New Results on Single User Massive MIMO

Abstract

Achieving high bit rates is the main goal of wireless technologies like 5G and beyond. This translates to obtaining high spectral efficiencies using large number of antennas at the transmitter and receiver (single user massive multiple input multiple output or SU-MMIMO). It is possible to have a large number of antennas in the mobile handset at mm-wave frequencies in the range 30–300 GHz due to the small antenna size. In this work, we investigate the bit-error-rate (BER) performance of SU-MMIMO in two scenarios (a) using serially concatenated turbo code (SCTC) in uncorrelated channel and (b) parallel concatenated turbo code (PCTC) in correlated channel. Computer simulation results indicate that the BER is quite insensitive to re-transmissions and wide variations in the number of transmit and receive antennas. Moreover, we have obtained a BER of 10−5 at an average signal-to-interference plus noise ratio (SINR) per bit of just 1.25 dB with 512 transmit and receive antennas (512 × 512 SU-MMIMO system) with a spectral efficiency of 256 bits/transmission or 256 bits/sec/Hz in an uncorrelated channel. Similar BER results have been obtained for SU-MMIMO using PCTC in correlated channel. A semi-analytic approach to estimating the BER of a turbo code has been derived.

1. Introduction

As wireless technologies evolve beyond 5G [1, 2, 3], there is a growing need to attain peak data rates of about gigabits per second per user, which is required for high definition video, remote surgery, autonomous vehicles, gaming and so on, while at the same time consuming minimum transmit power. This can only be achieved by using multiple antennas at the transmitter and receiver [4, 5, 6, 7, 8], small constellations like quadrature shift keying (QPSK) and powerful error correcting codes like turbo or low density parity check (LDPC) codes. Having a large number of antennas in the mobile handset is feasible in mm-wave frequencies [9, 10, 11, 12] (30–300 GHz) due to the small antenna size. The main concern about mm wave communications has been its rather high attenuation in outdoor environments with rain and snow [13]. Therefore, at least in the initial stages, mm wave could be deployed indoors. The second issue relates to the poor penetration characteristics of mm wave through walls, doors, windows and other materials. This points towards to usage of mm wave [9] in a single room, say a big auditorium or underground parking and so on. Reconfigurable intelligent surface (RIS) [11, 12, 13, 14, 15, 16, 17] could be used to boost the propagation of mm waves, both indoors and outdoors. Most of the massive MIMO systems discussed in the literature are multi-user (MU) [18, 19, 20, 21, 22, 23, 24, 25, 26], that is, the base station has a large number of antennas and the mobile handset has only a single antenna ($N_t=1$). A large number of users are served simultaneously by the base station. A comparison between MU-MMIMO and SU-MMIMO is given in Table 1 [27, 28].

MU-MIMO	SU-MMIMO
Beamforming possible in downlink	Beamforming possible in uplink & downlink
Spatial multiplexing not possible	Spatial multiplexing possible in uplink & downlink
Low spectral efficiency per user	High spectral efficiency per user
High directivity in downlink in beamforming mode	High directivity in uplink & downlink in beamforming mode

Table 1.

Comparison of MU-MMIMO and SU-MMIMO.

The base station in MU-MMIMO uses beamforming to improve the signal-to-noise ratio at the mobile handset. On the other hand, SU-MMIMO uses spatial multiplexing to improve the spectral efficiency in the downlink and uplink. The comparison between beamforming and spatial multiplexing is given in Table 2 [27, 28]. The total transmit power of SU-MMIMO using uncoded QPSK versus MU-MMIMO using $M$-ary QAM is shown in Table 3. The minimum Euclidean distance between symbols of all constellations is taken to be 2. The peak-to-average power ratio (PAPR) for SU-MMIMO using QPSK is compared with MU-MMIMO using $M$-ary QAM in Table 4 [27]. Of course in the case of frequency selective fading channels, OFDM needs to be used, which would result in PAPR greater than 0 dB even for QPSK signaling. It is clear from Tables 1–4 that technologies that use SU-MMIMO have a lot to gain.

Beamforming	Spatial multiplexing
High directivity	Little or no directivity
Line-of-sight communication required	Rich scattering channel required
Low spectral efficiency per user since the same signal is transmitted from each antenna element	High spectral efficiency per user since different signals are transmitted from each antenna element
Spectral efficiency can be improved by increasing the constellation size resulting in high PAPR	QPSK constellations with PAPR 0 dB can be used
Difficult to turbo/LDPC code large constellations	Easy to turbo/LDPC code QPSK
Large BER at average SINR per bit close to 0 dB	Small BER at average SINR per bit close to 0 dB

Table 2.

Comparison of beamforming and spatial multiplexing.

Spectral efficiency (bits/sec/Hz)	QPSK		M-ary QAM
	Transmit antennas Nt	Total average transmit power	M-ary QAM	Nt = 1 average transmit power
4	2	4	16-QAM	10
6	3	6	64-QAM	42
8	4	8	256-QAM	170
10	5	10	1024-QAM	682

Table 3.

SU-MMIMO using QPSK vs. MU-MMIMO using M-ary.

Spectral efficiency (bits/sec/Hz)	QPSK		M-ary Nt = 1
	Transmit antennas Nt	PAPR (dB)	M	PAPR (dB)
4	2	0	16-QAM	2.5
6	3	0	64-QAM	3.7
8	4	0	256-QAM	4.23
10	5	0	1024-QAM	4.5

Table 4.

PAPR of SU-MMIMO using QPSK vs. MU-MMIMO using M-ary.

Moreover, since all transmit antennas use the same carrier frequency, there is no increase in bandwidth. SU-MMIMO with equal number of transmit and receive antennas is given in [29, 30]. The probability of erasure in MIMO-OFDM is presented in [31]. A practical SU-MMIMO receiver with estimated channel, carrier frequency offset and timing is described in [32, 33]. SU-MMIMO with unequal number of transmit and receive antennas and precoding is discussed in [34, 35] and the case without precoding in [36, 37]. All the earlier research on SU-MMIMO involved the use of a parallel concatenated turbo code (PCTC) and uncorrelated channel. In this work, we investigate the performance of SU-MMIMO using (a) serial concatenated turbo code (SCTC) in uncorrelated channel and (b) PCTC in correlated channel. Throughout this article we assume that the channel is known perfectly at the receiver. Perfect carrier and timing synchronization is also assumed.

This work is organized as follows. Section II discusses SU-MMIMO with SCTC in uncorrelated channel, the procedure for bit-error-rate (BER) estimation and computer simulation results. Section III deals with SU-MMIMO using PCTC in correlated channel with and without precoding along with computer simulation results. Section IV presents the conclusions and scope for future work.

2. SU-MMIMO with SCTC

2.1 System model

Consider the block diagram in Figure 1 [36, 38]. The input bits $a_i$, $1\le i\le L_{d1}$ is passed through an outer rate-$1/2$ recursive systematic convolutional (RSC) encoder to obtain the coded bit stream $b_i$, $1\le i\le L_d$, where

Figure 1.

SU-MMIMO with serially concatenated turbo code.

$$L_d=2L_{d1}.$$E1

Now $b_i$ is input to an interleaver to generate $c_i$, $1\le i\le L_d$. Next $c_i$ is passed through an inner rate-$1/2$ RSC encoder and mapped to symbols $S_i$, $1\le i\le L_d$, in a quadrature phase shift keyed (QPSK) constellation having symbol coordinates $\pm 1\pm \mathrm{j}$, where $\mathrm{j}=\sqrt{-1}$. Throughout this article we assume that bit “0” maps to $+1$ and bit “1” maps to $-1$. The set of $L_d$ QPSK symbols constitute a “frame” and are transmitted using $N_t$ antennas. We assume that

$$\frac{L_d}{N_t}=\text{an integer}$$E2

so that all symbols in the frame are transmitted using $N_t$ antennas. The set of QPSK symbols transmitted simultaneously using $N_t$ antennas constitute a “block”. The generator matrix for both the inner and outer rate-$1/2$ RSC encoder is given by

$$\mathbf{G}\left(D\right)=\left[1\frac{1+D^2}{1+D+D^2}\right].$$E3

Hence, both encoders have $S_E=4$ states in the trellis. Assuming uncorrelated Rayleigh flat fading, the received signal for the $k^{\mathit{th}}$ re-transmission ($0\le k\le N_{\mathit{rt}}-1$, $k$ is an integer) is given by (2) of [36], which is repeated here for convenience

$${\stackrel{\sim }{\mathbf{R}}}_k={\stackrel{\sim }{\mathbf{H}}}_k\mathbf{S}+{\stackrel{\sim }{\mathbf{W}}}_k$$E4

where $\mathbf{S}\in {\mathbb{C}}^{N_t\times 1}$ whose elements are drawn from the QPSK constellation, ${\stackrel{\sim }{\mathbf{H}}}_k\in {\mathbb{C}}^{N_r\times N_t}$ whose elements are mutually independent and $\mathcal{CN}\left(0,2{\sigma }_H^2\right)$ and ${\stackrel{\sim }{\mathbf{W}}}_k\in {\mathbb{C}}^{N_r\times 1}$ is the additive white Gaussian noise (AWGN) vector whose elements are mutually independent and $\mathcal{CN}\left(0,2{\sigma }_W^2\right)$. Note that ${\sigma }_H^2,{\sigma }_W^2$ denote the variance per dimension (real part or imaginary part) and $N_r$ is the number of receive antennas. We assume that ${\stackrel{\sim }{\mathbf{H}}}_k$ and ${\stackrel{\sim }{\mathbf{W}}}_k$ are independent across blocks and re-transmissions, hence (4) in [29] is valid with $N$ replaced by $N_t$. Recall that (see also (16) of [36])

$$N_{\mathrm{tot}}=N_t+N_r.$$E5

Following the procedure given in Section 4 of [36] we get (see (36) of [36])

$${\stackrel{\sim }{Y}}_i=F_i{S}_i+{\stackrel{\sim }{U}}_i\text{for}1\le i\le N_t.$$E6

