In this chapter, we have proposed two criteria for the design of the precoding matrix in a multi-antenna broadcast system. First, we maximized the sum rate, and then we showed how to maximize the minimum rate among all users. The latter problem is shown to be quasi convex and solved exactly. The precoding techniques are constrained to linear preprocessing at the transmitter. In addition to precoding, we have employed a signal scaling scheme that minimizes the average BER of the users. The signal scaling scheme is posed as a convex optimization problem, and solved exactly via interior-point methods.
Finally, we have combined the precoding with signal scaling. The combined scheme can be efficiently applied in practice. In terms of the achievable sum rate, the proposed technique
10 15 20 25 10−4
10−3 10−2 10−1
1/σ2 [db]
bit error rate
Method 2.1 Method 2 reg. pseudo inverse Method 2.2
Figure 4.6: Comparison of BER, M=6 antennas/users, 8PSK-Method 2, 8PSK-Method 2.1, 4-PSK-Method 2.2, 4PSK-regularized pseudo-inverse
10 15 20 25
11 12 13 14 15 16 17 18
1/σ2 [db]
bits per channel use
Method 2, Method 2.1 reg. pseudo inverse, Method 2.2
Figure 4.7: Comparison of rates, M=6 antennas/users, 8PSK-Method 2, 8PSK-Method 2.1, 4PSK-MEthod 2.2, 4PSK-regularized pseudo-inverse
10 12 14 16 18 20 22 24 26 28 30 10−4
10−3 10−2 10−1 100
1/σ2 [db]
bit error rate
Method 3 reg. pseudo inverse
Figure 4.8: Comparison of BER, M=20 antennas/users, 8PSK-Method 3, 8PSK-regularized pseudo-inverse
10 15 20 25 30 35
10−4 10−3 10
−2 10−1 100
1/σ2 [db]
bit error rate
Method 4 Method 2 reg. pseudo inverse
Figure 4.9: Comparison of BER, M=6 antennas/users, 16PSK-Method 4, 16PSK-Method 2, 8PSK-regularized pseudo-inverse
10 15 20 25 30 35 16
17 18 19 20 21 22 23 24 25
1/σ2 [db]
bits per channel use
Method 2, Method 4 reg. pseudo inverse
Figure 4.10: Comparison of rates, M=6 antennas/users, 16PSK-Method 4, 16PSK-Method 2, 8PSK-regularized pseudo-inverse
significantly outperforms traditional channel inversion methods, while having comparable (in fact, often superior) BER performance.
Chapter 5
Gaussian Broadcast Channel — Asymptotic Analysis of a
Particular Nonlinear Scheme
As we have said in the previous chapter, the sum-rate capacity of the multi-antenna Gaus- sian broadcast channel has recently been computed. However, the search for computation- ally efficient practical schemes that achieve it is still in progress. When the channel state information is fully available at the transmitter, the dirty-paper coding (DPC) technique is known to achieve the maximal throughput, but is computationally infeasible. In this chapter, we analyze the asymptotic behavior of one of its alternatives — the recently sug- gested so-called vector-perturbation technique. We show that for a square channel, where the number of users is large and equal to the number of transmit antennas, its sum rate approaches that of the DPC technique. More precisely, we show that at both low and high signal-to-noise ratio (SNR), the scheme under consideration is asymptotically optimal.
Furthermore, we obtain similar results in the case where the number of users is much larger than the number of transmit antennas.
5.1 Introduction
The limits of performance of multi-antenna Gaussian broadcast channel are currently the subject of extensive research (see, e.g., [14], [101], and the references therein). As we have mentioned in the previous chapter, when the channel state information (CSI) is fully
available at the transmitter, the so-called dirty-paper coding (DPC) technique achieves the capacity of multi-antenna broadcast channel [105]. However, the DPC scheme is exponen- tially complex and appears to be difficult to implement in practical systems. To this end, various heuristics with suboptimal performance but efficient implementation have recently been proposed. In [35], vector quantization is used in combination with powerful coding schemes to achieve a large fraction of the promised capacity. In [75], a technique referred to as the vector-perturbation technique (VPT) was proposed, and further considered in [108]. Simulation results presented there indicate that the proposed technique achieves performance close to the optimal one.
In this chapter, we analyze the theoretical limits of the VPT [75]. In particular, we show that when the number of users in the broadcast system is large, the sum rate achievable by the VPT approaches the sum rate achievable by the DPC scheme, both in the low- and the high-SNR regime. While the scheme introduced in [75] and further studied in [108] is practically feasible, the worst-case complexity of its implementation is still exponential. On the other hand, our proof for lower bounding the asymptotical sum-rate performance of the VPT is constructive and based on an algorithm that is polynomial in the number of users.
A complementary version of this chapter can be found in [91].
We assume the standard broadcast channel model similar to the one considered in the previous chapter,
y=Hs+v, (5.1)
where H is a K×M matrix whose entries are independent, identically distributed (i.i.d.) complex Gaussian random variables CN(0,1), K is the number of users,M is the number of transmit antennas,v is a K×1 noise vector whose entries are independent of entries in H and i.i.d. Gaussian random variables with zero-mean andσ2= 1/ρvariance, andsis an M×1 vector which is transmitted over the channel. Furthermore, we impose the constraint Eksk2 = 1; hence, the receivers do not need to know instantiations of the channel. (The caseksk2 = 1, considered in [75], can be treated similarly and leads to similar results.)
Since we focus on analyzing the asymptotic performance of the vector-perturbation
technique, we start by reviewing it in the next section.