**5.1. Single-user MIMO**

For the single-user case, the received signal at the *k*-th RE is given by

$$y\_{1,k} = \mathbf{h}\_{1,k}^{\dagger} \mathbf{p}\_{1,k} x\_{1,k} + z\_{1,k} \tag{23}$$

For EGT, the precoder vector is given by **<sup>p</sup>**1,*<sup>k</sup>* = <sup>√</sup> 1 2 <sup>1</sup> *<sup>h</sup>*21,*kh*<sup>∗</sup> 11,*k* |*h*21,*<sup>k</sup>* ||*h*11,*<sup>k</sup>* | *T* . So the received signal

after normalization by *<sup>h</sup>*11,*<sup>k</sup>* <sup>|</sup>*h*11,*<sup>k</sup>* <sup>|</sup> is given by

$$y\_{1,k}^{N} = \frac{1}{\sqrt{2}} \left( \left| h\_{11,k} \right| + \left| h\_{21,k} \right| \right) \ge\_{1,k} + \frac{h\_{11,k}}{\left| h\_{11,k} \right|} z\_{1,k} \tag{24}$$

where *y<sup>N</sup>* 1,*<sup>k</sup>* <sup>=</sup> *<sup>h</sup>*11,*<sup>k</sup>* |*h*11,*<sup>k</sup>* | *y*1,*k*. The max log MAP bit metric [4] for the bit *ck* ′ can be written as

$$
\Lambda\_1^i \left( y\_{k'} c\_{k'} \right) \approx \min\_{\mathbf{x}\_1 \in \mathcal{X}\_{1:\ell\_{k'}}^i} \left[ \frac{1}{N\_0} \left| y\_{1,k}^N - \frac{1}{\sqrt{2}} \left( \left| h\_{11,k} \right| + \left| h\_{21,k} \right| \right) \mathbf{x}\_1 \right|^2 \right] \tag{25}
$$

The conditional PEP i.e *<sup>P</sup>* (**c**<sup>1</sup> <sup>→</sup> **<sup>c</sup>**ˆ1|**h**1) is given as

$$P\left(\mathfrak{s}\_1 \rightarrow \mathfrak{s}\_1 | \mathbf{H}\_1\right) = P\left(\sum\_{k'} \min\_{\mathbf{x}\_1 \in \chi\_{1\_{k'}}^\circ} \frac{1}{N\_0} \left| y\_{1,k}^N - \frac{1}{\sqrt{2}} \left( |h\_{11,k}| + |h\_{21,k}| \right) \mathbf{x}\_1 \right|^2\right.$$

$$\geq \sum\_{k'} \min\_{\mathbf{x}\_1 \in \chi\_{1\_{k'}}^\circ} \frac{1}{N\_0} \left| y\_{1,k}^N - \frac{1}{\sqrt{2}} \left( |h\_{11,k}| + |h\_{21,k}| \right) \mathbf{x}\_1 \right|^2 \left| \overline{\mathbf{H}}\_1 \right|\tag{26}$$

$$\hat{\mathbf{x}}\_{1,k} = \arg\min\_{\mathbf{x}\_1 \in \chi\_{1\_{\hat{\mathcal{K}}\_{\hat{\mathcal{K}}}}'}} \frac{1}{N\_0} \left| y\_{1,k}^N - \frac{1}{\sqrt{2}} \left( |h\_{11,k}| + |h\_{21,k}| \right) \mathbf{x}\_1 \right|^2$$

$$\hat{\mathbf{x}}\_{1,k} = \arg\min\_{\mathbf{x}\_1 \in \chi\_{1\_{\hat{\mathcal{K}}\_{\hat{\mathcal{K}}}}'}} \frac{1}{N\_0} \left| y\_{1,k}^N - \frac{1}{\sqrt{2}} \left( |h\_{11,k}| + |h\_{21,k}| \right) \mathbf{x}\_1 \right|^2 \tag{27}$$

$$P\left(\hat{\mathbf{c}}\_{1}\rightarrow\hat{\mathbf{c}}\_{1}\right)\leq\frac{1}{2}E\_{\overline{\mathbf{H}}\_{1}}\prod\_{k,d\_{fnc}}\exp\left(-\frac{1}{8N\_{0}}\left(|h\_{11,k}|+|h\_{21,k}|\right)^{2}d\_{1,\text{min}}^{2}\right)$$