After concatenation over blocks, ${\stackrel{\sim }{Y}}_i$ in (6) for $1\le i\le L_d$ is sent to the turbo decoder (see also the sentence after (25) in [29]). For the sake of consistency with earlier work [38], we re-index $i$ as $0\le i\le L_d-1$ and use the same index $i$ for $a_i$, $b_i$, $c_i$ and $Y_i$ without any ambiguity. In the next subsection, we discuss the turbo decoding (BCJR) algorithm [39, 40] for the inner code.

2.2 BCJR for the inner code

Let ${\mathcal{D}}_n$ denote the set of states that diverge from state $n$ in the trellis [38, 40]. Similarly, let ${\mathcal{C}}_n$ denote the set of states that converge to state $n$. Let ${\alpha }_{i,n}$ denote the forward sum-of-products (SOP) at time $i$, $0\le i\le L_d-2$, at state $n$, $0\le n\le S_E-1$. Then the forward SOP can be recursively computed as follows (see also (30) of [38]):

$${\alpha }_{i+1,n}^{\prime }=\sum _{m\in {\mathcal{C}}_n}{\alpha }_{i,m}{\gamma }_{i,m,n}P\left(c_{i,m,n}\right);{\alpha }_{0,n}=1;{\alpha }_{i+1,n}=\raisebox{1ex}{${\alpha }_{i+1,n}^{\prime }$}\!\left/ \!\raisebox{-1ex}{$\left(\sum _{n=0}^{S_E-1}\underset{i+1,n}{\overset{\prime }{\alpha }}\right)$}\right.$$E7

where $P\left(c_{i,m,n}\right)$ denotes the a priori probability of the systematic bit corresponding to the transition from encoder state $m$ to $n$, at time $i$ (this is set to 0.5 at the beginning of the first iteration). The last equation in (7) is required to prevent numerical instabilities [40]. We have

$${\gamma }_{i,m,n}=exp\left(-\frac{{\left({\stackrel{\sim }{Y}}_i-S_{m,n}\right)}^2}{2{\sigma }_U^2}\right)$$E8

where ${\stackrel{\sim }{Y}}_i$ is given by (6), $S_{m,n}$ is the QPSK symbol corresponding to the transition from encoder state $m$ to $n$ and ${\sigma }_U^2$ is given by (38) of [36] which is repeated here for convenience:

$$\begin{array}{rr}E\left[{\left|{\stackrel{\sim }{U}}_i\right|}^2\right]& =\frac{8{\sigma }_H^4{N}_r\left(N_t-1\right)+4{\sigma }_W^2{\sigma }_H^2{N}_r}{N_{\mathit{rt}}}\\ & \end{array}\stackrel{\Delta }{=}{\sigma }_U^2.$$E9

Robust turbo decoding (see Section 4.2 of [41]) can be employed to compute ${\gamma }_{i,m,n}$ in (8). Similarly, let ${\beta }_{i,m}$ denote the backward SOP at time $i$, $1\le i\le L_d-1$, at state $m$, $0\le m\le S_E-1$. Then the backward SOP can be recursively computed as (see also (33) of [38]):

$${\beta }_{i,m}^{\prime }=\sum _{n\in {\mathcal{D}}_m}{\beta }_{i+1,n}{\gamma }_{i,m,n}P\left(c_{i,m,n}\right);{\beta }_{L_{d,m}}=1;{\beta }_{i,m}=\raisebox{1ex}{${\beta }_{i,m}^{\prime }$}\!\left/ \!\raisebox{-1ex}{$\left({\sum }_{m=0}^{S_E-1}\underset{i,m}{\overset{\prime }{\beta }}\right)$}\right..$$E10

Let ${\rho }^{+}\left(n\right)$ denote the state that is reached from encoder state $n$ when the input symbol is $+1$. Similarly let ${\rho }^{-}\left(n\right)$ denote the state that can be reached from encoder state $n$ when the input symbol is $-1$. Then for $0\le i\le L_d-1$ we compute

$$C_{i+}=\sum _{n=0}^{S_E-1}{\alpha }_{i,n}{\gamma }_{i,n,{\rho }^{+}\left(n\right)}{\beta }_{i+1,{\rho }^{+}\left(n\right)};C_{i-}=\sum _{n=0}^{S_E-1}{\alpha }_{i,n}{\gamma }_{i,n,{\rho }^{-}\left(n\right)}{\beta }_{i+1,{\rho }^{-}\left(n\right)}.$$E11

Finally, the extrinsic information that is fed to the BCJR algorithm for the outer code is computed as, for $0\le i\le L_d-1$, (see (36) of [38]):

$$E\left(c_i=+1\right)=\raisebox{1ex}{$C_{i+}$}\!\left/ \!\raisebox{-1ex}{$\left(C_{i+}+C_{i-}\right)$}\right.;E\left(c_i=-1\right)=C_{i-}/\left(C_{i+}+C_{i-}\right).$$E12

Next, we describe the BCJR for the outer code.

2.3 BCJR for the outer code

Let ${\alpha }_{i,n}$ denote the forward SOP at time $i$, $0\le i\le L_{d1}-2$, at state $n$, $0\le n\le S_E-1$. Then the forward SOP is recursively computed as follows:

$${\alpha }_{i+1,n}^{\prime }=\sum _{m\in {\mathcal{C}}_n}{\alpha }_{i,m}{\gamma }_{\mathit{sys},i,m,n}{\gamma }_{\mathit{par},i,m,n}P\left({\alpha }_{i,m,n}\right);{\alpha }_{0,n}=1;{\alpha }_{i+1,n}=\raisebox{1ex}{${\alpha }_{i+1,n}^{\prime }$}\!\left/ \!\raisebox{-1ex}{$\left(\sum _{n=0}^{S_E-1}{\alpha }_{i+1,n}^{\prime }\right)$}\right.$$E13

where $P\left(a_{i,m,n}\right)$ denotes the a priori probability of the systematic bit corresponding to the transition from state $m$ to state $n$, at time $i$. In the absence of any other information, we assume $\left(a_{i,m,n}\right)=0.5$ [42]. We also have for $0\le i\le L_{d1}-1$ (similar to (38) of [38])

$${\gamma }_{\mathrm{sys},i,m,n}=\left\{\begin{array}{ll}E\left(c_{\pi \left(2i\right)}=+1\right)& \text{if}{\mathcal{H}}_1\\ E\left(c_{\pi \left(2i\right)}=-1\right)& \text{if}{\mathcal{H}}_2\end{array}\right.;{\gamma }_{\mathrm{par},i,m,n}=\left\{\begin{array}{ll}E\left(c_{\pi \left(2i+1\right)}=+1\right)& \text{if}{\mathcal{H}}_3\\ E\left(c_{\pi \left(2i+1\right)}=-1\right)& \text{if}{\mathcal{H}}_4\end{array}\right.$$E14

where $\pi \left(·\right)$ denotes the interleaver map and

${\mathcal{H}}_1:\text{systematic}\mathrm{bit}\text{from state}m\text{to}n\text{is}+1$; ${\mathcal{H}}_2:\text{systematic}\mathrm{bit}\text{from state}m\text{to}n\text{is}-1$

$${\mathcal{H}}_3:\text{parity}\mathrm{bit}\text{from state}m\text{to}n\text{is}+1;{\mathcal{H}}_4:\text{parity}\mathrm{bit}\text{from state}m\text{to}n\text{is}-1.$$E15

