**3. SU-MMIMO using PCTC in correlated channel**

#### **3.1 System model**

The block diagram of the system is identical to **Figure 2** in [36] and the received signal is given by (4). Note that in (4), the channel autocorrelation matrix is given by

$$\mathbf{R}\_{\tilde{\mathbf{H}}\tilde{\mathbf{H}}} = \frac{1}{2} E \left[ \tilde{\mathbf{H}}\_k^H \tilde{\mathbf{H}}\_k \right] = N\_r \mathbf{I}\_{N\_t} \tag{38}$$

**Figure 5.** *Simulation results for Ntot = 32.*

**Figure 6.** *Simulation results for Ntot = 2, Nt = 1.*

where the superscript "*H*" denotes Hermitian and **I***Nt* denotes the *Nt Nt* identity matrix. In this section, we investigate the situation where **RH**<sup>~</sup>**H**<sup>~</sup> is not an identity matrix, but is a valid autocorrelation matrix [40]. As mentioned in [36], the elements of **H**~ *<sup>k</sup>* – given by *H*~ *<sup>k</sup>*,*i*,*<sup>j</sup>* for the *kth* re-transmission, *i th* row, *j th* column of **H**~ *<sup>k</sup>* – are

*New Results on Single User Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112469*

zero-mean, complex Gaussian random variables with variance per dimension equal to *σ*2 *<sup>H</sup>*. The in-phase and quadrature components of *<sup>H</sup>*<sup>~</sup> *<sup>k</sup>*,*i*,*<sup>j</sup>* – denoted by *Hk*,*i*,*j*,*<sup>I</sup>* and *Hk*,*i*,*j*,*<sup>Q</sup>* respectively – are statistically independent. Moreover, we assume that the rows of **H**~ *<sup>k</sup>* are statistically independent. Following the procedure in [36] for the case without precoding, we now find the expression for the average SINR per bit before and after averaging over re-transmissions (*k*). All symbols and notations have the usual meaning, as given in [36].

#### **3.2 SINR analysis**

The *i th* element of **H**~ *<sup>H</sup> <sup>k</sup>* **<sup>R</sup>**<sup>~</sup> *<sup>k</sup>* is given by (25) of [36] which is repeated here for convenience

$$
\tilde{Y}\_{k,i} = \tilde{F}\_{k,i,i} \mathbf{S}\_i + \tilde{I}\_{k,i} + \tilde{V}\_{k,i} \quad \text{for } 1 \le i \le N\_t \tag{39}
$$

where

$$
\tilde{\mathbf{V}}\_{k,i} = \sum\_{j=1}^{N\_r} \tilde{H}\_{k,j,i}^\* \tilde{\mathbf{W}}\_{k,j}; \tilde{\mathbf{I}}\_{k,i} = \sum\_{j=1 \atop j \neq i}^{N\_t} \tilde{F}\_{k,i,j} \mathbf{S}\_j; \tilde{F}\_{k,i,j} = \sum\_{l=1}^{N\_r} \tilde{H}\_{k,l,i}^\* \tilde{H}\_{k,l,i}. \tag{40}
$$

We have

$$\begin{split} E\left[\tilde{\boldsymbol{F}}\_{k,i,i}^{2}\right] &= E\left[\sum\_{l=1}^{N\_r} \left|\tilde{\boldsymbol{H}}\_{k,l,i}\right|^2 \sum\_{m=1}^{N\_r} \left|\tilde{\boldsymbol{H}}\_{k,m,i}\right|^2\right] \\ &= E\left[\sum\_{l=1}^{N\_r} \left(H\_{k,l,i,l}^2 + H\_{k,l,i,Q}^2\right) \sum\_{m=1}^{N\_r} \left(H\_{k,m,i,l}^2 + H\_{k,m,i,Q}^2\right)\right] = 4\sigma\_H^4 N\_r (N\_r + 1) \end{split} \tag{41}$$

which is identical to (27) in [36] and we have used the following properties


The interference power is

$$E\left[\left|\tilde{I}\_{k,i}\right|^2\right] = E\left[\sum\_{\substack{j=1\\j\neq i}}^{N\_t} \tilde{F}\_{k,i,j} \mathbf{S}\_j \sum\_{l=1}^{N\_t} \tilde{F}\_{k,i,l}^\* \mathbf{S}\_l^\*\right] \\ = \sum\_{\substack{j=1\\j\neq i}}^{N\_t} \sum\_{l=1}^{N\_t} E\left[\tilde{F}\_{k,i,j} \tilde{F}\_{k,i,l}^\*\right] E\left[\mathbf{S}\_j \mathbf{S}\_l^\*\right] \\ = P\_{\text{av}} \sum\_{\substack{j=1\\j\neq i}}^{N\_t} E\left[\left|\tilde{F}\_{k,i,j}\right|^2\right]. \tag{42}$$

where we have used (9) in. Similarly the noise power is

$$\begin{split} E\left[\left|\bar{V}\_{k,i}\right|^2\right] &= E\left[\sum\_{j=1}^{N\_r} \bar{H}\_{k,j,i}^\* \bar{\mathcal{W}}\_{k,j} \sum\_{m=1}^{N\_r} \bar{H}\_{k,m,i} \bar{\mathcal{W}}\_{k,m}^\* \right] \\ &= \sum\_{j=1}^{N\_r} \sum\_{m=1}^{N\_r} E\left[\bar{H}\_{k,j,i}^\* \bar{H}\_{k,m,i}\right] E\left[\bar{\mathcal{W}}\_{k,m}^\* \bar{\mathcal{W}}\_{k,j}\right] \\ &= \sum\_{j=1}^{N\_r} \sum\_{m=1}^{N\_r} 2\sigma\_H^2 \delta\_K(j-m) 2\sigma\_W^2(j-m) = 4N\_r \sigma\_H^2 \sigma\_W^2 \end{split} \tag{43}$$

which is identical to (29) in [36] and we have used the following properties:

1.Rows of **H**~ *<sup>k</sup>* are independent.

