**Appendix A Mutual information for finite alphabets**

26 Recent Trends in Multiuser MIMO Communications

100

10−2

Extended Typical Urban model (ETU).

**8. Conclusions**

10−1

FER

2 4 6 8 10 12 14

SNR

**Figure 9.** Performance of the proposed precoder codebook in 3GPP LTE channel models [18]. Black continuous lines show the Extended Pedestrian A model (EPA), blue dashed lines show Extended Vehicular A model (EVA), while red dotted lines show

Fig. 9 shows the case where we have considered 3GPP LTE channel model introduced in [18] for three representative scenarios, i.e. pedestrian, vehicular and typical urban scenario. The transmission chain is dominantly LTE compliant with 15 KHz subcarrier-spacing and 20 MHz system bandwidth. The results confirm the earlier findings of the improved performance of proposed codebook design (enhanced levels of transmission) for multi-user transmission mode. Pedestrian channel offers less diversity in the channel as compared to the vehicular channel, so the performance of LTE precoders for multi-user MIMO in severely degraded in the former case. However as the proposed precoder design recovers the lost order of

In this chapter, we have looked at the feasibility of the multi-user MIMO for future wireless systems which are characterized by low-level quantization of CSIT. We have shown that multi-user MIMO can deliver its promised gains if the UEs resort to intelligent detection rather than the sub-optimal single-user detection. To this end, we have proposed a low-complexity interference-aware receiver structure which is characterized by the exploitation of the structure of residual interference. We have further investigated the impact of low-level fixed rate feedback on the performance of multi-user MIMO in LTE systems. We have analyzed two important characteristics of the LTE precoders, i.e. low resolution and EGT. We have shown that the performance loss of the LTE precoders in the multi-user MIMO mode is attributed to their characteristic of EGT rather than their low resolution. We have proposed a feedback and precoding design and have shown that the performance in multi-user MIMO significantly improves once strategy of more levels of transmission is resorted to as compared to the case of increased angular resolution. The work presented in this chapter is not merely confined to the framework of LTE, rather it gives the receiver

MU MIMO − Proposed Precoders SU MIMO − Proposed Precoders SU MIMO − LTE Precoders MU MIMO − LTE Precoders

diversity, there is an improvement of 6dB at the target FER of 10<sup>−</sup>1.

structure and precoding design guidelines for modern wireless systems.

4bps/Hz

The mutual information for UE-1 for finite size QAM constellation with |*χ*1| = *<sup>M</sup>*<sup>1</sup> takes the form as

$$\begin{split} \mathcal{H}\left(\mathbf{Y}\_{1};\mathbf{X}\_{1}|\mathbf{h}\_{1}^{\dagger},\mathbf{P}\right) &= \mathcal{H}\left(\mathbf{X}\_{1}|\mathbf{h}\_{1}^{\dagger},\mathbf{P}\right) - \mathcal{H}\left(\mathbf{X}\_{1}|\mathbf{Y}\_{1},\mathbf{h}\_{1}^{\dagger},\mathbf{P}\right) \\ &= \log M\_{1} - \mathcal{H}\left(\mathbf{X}\_{1}|\mathbf{Y}\_{1},\mathbf{h}\_{1}^{\dagger},\mathbf{P}\right) \end{split} \tag{49}$$

where H (.) = −*E* log *p* (.) is the entropy function. The second term of (49) is given as

$$\begin{split} \mathcal{H}\left(\mathbf{X}\_{1}|\mathbf{Y}\_{1},\mathbf{h}\_{1}^{\dagger}\mathbf{P}\right) &= \sum\_{\mathbf{x}\_{1}} \int\_{\mathcal{Y}\_{1}} \int\_{\mathbf{h}\_{1}^{\dagger}\mathbf{P}} p\left(\mathbf{x}\_{1},y\_{1},\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{1},\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}\right) \log \frac{1}{p\left(\mathbf{x}\_{1}|y\_{1},\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{1},\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}\right)} dy\_{1} d(\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{1}) d(\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}) \\ &= \sum\_{\mathbf{x}\_{1}} \sum\_{\mathbf{x}\_{2}} \int\_{\mathcal{Y}\_{2}} \int\_{\mathbf{h}\_{1}^{\dagger}\mathbf{P}} p\left(\mathbf{x}\_{1},x\_{2},y\_{1},\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{1},\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}\right) \log \frac{\sum\_{\mathbf{x}\_{1}'}\sum\_{\mathbf{x}\_{2}'} p\left(y\_{1}|\mathbf{x}\_{1}',\mathbf{x}\_{2}',\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{1},\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}\right)}{\sum\_{\mathbf{x}\_{2}'}p\left(y\_{1}|\mathbf{x}\_{1},x\_{2}',\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{1},\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}\right)} dy\_{1} d(\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{1}) d(\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}) \end{split} \tag{50}$$