Observe that in (14) and (15) it is assumed that after the parallel-to-serial conversion in Figure 1, $b_{2i}$ corresponds to the systematic (data) bits and $b_{2i+1}$ corresponds to the parity bits for $0\le i\le L_{d1}-1$. Similarly, let ${\beta }_{i,m}$ denote the backward SOP at time $i$, $1\le i\le L_{d1}-1$, at state $m$, $0\le m\le S_E-1$. Then the backward SOP can be recursively computed as:

$${\beta }_{i,m}^{\prime }=\sum _{n\in {\mathcal{D}}_m}{\beta }_{i+1,n}{\gamma }_{\mathit{sys},i,m,n}{\gamma }_{\mathit{par},i,m,n}P\left(a_{i,m,n}\right);{\beta }_{L_{d1,m}}=1;{\beta }_{i,m}=\raisebox{1ex}{${\beta }_{i,m}^{\prime }$}\!\left/ \!\raisebox{-1ex}{$\left(\sum _{m=0}^{S_E-1}{\beta }_{i,m}^{\prime }\right).$}\right.$$E16

Next, for $0\le i\le L_{d1}-1$ we compute

$$B_{2i+}=\sum _{n=0}^{S_E-1}{\alpha }_{i,n}{\gamma }_{\mathrm{par},i,n,{\rho }^{+}\left(n\right)}{\beta }_{i+1,{\rho }^{+}\left(n\right)};B_{2i-}=\sum _{n=0}^{S_E-1}{\alpha }_{i,n}{\gamma }_{\mathrm{par},i,n,{\rho }^{-}\left(n\right)}{\beta }_{i+1,{\rho }^{-}\left(n\right)}.$$E17

Let ${\mu }^{+}\left(n\right),$ and ${\mu }^{-}\left(n\right)$ denote the states that are reached from state $n$ when the parity bit is $+1$ and $-1$respectively. Similarly for $0\le i\le L_{d1}-1$ compute

$$B_{2i+1+}=\sum _{n=0}^{S_E-1}{\alpha }_{i,n}{\gamma }_{\mathrm{sys},i,n,{\mu }^{+}\left(n\right)}{\beta }_{i+1,{\mu }^{+}\left(n\right)};B_{2i+1-}=\sum _{n=0}^{S_E-1}{\alpha }_{i,n}{\gamma }_{\mathrm{sys},i,n,{\mu }^{-}\left(n\right)}{\beta }_{i+1,{\mu }^{-}\left(n\right)}.$$E18

The extrinsic information that is sent to the inner decoder for $0\le i\le L_d-1$ is computed as

$$E\left(b_i=+1\right)=\raisebox{1ex}{$B_{i+}$}\!\left/ \!\raisebox{-1ex}{$\left(B_{i+}+B_{i-}\right)$}\right.;E\left(b_i=-1\right)=B_{i-}/\left(B_{i+}+B_{i-}\right)$$E19

where $B_{i+},B_{i-}$ are given by (17)or (18) depending on whether $i$ is even or odd respectively. Note that $P\left(c_{i,m,n}\right)$ for $0\le i\le L_d-1$ in (7) and (10) is equal to

$$P\left(c_{i,m,n}\right)=\left\{\begin{array}{ll}E\left(b_{{\pi }^{-1}\left(i\right)}=+1\right)& \text{if}{\mathcal{H}}_1\\ E\left(b_{{\pi }^{-1}\left(i\right)}=-1\right)& \text{if}{\mathcal{H}}_2\end{array}\right.$$E20

where ${\pi }^{-1}\left(·\right)$ denotes the inverse interleaver map. Note that $c_{i,m,n}$ are the systematic (data) bits for the inner encoder.

After the convergence of the BCJR algorithm in the last iteration, the final a posteriori probabilities of $a_i$ for $0\le i\le L_{d1}-1$ is given by

$$P\left(a_i=+1\right)=E\left(b_{2i}=+1\right)E\left(c_{\pi \left(2i\right)}=+1\right);P\left(a_i=-1\right)=E\left(b_{2i}=-1\right)E\left(c_{\pi \left(2i\right)}=-1\right)$$E21

where $E\left(c_i=\pm 1\right)$ and $E\left(b_i=\pm 1\right)$ are given by (12) and (19) respectively. Finally note that for $0\le i\le L_{d1}-1$

$$\begin{array}{r}{a}_i=b_{2i}=c_{\pi \left(2i\right)}.\end{array}$$E22

In the next section we present the estimation of the bit-error-rate (BER) of the SCTC.

2.4 Estimation of BER

The estimation of BER of SCTC is based on the following propositions:

Proposition1.The extrinsic information as computed in(12)and(19)lies in the range$\left[0,1\right]$(0 and 1 included). The extrinsic information in the range$\left(0,1\right)$, 0 and 1 excluded, is Gaussian distributed [43] for each frame.

This is illustrated in Figure 2 for different values of the frame length $L_{d1}$, over many frames ($F$). We find that for large values of $L_{d1}$, the histogram better approximates the Gaussian characteristic. It may be noted that the extrinsic information at the output of one decoder is equal to the a priori probabilities for the other decoder.

Figure 2.

Normalized histogram for N_tot = 1024, N_t = 512, N_rt = 2 (a) L_d1 = 1024, SNR_{av, b} = 1.25 dB, F = 10⁵ frames (b) L_d1 = 50,176, SNR_{av, b} = 0.3 dB, F = 2000 frames (c) expanded view of (around) r_3,i = 0 and (d) L_d1 = 50,176, SNR_{av, b} = 0.5 dB, F = 2000 frames.

Proposition 2.After convergence of the BCJR algorithm in the final iteration, the extrinsic information at a decoder output has the same mean and variance as that of the a priori probability at its input.

Proposition 3.The mean and variance of the Gaussian distribution may vary from frame to frame.

This is illustrated in Figure 3 over two frames, that is, $F=2$.

$$P\left(e\right)=\frac{1}{2}erfc\left(\sqrt{\frac{A^2}{{\sigma }^2}}\right).$$E23

Figure 3.

Normalized histogram over two frames (F = 2) for N_tot = 1024, N_t = 512, N_rt = 2 (a) L_d1 = 1024, SNR_{av, b} = 1.25 dB and (d) L_d1 = 50,176, SNR_{av, b} = 0.5 dB.

Based on Propositions 1 & 2 and (22), after convergence of the BCJR algorithm, we can write for $0\le i\le L_{d1}-1$

$$E\left(b_{2i}=+1\right)=\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-{\left({\mathrm{r}}_{1,\mathrm{i}}-\mathrm{A}\right)}^2/\left(2{\mathrm{\sigma }}^2\right)};E\left(c_{\pi \left(2i\right)}=+1\right)=\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-\raisebox{1ex}{${\left({\mathrm{r}}_{2,\mathrm{i}}-\mathrm{A}\right)}^2$}\!\left/ \!\raisebox{-1ex}{$\left(2{\mathrm{\sigma }}^2\right)$}\right.}$$E24

where it is assumed that bit “0” maps to $A$ and bit “1” maps to $-A$ and

$$r_{1,i}=\pm A+w_{1,i};r_{2,i}=\pm A+w_{2,i}$$E25

where $w_{1,i},w_{2,i}$ denote real-valued samples of zero-mean additive white Gaussian noise (AWGN) with variance ${\sigma }^2$. Similarly we have

$$E\left(b_{2i}=-1\right)=\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-{\left({\mathrm{r}}_{1,\mathrm{i}}+\mathrm{A}\right)}^2/\left(2{\mathrm{\sigma }}^2\right)};E\left(c_{\pi \left(2i\right)}=-1\right)=\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-\raisebox{1ex}{${\left({\mathrm{r}}_{2,\mathrm{i}}+\mathrm{A}\right)}^2$}\!\left/ \!\raisebox{-1ex}{$\left(2{\mathrm{\sigma }}^2\right)$}\right.}.$$E26

Clearly

$$ln\left(\frac{E\left(b_{2i}=+1\right)}{E\left(b_{2i}=-1\right)}\right)=\frac{2A}{{\sigma }^2}{r}_{1,i};ln\left(\frac{E\left(c_{\pi \left(2i\right)}=+1\right)}{E\left(c_{\pi \left(2i\right)}=-1\right)}\right)=\frac{2A}{{\sigma }^2}{r}_{2,i}.$$E27

From (21) and (26) we have for $0\le i\le L_{d1}-1$

$$ln\left(\frac{P\left(a_i=+1\right)}{P\left(a_i=-1\right)}\right)=\frac{2A}{{\sigma }^2}\left(r_{1,i}+r_{2,i}\right)\stackrel{\Delta }{=}\frac{2A}{{\sigma }^2}{r}_{3,i}.$$E28

Consider the average

$$\mathcal{Y}=\frac{2A}{{\sigma }^2{L}_{d2}}\sum _{i=0}^{L_{d2}-1}{a}_i{r}_{3,i}=\frac{4A^2}{{\sigma }^2}+\mathcal{Z}$$E29

where

$$\mathcal{Z}=\frac{2A}{{\sigma }^2{L}_{d2}}\sum _{i=0}^{L_{d2}-1}{a}_i\left(w_{1,i}+w_{2,i}\right);.L_{d2}<L_{d1}$$E30

