Early limited feedback studies considered Rayleigh i.i.d. channels and later the focus of the research shifted towards more practical spatially and tempo- rally correlated channels. It is important to design a codebook that takes into account the effects of the propagation environment, as different propagation environments may need different codebooks. In Chapter 2, we discussed analyt- ical MIMO channels: Rayleigh i.i.d., Rician, spatially correlated and temporally correlated MIMO channels.
The distribution of the right singular vectors (corresponding to the dominant singular value of the channel) on a unit sphere for 2× 2 MIMO channels, is
Rayleigh i.i.d. channels Spatially corr. channels
Figure 3.3: The distribution of right singular vectors on the unit sphere for the Rayleigh i.i.d. and spatially correlated 2× 2 MIMO channels.
illustrated in Fig. 3.3 for the Rayleigh i.i.d. and spatially correlated channels. In Fig. 3.3, it is seen that the right singular vectors of the channel are uniformly distributed across the entire unit sphere in the case of Rayleigh i.i.d. channel, however, they form clusters for spatially correlated channels with zt = zr= 0.95
in (2.9) and (2.10). The figure helps to understand what sort of codebooks are required to deal with such channels. For example, in Rayleigh i.i.d. channels (blue circles) in Fig. 3.3, it is evident that in order to effectively quantize the channel, a codebook structure should have codewords that are uniformly distributed on the sphere. For spatially correlated channels (red circles), in Fig. 3.3, the effective codebook should have a cap like structure with clustered
Temporally corr. channels
Figure 3.4: The distribution of right singular vectors on the unit sphere for the temporally correlated Rayleigh 2× 2 MIMO channel with 100 time samples.
3.8. SUMMARY 34
WINNER II channels
Figure 3.5: The distribution of right singular vectors on the unit sphere for the WINNER II 2× 2 MIMO channel with 100 time samples.
codewords.
The right singular vectors for the temporally correlated channel are plotted in Fig. 3.4 for 100 consecutive time intervals with ǫ = 0.9987 for carrier fre- quency 2.5 GHz and velocity, v = 1 km/h, in (2.12). The vectors generally lie very close to previous right singular vectors and follow a trajectory on the unit sphere as seen in Fig. 3.4. In Fig. 3.4, the channel lies close to the previous channel in time, therefore in order to successfully track the slowly varying chan- nel, a codebook should be able to transform itself after each feedback interval. Similarly, it is noted in Fig. 3.5 for the WINNER II channel, with velocity v = 1 km/h and 0.5λ0 antenna spacing, that the right singular vectors lie close to each
other and similar to Fig. 3.4, the entries also follow a trajectory. The main goal of an effective codebook design is to reduce quantization errors by taking into account factors such as propagation environment, antenna polarizations and channel correlation statistics.
3.8
Summary
In this chapter, we briefly discussed important concepts of limited feedback MIMO/MISO systems. For SU systems, we discussed beamforming and spatial multiplexing. We have provided an overview of various precoding schemes for limited feedback MU MISO systems as used in this thesis.
4
MIMO Capacity Analysis
In this chapter, the capacity gains of the MIMO system over the SISO system with full and no CSI at the transmitter are discussed. We then consider a limited feedback SU MIMO system and derive capacity loss expressions for various linear receivers; the capacity loss is defined as the difference between the capacity with perfect CSI feedback and the capacity with codebook feedback. The capacity loss analysis considers both analytical and standardized MIMO channel models discussed in the previous chapter. The capacity loss study is focused on single- stream and two-stream MIMO transmission modes. The results in this chapter serves as a motivation for various codebook designs in the following chapters.
4.1
MIMO Capacity Gains
In [84], we investigate the MIMO capacity gains1 relative to the SISO capacity using two metrics: the ratio between the expected value of the two capacities and the difference between the two capacities. We derive limiting values of the two capacity metrics for low and high SNR regimes and show via simulation their behavior at other values of SNR. We also examine a range of correlation conditions using the spatially correlated Kronecker channel model described in Chapter 2 Section 2.2.3. In [84], we do not consider a limited feedback system and assume either perfect CSI or no CSI is available at the transmitter.