where *x* ′ <sup>1</sup> <sup>∈</sup> *<sup>χ</sup>*<sup>1</sup> and *<sup>x</sup>* ′ <sup>2</sup> <sup>∈</sup> *<sup>χ</sup>*2. Conditioned on the channel and the precoder, there is one source of randomness, i.e. noise. So (50) can be extended as

$$\begin{split} \mathcal{H}\left(\mathbf{X}\_{1}|Y\_{1},\mathbf{h}\_{1}^{\dagger},\mathbf{P}\right) &= \frac{1}{M\_{\mathrm{1}}M\_{\mathrm{2}}}\sum\_{\mathbf{x}}{\mathrm{E}\_{z\_{1}}\mathrm{log}}\frac{\sum\_{\mathbf{x}'}\exp\left[-\frac{1}{N\_{\mathrm{0}}}\left|\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{1}\mathbf{x}\_{1} + \mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}\mathbf{x}\_{2} + z\_{\mathrm{i}} - \mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{1}\mathbf{x}\_{1}^{\prime} - \mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}\mathbf{x}\_{2}^{\prime}\right|^{2}\right]}{\sum\_{\mathbf{x}\_{2}'}\exp\left[-\frac{1}{N\_{\mathrm{0}}}\left|\mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}\mathbf{x}\_{2} + z\_{\mathrm{i}} - \mathbf{h}\_{1}^{\dagger}\mathbf{p}\_{2}\mathbf{x}\_{2}^{\prime}\right|^{2}\right]} \\ &= \frac{1}{M\_{\mathrm{1}}M\_{\mathrm{2}}}\sum\_{\mathbf{x}}{\mathrm{E}\_{z\_{1}}\mathrm{log}}\frac{\sum\_{\mathbf{x}'}\exp\left[-\frac{1}{N\_{\mathrm{0}}}\left|\mathbf{h}\_{1}^{\dagger}\mathbf{P}\left(\mathbf{x} - \mathbf{x}^{\prime}\right) + z\_{\mathrm{i}}\right|^{2}\right]}{\sum\_{\mathbf{x}\_{2}'}\exp\left[-\frac{1}{N\_{\mathrm{0}}}\left|\mathbf{h}\_{1}^{\dagger}\mathbf{P}\left(\mathbf{x} - \mathbf{x}\_{2}^{\prime}\right) + z\_{\mathrm{i}}\right|^{2}\right]} \end{split} \tag{51}$$

where *<sup>M</sup>*<sup>2</sup> = |*χ*2|, **<sup>x</sup>** = [*x*<sup>1</sup> *<sup>x</sup>*2] *<sup>T</sup>*, **x** ′ = *x* ′ 1 *x* ′ 2 *T* and **x** ′ <sup>2</sup> = *x*<sup>1</sup> *x* ′ 2 *T* . The mutual information for UE-1 can be rewritten as

$$I\left(Y\_1; X\_1 | \mathbf{h}\_1^\dagger, \mathbf{P}\right) = \log M\_1 - \frac{1}{M\_1 M\_2} \sum\_{\mathbf{x}} E\_{\mathbf{z}\_1} \log \frac{\sum\_{\mathbf{x}'} p\left(y\_1 | \mathbf{x}', \mathbf{h}\_{1'}^\dagger, \mathbf{P}\right)}{\sum\_{\mathbf{x}\_2'} p\left(y\_1 | \mathbf{x}\_{2'}', \mathbf{h}\_{1'}^\dagger, \mathbf{P}\right)} \tag{52}$$

The above quantities can be easily approximated using sampling (Monte-Carlo) methods with *Nz* realizations of noise and *Nh*<sup>1</sup> realizations of the channel **<sup>h</sup>**† <sup>1</sup> where the precoding matrix depends on the channel. So we can rewrite (52) as (53)