Note that the average in (28) is done over less than $L_{d1}$ terms to avoid situations like

$$P\left(a_i=\pm 1\right)=1\text{or}0.$$E31

In fact, only those time instants $i$ have been considered in the summation of (28) for which

$$P\left(a_i=\pm 1\right)>{\mathrm{e}}^{-500}.$$E32

Now

$$E\left[\mathcal{Z}\right]=0;E\left[{\mathcal{Z}}^2\right]=\frac{4A^2}{{\sigma }^4{L}_{d2}^2}\sum _{i=0}^{L_{d2}-1}2{\sigma }^2=\frac{8A^2}{{\sigma }^2{L}_{d2}}$$E33

where we have used the fact that $w_{1,i},w_{2,i}$ are independent. Now, we know that the probability of error for the BPSK signal in (27), that is equal to [40].

$$r_{3,i}=r_{1,i}+r_{2,i}=\pm 2A+w_{1,i}+w_{2,i}$$E34

Therefore from (28), (32) and (34) we have

$$P_f\left(e\right)\approx \frac{1}{2}erfc\left(\sqrt{\frac{\left|\mathcal{Y}\right|}{4}}\right)$$E35

where $P_f\left(e\right)$ denotes the probability of bit error for frame “$f$” and

$$E\left[{\mathcal{Z}}^2\right]\to 0\text{for}{L}_{d2}\gg 1.$$E36

Observe that it is necessary to take the absolute value of $\mathcal{Y}$ in (35) since there is a possibility that it can be negative. The average probability of bit error over $F$ frames is given by

$$P\left(e\right)=\frac{1}{F}\sum _{f=0}^{F-1}{P}_f\left(e\right).$$E37

In the next section we present computer simulation results for SU-MMIMO using SCTC in uncorrelated channel.

2.5 Simulation results

The simulation parameters are given in Table 5. We can make the following observations from Figures 4–6 [36]:

Parameter	Value(s)
Modulation	QPSK
Total Antennas (Ntot = Nt + Nr)	1024		32			2
Transmit antennas (Nt)	400	512	7	12	16	1
Frame length (Ld1)	1200 50,400	1024 50,176	1001	1008	1024	1001
Frames simulated (F)	10^4. 10^5 for Ld1 range 1001 to 1200, 200, 2000 for Ld1 = 50,176, 50,400
Spectral eff. For Nrt = 1 (bits/sec/Hz)	200	256	3.5	6	8	0.5

Table 5.

Simulation parameters for results in Figures 4–6.

Figure 4.

Simulation results for N_tot = 1024.

Figure 5.

Simulation results for N_tot = 32.

Figure 6.

Simulation results for N_tot = 2, N_t = 1.

The theoretical prediction of BER closely matches with simulations.

• For N_tot = 32, 1024, the BER is quite insensitive to wide variations in the total number of antennas N_tot, transmit antennas N_t and retransmissions N_rt.

• For N_tot = 2, the BER improves significantly with increasing retransmissions.

In Figure 4(c) we observe that there is more than 1 dB improvement in SINR compared to Figures 4–6(a, b). However, large values of L_d1 may introduce more latency which is contrary to the requirements of 5G and beyond. In the next section we present SU-MMIMO using PCTC in correlated channel.

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

$${\mathbf{R}}_{\stackrel{\sim }{\mathbf{H}}\stackrel{\sim }{\mathbf{H}}}=\frac{1}{2}E\left[{\stackrel{\sim }{\mathbf{H}}}_k^H{\stackrel{\sim }{\mathbf{H}}}_k\right]=N_r{\mathbf{I}}_{N_t}$$E38

where the superscript “$H$” denotes Hermitian and ${\mathbf{I}}_{N_t}$ denotes the $N_t\times N_t$ identity matrix. In this section, we investigate the situation where ${\mathbf{R}}_{\stackrel{\sim }{\mathbf{H}}\stackrel{\sim }{\mathbf{H}}}$ is not an identity matrix, but is a valid autocorrelation matrix [40]. As mentioned in [36], the elements of ${\stackrel{\sim }{\mathbf{H}}}_k$ – given by ${\stackrel{\sim }{H}}_{k,i,j}$ for the $k^{\mathit{th}}$ re-transmission, $i^{\mathit{th}}$ row, $j^{\mathit{th}}$ column of ${\stackrel{\sim }{\mathbf{H}}}_k$ – are zero-mean, complex Gaussian random variables with variance per dimension equal to ${\sigma }_H^2$. The in-phase and quadrature components of ${\stackrel{\sim }{H}}_{k,i,j}$ – denoted by $H_{k,i,j,I}$ and $H_{k,i,j,Q}$ respectively – are statistically independent. Moreover, we assume that the rows of ${\stackrel{\sim }{\mathbf{H}}}_k$ 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 $i^{\mathit{th}}$ element of ${\stackrel{\sim }{\mathbf{H}}}_k^H{\stackrel{\sim }{\mathbf{R}}}_k$ is given by (25) of [36] which is repeated here for convenience

$${\stackrel{\sim }{Y}}_{k,i}={\stackrel{\sim }{F}}_{k,i,i}{S}_i+{\stackrel{\sim }{I}}_{k,i}+{\stackrel{\sim }{V}}_{k,i}\text{for}1\le i\le N_t$$E39

where

$${\stackrel{\sim }{V}}_{k,i}=\sum _{j=1}^{N_r}{\stackrel{\sim }{H}}_{k,j,i}^{\ast }{\stackrel{\sim }{W}}_{k,j};{\stackrel{\sim }{I}}_{k,i}=\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\stackrel{\sim }{F}}_{k,i,j}{S}_j;{\stackrel{\sim }{F}}_{k,i,j}=\sum _{l=1}^{N_r}{\stackrel{\sim }{H}}_{k,l,i}^{\ast }{\stackrel{\sim }{H}}_{k,l,j}.$$E40

We have

$$\begin{array}{l}E\left[{\stackrel{\sim }{F}}_{k,i,i}^2\right]=E\left[\sum _{l=1}^{N_r}{\left|{\stackrel{\sim }{H}}_{k,l,i}\right|}^2\sum _{m=1}^{N_r}{\left|{\stackrel{\sim }{H}}_{k,m,i}\right|}^2\right]\\ =E\left[\sum _{l=1}^{N_r}\left(H_{k,l,i,I}^2+H_{k,l,i,Q}^2\right)\sum _{m=1}^{N_r}\left(H_{k,m,i,I}^2+H_{k,m,i,Q}^2\right)\right]=4{\sigma }_H^4{N}_r\left(N_r+1\right)\end{array}$$E41

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

1. The in-phase and quadrature components of ${\stackrel{\sim }{H}}_{k,i,j}$ are independent.

2. The rows of ${\stackrel{\sim }{\mathbf{H}}}_k$ are independent.

3. For zero-mean, real-valued Gaussian random variable $X,.$ with variance equal to ${\sigma }_X^2,E\left[X^4\right]=3{\sigma }_X^4$.

The interference power is

$$E\left[{\left|{\stackrel{\sim }{I}}_{k,i}\right|}^2\right]=E\left[\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\stackrel{\sim }{F}}_{k,i,j}{S}_j\sum _{\begin{array}{c}l=1\\ l\ne i\end{array}}^{N_t}{\stackrel{\sim }{F}}_{k,i,l}^{\ast }{S}_l^{\ast }\right]=\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}\sum _{\begin{array}{c}l=1\\ l\ne i\end{array}}^{N_t}E\left[{\stackrel{\sim }{F}}_{k,i,j}{\stackrel{\sim }{F}}_{k,i,l}^{\ast }\right]E\left[S_j{S}_l^{\ast }\right]=P_{\mathrm{av}}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}E\left[{\left|{\stackrel{\sim }{F}}_{k,i,j}\right|}^2\right].$$E42

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

$$\begin{array}{rcl}E\left[{\left|{\stackrel{\sim }{V}}_{k,i}\right|}^2\right]& =& E\left[\sum _{j=1}^{N_r}{\stackrel{\sim }{H}}_{k,j,i}^{\ast }{\stackrel{\sim }{W}}_{k,j}\sum _{m=1}^{N_r}{\stackrel{\sim }{H}}_{k,m,i}{\stackrel{\sim }{W}}_{k,m}^{\ast }\right]\\ & =& \sum _{j=1}^{N_r}\sum _{m=1}^{N_r}E\left[{\stackrel{\sim }{H}}_{k,j,i}^{\ast }{\stackrel{\sim }{H}}_{k,m,i}\right]E\left[{\stackrel{\sim }{W}}_{k,m}^{\ast }{\stackrel{\sim }{W}}_{k,j}\right]\\ & =& \sum _{j=1}^{N_r}\sum _{m=1}^{N_r}2{\sigma }_H^2{\delta }_K\left(j-m\right)2{\sigma }_W^2\left(j-m\right)=4N_r{\sigma }_H^2{\sigma }_W^2\end{array}$$E43

