of sight (NLOS) between the base station and user terminals. Each user can be reached through a finite number of multipath components with a certain mean angle of departure (AoD) from the antenna broadside and a certain angle spread. The mean of the angles of departure are fixed and thus no user mobility is considered.

In this chapter, we highlight the importance of cell statistics for codebook design in MIMO broadcast channels with limited feedback. Firstly, the im-portance of exploiting spatial correlations is addressed. The average sum-rate distortion in a system with joint linear beamforming and multiuser schedul-ing is minimized, exploitschedul-ing the information on the macroscopic nature of the underlying channel. In this first part, a non-geometrical stochastic channel model is considered, in which each user can be reached in different spatial directions and with different angle spread. Based on this model, comparisons with limited feedback approaches relying on random codebooks are provided in order to illustrate the importance of matching the codebook design to the cell statistics.

Secondly, we address the problem of exploiting the temporal correlations in the system. We present a scheme that uses Predictive Vector Quantization (PVQ) [78] to exploit the correlation between successive channel realizations in order to improve the quantization, and thus to improve the sum rate of the system. Further, our scheme does not make any assumptions on the scheduling function and on the transmission strategy, thus providing a high flexibility.

Numerical results illustrate the benefits of effectively designing quantiza-tion codebooks in MIMO broadcast channels with spatial or temporal corre-lations. The proposed optimized codebooks outperform existing techniques that rely on separate CDI and CQI feedback, in a system with a sum-rate feedback constraint.

5.2 System Description

We consider a multiple antenna broadcast channel consisting ofM antennas at the transmitter and K ≥ M single-antenna receivers in a single cell sce-nario. The system model is equivalent to the one described in (3.1). Let S denote an arbitrary set of users with cardinality|S|=M. Given a set of M users scheduled for transmission, the signal received at thek-th user terminal is given by

y_k =h^H_kw_ks_k+ %

i∈S,i=k

h^H_k w_is_i+n_k (5.1) whereh_k ∈C^M×1,w_k ∈C^M×1,s_k andn_k are the channel vector, beamform-ing vector, transmitted signal and additive white Gaussian noise at receiver k, respectively. The first term in the above equation is the useful signal, while the second term corresponds to the interference. We assume that the vari-ance of the transmitted signals_k is normalized to one andn_k is independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variable with zero mean and variance σ². We assume that the re-ceivers have achieved perfect CSI through the use of pilots. The users then quantize the channel to an element of a common codebookV, and feed back the corresponding index to the base station. The base station then decides, based on the received feedback, which set of users to serve, and forms the appropriate beamforming vector. The data is transmitted in a block-wise fashion. We assume a data-rate limited feedback link that can send back B bits at the beginning of each block. Further, the feedback is assumed to be instantaneous and error-free.