*I Y*; *X*1|**h**† <sup>1</sup>**p**1, **<sup>h</sup>**† <sup>1</sup>**p**<sup>2</sup> <sup>=</sup> log *<sup>M</sup>*<sup>1</sup> <sup>−</sup> <sup>1</sup> *M*1*M*2*NzNh*<sup>1</sup> ∑*x*1 ∑*x*2 *Nh*<sup>1</sup> ∑ **h**1 *Nz* ∑*z*1 log ∑*x* ′ 1 ∑*x* ′ 2 exp − <sup>1</sup> *N*0 *<sup>y</sup>*<sup>1</sup> <sup>−</sup> **<sup>h</sup>**† <sup>1</sup>**p**<sup>1</sup> *x* ′ <sup>1</sup> <sup>−</sup> **<sup>h</sup>**† <sup>1</sup>**p**<sup>2</sup> *x* ′ 2 2 ∑*x* ′ 2 exp − <sup>1</sup> *N*0 *<sup>y</sup>*<sup>1</sup> <sup>−</sup> **<sup>h</sup>**† <sup>1</sup>**p**<sup>1</sup> *<sup>x</sup>*<sup>1</sup> <sup>−</sup> **<sup>h</sup>**† <sup>1</sup>**p**<sup>2</sup> *x* ′ 2 2 <sup>=</sup> log *<sup>M</sup>*<sup>1</sup> <sup>−</sup> <sup>1</sup> *M*1*M*2*NzNh*<sup>1</sup> ∑*x*1 ∑*x*2 *Nh*<sup>1</sup> ∑ **h**1 *Nz* ∑*z*1 log ∑*x* ′ 1 ∑*x* ′ 2 exp − <sup>1</sup> *N*0 **h**† <sup>1</sup>**p**<sup>1</sup> *<sup>x</sup>*1+**h**† <sup>1</sup>**p**<sup>2</sup> *<sup>x</sup>*2+*z*1−**h**† <sup>1</sup>**p**<sup>1</sup> *x* ′ 1−**h**† <sup>1</sup>**p**<sup>2</sup> *x* ′ 2 2 ∑*x* ′ 2 exp − <sup>1</sup> *N*0 **h**† <sup>1</sup>**p**<sup>2</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>1</sup> <sup>−</sup> **<sup>h</sup>**† <sup>1</sup>**p**<sup>2</sup> *x* ′ 2 2 (53)

89

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

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Similarly the mutual information for UE-2 is given as

$$I\left(Y\_2; \mathbf{X}\_2 | \mathbf{h}\_{2'}^\dagger \mathbf{P}\right) = \log M\_2 - \frac{1}{M\_1 M\_2} \sum\_{\mathbf{x}} E\_{z\_2} \log \frac{\sum\_{\mathbf{x}'} p\left(y\_2 | \mathbf{x}', \mathbf{h}\_{2'}^\dagger \mathbf{P}\right)}{\sum\_{\mathbf{x}\_1'} p\left(y\_2 | \mathbf{x}\_1', \mathbf{h}\_{2'}^\dagger \mathbf{P}\right)} \tag{54}$$

where **x** ′ <sup>1</sup> = *x* ′ <sup>1</sup> *x*<sup>2</sup> *T* .

For the case of single-user MIMO mode, the mutual information is given by

$$I\left(Y\_1; X\_1 \middle| \mathbf{h}\_1^\dagger, \mathbf{p}\_1\right) = \log M\_1 - \mathcal{H}\left(X\_1 \middle| Y\_1, \mathbf{h}\_1^\dagger, \mathbf{p}\_1\right) \tag{55}$$

where the second term is given by

H *<sup>X</sup>*1|*Y*1, **<sup>h</sup>**† 1, **p**<sup>1</sup> = ∑*x*1 *y*1 **h**† <sup>1</sup>**p**<sup>1</sup> *p x*1, *y*1, **h**† <sup>1</sup>**p**<sup>1</sup> log <sup>1</sup> *p <sup>x</sup>*1|*y*1, **<sup>h</sup>**† <sup>1</sup>**p**<sup>1</sup> *dy*1*d*(**h**† <sup>1</sup>**p**1) =∑*x*1 *y*1 **h**† <sup>1</sup>**p**<sup>1</sup> *p x*1, *y*1, **h**† <sup>1</sup>**p**<sup>1</sup> log ∑*x* ′ 1 *p <sup>y</sup>*1|*<sup>x</sup>* ′ <sup>1</sup>, **<sup>h</sup>**† <sup>1</sup>**p**<sup>1</sup> *p <sup>y</sup>*1|*x*1, **<sup>h</sup>**† <sup>1</sup>**p**<sup>1</sup> *dy*1*d*(**h**† <sup>1</sup>**p**1) <sup>=</sup> <sup>1</sup> *M*1*NzNh*<sup>1</sup> ∑*x*1 *Nh*<sup>1</sup> ∑ **h**† 1 *Nz* ∑*z*1 log ∑*x* ′ 1 exp − 1 *N*<sup>0</sup> *<sup>y</sup>*<sup>1</sup> <sup>−</sup> **<sup>h</sup>**† <sup>1</sup>**p**1*x* ′ 1 2 exp − 1 *N*<sup>0</sup> *<sup>y</sup>*<sup>1</sup> <sup>−</sup> **<sup>h</sup>**† <sup>1</sup>**p**1*x*<sup>1</sup> 2 (56)

where *Nh*<sup>1</sup> are the number of channel realizations of the channel **<sup>h</sup>**† 1. Note that the precoding vector **<sup>p</sup>**<sup>1</sup> is dependent on the channel **<sup>h</sup>**† 1.

### **Author details**

Rizwan Ghaffar1,<sup>⋆</sup>, Raymond Knopp2 and Florian Kaltenberger2

<sup>⋆</sup> Address all correspondence to: rizwan.ghaffar@eurecom.fr

1 Samsung Research America, USA 2 EURECOM, France