2. Sifting property of the Kronecker delta function.

3.Noise and channel coefficients are independent.

Now in (42)

$$\begin{aligned} E\left[\left|\bar{F}\_{k,j}\right|^2\right] &= E\left[\sum\_{l=1}^{N\_r} \bar{H}\_{k,l,i}^\* \bar{H}\_{k,l,j} \sum\_{m=1}^{N\_r} \bar{H}\_{k,m,i} \bar{H}\_{k,m,j}^\*\right] \\ &= \sum\_{l=1}^{N\_r} E\left[\bar{H}\_{k,l,i}^\* \bar{H}\_{k,l,j} \left(\bar{H}\_{k,l,i} \bar{H}\_{k,l,j}^\* + \sum\_{m=1}^{N\_r} \bar{H}\_{k,m,i} \bar{H}\_{k,m,j}^\*\right)\right] \\ &= \sum\_{l=1}^{N\_r} E\left[\left|\bar{H}\_{k,l,i}\right|^2 \left|\bar{H}\_{k,l,j}\right|^2 + \left(\sum\_{\substack{m=1\\m\neq l}}^{N\_r} \bar{H}\_{k,l,i}^\* \bar{H}\_{k,l,j} \bar{H}\_{k,m,i} \bar{H}\_{k,m,j}^\*\right)\right]. \end{aligned}$$

Now the first summation in (44) is equal to

$$E\_1 = E\left[\left|\bar{H}\_{k,l,i}\right|^2 \left|\bar{H}\_{k,l,j}\right|^2\right] = E\left[\left(H\_{k,l,i,l}^2 + H\_{k,l,i,Q}^2\right)\left(H\_{k,l,j,I}^2 + H\_{k,l,j,Q}^2\right)\right] = 4\sigma\_H^4 + 4R\_{\hat{H}\hat{H},j-\hat{H}}^2\tag{45}$$

where we have used the property that for real-valued, zero-mean Gaussian random variables *Xi*, 1≤*i* ≤4 [44, 45]

$$E[X\_1X\_2X\_3X\_4] = C\_{12}C\_{34} + C\_{13}C\_{24} + C\_{14}C\_{23} \tag{46}$$

where

*New Results on Single User Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112469*

$$\mathbf{C}\_{\vec{\eta}} = E\left[\mathbf{X}\_i \mathbf{X}\_j\right] \qquad \text{for } \mathbf{1} \le i, j \le 4 \tag{47}$$

and

$$R\_{\hat{H}\hat{H}j-i} = E\left[H\_{k,l,i,l}H\_{k,l,j,l}\right] = E\left[H\_{k,l,i,Q}H\_{k,l,j,Q}\right] = \frac{1}{2}E\left[\tilde{H}\_{k,l,i}^\*\tilde{H}\_{k,l,j}\right] = R\_{\hat{H}\hat{H},i-j} \tag{48}$$

is the real-valued autocorrelation of *H*~ *<sup>k</sup>*,*m*,*<sup>n</sup>* and we have made the assumption that the in-phase and quadrature components of *H*~ *<sup>k</sup>*,*m*,*<sup>n</sup>* are independent. The second summation in (44) can be written as

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>X</sup> *Nr m*¼1 *m*6¼*l E H*~ <sup>∗</sup> *k*,*l*,*i <sup>H</sup>*<sup>~</sup> *<sup>k</sup>*,*l*,*jH*<sup>~</sup> *<sup>k</sup>*,*m*,*iH*<sup>~</sup> <sup>∗</sup> *k*,*m*,*j* h i <sup>¼</sup> <sup>X</sup> *Nr m*¼1 *m*6¼*l E H*~ <sup>∗</sup> *k*,*l*,*i H*~ *<sup>k</sup>*,*l*,*<sup>j</sup>* h i*<sup>E</sup> <sup>H</sup>*<sup>~</sup> *<sup>k</sup>*,*m*,*iH*<sup>~</sup> <sup>∗</sup> *k*,*m*,*j* h i <sup>¼</sup> <sup>X</sup> *Nr m*¼1 *m*6¼*l* 4*R*<sup>2</sup> *<sup>H</sup>*~*H*<sup>~</sup> ,*j*�*<sup>i</sup>* <sup>¼</sup> <sup>4</sup>ð Þ *Nr* � <sup>1</sup> *<sup>R</sup>*<sup>2</sup> *<sup>H</sup>*~*H*<sup>~</sup> ,*j*�*<sup>i</sup>* (49)

where we have used the property that the rows of **H**~ *<sup>k</sup>* are independent. Therefore (44) becomes

$$E\left[\left|\check{F}\_{k,i,j}\right|^2\right] = N\_r(E\_1 + E\_2) = 4N\_r\left[\sigma\_H^4 + R\_{\check{H}\check{H}j-i}^2 + (N\_r - 1)R\_{\check{H}\check{H}j-i}^2\right] = 4N\_r\left[\sigma\_H^4 + N\_r R\_{\check{H}\check{H}j-i}^2\right].\tag{50}$$