$$=\frac{1}{2}\prod\_{k,d\_{fnc}}E\_{\mathbf{h}\_{1k}}\exp\left(\left(-\frac{d\_{1,\text{min}}^{2}}{4}\right)\frac{\left(|h\_{11,k}|+|h\_{21,k}|\right)^{2}\sigma\_{1}^{2}}{2N\_{0}}\right)\tag{30}$$

$$\pi - 2\sin^{-1}\left(\sqrt{\frac{\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{d\_{1\text{min}}^2}{4}}{2\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{d\_{1\text{min}}^2}{4}}}\right) = 2\cos^{-1}\left(\sqrt{\frac{\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{d\_{1\text{min}}^2}{4}}{2\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{d\_{1\text{min}}^2}{4}}\right) \tag{32}$$

$$\cos^{-1}\left(\sqrt{\mathbf{x}}\right) = \sqrt{2 - 2\sqrt{\mathbf{x}}} \sum\_{k=0}^{\infty} \frac{\left(1 - \sqrt{\mathbf{x}}\right)^{k} \left(1/2\right)\_{k}}{2^{k} \left(k! + 2kk\right)} \qquad \text{for } |-1 + \sqrt{\mathbf{x}}| < 2$$

$$\cos^{-1}\left(\sqrt{\frac{\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{\tilde{d}\_{1\text{min}}^2}{4}}{2\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{\tilde{d}\_{1\text{min}}^2}{4}}}\right) \approx \sqrt{2 - 2} \sqrt{\frac{\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{\tilde{d}\_{1\text{min}}^2}{4}}{2\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{\tilde{d}\_{1\text{min}}^2}{4}}}\tag{33}$$

$$\sqrt{\mathbf{x}} = 1 + \frac{\mathbf{x} - 1}{2} - \frac{\left(\mathbf{x} - 1\right)^2}{8} + \frac{\left(\mathbf{x} - 1\right)^3}{16} - \dotsb$$

$$\begin{split} \sqrt{2 - \sqrt{\frac{\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{\tilde{d}\_{1\text{min}}^2}{4}}{2\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{\tilde{d}\_{1\text{min}}^2}{4}}} & \approx \sqrt{2 - 2\left(1 + \frac{\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{\tilde{d}\_{1\text{min}}^2}{4}}{2} - 1\right)} \\ &= \sqrt{-\left(\frac{\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{\tilde{d}\_{1\text{min}}^2}{4}}{2\left(\frac{\sigma\_1^2}{N\_0}\right)^{-1} + \frac{\tilde{d}\_{1\text{min}}^2}{4}} - 1\right)} \\ &= \frac{1}{\sqrt{2 + \frac{\tilde{d}\_{1\text{min}}^2}{4} \left(\frac{\sigma\_1^2}{N\_0}\right)}} \end{split} \tag{34}$$

$$\begin{split} P\left(\underline{\mathbf{s}\_{1}}\rightarrow\underline{\mathbf{s}\_{1}}\right)\leq\frac{1}{2}\prod\_{d\_{f\neq\upsilon}}\left(\frac{2}{\left(2+\frac{\tilde{d}\_{1\text{min}}^{2}}{4}\left(\frac{\sigma\_{1}^{2}}{N\_{0}}\right)\right)^{2}}+\frac{\tilde{d}\_{1\text{min}}^{2}\left(\frac{\sigma\_{1}^{2}}{N\_{0}}\right)}{4\left(2+\frac{\tilde{d}\_{1\text{min}}^{2}}{4}\left(\frac{\sigma\_{1}^{2}}{N\_{0}}\right)\right)^{2}}\right)\\ &-\frac{2\left(\frac{\tilde{d}\_{1\text{min}}^{2}}{2\sqrt{2}}\right)\left(\frac{\sigma\_{1}^{2}}{N\_{0}}\right)}{\left(2+\frac{\tilde{d}\_{1\text{min}}^{2}}{2}\left(\frac{\sigma\_{1}^{2}}{N\_{0}}\right)\right)^{3/2}\left(2+\frac{\tilde{d}\_{1\text{min}}^{2}}{4}\left(\frac{\sigma\_{1}^{2}}{N\_{0}}\right)\right)^{1/2}}\\ &+\frac{\left(4+\frac{\tilde{d}\_{1\text{min}}^{2}}{2}\left(\frac{\sigma\_{1}^{2}}{N\_{0}}\right)\right)}{\left(2+\frac{\tilde{d}\_{1\text{min}}^{2}}{4}\left(\frac{\sigma\_{1}^{2}}{N\_{0}}\right)\right)^{2}\left(2+\frac{\tilde{d}\_{1\text{min}}^{2}}{2}\left(\frac{\sigma\_{1}^{2}}{N\_{0}}\right)\right)}\end{split} \tag{35}$$