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

1. Rows of ${\stackrel{\sim }{\mathbf{H}}}_k$ are independent.

2. Sifting property of the Kronecker delta function.

3. Noise and channel coefficients are independent.

Now in (42)

$$\begin{array}{rcl}E\left[{\left|{\stackrel{\sim }{F}}_{k,i,j}\right|}^2\right]& =& E\left[\sum _{l=1}^{N_r}{\stackrel{\sim }{H}}_{k,l,i}^{\ast }{\stackrel{\sim }{H}}_{k,l,j}\sum _{m=1}^{N_r}{\stackrel{\sim }{H}}_{k,m,i}{\stackrel{\sim }{H}}_{k,m,j}^{\ast }\right]\\ & =& \sum _{l=1}^{N_r}E\left[{\stackrel{\sim }{H}}_{k,l,i}^{\ast }{\stackrel{\sim }{H}}_{k,l,j}\left({\stackrel{\sim }{H}}_{k,l,i}{\stackrel{\sim }{H}}_{k,l,j}^{\ast }+\sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^{N_r}{\stackrel{\sim }{H}}_{k,m,i}{\stackrel{\sim }{H}}_{k,m,j}^{\ast }\right)\right]\end{array}$$

$$=\sum _{l=1}^{N_r}E\left[{\left|{\stackrel{\sim }{H}}_{k,l,i}\right|}^2{\left|{\stackrel{\sim }{H}}_{k,l,j}\right|}^2+\left(\sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^{N_r}{\stackrel{\sim }{H}}_{k,l,i}^{\ast }{\stackrel{\sim }{H}}_{k,l,j}{\stackrel{\sim }{H}}_{k,m,i}{\stackrel{\sim }{H}}_{k,m,j}^{\ast }\right)\right].$$E44

Now the first summation in (44) is equal to

$$E_1=E\left[{\left|{\stackrel{\sim }{H}}_{k,l,i}\right|}^2{\left|{\stackrel{\sim }{H}}_{k,l,j}\right|}^2\right]=E\left[\left(H_{k,l,i,I}^2+H_{k,l,i,Q}^2\right)\left(H_{k,l,j,I}^2+H_{k,l,j,Q}^2\right)\right]=4{\sigma }_H^4+4R_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2$$E45

where we have used the property that for real-valued, zero-mean Gaussian random variables $X_i$, $1\le i\le 4$ [44, 45]

$$E\left[X_1{X}_2{X}_3{X}_4\right]=C_{12}{C}_{34}+C_{13}{C}_{24}+C_{14}{C}_{23}$$E46

where

$$C_{\mathit{ij}}=E\left[X_i{X}_j\right]\text{for}1\le i,j\le 4$$E47

and

$$R_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}=E\left[H_{k,l,i,I}{H}_{k,l,j,I}\right]=E\left[H_{k,l,i,Q}{H}_{k,l,j,Q}\right]=\frac{1}{2}E\left[{\stackrel{\sim }{H}}_{k,l,i}^{\ast }{\stackrel{\sim }{H}}_{k,l,j}\right]=R_{\stackrel{\sim }{H}\stackrel{\sim }{H},i-j}$$E48

is the real-valued autocorrelation of ${\stackrel{\sim }{H}}_{k,m,n}$ and we have made the assumption that the in-phase and quadrature components of ${\stackrel{\sim }{H}}_{k,m,n}$ are independent. The second summation in (44) can be written as

$$\begin{array}{rcl}{E}_2& =& \sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^{N_r}E\left[{\stackrel{\sim }{H}}_{k,l,i}^{\ast }{\stackrel{\sim }{H}}_{k,l,j}{\stackrel{\sim }{H}}_{k,m,i}{\stackrel{\sim }{H}}_{k,m,j}^{\ast }\right]\\ & =& \sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^{N_r}E\left[{\stackrel{\sim }{H}}_{k,l,i}^{\ast }{\stackrel{\sim }{H}}_{k,l,j}\right]E\left[{\stackrel{\sim }{H}}_{k,m,i}{\stackrel{\sim }{H}}_{k,m,j}^{\ast }\right]=\sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^{N_r}4R_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2=4\left(N_r-1\right)R_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2\end{array}$$E49

where we have used the property that the rows of ${\stackrel{\sim }{\mathbf{H}}}_k$ are independent. Therefore (44) becomes

$$E\left[{\left|{\stackrel{\sim }{F}}_{k,i,j}\right|}^2\right]=N_r\left(E_1+E_2\right)=4N_r\left[{\sigma }_H^4+R_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2+\left(N_r-1\right)R_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2\right]=4N_r\left[{\sigma }_H^4+N_r{R}_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2\right].$$E50

The total power of interference plus noise is

$$E\left[{\left|{\stackrel{\sim }{I}}_{k,i}+{\stackrel{\sim }{V}}_{k,i}\right|}^2\right]=E\left[{\left|{\stackrel{\sim }{I}}_{k,i}\right|}^2\right]+E\left[{\left|{\stackrel{\sim }{V}}_{k,i}\right|}^2\right]=4P_{\mathrm{av}}{N}_r\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}\left[{\sigma }_H^4+N_r{R}_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2\right]+4N_r{\sigma }_H^2{\sigma }_W^2$$E51

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

$${\text{SINR}}_{\mathrm{av},b,i}=\frac{E\left[{\left|{\stackrel{\sim }{F}}_{k,i,i}{S}_i\right|}^2\right]\times 2N_{\mathit{rt}}}{E\left[{\left|{\stackrel{\sim }{I}}_{k,i}+{\stackrel{\sim }{V}}_{k,i}\right|}^2\right]}\text{for}1\le i\le N_t$$E52

into which (41) and (51) have to be substituted. The upper bound on the average SINR per bit for the $i^{\mathit{th}}$ transmit antenna is obtained by setting ${\sigma }_W^2=0$ in (51), (52) and is given by, for $1\le i\le N_t$

$${\text{SINR}}_{\mathrm{av},b,\mathrm{UB},i}=\frac{{\sigma }_H^4\left(1+N_r\right)\times 2N_{\mathit{rt}}}{{\sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}\left[{\sigma }_H^4+N_r{R}_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2\right]}.$$E53

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])

$$F_i=\frac{1}{N_{\mathit{rt}}}\sum _{k=0}^{N_{\mathit{rt}}-1}{\stackrel{\sim }{F}}_{k,i,i}$$

$$\stackrel{\sim }{U}=\frac{1}{N_{\mathit{rt}}}\sum _{k=0}^{N_{\mathit{rt}}-1}\left({\stackrel{\sim }{I}}_{k,i}+{\stackrel{\sim }{V}}_{k,i}\right)=\frac{1}{N_{\mathit{rt}}}\sum _{k=0}^{N_{\mathit{rt}}-1}{{\stackrel{\sim }{U}}_{k,i}}^{\prime }\left(\mathrm{say}\right)$$E54

where ${\stackrel{\sim }{F}}_{k,i,i}$, ${\stackrel{\sim }{I}}_{k,i}$ and ${\stackrel{\sim }{V}}_{k,i}$ are given in (39). The power of the signal component of (6) is

$$\begin{array}{rcl}E\left[{\left|S_i\right|}^2{F}_i^2\right]& =& P_{\mathrm{av}}E\left[F_i^2\right]=\frac{P_{\mathrm{av}}}{N_{\mathit{rt}}^2}E\left[\sum _{k=0}^{N_{\mathit{rt}}-1}{\stackrel{\sim }{F}}_{k,i,i}\sum _{l=0}^{N_{\mathit{rt}}-1}{\stackrel{\sim }{F}}_{l,i,i}\right]\\ & =& \frac{P_{\mathrm{av}}}{N_{\mathit{rt}}^2}\sum _{k=0}^{N_{\mathit{rt}}-1}\left[\sum _{\begin{array}{c}l=0\\ l\ne k\end{array}}^{N_{\mathit{rt}}-1}E\left[{\stackrel{\sim }{F}}_{k,i,i}\right]E\left[{\stackrel{\sim }{F}}_{l,i,i}\right]+E\left[{\left|{\stackrel{\sim }{F}}_{k,i,i}\right|}^2\right]\right]\end{array}$$E55

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

$$E\left[{\stackrel{\sim }{F}}_{k,i,i}{\stackrel{\sim }{F}}_{l,i,i}\right]=E\left[{\stackrel{\sim }{F}}_{k,i,i}\right]E\left[{\stackrel{\sim }{F}}_{l,i,i}\right]\text{for}k\ne l.$$E56

Now

$$E\left[{\stackrel{\sim }{F}}_{k,i,i}\right]=E\left[\sum _{l=1}^{N_r}{\left|{\stackrel{\sim }{H}}_{k,l,i}\right|}^2\right]=2N_r{\sigma }_H^2.$$E57

