**6. The proposed feedback and codebook design**

20 Recent Trends in Multiuser MIMO Communications

*P* �

where ∆∆† is a 4 <sup>×</sup> 4 matrix while <sup>∆</sup>4×*df ree* <sup>=</sup>

Chernoff bound, (44) is upper bounded by

The covariance matrix of the channel **h** is

*E***h** � 1 <sup>2</sup> exp � − 1 16*N*<sup>0</sup> **h**† ∆∆†**h** �� ≤

21,*<sup>k</sup> <sup>h</sup>*<sup>∗</sup>

**<sup>c</sup>**<sup>1</sup> <sup>→</sup> **<sup>c</sup>**ˆ1|**H**<sup>1</sup>

11,*<sup>k</sup>* <sup>−</sup> *qh*<sup>∗</sup>

constant for the duration of one codeword. So the PEP can be written as

� ≤ *Q*

*<sup>P</sup>* (**c**<sup>1</sup> → **<sup>c</sup>**ˆ1|**h**) ≤

*E* � **hh**† � = **R** =

Note that the minimizations in (42) ensure that in ∆, *x*ˆ

these factors will not affect the diversity order.

21,*k* �*T*

 ���� ∑ *k*,*df ree*

�� 1 8*N*<sup>0</sup> **h**† ∆∆†**h** �

1 <sup>2</sup> exp � − 1 16*N*<sup>0</sup>

�

 

Its rank is two with its two identical eigenvalues being 2. Using the moment generating function of a Hermitian quadratic form in complex Gaussian random variable, we get

> 2 det � **I** + <sup>1</sup> 16*N*<sup>0</sup>

2,*<sup>k</sup>* can be zero for *<sup>k</sup>* <sup>=</sup> 1, ··· , *df ree*. So in the worst case scenario, <sup>∆</sup> would have only

�16*N*<sup>0</sup> *σ*2

first two rows with non-zero elements. For the high SNR approximation, we get

1 2

where *r* is the rank and *µ<sup>k</sup>* are the eigenvalues of **R**∆∆†. The minimum rank is one thereby indicating the diversity order of one. Note that as the derivation has involved Chernoff bound, so the exact PEP expression would involve some additional multiplicative factors but

*<sup>P</sup>* (**c**<sup>1</sup> <sup>→</sup> **<sup>c</sup>**ˆ1) <sup>≤</sup>

′ 1,*<sup>k</sup>* <sup>−</sup> *<sup>x</sup>* ′

�*<sup>r</sup> <sup>r</sup>* ∏ *k*=1

1 *µk*

= *Q*

, **x***<sup>k</sup>* = �

1 8*N*<sup>0</sup> � � �

�*T*. We assume channel to be slow fading, i.e. the channel remains

**<sup>h</sup>***<sup>T</sup>* (**x***<sup>k</sup>* − **<sup>x</sup>**<sup>ˆ</sup> *<sup>k</sup>*)

**h**†∆∆†**h**

 

1

**R**∆∆†

*x*1,*<sup>k</sup> x*1,*<sup>k</sup> x*2,*<sup>k</sup> x*2,*<sup>k</sup>*

� � � 2 

**<sup>x</sup>**<sup>1</sup> <sup>−</sup> **<sup>x</sup>**ˆ1 **<sup>x</sup>**<sup>2</sup> <sup>−</sup> **<sup>x</sup>**ˆ2 ··· **<sup>x</sup>***k*,*df ree* <sup>−</sup> **<sup>x</sup>**<sup>ˆ</sup> *<sup>k</sup>*,*df ree*

�

�*<sup>T</sup>* and **<sup>x</sup>**<sup>ˆ</sup> *<sup>k</sup>* <sup>=</sup>

(44)

(45)

(46)

(48)

� . Using

� (47)

1,*<sup>k</sup>* is always non-zero where

� *h*∗ 11,*<sup>k</sup> qh*<sup>∗</sup>

*x*ˆ1,*<sup>k</sup> x*ˆ1,*<sup>k</sup> x*ˆ2,*<sup>k</sup> x*ˆ2,*<sup>k</sup>*

where **h***<sup>k</sup>* =

�

*x*ˆ ′ 2,*<sup>k</sup>* <sup>−</sup> *<sup>x</sup>* ′ It was shown in the PEP analysis that the multi-user MIMO mode in LTE suffers from a loss of diversity. This loss is mainly attributed to the EGT characteristic of these precoders as will be shown in the next section. On the other hand, this transmission characteristic does not affect the diversity order in single-user MIMO mode. Focusing on this result, we propose a design of LTE precoders to offset this diversity loss.