$$\begin{split} P\left(\mathfrak{S}\_1 \rightarrow \hat{\mathfrak{s}}\_1\right) &\leq \frac{1}{2} \prod\_{d\_{f\mid m}} \left( \frac{32}{\left( \hat{d}\_{1,\min}^2 \left( \frac{\sigma\_1^2}{N\_0} \right) \right)^2} + \frac{16}{\left( \hat{d}\_{1,\min}^2 \left( \frac{\sigma\_1^2}{N\_0} \right) \right)^2} \right) \\ &= \frac{1}{2} \prod\_{d\_{f\mid m}} \left( \frac{48}{\left( \hat{d}\_{1,\min}^2 \left( \frac{\sigma\_1^2}{N\_0} \right) \right)^2} \right) \end{split} \tag{36}$$

$$\sum\_{k,d\_{free}} \left| y\_{1,k}^N - \frac{1}{\sqrt{2}} \left( |h\_{11,k}| + |h\_{21,k}| \right) \mathbf{x}\_{1,k} \right|^2 = \left\| \mathbf{y}\_1^N - \frac{1}{\sqrt{2}} \left( |h\_{11}| + |h\_{21}| \right) \mathbf{x}\_1 \right\|^2$$

So conditional PEP is given as

$$\begin{split} P\left(\mathbf{\hat{g}}\_{1}\rightarrow\mathbf{\hat{g}}\_{1}|\overline{\mathbf{H}}\_{1}\right) &\leq P\left(\left\|\mathbf{y}\_{1}^{N}-\frac{1}{\sqrt{2}}\left(|h\_{11}|+|h\_{21}|\right)\mathbf{x}\_{1}\right\|^{2} \geq \left\|\mathbf{y}\_{1}^{N}-\frac{1}{\sqrt{2}}\left(|h\_{11}|+|h\_{21}|\right)\mathbf{\hat{x}}\_{1}\right\|^{2}\right) \\ &= P\left(\mathfrak{R}\left(\left(\frac{1}{\sqrt{2}}\left(|h\_{11}|+|h\_{21}|\right)\mathbf{x}+\mathbf{z}\_{1}\right)^{\dagger}\left(\mathbf{x}\_{1}-\mathbf{\hat{x}}\_{1}\right)\right) \leq 0\right) \\ &= P\left(\frac{1}{\sqrt{2}}\left(|h\_{11}|+|h\_{21}|\right)\left(\left\|\mathbf{x}\_{1}\right\|^{2}-\mathfrak{R}\left(\mathbf{x}\_{1}^{\dagger}\mathbf{\hat{x}}\_{1}\right)\right)+\mathfrak{R}\left(\mathbf{z}\_{1}^{\dagger}\left(\mathbf{x}\_{1}-\mathbf{\hat{x}}\_{1}\right)\right)\leq 0\right) \end{split} \tag{37}$$

10.5772/57134

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

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

2

2

2

� � � �

*<sup>x</sup>*2,*k*−*x*ˆ2,*<sup>k</sup>*

2

�� � 2 

(43)

2

21,*k*.

79

(41)

(42)

**5.2. Multi-user MIMO**

The max log MAP bit metric is written as

Λ*i* 1 � *y*1,*k*, *ck* ′ �

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

� = *P* ∑ *k* ′

> 1 *N*0 � � � �

*x*˜1,*k*, *x*˜2,*<sup>k</sup>* =arg min *x*1∈*χ<sup>i</sup>* 1,*c k* ′ ,*x*2∈*χ*<sup>2</sup>

*x*ˆ1,*k*, *x*ˆ2,*<sup>k</sup>* =arg min *x*1∈*χ<sup>i</sup>* 1,*c*¯ *k* ′ ,*x*2∈*χ*<sup>2</sup>