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

$$E\left[{\left|S_i\right|}^2{F}_i^2\right]=\frac{4N_r{P}_{\mathrm{av}}{\sigma }_H^4}{N_{\mathit{rt}}}\left(1+N_r{N}_{\mathit{rt}}\right).$$E58

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

$$E\left[{\left|{\stackrel{\sim }{U}}_i\right|}^2\right]=\frac{1}{N_{\mathit{rt}}^2}E\left[\sum _{k=0}^{N_{\mathit{rt}}-1}\left({\stackrel{\sim }{I}}_{k,i}+{\stackrel{\sim }{V}}_{k,i}\right)\sum _{l=0}^{N_{\mathit{rt}}-1}\left({\stackrel{\sim }{I}}_{l,i}^{\ast }+{\stackrel{\sim }{V}}_{l,i}^{\ast }\right)\right]=\frac{1}{N_{\mathit{rt}}^2}\sum _{k=0}^{N_{\mathit{rt}}-1}\sum _{l=0}^{N_{\mathit{rt}}-1}E\left[{\stackrel{\sim }{I}}_{k,i}{\stackrel{\sim }{I}}_{l,i}^{\ast }\right]+E\left[{\stackrel{\sim }{V}}_{k,i}{\stackrel{\sim }{V}}_{l,i}^{\ast }\right]$$E59

where we have used the following properties from (40)

$$E\left[{\stackrel{\sim }{I}}_{k,i}\right]=E\left[{\stackrel{\sim }{V}}_{k,i}\right]=0;E\left[{\stackrel{\sim }{I}}_{k,i}{\stackrel{\sim }{V}}_{l,i}^{\ast }\right]=E\left[{\stackrel{\sim }{V}}_{k,i}{\stackrel{\sim }{I}}_{l,i}^{\ast }\right]=0\text{for}\mathrm{all}k,l$$E60

since $S_j$ and ${\stackrel{\sim }{W}}_{k,j}$ are mutually independent with zero-mean. Now

$$\begin{array}{l}E\left[{\stackrel{\sim }{I}}_{k,i}{\stackrel{\sim }{I}}_{l,i}^{\ast }\right]=E\left[\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\stackrel{\sim }{F}}_{k,i,j}{S}_j\sum _{\begin{array}{c}n=1\\ n\ne i\end{array}}^{N_t}{\stackrel{\sim }{F}}_{l,i,n}^{\ast }{S}_n^{\ast }\right]=\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}\sum _{\begin{array}{c}n=1\\ n\ne i\end{array}}^{N_t}E\left[{\stackrel{\sim }{F}}_{k,i,j}{\stackrel{\sim }{F}}_{l,i,n}^{\ast }\right]E\left[S_j{S}_n^{\ast }\right]\\ =\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}\sum _{\begin{array}{c}n=1\\ n\ne i\end{array}}^{N_t}E\left[{\stackrel{\sim }{F}}_{k,i,j}{\stackrel{\sim }{F}}_{l,i,n}^{\ast }\right]P_{\mathrm{av}}{\delta }_K\left(j-n\right)\\ =P_{\mathrm{av}}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}E\left[{\stackrel{\sim }{F}}_{k,i,j}{\stackrel{\sim }{F}}_{l,i,j}^{\ast }\right]\end{array}$$E61

where we have used the property that the symbols are uncorrelated and ${\delta }_K\left(·\right)$ is the Kronecker delta function [40]. When $k=l$, (61) is given by (42) and (50). When $k\ne l$, (61) is given by

$$E\left[{\stackrel{\sim }{I}}_{k,i}{\stackrel{\sim }{I}}_{l,i}^{\ast }\right]=P_{\mathrm{av}}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}E\left[{\stackrel{\sim }{F}}_{k,i,j}\right]E\left[{\stackrel{\sim }{F}}_{l,i,j}^{\ast }\right]=P_{\mathrm{av}}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}4N_r^2{R}_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2$$E62

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

$$E\left[{\stackrel{\sim }{V}}_{k,i}{\stackrel{\sim }{V}}_{l,i}^{\ast }\right]=4N_r{\sigma }_H^2{\sigma }_W^2{\delta }_K\left(k-l\right)$$E63

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

$$\begin{array}{rcl}E\left[{\left|{\stackrel{\sim }{U}}_i\right|}^2\right]& =& \frac{1}{N_{\mathit{rt}}^2}\left[4P_{\mathrm{av}}{N}_r{N}_{\mathit{rt}}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}\left({\sigma }_H^4+N_r{R}_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2\right)\right.+\left.4P_{\mathrm{av}}{N}_r^2{N}_{\mathit{rt}}\left(N_{\mathit{rt}}-1\right)\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{R}_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2\right]\\ & & +\frac{4N_r}{N_{\mathit{rt}}}{\sigma }_H^2{\sigma }_W^2\\ & =& \frac{1}{N_{\mathit{rt}}}\left[4P_{\mathrm{av}}{N}_r\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}\left({\sigma }_H^4+N_r{R}_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2\right)\right.+4P_{\mathrm{av}}{N}_r^2\left(N_{\mathit{rt}}-1\right)\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{R}_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}^2]\\ & & +\frac{4N_r}{N_{\mathit{rt}}}{\sigma }_H^2{\sigma }_W^2.\end{array}$$E64

The average SINR per bit for the $i^{\mathit{th}}$ transmit antenna, after averaging over retransmissions (also referred to as “combining” [36]) is given by

$${\text{SINR}}_{\mathrm{av},b,C,i}=\frac{2P_{\mathrm{av}}E\left[F_i^2\right]}{E\left[{\left|{\stackrel{\sim }{U}}_i\right|}^2\right]}$$E65

into which (58) and (64) have to be substituted. The upper bound on the average SINR per bit after “combining” for the $i^{\mathit{th}}$ transmit antenna is given by

$${\text{SINR}}_{\mathrm{av},b,C,\mathrm{UB},i}={\left.{\text{SINR}}_{\mathrm{av},b,C,i}\right|}_{{\sigma }_W^2=0}.$$E66

The plots of the average SINR per bit for the $i^{\mathit{th}}$ transmit antenna before and after “combining” are shown in Figures 7 and 8 respectively for $N_{\mathrm{tot}}=1024$ and $N_{\mathit{rt}}=2$. The channel correlation is given by

Figure 7.

Plot of SINR_av,b,UB,i for N_tot = 1024, N_rt = 2. (a) Back view. (b) Sideview. (c) Front view.

Figure 8.

Plot of SINR_av,b,C,UB,i for N_tot = 1024, N_rt = 2. (a) Back view. (b) Side view. (c) Front view.

$$R_{\stackrel{\sim }{H}\stackrel{\sim }{H},j-i}={0.9}^{\left|j-i\right|}{\sigma }_H^2$$E67

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 $a=-0.9$ (see (30) of [46]).

We observe in Figures 7 and 8 that

The upper bound on the average SINR per bit decreases rapidly with increasing transmit antennas $N_t$ and falls below 0 dB for $N_t>5$ (see Figures 7(b) and 8(b)). Since the spectral efficiency of the system is $N_t/\left(2N_{\mathit{rt}}\right)$ 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 $N_t>5$.

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

$${\stackrel{\sim }{\mathbf{R}}}_k={\stackrel{\sim }{\mathbf{H}}}_k\stackrel{\sim }{\mathbf{B}}\mathbf{S}+{\stackrel{\sim }{\mathbf{W}}}_k$$E68

where

$$\stackrel{\sim }{\mathbf{B}}={\left[\begin{array}{cccc}1& 0& \cdots & 0\\ {\stackrel{\sim }{a}}_{1,1}& 1& \cdots & 0\\ ⋮& \cdots & \cdots & ⋮\\ {\stackrel{\sim }{a}}_{N_t-1,N_t-1}& \cdots & {\stackrel{\sim }{a}}_{N_t-1,1}& 1\end{array}\right]}^T\stackrel{\Delta }{=}{\stackrel{\sim }{\mathbf{A}}}^T$$E69

where ${\left(·\right)}^T$ denotes transpose. In (69), $\stackrel{\sim }{\mathbf{A}}$ is an $N_t\times N_t$ lower triangular matrix with diagonal elements equal to unity and ${\stackrel{\sim }{a}}_{i,j}$ denotes the $j^{\mathit{th}}$ coefficient of the optimum $i^{\mathit{th}}$-order forward prediction filter [40] and $\stackrel{\sim }{\mathbf{B}}$ is the precoding matrix. Let

$${\stackrel{\sim }{\mathbf{Y}}}_k={\stackrel{\sim }{\mathbf{B}}}^H{\stackrel{\sim }{\mathbf{H}}}_k^H{\stackrel{\sim }{\mathbf{R}}}_k={\stackrel{\sim }{\mathbf{B}}}^H{\stackrel{\sim }{\mathbf{H}}}_k^H{\stackrel{\sim }{\mathbf{H}}}_k\stackrel{\sim }{\mathbf{B}}\mathbf{S}+{\stackrel{\sim }{\mathbf{B}}}^H{\stackrel{\sim }{\mathbf{H}}}_k^H{\stackrel{\sim }{\mathbf{W}}}_k.$$E70