The total power of interference plus noise is

$$E\left[\left|\bar{I}\_{k,i} + \bar{V}\_{k,i}\right|^2\right] = E\left[\left|\bar{I}\_{k,i}\right|^2\right] + E\left[\left|\bar{V}\_{k,i}\right|^2\right] = 4\mathcal{P}\_{\text{av}}N\_r\sum\_{j=1}^{N\_t} \left[\sigma\_H^4 + N\_r\mathcal{R}\_{HH,j-i}^2\right] + 4\mathcal{N}\_r\sigma\_H^2\sigma\_W^2\tag{51}$$

$$j \neq i \tag{51}$$

where we have made the assumption that noise and symbols are independent. The average SINR per bit for the *i th* transmit antenna is similar to (31) of [36] which is repeated here for convenience

$$\text{SINR}\_{\text{av},b,i} = \frac{E\left[\left|\tilde{F}\_{k,i,i}\mathbb{S}\_{i}\right|^{2}\right] \times 2N\_{rt}}{E\left[\left|\tilde{I}\_{k,i} + \tilde{V}\_{k,i}\right|^{2}\right]} \qquad \text{for } 1 \le i \le N\_{t} \tag{52}$$

into which (41) and (51) have to be substituted. The upper bound on the average SINR per bit for the *i th* transmit antenna is obtained by setting *σ*<sup>2</sup> *<sup>W</sup>* ¼ 0 in (51), (52) and is given by, for 1≤ *i*≤ *Nt*

$$\text{SINR}\_{\text{av},b,\text{UB},i} = \frac{\sigma\_H^4 (\mathbf{1} + \mathbf{N}\_r) \times \mathbf{2N}\_{rt}}{\sum\_{\substack{j=1\\j\neq i}}^{N\_t} \left[ \sigma\_H^4 + N\_r \mathbf{R}\_{\hat{H}\hat{H},j-i}^2 \right]}. \tag{53}$$

Observe that in contrast to (31) and (32) in [36], the average SINR per bit and its upper bound depend on the transmit antenna. Let us now compute the average SINR per bit after averaging over retransmissions. The received signal after averaging over retransmissions is given by (6) with (see also (20) of [36])

$$F\_i = \frac{1}{N\_{rt}} \sum\_{k=0}^{N\_{rt}-1} \tilde{F}\_{k,i,i}$$

$$\tilde{U} = \frac{1}{N\_{rt}} \sum\_{k=0}^{N\_{rt}-1} (\tilde{I}\_{k,i} + \tilde{V}\_{k,i}) = \frac{1}{N\_{rt}} \sum\_{k=0}^{N\_{rt}-1} \tilde{U}\_{k,i} \,' \qquad \text{(say)} \tag{54}$$

where *F*~*<sup>k</sup>*,*i*,*<sup>i</sup>*, ~*Ik*,*<sup>i</sup>* and *V*~ *<sup>k</sup>*,*<sup>i</sup>* are given in (39). The power of the signal component of (6) is

$$E\left[|\mathcal{S}\_{i}|^{2}\bar{F}\_{i}^{2}\right] = P\_{\rm av}E\left[F\_{i}^{2}\right] = \frac{P\_{\rm av}}{N\_{\rm rr}^{2}}E\left[\sum\_{k=0}^{N\_{\rm r}-1}\bar{F}\_{k,i,i}\sum\_{l=0}^{N\_{\rm r}-1}\bar{F}\_{l,i,i}\right]$$

$$= \frac{P\_{\rm av}}{N\_{\rm r}^{2}}\sum\_{k=0}^{N\_{\rm r}-1}\left[\sum\_{l=0}^{N\_{\rm r}-1}E\left[\bar{F}\_{k,i,i}\right]E\left[\bar{F}\_{l,i,i}\right] + E\left[\left|\bar{F}\_{k,i,i}\right|^{2}\right]\right] \tag{55}$$

where we have used the fact that the channel is independent across retransmissions, therefore

$$E\left[\tilde{F}\_{k,i,i}\tilde{F}\_{l,i,i}\right] = E\left[\tilde{F}\_{k,i,i}\right]E\left[\tilde{F}\_{l,i,i}\right] \qquad \text{for } k \neq l. \tag{56}$$

Now

$$E\left[\tilde{F}\_{k,i,i}\right] = E\left[\sum\_{l=1}^{N\_r} \left|\tilde{H}\_{k,l,i}\right|^2\right] = 2N\_r\sigma\_H^2. \tag{57}$$

Substituting (41) and (57) in (55) we get

$$E\left[\left|\mathbf{S}\_{i}\right|^{2}F\_{i}^{2}\right] = \frac{4N\_{r}P\_{\text{av}}\sigma\_{H}^{4}}{N\_{\text{rt}}}(\mathbf{1}+N\_{r}N\_{\text{rt}}).\tag{58}$$

The power of the interference component in (6) and (54) is

$$E\left[\left|\tilde{U}\_{i}\right|^{2}\right] = \frac{1}{N\_{rt}^{2}}E\left[\sum\_{k=0}^{N\_{\pi}-1} (\check{I}\_{k,i} + \check{V}\_{k,i})\sum\_{l=0}^{N\_{\pi}-1} \left(\check{I}\_{l,i}^{\*} + \check{V}\_{l,i}^{\*}\right)\right] = \frac{1}{N\_{rt}^{2}}\sum\_{k=0}^{N\_{\pi}-1}\sum\_{l=0}^{N\_{\pi}-1} E\left[\check{I}\_{k,i}\check{I}\_{l,i}^{\*}\right] + E\left[\check{V}\_{k,i}\check{V}\_{l,i}^{\*}\right] \tag{59}$$

where we have used the following properties from (40)