LTE precoders are characterized by two features, i.e. angular resolution and EGT. Limited increase in the feedback can be either employed to increase the angular resolution of these structured precoders or it can be used to enhance the levels of transmission. Increasing the levels of transmission implies that additional feedback bits can be used to indicate an increase of the power level on either of the two antennas, i.e. creating more circles with different radii. For this we resorted to numerical optimization for fixing the radii of two circles and the precoders turn out to be [1 2 exp(*jθ*)] *T* or [2 exp(*jθ*) 1] *<sup>T</sup>* where *<sup>θ</sup>* ∈ {0, ±90◦, 180◦}. This approach gives 8 additional codebook entries, and 12 in total. Improving angular resolution is trivial, i.e. increasing equally angular spaced points on the unit circle but restricting to EGT, i.e. precoder is given as [1 exp(*jθ*)] *<sup>T</sup>*, where *θ* = 2*πl*/12, *l* = 0, . . . , 11. These two different codebook options have been illustrated in Fig. 4.

To quantize the proposed codebooks of size 12, ⌈log2(12)⌉ <sup>=</sup> 4 bits are needed. That means that we could add 4 more additional codebook entries for free, but it is not obvious how those extra entries should be designed in the case of the codebook with the additional transmission levels. On the other hand it can be argued that several PMI feedbacks (for example for different subbands) can be bundled to optimize the feedback rate.

83

http://dx.doi.org/10.5772/57134

6 7 8 9 10 11 12 13 14 15

4bps/Hz

Mode 5 − IA Mode 5 − SU Mode 4 Mode 6 Mode 2

Multi-user MIMO in LTE and LTE-Advanced - Receiver Structure and Precoding Design

SNR

**Figure 6.** Downlink fast fading channel with the dual-antenna eNodeB and 2 dual-antenna UEs. IA indicates the low-complexity interference-aware receiver while SU indicates the single-user receiver. 3GPP LTE rate 1/3 turbo code is used

low-complexity interference-aware receivers. To analyze the degradation caused by the low-resolution and EGT of LTE precoders, we also look at the system performance employing the unquantized MF and unquantized MF EGT precoders. To be fair in the comparison of the LTE multi-user MIMO mode (mode 5) employing the geometric scheduling algorithm with the multi-user MIMO mode employing unquantized MF and MF EGT precoders, we consider the geometric scheduling algorithm (Section 4) based on the spatial angle between the two channels (equation (22)). Perfect CSIT is assumed for the case of MF and MF EGT precoding while error free feedback of 2 bits (PMI) to the eNodeB is assumed for LTE precoders. It is assumed that the UE has knowledge of the constellation of co-scheduled UE in the multi-user MIMO mode. It is further assumed that the UE knows its own channel from the eNodeB. So in multi-user MIMO mode, the UE can find the effective interference channel based on the fact that the eNodeB schedules the second UE on the same RE whose precoder is 180◦ out of phase of the precoder of the first UE. Fig. 5 shows the results for the case of single-antenna UEs. It shows enhanced performance of the multi-user MIMO mode once the UEs resort to intelligent detection by employing the low-complexity interference-aware receivers. The performance is severely degraded once the UEs resort to single-user detection. An interesting result is almost the equivalent performance of the unquantized MF EGT and low-resolution LTE precoders which shows that the loss with respect to the unquantized CSIT is attributed

Fig. 6 shows the results for the case of dual-antenna UEs and focuses on different LTE modes employing LTE precoders. It shows the degraded performance of single-user detection which is due to the fact that the rate with single-user detection gets saturated at high SNR due to the increased interference strength as was shown in Section. 4. So the performance of single-user detection is degraded as the spectral efficiency is higher than the rate or mutual information of the single-user detection. For single-user MIMO (Mode 6), there is no saturation of the rate at high SNR as there is no interference. So mode 6 performs better than mode 5 at

10−4

to the EGT rather than the low-resolution of LTE precoders.

10−3

10−2

FER

with different puncturing patterns.

10−1

100

**Figure 5.** Downlink fast fading channel with the dual-antenna eNodeB and 2 single-antenna UEs. IA Rx indicates the low-complexity interference-aware receiver while SU Rx indicates the single-user receiver. MU MIMO and SU MIMO indicate multi-user and single-user MIMO respectively. To be fair in comparison amongst different schemes, sum rates are fixed, i.e. if 2 users are served with QPSK with rate 1/2 in the multi-user mode, then one user is served with QAM16 with rate 1/2 in the single-user mode thereby equating the sum rate in both cases to 2bps/Hz. 3GPP LTE rate 1/3 turbo code is used with different puncturing patterns.