Define

$${\stackrel{\sim }{\mathbf{Z}}}_k={\stackrel{\sim }{\mathbf{H}}}_k\stackrel{\sim }{\mathbf{B}}=\left[\begin{array}{ccc}{\stackrel{\sim }{Z}}_{k,1,1}& \cdots & {\stackrel{\sim }{Z}}_{k,1,N_t}\\ ⋮& \cdots & ⋮\\ {\stackrel{\sim }{Z}}_{k,N_r,1}& \cdots & {\stackrel{\sim }{Z}}_{k,N_r,N_t}\end{array}\right].$$E71

Now [40]

$$\frac{1}{2}E\left[{\stackrel{\sim }{\mathbf{Z}}}_k^H{\stackrel{\sim }{\mathbf{Z}}}_k\right]=N_r\left[\begin{array}{cccc}{\sigma }_{Z,1}^2& 0& \cdots & 0\\ 0& {\sigma }_{Z,2}^2& \cdots & 0\\ ⋮& \cdots & \cdots & ⋮\\ 0& \cdots & 0& {\sigma }_{Z,N_t}^2\end{array}\right]\stackrel{\Delta }{=}{\stackrel{\sim }{\mathbf{R}}}_{\stackrel{\sim }{\mathbf{Z}}\stackrel{\sim }{\mathbf{Z}}}$$E72

is an $N_t\times N_t$ diagonal matrix and ${\sigma }_{Z,i}^2$ denotes the variance per dimension of the optimum ${\left(i-1\right)}^{\mathit{th}}$-order forward prediction filter. Note that [40]

$${\sigma }_{Z,1}^2={\sigma }_H^2;{\sigma }_{Z,i}^2\ge {\sigma }_{Z,j}^2\text{for}i<j.$$E73

Let

$${\stackrel{\sim }{\mathbf{V}}}_k={\stackrel{\sim }{\mathbf{Z}}}_k^H{\stackrel{\sim }{\mathbf{W}}}_k={\left[\begin{array}{ccc}{\stackrel{\sim }{V}}_{k,1}& \cdots & {\stackrel{\sim }{V}}_{k,N_t}\end{array}\right]}^T$$E74

which is an $N_t\times 1$ vector. Now

$$E\left[{\stackrel{\sim }{V}}_{k,i}{\stackrel{\sim }{V}}_{k,m}^{\ast }\right]=E\left[\sum _{j=1}^{N_r}{\stackrel{\sim }{Z}}_{k,j,i}^{\ast }{\stackrel{\sim }{W}}_{k,j}\sum _{l=1}^{N_r}{\stackrel{\sim }{Z}}_{k,l,m}{\stackrel{\sim }{W}}_{k,l}^{\ast }\right]=\sum _{j=1}^{N_r}\sum _{l=1}^{N_r}E\left[{\stackrel{\sim }{Z}}_{k,l,m}{\stackrel{\sim }{Z}}_{k,j,i}^{\ast }\right]E\left[{\stackrel{\sim }{W}}_{k,j}{\stackrel{\sim }{W}}_{k,l}^{\ast }\right]$$

$$=\sum _{j=1}^{N_r}\sum _{l=1}^{N_r}2{\sigma }_{Z,i}^2{\delta }_K\left(i-m\right){\delta }_K\left(j-l\right)\times 2{\sigma }_W^2{\delta }_K\left(j-l\right)=4N_r{\sigma }_{Z,i}^2{\sigma }_W^2{\delta }_K\left(i-m\right)$$E75

where we have used (72). Let

$${\stackrel{\sim }{\mathbf{F}}}_k={\stackrel{\sim }{\mathbf{Z}}}_k^H{\stackrel{\sim }{\mathbf{Z}}}_k$$E76

which is an $N_t\times N_t$ matrix. Substituting (76) in (70) we get

$${\stackrel{\sim }{\mathbf{Y}}}_k={\stackrel{\sim }{\mathbf{F}}}_k\mathbf{S}+{\stackrel{\sim }{\mathbf{V}}}_k.$$E77

Similar to (39), the $i^{\mathit{th}}$ element of ${\stackrel{\sim }{\mathbf{Y}}}_k$ in (77) is given by

$${\stackrel{\sim }{y}}_{k,i}\mathbf{=}{\stackrel{\sim }{F}}_{k,i,i}{S}_i\mathbf{+}{\stackrel{\sim }{I}}_{k,i}\mathbf{+}{\stackrel{\sim }{V}}_{k,i}\text{for}1\le i\le N_t$$E78

where

$${\stackrel{\sim }{V}}_{k,i}\mathbf{=}\sum _{j=1}^{N_r}{\stackrel{\sim }{Z}}_{k,j,i}^{\ast }{\stackrel{\sim }{W}}_{k,j};{\stackrel{\sim }{I}}_{k,i}\mathbf{=}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\stackrel{\sim }{F}}_{k,i,j}{S}_j;{\stackrel{\sim }{F}}_{k,i,j}\sum _{l=1}^{N_r}{\stackrel{\sim }{Z}}_{k,l,i}^{\ast }{\stackrel{\sim }{Z}}_{k,l,j}.$$E79

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

$$E\left[{\stackrel{\sim }{F}}_{k,i,i}\right]=2N_r{\sigma }_{Z,i}^2$$E80

Now

$$\begin{array}{l}E\left[{\stackrel{\sim }{F}}_{k,i,i}^2\right]=E\left[\sum _{l=1}^{N_r}{\left|{\stackrel{\sim }{Z}}_{k,l,i}\right|}^2\sum _{m=1}^{N_r}{\left|{\stackrel{\sim }{Z}}_{k,m,i}\right|}^2\right]=\sum _{l=1}^{N_r}{\left|{\stackrel{\sim }{Z}}_{k,l,i}\right|}^4+\\ +\sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^{N_r}E\left[{\left|{\stackrel{\sim }{Z}}_{k,l,i}\right|}^2\right]E\left[{\left|{\stackrel{\sim }{Z}}_{k,m,i}\right|}^2\right]=4N_r\left(N_r+1\right){\sigma }_{Z,i}^2.\end{array}$$E81

Similarly

$$E\left[{\left|{\stackrel{\sim }{I}}_{k,i}\right|}^2\right]=E\left[\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\stackrel{\sim }{F}}_{k,i,j}{S}_j\sum _{\begin{array}{c}l=1\\ l\ne i\end{array}}^{N_t}{\stackrel{\sim }{F}}_{k,i,l}^{\mathbf{\ast }}{S}_l^{\mathbf{\ast }}\right]=P_{\mathrm{av}}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}E\left[{\left|{\stackrel{\sim }{F}}_{k,i,j}\right|}^2\right].$$E82

Now

$$E\left[{\left|{\stackrel{\sim }{F}}_{k,i,j}\right|}^2\right]=E\left[\sum _{l=1}^{N_r}{\stackrel{\sim }{Z}}_{k,l,i}^{\ast }{\stackrel{\sim }{Z}}_{k,l,j}\sum _{m=1}^{N_r}{\stackrel{\sim }{Z}}_{k,m,i}{\stackrel{\sim }{Z}}_{k,m,j}^{\ast }\right]=\sum _{l=1}^{N_r}\sum _{m=1}^{N_r}4{\sigma }_{Z,i}^2{\sigma }_{Z,j}^2{\delta }_K\left(l-m\right)=4N_r{\sigma }_{Z,i}^2{\sigma }_{Z,j}^2$$E83

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

$$E\left[{\left|{\stackrel{\sim }{I}}_{k,i}\right|}^2\right]=4P_{\mathrm{av}}{N}_r{\sigma }_{Z,i}^2\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\sigma }_{Z,j}^2.$$E84

Note that

$$E\left[{\left|{\stackrel{\sim }{I}}_{k,i}+{\stackrel{\sim }{V}}_{k,i}\right|}^2\right]=E\left[{\left|{\stackrel{\sim }{I}}_{k,i}\right|}^2\right]+E\left[{\left|{\stackrel{\sim }{V}}_{k,i}\right|}^2\right].$$E85

The average SINR per bit for the $i^{\mathit{th}}$ transmit antenna is given by (52) and is equal to

$${\text{SINR}}_{\mathrm{av},b,i}=\frac{E\left[{\left|{\stackrel{\sim }{F}}_{k,i,i}{S}_i\right|}^2\times 2N_{\mathit{rt}}\right]}{E\left[{\left|{\stackrel{\sim }{I}}_{k,i}+{\stackrel{\sim }{V}}_{k,i}\right|}^2\right]}=\frac{P_{\mathrm{av}}\left(N_r+1\right){\sigma }_{Z,i}^2\times 2N_{\mathit{rt}}}{P_{\mathrm{av}}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\sigma }_{Z,j}^2+{\sigma }_W^2}$$E86

where we have used (75), (81) and (84). The upper bound on the average SINR per bit for the $i^{\mathit{th}}$ transmit antenna is obtained by setting ${\sigma }_W^2=0$ in (86) and is equal to

$${\text{SINR}}_{\mathrm{av},b,\mathrm{UB},i}=\frac{\left(N_r+1\right){\sigma }_{Z,i}^2\times 2N_{\mathit{rt}}}{\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\sigma }_{Z,j}^2}$$E87

which is illustrated in Figure 9 for $N_{\mathrm{tot}}=1024$ and $N_{\mathit{rt}}=2$. The value of the upper bound on the average SINR per bit for $N_t=i=50$ is 18.6 dB. The channel correlation is given by (67). Note that a first-order prediction filter completely decorrelates the channel with [40]

Figure 9.