$$E\left[\tilde{I}\_{k,i}\right] = E\left[\tilde{V}\_{k,i}\right] = \mathbf{0}; \mathbf{E}\left[\tilde{I}\_{k,i}\tilde{V}\_{l,i}^\*\right] = \mathbf{E}\left[\tilde{V}\_{k,i}\tilde{I}\_{l,i}^\*\right] = \mathbf{0} \qquad \text{for all } k, l \tag{60}$$

since *Sj* and *W*~ *<sup>k</sup>*,*<sup>j</sup>* are mutually independent with zero-mean. Now

$$\begin{split} E\left[\bar{\mathbf{I}}\_{k,i}\bar{I}\_{l,j}^{\*}\right] &= E\left[\sum\_{j=1}^{N\_{t}} \bar{F}\_{k,i,j}\mathbf{S}\_{l}^{\*} \sum\_{n=1}^{N\_{t}} \tilde{F}\_{l,i,n}^{\*}\mathbf{S}\_{n}^{\*}\right] = \sum\_{j=1}^{N\_{t}} \sum\_{n=1}^{N\_{t}} E\left[\bar{F}\_{k,i,j}\tilde{F}\_{l,i,n}^{\*}\right] E\left[\mathbf{S}\_{j}\mathbf{S}\_{n}^{\*}\right] \\ &= \sum\_{j=1}^{N\_{t}} \sum\_{n=1}^{N\_{t}} E\left[\bar{F}\_{k,i,j}\tilde{F}\_{l,i,n}^{\*}\right] \mathbf{P}\_{\text{av}}\delta\_{K}(j-n) \\ &= \mathbf{P}\_{\text{av}} \sum\_{j=1}^{n \neq i} \mathbf{E}\left[\bar{F}\_{k,i,j}\tilde{F}\_{l,i,j}^{\*}\right] \\ &= \mathbf{P}\_{\text{av}} \sum\_{j=1}^{N\_{t}} \mathbf{E}\left[\bar{F}\_{k,i,j}\tilde{F}\_{l,i,j}^{\*}\right] \end{split} \tag{61}$$

where we have used the property that the symbols are uncorrelated and *δK*ð Þ� is the Kronecker delta function [40]. When *k* ¼ *l*, (61) is given by (42) and (50). When *k* 6¼ *l*, (61) is given by

$$E\left[\tilde{I}\_{k,i}\tilde{I}\_{l,i}^{\*}\right] = P\_{\rm av} \sum\_{j=1 \atop j \neq i}^{N\_t} E\left[\tilde{F}\_{k,i,j}\right] E\left[\tilde{F}\_{l,i,j}^{\*}\right] = P\_{\rm av} \sum\_{j=1 \atop j \neq i}^{N\_t} 4N\_r^2 R\_{\tilde{H}\tilde{H}, j-i}^2 \tag{62}$$

where we have used (40) and (48). Similarly, we have

$$E\left[\tilde{\mathcal{V}}\_{k,i}\tilde{\mathcal{V}}\_{l,i}^\*\right] = 4\mathcal{N}\_r\sigma\_H^2\sigma\_W^2\delta\_K(k-l) \tag{63}$$

where we have used (43). Substituting (42), (50), (62) and (63) in (59) we get

$$\begin{split} E\left[\left|\bar{U}\_{i}\right|^{2}\right] &= \frac{1}{N\_{\text{rt}}^{2}} \left[ 4P\_{\text{av}}N\_{\text{r}}N\_{\text{rt}} \sum\_{j=1}^{N\_{\text{r}}} \left(\sigma\_{H}^{4} + N\_{\text{r}}R\_{\hat{H}\hat{H}j-i}^{2}\right) + 4P\_{\text{av}}N\_{\text{r}}^{2}N\_{\text{rt}}(N\_{\text{rt}}-1) \sum\_{j=1}^{N\_{\text{r}}} R\_{\hat{H}\hat{H}j-i}^{2} \right] \\ &+ \frac{4N\_{\text{r}}}{N\_{\text{rt}}} \sigma\_{H}^{2}\sigma\_{W}^{2} \\ &= \frac{1}{N\_{\text{rt}}} \left[ 4P\_{\text{av}}N\_{\text{r}}\sum\_{j=1}^{N\_{\text{r}}} \left(\sigma\_{H}^{4} + N\_{\text{r}}R\_{\hat{H}\hat{H}j-i}^{2}\right) + 4P\_{\text{av}}N\_{\text{r}}^{2}(N\_{\text{rt}}-1) \sum\_{j=1}^{N\_{\text{r}}} R\_{\hat{H}\hat{H}j-i}^{2} \right] \\ &+ \frac{4N\_{\text{r}}}{N\_{\text{rt}}} \sigma\_{H}^{2}\sigma\_{W}^{2}. \end{split}$$

The average SINR per bit for the *i th* transmit antenna, after averaging over retransmissions (also referred to as "combining" [36]) is given by

$$\text{SINR}\_{\text{av},b,C,i} = \frac{2P\_{\text{av}}E\left[F\_i^2\right]}{E\left[\left|\bar{U}\_i\right|^2\right]}\tag{65}$$

into which (58) and (64) have to be substituted. The upper bound on the average SINR per bit after "combining" for the *i th* transmit antenna is given by

$$\text{SINR}\_{\text{av},b,C,\text{UB},i} = \text{SINR}\_{\text{av},b,C,i}|\_{\sigma\_W^2 = 0}. \tag{66}$$

The plots of the average SINR per bit for the *i th* transmit antenna before and after "combining" are shown in **Figures 7** and **8** respectively for *N*tot ¼ 1024 and *Nrt* ¼ 2. The channel correlation is given by

$$R\_{\hat{H}\hat{H}\_{\hat{y}}-i} = \mathbf{0}.\mathcal{Y}^{|\boldsymbol{j}-\boldsymbol{i}|}\sigma\_{\boldsymbol{H}}^{2} \tag{67}$$

in (48), which is obtained by passing samples of white Gaussian noise through a unit-energy, first-order infinite impulse response (IIR) lowpass filter with *a* ¼ �0*:*9 (see (30) of [46]).