(*h*1,*kx*1,*k*+*h*2,*kx*2,*k*)

min *x*1∈*χ<sup>i</sup>* 1,*c*ˆ *k* ′ ,*x*2∈*χ*<sup>2</sup>

Conditional PEP is given as

*P* �

Let's denote

Note that

≥ ∑ *k* ′

> � � � �

So conditional PEP is given as

*P* � *<sup>y</sup>*1,*k*<sup>−</sup> <sup>1</sup> √4

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

� ≤*Q* 

 ���� ∑ *k*,*df ree*

 ���� ∑ *k*,*df ree*

= *Q*

Let **p**<sup>1</sup> = [1 *q*]

be [1 − *q*]

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

We now focus on the PEP of UE-1 in the multi-user MIMO mode as per system equation (3).

be served in the multi-user MIMO mode [8], scheduling at the eNodeB would ensure **p**<sup>2</sup> to

21,*<sup>k</sup>* whereas the channel seen by the interference stream *<sup>x</sup>*2,*<sup>k</sup>* is *<sup>h</sup>*2,*<sup>k</sup>* <sup>=</sup> *<sup>h</sup>*<sup>∗</sup>

1 *N*0 � � � �

min *x*1∈*χ<sup>i</sup>* 1,*c k* ′ ,*x*2∈*χ*<sup>2</sup>

> 1 *N*0 � � � �

> 1 *N*0 � � � �

> > � � � �

1 8*N*<sup>0</sup> � �*h*1,*<sup>k</sup>* �

> 1 8*N*<sup>0</sup> � � � **h***T*

2 ≥ � � � �

*<sup>y</sup>*1,*k*<sup>−</sup> <sup>1</sup> √4

≈ min *x*1∈*χ<sup>i</sup>* 1,*c k* ′ ,*x*2∈*χ*<sup>2</sup>

*<sup>T</sup>* where *q* ∈ {±1, ±*j*}. To have good channel separation between the UEs to

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

*<sup>T</sup>*. The effective channel seen by the desired stream *<sup>x</sup>*1,*<sup>k</sup>* at UE-1 is given as *<sup>h</sup>*1,*<sup>k</sup>* <sup>=</sup>

*<sup>y</sup>*1,*k*<sup>−</sup> <sup>1</sup> √4

> 1 *N*0 � � � �

*<sup>h</sup>*1,*kx*1<sup>−</sup> <sup>1</sup> √4 *h*2,*kx*<sup>2</sup> � � � �

> *<sup>y</sup>*1,*k*<sup>−</sup> <sup>1</sup> √4

> *<sup>y</sup>*1,*k*<sup>−</sup> <sup>1</sup> √4

> > *<sup>y</sup>*1,*k*<sup>−</sup> <sup>1</sup> √4

> > > *<sup>x</sup>*1,*k*−*x*ˆ1,*<sup>k</sup>*

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

� +*h*2,*<sup>k</sup>* �

� � � 2 

*<sup>y</sup>*1,*k*<sup>−</sup> <sup>1</sup> √4

*<sup>h</sup>*1,*kx*1<sup>−</sup> <sup>1</sup> √4 *h*2,*kx*<sup>2</sup> � � � �

> 2

*<sup>h</sup>*1,*kx*1<sup>−</sup> <sup>1</sup> √4 *h*2,*kx*<sup>2</sup> � � � �

*<sup>h</sup>*1,*kx*1<sup>−</sup> <sup>1</sup> √4 *h*2,*kx*<sup>2</sup> � � � �

(*h*1,*kx*˜1,*k*+*h*2,*kx*˜2,*k*)

*<sup>h</sup>*1,*kx*1<sup>−</sup> <sup>1</sup> √4 *h*2,*kx*<sup>2</sup> � � � �

Using �**x**<sup>1</sup> − **<sup>x</sup>**ˆ1�<sup>2</sup> = �**x**1�<sup>2</sup> + �**x**ˆ1�<sup>2</sup> − <sup>2</sup>ℜ (**x**ˆ1**x**) we get