The difference, D, between the expected MIMO capacity and the expected
1The analysis of MIMO capacity gain has been adopted from [84], this analysis serves as
4.1. MIMO CAPACITY GAINS 36
SISO capacity is defined as
D = E [CM]− mE [CS] , (4.1)
where CMand CS are the capacities of MIMO and SISO systems given in (2.22)
and (2.23), respectively. m is given by m = min(nt, nr). Similarly, the ratio of
the MIMO capacity to the SISO capacity, denoted by R is defined as R = E [CM]
E [CS]
(4.2) The results derived in [84] are summarized in Table 4.1, for limiting values of SNR, ρ = Pt/σ2, under the i.i.d. Rayleigh channel and the fully correlated
Kronecker channel (2.8), such that zt = zr = 1. Here ˜D and ˜R represent
the capacity difference and capacity ratio when full CSI is available at the transmitter, given by ˜ D = EhC˜M i − mE [CS] (4.3) and ˜ R = EhC˜Mi E [CS] , (4.4)
respectively, where ˜CM is given by (2.20). In Table 4.1, D∞ is a constant that
could be positive or negative depending on the channel statistics and is shown in [84] to be, D∞, E [log2((nt|h|2)−mdet (Υ))], where Υ is given in (2.21).
Table 4.1: Capacity gain results: ratio, R, and difference, D [84]. i.i.d. Rayleigh Channel Fully Correlated Channel
ρ→ 0 ρ→ ∞ ρ→ 0 ρ→ ∞
No CSI R nr m nr 1
D 0 D∞ 0 −∞
Full CSI R˜ E(λ1) m ntnr 1
˜
D 0 D∞+ m log2nt
m 0 log2ntnr (m = 1),−∞ (m > 1)
Figures 4.1 and 4.2 show the capacity difference, D, and ratio, R, for i.i.d. Rayleigh and spatially correlated channels when each eigenchannel is given equal power. The antenna configurations are denoted as nt× nr in the figures. The
results with i.i.d Rayleigh channels agree with the limits given in Table 4.1. Troughs in R for 2× 2 and 4 × 4 systems are visible in Fig. 4.2 indicating an intermediate drop in the value of R before returning to the limit. A peak in R is observed for a 4× 2 system, in the −5 to 20 dB SNR range. The capacity scaling of m in MIMO relative to SISO is not achieved for nt = nr systems and
SNR ρ (dB) -40 -20 0 20 40 60 80 C ap ac it y D iff er en ce ,D = CM − mC S -4 -3 -2 -1 0 1 2 3 i.i.d. zt = zr=0.3 zt = zr=0.5 2 x 4 2 x 2 4 x 4 4 x 2
Figure 4.1: D with equal power and spatial correlation at both transmitter and receiver. SNR ρ (dB) -60 -40 -20 0 20 40 60 80 100 C ap ac it y R at io ,R = CM /C S 1.5 2 2.5 3 3.5 4 i.i.d. zt = zr=0.3 zt = zr =0.5 4 x 2 2 x 2 2 x 4 4 x 4
Spatial DOF gain Transition
Power gain
Figure 4.2: R with equal power and spatial correlation at both transmitter and receiver.
two kinds of gain; the power gain regime at low SNR and the spatial degrees of freedom (DOF) gain at high SNR.
Figures 4.3 and 4.4 show the capacity difference and ratio, respectively, with waterfilling power allocation for i.i.d. Rayleigh and spatially correlated channels. We note that the difference, ˜D, in Fig. 4.3 with waterfilling is higher for nt6= nr
systems compared to nt = nr systems, at high SNR values. On the other hand,
the ratio, ˜R, in Fig. 4.4 with waterfilling yields higher gains at low SNR values compared to the R (in Fig. 4.2), but eventually drops down to R, for high SNR values.
4.2. MIMO CAPACITY LOSS ANALYSIS WITH LIMITED FEEDBACK 38