We observe in **Figures 7** and **8** that

**Figure 7.** *Plot of SINRav*,b,*UB*,i *for* N*tot = 1024,* Nrt *= 2. (a) Back view. (b) Sideview. (c) Front view.*

**Figure 8.** *Plot of SINRav*,b,C,*UB*,i *for* N*tot = 1024,* Nrt *= 2. (a) Back view. (b) Side view. (c) Front view.*

The upper bound on the average SINR per bit decreases rapidly with increasing transmit antennas *Nt* and falls below 0 dB for *Nt* >5 (see **Figures 7(b)** and **8(b)**). Since the spectral efficiency of the system is *Nt=*ð Þ 2*Nrt* bits/sec/Hz (see (33) of [36]), the system would be of no practical use, since the BER would be close to 0.5 for *Nt* > 5.

The upper bound on the average SINR per bit after "combining" is *less* than that before "combining". Therefore retransmissions are ineffective.

In view of the above observation, it becomes necessary to design a better receiver using precoding. This is presented in the next section.

#### **3.3 Precoding**

Similar to (4) consider the modified received signal given by

$$
\tilde{\mathbf{R}}\_k = \tilde{\mathbf{H}}\_k \tilde{\mathbf{B}} \mathbf{S} + \tilde{\mathbf{W}}\_k \tag{68}
$$

where

$$
\tilde{\mathbf{B}} = \begin{bmatrix}
\mathbf{1} & \mathbf{0} & \cdots & \mathbf{0} \\
& \tilde{a}\_{1,1} & \mathbf{1} & \cdots & \mathbf{0} \\
& \vdots & \cdots & \cdots & \vdots \\
\tilde{a}\_{N\_t-1, N\_t-1} & \cdots & \tilde{a}\_{N\_t-1,1} & \mathbf{1}
\end{bmatrix}^T \triangleq \tilde{\mathbf{A}}^T \tag{69}
$$

where ð Þ� *<sup>T</sup>* denotes transpose. In (69), **<sup>A</sup>**<sup>~</sup> is an *Nt* � *Nt* lower triangular matrix with diagonal elements equal to unity and *a*~*<sup>i</sup>*,*<sup>j</sup>* denotes the *j th* coefficient of the optimum *i th*order forward prediction filter [40] and **B**~ is the precoding matrix. Let

*MIMO Communications – Fundamental Theory, Propagation Channels, and Antenna Systems*

$$
\tilde{\mathbf{Y}}\_k = \tilde{\mathbf{B}}^H \tilde{\mathbf{H}}\_k^H \tilde{\mathbf{R}}\_k = \tilde{\mathbf{B}}^H \tilde{\mathbf{H}}\_k^H \tilde{\mathbf{H}}\_k \tilde{\mathbf{B}} \mathbf{S} + \tilde{\mathbf{B}}^H \tilde{\mathbf{H}}\_k^H \tilde{\mathbf{W}}\_k. \tag{70}
$$

Define

$$
\tilde{\mathbf{Z}}\_k = \mathbf{\tilde{H}}\_k \tilde{\mathbf{B}} = \begin{bmatrix}
\tilde{Z}\_{k,1,1} & \cdots & \tilde{Z}\_{k,1,N\_t} \\
\vdots & \cdots & \vdots \\
\tilde{Z}\_{k,N\_r,1} & \cdots & \tilde{Z}\_{k,N\_r,N\_t}
\end{bmatrix}.
\tag{71}
$$

Now [40]

$$\frac{1}{2}E\left[\bar{\mathbf{Z}}\_k^H \bar{\mathbf{Z}}\_k\right] = N\_r \begin{bmatrix} \sigma\_{Z,1}^2 & 0 & \cdots & 0\\ 0 & \sigma\_{Z,2}^2 & \cdots & 0\\ \vdots & \cdots & \cdots & \vdots\\ 0 & \cdots & 0 & \sigma\_{Z,N\_t}^2 \end{bmatrix} \triangleq \bar{\mathbf{R}}\_{22} \tag{72}$$

is an *Nt* � *Nt* diagonal matrix and *<sup>σ</sup>*<sup>2</sup> *<sup>Z</sup>*,*<sup>i</sup>* denotes the variance per dimension of the optimum ð Þ *<sup>i</sup>* � <sup>1</sup> *th*-order forward prediction filter. Note that [40]

$$
\sigma\_{Z,1}^2 = \sigma\_H^2; \sigma\_{Z,i}^2 \ge \sigma\_{Z,j}^2 \qquad \text{for } i < j. \tag{73}
$$