$$\begin{split} P\left(\mathbf{\hat{c}}\_{1}\rightarrow\mathbf{\hat{c}}\_{1}|\overline{\mathbf{H}}\_{1}\right)\leq P\left(|\left|h\_{11}\right|+\left|h\_{21}\right|\right)\left(\frac{3}{2\sqrt{2}}\left\||\mathbf{x}\_{1}\right\|^{2}-\frac{1}{2\sqrt{2}}\left\Vert\mathbf{x}\_{1}-\hat{\mathbf{x}}\_{1}\right\Vert^{2}+\frac{1}{2\sqrt{2}}\left\Vert\hat{\mathbf{x}}\_{1}\right\Vert^{2}\right)\#\mathcal{R}\left(\mathbf{z}\_{1}^{\dagger}\left\Vert\mathbf{x}\_{1}-\hat{\mathbf{x}}\_{1}\right\Vert\right)\leq 0\\ = P\left(\mathbf{x}\left(\left|h\_{11}\right|+\left|h\_{21}\right|\right)+z\_{1}^{'}\leq 0\right) \end{split}$$

where *κ* = <sup>3</sup> 2 √<sup>2</sup> �**x**1�<sup>2</sup> <sup>−</sup> <sup>1</sup> 2 √<sup>2</sup> �**x**<sup>1</sup> <sup>−</sup> **<sup>x</sup>**ˆ1�<sup>2</sup> <sup>+</sup> <sup>1</sup> 2 √<sup>2</sup> �**x**ˆ1�<sup>2</sup> . *z* ′ <sup>1</sup> <sup>=</sup> <sup>ℜ</sup> **z**† <sup>1</sup> (**x**<sup>1</sup> <sup>−</sup> **<sup>x</sup>**ˆ1) is circularly symmetric complex while Gaussian noise of variance *<sup>N</sup>*<sup>0</sup> <sup>2</sup> �**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**ˆ1�<sup>2</sup> . So the PEP is upperbounded as

$$P\left(\underline{\mathfrak{g}}\_{1}\rightarrow\underline{\mathfrak{g}}\_{1}|\overline{\mathbf{H}}\_{1}\right)\leq P\left(\kappa\left(|h\_{11}|+|h\_{21}|\right)+z\_{1}^{'}\leq 0\right)\tag{38}$$

As per ours notations, the decision variable *γ* as per (5) and (13) in [29] is given as

$$\gamma = \kappa \left( |h\_{11}| + |h\_{21}| \right) + z\_1^{'} \tag{39}$$

So the probability of error which is given as *P* (*γ* ≤ 0) is given as

$$P\left(\underline{\mathbf{c}}\_{1}\rightarrow\hat{\underline{\mathbf{c}}}\_{1}\right) = \frac{1}{2}\left\{1 - \frac{\sqrt{\rho\_{11}\left(\rho\_{11} + 2\kappa\right)} + \sqrt{\rho\_{21}\left(\rho\_{21} + 2\kappa\right)}}{\rho\_{11} + \rho\_{21} + 2\kappa}\right\} \tag{40}$$

where *ρij* = **E** *hij* 2 /*N*<sup>0</sup> is the SNR at the individual branch and *κ* is a constant that will depend on the constellation. (40) shows the full diversity order of 2, a result earlier derived for EGT in single-user MIMO systems in [14] using the approach of metrics of diversity order.

## **5.2. Multi-user MIMO**

18 Recent Trends in Multiuser MIMO Communications

 ≤ *P* **y***N* <sup>1</sup> <sup>−</sup> <sup>1</sup> √2

= *P* ℜ 1 √2

=*P* 1 √2

Using �**x**<sup>1</sup> − **<sup>x</sup>**ˆ1�<sup>2</sup> = �**x**1�<sup>2</sup> + �**x**ˆ1�<sup>2</sup> − <sup>2</sup>ℜ (**x**ˆ1**x**) we get

(|*h*11|+|*h*21|)

2 √ (|*h*11|+|*h*21|) **<sup>x</sup>**<sup>1</sup>

− 1 2 √2

> 2 √<sup>2</sup> �**x**ˆ1�<sup>2</sup>

′ <sup>1</sup> <sup>≤</sup> <sup>0</sup> 

<sup>2</sup> �**x**<sup>1</sup> <sup>−</sup> **<sup>x</sup>**ˆ1�<sup>2</sup> <sup>+</sup> <sup>1</sup>

 ≤ *P* 

As per ours notations, the decision variable *γ* as per (5) and (13) in [29] is given as

*<sup>γ</sup>* = *<sup>κ</sup>* (|*h*11| + |*h*21|) + *<sup>z</sup>*

depend on the constellation. (40) shows the full diversity order of 2, a result earlier derived for EGT in single-user MIMO systems in [14] using the approach of metrics of diversity order.

(|*h*11|+|*h*21|)

 3 2 √2 �**x**1�<sup>2</sup>

*<sup>κ</sup>* (|*h*11| + |*h*21|) + *<sup>z</sup>*

symmetric complex while Gaussian noise of variance *<sup>N</sup>*<sup>0</sup>

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

So the probability of error which is given as *P* (*γ* ≤ 0) is given as

2 1−

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

(|*h*11| + |*h*21|) **<sup>x</sup>** + **<sup>z</sup>**<sup>1</sup>

�**x**1�2−ℜ

  2 ≥ **y***N* <sup>1</sup> <sup>−</sup> <sup>1</sup> √2

 **x**† 1**x**ˆ1 +ℜ **z**†

�**x**1−**x**ˆ1�<sup>2</sup>

. *z* ′ <sup>1</sup> <sup>=</sup> <sup>ℜ</sup>

*<sup>κ</sup>* (|*h*11| + |*h*21|) + *<sup>z</sup>*

′

*ρ*<sup>11</sup> (*ρ*11+2*κ*)+*ρ*<sup>21</sup> (*ρ*21+2*κ*) *ρ*<sup>11</sup> + *ρ*<sup>21</sup> + 2*κ*

/*N*<sup>0</sup> is the SNR at the individual branch and *κ* is a constant that will

+ 1 2 √2 �**x**ˆ1�<sup>2</sup> +ℜ **z**† <sup>1</sup>(**x**1−**x**ˆ1) ≤0 

**z**†

<sup>2</sup> �**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**ˆ1�<sup>2</sup>

′ <sup>1</sup> <sup>≤</sup> <sup>0</sup> 

<sup>1</sup> (**x**<sup>1</sup> <sup>−</sup> **<sup>x</sup>**ˆ1)

<sup>1</sup> (39)

†

(**x**<sup>1</sup> − **<sup>x</sup>**ˆ1)

 ≤ 0 

(|*h*11|+|*h*21|) **<sup>x</sup>**ˆ1

<sup>1</sup> (**x**1−**x**ˆ1)

 ≤0 

  2 

is circularly

(38)

(40)

. So the PEP is

(37)

So conditional PEP is given as

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

*P* 

*P* 

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

where *κ* = <sup>3</sup>

upperbounded as

where *ρij* = **E**

 *hij* 2 

 ≤*P* 

2 √ = *P* 

<sup>2</sup> �**x**1�<sup>2</sup> <sup>−</sup> <sup>1</sup>

*P*  We now focus on the PEP of UE-1 in the multi-user MIMO mode as per system equation (3). Let **p**<sup>1</sup> = [1 *q*] *<sup>T</sup>* where *q* ∈ {±1, ±*j*}. To have good channel separation between the UEs to be served in the multi-user MIMO mode [8], scheduling at the eNodeB would ensure **p**<sup>2</sup> to be [1 − *q*] *<sup>T</sup>*. The effective channel seen by the desired stream *<sup>x</sup>*1,*<sup>k</sup>* at UE-1 is given as *<sup>h</sup>*1,*<sup>k</sup>* <sup>=</sup> *h*∗ 11,*<sup>k</sup>* <sup>+</sup> *qh*<sup>∗</sup> 21,*<sup>k</sup>* whereas the channel seen by the interference stream *<sup>x</sup>*2,*<sup>k</sup>* is *<sup>h</sup>*2,*<sup>k</sup>* <sup>=</sup> *<sup>h</sup>*<sup>∗</sup> 11,*<sup>k</sup>* <sup>−</sup> *qh*<sup>∗</sup> 21,*k*. The max log MAP bit metric is written as

$$\Lambda\_1^i \left( y\_{1,k'} c\_{k'} \right) \approx \min\_{\mathbf{x}\_1 \in \chi\_{1,\boldsymbol{\varepsilon}\_{k'}}^i, \mathbf{x}\_2 \in \chi\_2} \frac{1}{N\_0} \left| y\_{1,k} - \frac{1}{\sqrt{4}} h\_{1,k} \mathbf{x}\_1 - \frac{1}{\sqrt{4}} h\_{2,k} \mathbf{x}\_2 \right|^2$$