Let

$$
\tilde{\mathbf{V}}\_k = \tilde{\mathbf{Z}}\_k^H \tilde{\mathbf{W}}\_k = \begin{bmatrix} \mathbb{V}\_{k,1} & \cdots & \mathbb{V}\_{k, \mathbb{V}\_t} \end{bmatrix}^T \tag{74}
$$

which is an *Nt* � 1 vector. Now

$$E\left[\bar{\boldsymbol{V}}\_{k,i}\bar{\boldsymbol{V}}\_{k,m}^{\*}\right] = E\left[\sum\_{j=1}^{N\_r} \bar{\boldsymbol{Z}}\_{k,j,i}^{\*}\bar{\boldsymbol{W}}\_{k,j}\sum\_{l=1}^{N\_r} \bar{\boldsymbol{Z}}\_{k,l,m}\bar{\boldsymbol{W}}\_{k,l}^{\*}\right] = \sum\_{j=1}^{N\_r} \sum\_{l=1}^{N\_r} E\left[\bar{\boldsymbol{Z}}\_{k,l,m}\bar{\boldsymbol{Z}}\_{k,j,i}^{\*}\right] E\left[\bar{\boldsymbol{W}}\_{k,j}\bar{\boldsymbol{W}}\_{k,l}^{\*}\right]$$

$$= \sum\_{j=1}^{N\_r} \sum\_{l=1}^{N\_r} 2\sigma\_{\mathbf{Z},i}^{2}\delta\_{\mathbf{K}}(i-m)\delta\_{\mathbf{K}}(j-l) \times 2\sigma\_{\mathbf{W}}^{2}\delta\_{\mathbf{K}}(j-l) = 4N\_r\sigma\_{\mathbf{Z},i}^{2}\sigma\_{\mathbf{W}}^{2}\delta\_{\mathbf{K}}(i-m) \tag{75}$$

where we have used (72). Let

$$
\tilde{\mathbf{F}}\_k = \tilde{\mathbf{Z}}\_k^H \tilde{\mathbf{Z}}\_k \tag{76}
$$

which is an *Nt* � *Nt* matrix. Substituting (76) in (70) we get

$$
\tilde{\mathbf{Y}}\_k = \tilde{\mathbf{F}}\_k \mathbf{S} + \tilde{\mathbf{V}}\_k. \tag{77}
$$

Similar to (39), the *i th* element of **Y**~ *<sup>k</sup>* in (77) is given by

$$
\tilde{y}\_{k,i} = \tilde{F}\_{k,i,i} \mathbf{S}\_i + \tilde{I}\_{k,i} + \tilde{V}\_{k,i} \text{for } \mathbf{1} \le i \le N\_t \tag{78}
$$

where

*New Results on Single User Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112469*

$$
\tilde{\boldsymbol{V}}\_{k,i} = \sum\_{j=1}^{N\_r} \tilde{\boldsymbol{Z}}\_{k,j,i}^\* \tilde{\boldsymbol{W}}\_{k,j}; \tilde{\boldsymbol{I}}\_{k,i} = \sum\_{\substack{j=1\\j\neq i}}^{N\_t} \tilde{\boldsymbol{F}}\_{k,i,j} \mathbf{S}\_j; \tilde{\boldsymbol{F}}\_{k,i,j} \sum\_{l=1}^{N\_r} \tilde{\boldsymbol{Z}}\_{k,l,i}^\* \tilde{\boldsymbol{Z}}\_{k,l,j}.\tag{79}
$$

Note that from (72) and (76) we have

$$E\left[\tilde{F}\_{k,i,i}\right] = 2N\_r \sigma\_{Z,i}^2\tag{80}$$

Now

$$\begin{split} E\left[\bar{\tilde{F}}\_{k,i,i}^{2}\right] &= E\left[\sum\_{l=1}^{N\_r} \left|\tilde{Z}\_{k,l,i}\right|^2 \sum\_{m=1}^{N\_r} \left|\tilde{Z}\_{k,m,i}\right|^2\right] = \sum\_{l=1}^{N\_r} \left|\tilde{Z}\_{k,l,i}\right|^4 + \\ &\quad + \sum\_{m=1}^{N\_r} E\left[\left|\tilde{Z}\_{k,l,i}\right|^2\right] E\left[\left|\tilde{Z}\_{k,m,i}\right|^2\right] = 4N\_r(N\_r+1)\sigma\_{Z,i}^2. \end{split} \tag{81}$$

Similarly

$$E\left[\left|\tilde{I}\_{k,i}\right|^2\right] = E\left[\sum\_{\substack{j=1\\j\neq i}}^{N\_t} \tilde{F}\_{k,i,j} \mathbf{S}\_j \sum\_{l=1\\j\neq i}^{N\_t} \tilde{F}\_{k,i,l}^\* \mathbf{S}\_l^\*\right] = P\_{\text{av}} \sum\_{\substack{j=1\\j\neq i}}^{N\_t} E\left[\left|\tilde{F}\_{k,i,j}\right|^2\right].\tag{82}$$

Now

$$E\left[\left|\bar{F}\_{k,i,j}\right|^2\right] = E\left[\sum\_{l=1}^{N\_r} \bar{Z}\_{k,l,i}^\* \bar{Z}\_{k,l,j} \sum\_{m=1}^{N\_r} \bar{Z}\_{k,m,j} \bar{Z}\_{k,m,j}^\*\right] = \sum\_{l=1}^{N\_r} \sum\_{m=1}^{N\_r} 4\sigma\_{Z,i}^2 \sigma\_{Z,j}^2 \delta\_{\mathcal{K}}(l-m) = 4N\_r \sigma\_{Z,i}^2 \sigma\_{Z,j}^2\tag{83}$$

where we have used (72). Substituting (83) in (82) we get

$$E\left[\left|\tilde{I}\_{k,i}\right|^2\right] = 4P\_{\rm av}N\_r\sigma\_{Z,i}^2\sum\_{j=1 \atop j\neq i}^{N\_t}\sigma\_{Z,j}^2. \tag{84}$$

Note that

$$E\left[\left|\ddot{I}\_{k,i} + \ddot{\mathbf{V}}\_{k,i}\right|^2\right] = E\left[\left|\ddot{I}\_{k,i}\right|^2\right] + E\left[\left|\ddot{\mathbf{V}}\_{k,i}\right|^2\right].\tag{85}$$

The average SINR per bit for the *i th* transmit antenna is given by (52) and is equal to

$$\text{SINR}\_{\text{av},b,i} = \frac{E\left[\left|\bar{F}\_{k,i,i}S\_i\right|^2 \times 2N\_{rt}\right]}{E\left[\left|\bar{I}\_{k,i} + \bar{V}\_{k,i}\right|^2\right]} = \frac{P\_{\text{av}}(N\_r + 1)\,\sigma\_{Z,i}^2 \times 2N\_{rt}}{P\_{\text{av}}\sum\_{j=1 \atop j\neq i}^{N\_t} \sigma\_{Z,j}^2 + \sigma\_W^2} \tag{86}$$

where we have used (75), (81) and (84). The upper bound on the average SINR per bit for the *i th* transmit antenna is obtained by setting *σ*<sup>2</sup> *<sup>W</sup>* ¼ 0 in (86) and is equal to

$$\text{SINR}\_{\text{av},b,\text{UB},i} = \frac{(N\_r + 1)\sigma\_{Z,i}^2 \times 2N\_{rt}}{\sum\_{j=1 \atop j \neq i}^{N\_t} \sigma\_{Z,j}^2} \tag{87}$$

which is illustrated in **Figure 9** for *N*tot ¼ 1024 and *Nrt* ¼ 2. The value of the upper bound on the average SINR per bit for *Nt* ¼ *i* ¼ 50 is 18.6 dB. The channel correlation is given by (67). Note that a first-order prediction filter completely decorrelates the channel with [40]

$$
\tilde{a}\_{i,1} = -0.9 \text{ for } 1 \le i \le N\_t - 1; \\
\tilde{a}\_{i,j} = 0 \text{ for } 2 \le i \le N\_t - 1, 2 \le j \le i. \tag{88}
$$

We also have [40]

$$
\sigma\_{\mathbf{Z},i}^2 = \sigma\_{\mathbf{Z},2}^2 = \left(\mathbf{1} - \left|-\mathbf{0}.\mathbf{9}\right|^2\right) \sigma\_{\mathbf{Z},1}^2 = \mathbf{0}.\mathbf{19}\sigma\_{\mathbf{Z},1}^2 \text{ for } i > 2.\tag{89}
$$

Therefore we see in **Figure 9** that the first transmit antenna *i* ¼ 1 has a high SINRav,*<sup>b</sup>*,UB,*<sup>i</sup>* due to low interference power from remaining transmit antennas,

**Figure 9.** *Plot of SINRav*,b,*UB*,i *for* N*tot = 1024,* Nrt *= 2 after precoding. (a) Back view. (b) Sideview. (c) Front view.*

whereas for *i* 6¼ 1 the SINRav,*<sup>b</sup>*,UB,*<sup>i</sup>* is low due to high interference power from the first transmit antenna (*i* ¼ 1). The received signal after "combining" is given by (6) and (54). Note that from (54) and (79)

*E F*~<sup>2</sup> *i* h i <sup>¼</sup> <sup>1</sup> *N*2 *rt E N* X*rt*�<sup>1</sup> *k*¼0 *F*~*k*,*i*,*<sup>i</sup> N* X*rt*�<sup>1</sup> *l*¼0 *F*~*l*,*i*,*<sup>i</sup>* " # ¼ 1 *N*2 *rt N* X*rt*�<sup>1</sup> *k*¼0 *E F*~*k*,*i*,*<sup>i</sup>* � � � � <sup>2</sup> h i<sup>þ</sup> þ *N* X*rt*�<sup>1</sup> *l*¼0 *l*6¼*k E F*~*k*,*i*,*iF*~*l*,*i*,*<sup>i</sup>* � � <sup>¼</sup> <sup>4</sup>*Nrσ*<sup>2</sup> *Z*,*i N*2 *rt N* X*rt*�<sup>1</sup> *k*¼0 ð Þþ *Nr* þ 1 ð Þ *Nrt* � 1 *Nr* <sup>¼</sup> <sup>4</sup>*Nrσ*<sup>2</sup> *Z*,*i N*2 *rt* ð Þ 1 þ *NrNrt* (90)

where we have used (56), (80) and (81). Similarly from (54), (75), (84) and (85) we have

$$\begin{split} E\left[\bar{U}\_{i}^{2}\right] &= \frac{1}{N\_{\pi}^{2}} E\left[\sum\_{k=0}^{N\_{\pi}-1} \bar{U}\_{k,i}' \sum\_{l=0}^{N\_{\pi}-1} \left(\bar{U}\_{l,i}'\right)^{\*}\right] = \frac{1}{N\_{\pi}^{2}} \sum\_{k=0}^{N\_{\pi}-1} E\left[\bar{U}\_{k,i}' \left|\bar{U}\_{l,i}'\right|^{\*}\right] \\ &= \frac{1}{N\_{\pi}^{2}} \sum\_{k=0}^{N\_{\pi}-1} \sum\_{l=0}^{N\_{\pi}-1} E\left[\left|\bar{U}\_{k,i}'\right|^{2}\right] \delta\_{K}(k-l) \\\\ &= \frac{1}{N\_{\pi}} E\left[\left|\bar{U}\_{k,i}'\right|^{2}\right] = \frac{1}{N\_{\pi}} \left[E\left|\left|\bar{I}\_{k,i}\right|^{2}\right] + E\left[\left|\bar{V}\_{k,i}\right|^{2}\right]\right] = \frac{4N\_{\pi}\sigma\_{Z,i}^{2}}{N\_{\pi}} \left[P\_{\text{av}} \sum\_{\begin{subarray}{c} j=1\\ j\neq i \end{subarray}}^{N\_{\pi}} \sigma\_{Z,j}^{2} + \sigma\_{W}^{2}\right]. \end{split} \tag{91}$$