Conditional PEP is given as

$$\begin{split} P\left(\mathfrak{S}\_{1}\rightarrow\hat{\mathfrak{S}}\_{1}|\overline{\mathbf{H}}\_{1}\right) &= P\left(\sum\_{k'}\min\_{\mathbf{x}\_{1}\in\chi^{i}\_{1:\chi^{i}\_{k'},\mathbf{x}\_{2}\in\chi\_{2}}}\frac{1}{N\_{0}}\left|y\_{1,k}-\frac{1}{\sqrt{4}}h\_{1,k}\mathbf{x}\_{1}-\frac{1}{\sqrt{4}}h\_{2,k}\mathbf{x}\_{2}\right|^{2} \\ &\geq\sum\_{k'}\min\_{\mathbf{x}\_{1}\in\chi^{i}\_{1:\chi^{i}\_{k'},\mathbf{x}\_{2}\in\chi\_{2}}}\frac{1}{N\_{0}}\left|y\_{1,k}-\frac{1}{\sqrt{4}}h\_{1,k}\mathbf{x}\_{1}-\frac{1}{\sqrt{4}}h\_{2,k}\mathbf{x}\_{2}\right|^{2}\right) \end{split} \tag{41}$$

Let's denote

$$\tilde{\mathbf{x}}\_{1,k}, \tilde{\mathbf{x}}\_{2,k} = \underset{\mathbf{x}\_1 \in \chi\_{1\_{\mathcal{I}\_{\tilde{\mathbf{1}}}}}^{\mathrm{l}}}{\mathrm{arg}} \min\_{\mathbf{x}\_2 \in \chi\_2} \frac{1}{N\_0} \left| y\_{1,k} - \frac{1}{\sqrt{4}} h\_{1,k} \mathbf{x}\_1 - \frac{1}{\sqrt{4}} h\_{2,k} \mathbf{x}\_2 \right|^2$$

$$\hat{\mathbf{x}}\_{1,k}, \hat{\mathbf{x}}\_{2,k} = \underset{\mathbf{x}\_1 \in \chi\_{1\_{\mathcal{I}\_{\tilde{\mathbf{1}}}}}^{\mathrm{l}}}{\mathrm{arg}} \min\_{\mathbf{x}\_2 \in \chi\_2} \frac{1}{N\_0} \left| y\_{1,k} - \frac{1}{\sqrt{4}} h\_{1,k} \mathbf{x}\_1 - \frac{1}{\sqrt{4}} h\_{2,k} \mathbf{x}\_2 \right|^2 \tag{42}$$

Note that

$$\left| y\_{1,k} - \frac{1}{\sqrt{4}} (h\_{1,k} \mathbf{x}\_{1,k} + h\_{2,k} \mathbf{x}\_{2,k}) \right|^2 \ge \left| y\_{1,k} - \frac{1}{\sqrt{4}} (h\_{1,k} \tilde{\mathbf{x}}\_{1,k} + h\_{2,k} \tilde{\mathbf{x}}\_{2,k}) \right|^2$$

So conditional PEP is given as

$$\begin{split} P\left(\mathfrak{s}\_{1}\rightarrow\mathfrak{s}\_{1}|\mathbf{H}\_{1}\right) &\leq Q\left(\sqrt{\sum\_{k,d\_{fuc}}\frac{1}{8N\_{0}}\left|h\_{1,k}\left(\mathbf{x}\_{1,k}-\hat{\mathbf{x}}\_{1,k}\right)+h\_{2,k}\left(\mathbf{x}\_{2,k}-\hat{\mathbf{x}}\_{2,k}\right)\right|^{2}}\right) \\ &= Q\left(\sqrt{\sum\_{k,d\_{fuc}}\frac{1}{8N\_{0}}\left|\mathbf{h}\_{k}^{T}\left(\mathbf{x}\_{k}-\hat{\mathbf{x}}\_{k}\right)\right|^{2}}\right) \end{split} \tag{43}$$

where **h***<sup>k</sup>* = � *h*∗ 11,*<sup>k</sup> qh*<sup>∗</sup> 21,*<sup>k</sup> <sup>h</sup>*<sup>∗</sup> 11,*<sup>k</sup>* <sup>−</sup> *qh*<sup>∗</sup> 21,*k* �*T* , **x***<sup>k</sup>* = � *x*1,*<sup>k</sup> x*1,*<sup>k</sup> x*2,*<sup>k</sup> x*2,*<sup>k</sup>* �*<sup>T</sup>* and **<sup>x</sup>**<sup>ˆ</sup> *<sup>k</sup>* <sup>=</sup> � *x*ˆ1,*<sup>k</sup> x*ˆ1,*<sup>k</sup> x*ˆ2,*<sup>k</sup> x*ˆ2,*<sup>k</sup>* �*T*. We assume channel to be slow fading, i.e. the channel remains constant for the duration of one codeword. So the PEP can be written as

$$\begin{split}P\left(\mathfrak{s}\_{1}\rightarrow\hat{\mathfrak{s}}\_{1}|\overline{\mathbf{H}}\_{1}\right)\leq&Q\left(\sqrt{\sum\_{k,d\_{fuc}}\frac{1}{8N\_{0}}\left|\mathbf{h}^{T}\left(\mathbf{x}\_{k}-\hat{\mathbf{x}}\_{k}\right)\right|^{2}}\right)\\ =&Q\left(\sqrt{\frac{1}{8N\_{0}}\mathbf{h}^{\dagger}\mathbf{A}\mathbf{A}^{\dagger}\mathbf{h}}\right)\end{split}\tag{44}$$

10.5772/57134

81

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






*<sup>T</sup>* where *<sup>θ</sup>* ∈ {0, ±90◦, 180◦}. This

*<sup>T</sup>*, where *θ* = 2*πl*/12, *l* = 0, . . . , 11. These two

j

2j

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



(a) (b)

precoder entry for the first antenna while cross indicates the precoder entry for the second antenna.

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

design of LTE precoders to offset this diversity loss.

precoders turn out to be [1 2 exp(*jθ*)]

EGT, i.e. precoder is given as [1 exp(*jθ*)]

different codebook options have been illustrated in Fig. 4.

different subbands) can be bundled to optimize the feedback rate.

**Figure 4.** Two options of increasing the precoder codebook size. Fig.(a) corresponds to the option of increased angular resolution of LTE precoders while Fig.(b) corresponds to the option of enhanced levels of transmission. Square indicates the

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

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

or [2 exp(*jθ*) 1]

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

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

*T*

j

where ∆∆† is a 4 <sup>×</sup> 4 matrix while <sup>∆</sup>4×*df ree* <sup>=</sup> � **<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* � . Using Chernoff bound, (44) is upper bounded by

$$P\left(\mathbf{c}\_{1}\rightarrow\mathbf{\hat{c}}\_{1}|\mathbf{h}\right)\leq\frac{1}{2}\exp\left(-\frac{1}{16N\_{0}}\mathbf{h}^{\dagger}\Delta\mathbf{A}^{\dagger}\mathbf{h}\right)\tag{45}$$

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

$$E\left[\mathbf{h}\mathbf{h}^{\dagger}\right] = \mathbf{R} = \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & -1 \\ 1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{bmatrix} \tag{46}$$

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

$$E\_{\mathbf{h}}\left[\frac{1}{2}\exp\left(-\frac{1}{16N\_0}\mathbf{h}^{\dagger}\Delta\mathbf{A}^{\dagger}\mathbf{h}\right)\right] \leq \frac{1}{2\det\left(\mathbf{I} + \frac{1}{16N\_0}\mathbf{R}\Delta\mathbf{A}^{\dagger}\right)}\tag{47}$$

Note that the minimizations in (42) ensure that in ∆, *x*ˆ ′ 1,*<sup>k</sup>* <sup>−</sup> *<sup>x</sup>* ′ 1,*<sup>k</sup>* is always non-zero where *x*ˆ ′ 2,*<sup>k</sup>* <sup>−</sup> *<sup>x</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 first two rows with non-zero elements. For the high SNR approximation, we get

$$P\left(\mathfrak{g}\_1 \to \hat{\mathfrak{g}}\_1\right) \le \frac{1}{2} \left(\frac{16N\_0}{\sigma^2}\right)^r \prod\_{k=1}^r \frac{1}{\mu\_k} \tag{48}$$

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 these factors will not affect the diversity order.

**Figure 4.** Two options of increasing the precoder codebook size. Fig.(a) corresponds to the option of increased angular resolution of LTE precoders while Fig.(b) corresponds to the option of enhanced levels of transmission. Square indicates the precoder entry for the first antenna while cross indicates the precoder entry for the second antenna.