Substituting (90) and (91) in (65) we have, after simplification, for 1≤*i* ≤ *Nt*

$$\text{SINR}\_{\text{av},b,C,i} = \frac{2P\_{\text{av}}E\left[F\_i^2\right]}{E\left[\left|\tilde{U}\_i\right|^2\right]} = \frac{(N\_rN\_{rt}+1)\sigma\_{Z,i}^2 \times 2P\_{\text{av}}}{P\_{\text{av}}\sum\_{j=1 \atop j\neq i}^{N\_t} \sigma\_{Z,j}^2 + \sigma\_W^2}. \tag{92}$$

The upper bound on the average SINR per bit for the *i th* transmit antenna is obtained by substituting (92) in (66) and is equal to

$$\text{SINR}\_{\text{av},b,C,\text{UB},i} = \frac{(N\_r N\_{rt} + 1)\sigma\_{Z,i}^2 \times 2}{\sum\_{\substack{j=1\\j\neq i}}^{N\_t} \sigma\_{Z,j}^2} \approx \text{SINR}\_{\text{av},b,\text{UB},i} \tag{93}$$

#### **Figure 10.**

*Plot of SINRav,b,C,UB,i for Ntot = 1024, Nrt = 2 after precoding. (a) Back view. (b) Side view. (c) Front view.*

**Figure 11.** *Simulation results with precoding for Ntot = 1024.*

#### *New Results on Single User Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112469*

for 1≤*i*≤ *Nt*, *Nr* ≫ 1. This is illustrated in **Figure 10** for *N*tot ¼ 1024 and *Nrt* ¼ 2. We again observe that the first transmit antenna (*i* ¼ 1) has a high upper bound on the average SINR per bit, after "combining", compared to the remaining transmit antennas. The value of the upper bound on the average SINR per bit after "combining" for *Nt* <sup>¼</sup> *<sup>i</sup>* <sup>¼</sup> 50, *<sup>N</sup>*tot <sup>¼</sup> 1024 is 18.6 dB. After concatenation, *<sup>Y</sup>*~*<sup>i</sup>* for 0≤*i*≤ *Ld* � 1, in (6) and (54) is given to the turbo decoder [29, 40]. Let (see (26) of [29]):

$$\gamma\_{1,i,m,n} = \exp\left[-\frac{\left|\tilde{Y}\_i - F\_i \mathbf{S}\_{m,n}\right|^2}{2\sigma\_{U,i}^2}\right]; \gamma\_{2,i,m,n} = \exp\left[-\frac{\left|\tilde{Y}\_{i1} - F\_{i1} \mathbf{S}\_{m,n}\right|^2}{2\sigma\_{U,i}^2}\right] \tag{94}$$

$$\tilde{\mathbf{Y}}\_1 = \left[ \tilde{\mathbf{Y}}\_1 \cdots \tilde{\mathbf{Y}}\_{L\_{d1}-1} \right]; \tilde{\mathbf{Y}}\_2 = \left[ \tilde{\mathbf{Y}}\_{L\_{d1}} \cdots \tilde{\mathbf{Y}}\_{L\_{d}-1} \right]. \tag{95}$$

Thenwhere

$$i\mathbf{1} = i + L\_{d1} \qquad \text{for } 0 \le i \le L\_{d1} - \mathbf{1}.\tag{96}$$

The rest of the turbo decoding algorithm is similar to that discussed in [29, 40] will not be repeated here. In the next subsection we present the computer simulation results for correlated channel with precoding and PCTC.

**Figure 12.** *Simulation results with precoding for Ntot = 32.*

### **3.4 Simulation results**

The channel correlation is given by (67). The BER results for *N*tot ¼ 1024 with precoding are depicted in **Figure 11**. Incidentally, the value of the upper bound on the average SINR per bit before and after "combining" for *Nt* ¼ *i* ¼ 512, *N*tot ¼ 1024 is 6 dB. The BER results for *N*tot ¼ 32 with precoding are depicted in **Figure 12**. Note that since the average SINR per bit depends on the transmit antenna, the *minimum* average SINR per bit is indicated along the *x*-axis of **Figures 11** and **12**. We also observe from **Figures 11(a,b)** and **12** that there is a large difference between theory and simulations. This is probably because, the average SINR per bit is not identical for all transmit antennas. In particular, we observe from **Figures 9** and **10** that the first transmit antenna has a large average SINR per bit compared to the remaining antennas. However, in **Figure 11(c,d)** there is a close match between theory and simulations. This could be attributed to having a large number of blocks in a frame, as given by (2), resulting in better statistical properties. Even though the number of blocks is large in 12, the number of transmit antennas is small, resulting in inferior statistical properties. In order to improve the accuracy of the BER estimate for *N*tot ¼ 32, we propose to transmit "dummy data" from the first transmit antenna and "actual data" from the remaining antennas. The BER results shown in **Figure 13** indicates a good match between theory and practice. However, comparison of **Figures 11** and **14** demonstrates that "dummy data" is ineffective for large number of transmit antennas.

**Figure 13.** *Simulation results with precoding and dummy data for Ntot = 32.*

*New Results on Single User Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112469*

**Figure 14.** *Simulation results with precoding and dummy data for Ntot = 1024.*