Plot of SINR_av,b,UB,i for N_tot = 1024, N_rt = 2 after precoding. (a) Back view. (b) Sideview. (c) Front view.

$${\stackrel{\sim }{a}}_{i,1}\mathbf{=}\mathbf{-}0.9\text{for}1\le i\le N_t-1;{\stackrel{\sim }{a}}_{i,j}\mathbf{=}0\text{for}2\le i\le N_t-1,2\le j\le i.$$E88

We also have [40]

$${\sigma }_{Z,i}^2={\sigma }_{Z,2}^2=\left(1-{\left|-0.9\right|}^2\right){\sigma }_{Z,1}^2=0.19{\sigma }_{Z,1}^2\text{for}i>2.$$E89

Therefore we see in Figure 9 that the first transmit antenna $i=1$ has a high ${\text{SINR}}_{\mathrm{av},b,\mathrm{UB},i}$ due to low interference power from remaining transmit antennas, whereas for $i\ne 1$ the ${\text{SINR}}_{\mathrm{av},b,\mathrm{UB},i}$ is low due to high interference power from the first transmit antenna ($i=1$). The received signal after “combining” is given by (6) and (54). Note that from (54) and (79)

$$\begin{array}{l}E\left[{\stackrel{\sim }{F}}_i^2\right]=\frac{1}{N_{\mathit{rt}}^2}E\left[\sum _{k=0}^{N_{\mathit{rt}}-1}{\stackrel{\sim }{F}}_{k,i,i}\sum _{l=0}^{N_{\mathit{rt}}-1}{\stackrel{\sim }{F}}_{l,i,i}\right]=\frac{1}{N_{\mathit{rt}}^2}\sum _{k=0}^{N_{\mathit{rt}}-1}E\left[{\left|{\stackrel{\sim }{F}}_{k,i,i}\right|}^2\right]+\\ +\sum _{\begin{array}{c}l=0\\ l\ne k\end{array}}^{N_{\mathit{rt}}-1}E\left[{\stackrel{\sim }{F}}_{k,i,i}{\stackrel{\sim }{F}}_{l,i,i}\right]=\frac{4N_r{\sigma }_{Z,i}^2}{N_{\mathit{rt}}^2}\sum _{k=0}^{N_{\mathit{rt}}-1}\left(N_r+1\right)+\left(N_{\mathit{rt}}-1\right)N_r\\ =\frac{4N_r{\sigma }_{Z,i}^2}{N_{\mathit{rt}}^2}\left(1+N_r{N}_{\mathit{rt}}\right)\end{array}$$E90

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

$$\begin{array}{rcl}E\left[{\stackrel{\sim }{U}}_i^2\right]& =& \frac{1}{N_{\mathit{rt}}^2}E\left[\sum _{k=0}^{N_{\mathit{rt}}-1}{\stackrel{\sim }{U}}_{k,i}^{\prime }\sum _{l=0}^{N_{\mathit{rt}}-1}{\left({\stackrel{\sim }{U}}_{l,i}^{\prime }\right)}^{\ast }\right]=\frac{1}{N_{\mathit{rt}}^2}\sum _{k=0}^{N_{\mathit{rt}}-1}\sum _{l=0}^{N_{\mathit{rt}}-1}E\left[{\stackrel{\sim }{U}}_{k,i}^{\prime }{\left({\stackrel{\sim }{U}}_{l,i}^{\prime }\right)}^{\ast }\right]\\ & =& \frac{1}{N_{\mathit{rt}}^2}\sum _{k=0}^{N_{\mathit{rt}}-1}\sum _{l=0}^{N_{\mathit{rt}}-1}E\left[{\left|{\stackrel{\sim }{U}}_{k,i}^{\prime }\right|}^2\right]{\delta }_K\left(k-l\right)\end{array}$$

$$=\frac{1}{N_{\mathit{rt}}}E\left[{\left|{\stackrel{\sim }{U}}_{k,i}^{\prime }\right|}^2\right]=\frac{1}{N_{\mathit{rt}}}\left[E\left[{\left|{\stackrel{\sim }{I}}_{k,i}\right|}^2\right]+E\left[{\left|{\stackrel{\sim }{V}}_{k,i}\right|}^2\right]\right]=\frac{4N_r{\sigma }_{Z,i}^2}{N_{\mathit{rt}}}\left[P_{\mathrm{av}}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\sigma }_{Z,j}^2+{\sigma }_W^2\right].$$E91

Substituting (90) and (91) in (65) we have, after simplification, for $1\le i\le N_t$

$${\text{SINR}}_{\mathrm{av},b,C,i}=\frac{2P_{\mathrm{av}}E\left[F_i^2\right]}{E\left[{\left|{\stackrel{\sim }{U}}_i\right|}^2\right]}=\frac{\left(N_r{N}_{\mathit{rt}}+1\right){\sigma }_{Z,i}^2\times 2P_{\mathrm{av}}}{P_{\mathrm{av}}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\sigma }_{Z,j}^2+{\sigma }_W^2}.$$E92

The upper bound on the average SINR per bit for the $i^{\mathit{th}}$ transmit antenna is obtained by substituting (92) in (66) and is equal to

$${\text{SINR}}_{\mathrm{av},b,C,\mathrm{UB},i}=\frac{\left(N_r{N}_{\mathit{rt}}+1\right){\sigma }_{Z,i}^2\times 2}{{\sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_t}{\sigma }_{Z,j}^2}\approx {\text{SINR}}_{\mathrm{av},b,\mathrm{UB},i}$$E93

for $1\le i\le N_t$, $N_r\gg 1$. This is illustrated in Figure 10 for $N_{\mathrm{tot}}=1024$ and $N_{\mathit{rt}}=2$. We again observe that the first transmit antenna ($i=1$) 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 $N_t=i=50$, $N_{\mathrm{tot}}=1024$ is 18.6 dB. After concatenation, ${\stackrel{\sim }{Y}}_i$ for $0\le i\le L_d-1$, in (6) and (54) is given to the turbo decoder [29, 40]. Let (see (26) of [29]):

Figure 10.

Plot of SINR_av,b,C,UB,i for N_tot = 1024, N_rt = 2 after precoding. (a) Back view. (b) Side view. (c) Front view.

$${\gamma }_{1,i,m,n}=exp\left[-\frac{{\left|{\stackrel{\sim }{Y}}_i-F_i{S}_{m,n}\right|}^2}{2{\sigma }_{U,i}^2}\right];{\gamma }_{2,i,m,n}=exp\left[-\frac{{\left|{\stackrel{\sim }{Y}}_{i1}-F_{i1}{S}_{m,n}\right|}^2}{2{\sigma }_{U,i}^2}\right]$$E94

$${\stackrel{\sim }{\mathbf{Y}}}_1=\left[{\stackrel{\sim }{Y}}_1\cdots \cdots {\stackrel{\sim }{Y}}_{L_{d1}-1}\right];{\stackrel{\sim }{\mathbf{Y}}}_2=\left[{\stackrel{\sim }{Y}}_{L_{d1}}\cdots \cdots {\stackrel{\sim }{Y}}_{L_d-1}\right].$$E95

Thenwhere

$$i1=i+L_{d1}\text{for}0\le i\le L_{d1}-1.$$E96

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.

3.4 Simulation results

The channel correlation is given by (67). The BER results for $N_{\mathrm{tot}}=1024$ 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 $N_t=i=512$, $N_{\mathrm{tot}}=1024$ is 6 dB. The BER results for $N_{\mathrm{tot}}=32$ 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 $N_{\mathrm{tot}}=32$, 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 11.

Simulation results with precoding for N_tot = 1024.

Figure 12.

Simulation results with precoding for N_tot = 32.

Figure 13.

Simulation results with precoding and dummy data for N_tot = 32.

Figure 14.

Simulation results with precoding and dummy data for N_tot = 1024.

4. Conclusions and future work

This article presents the advantages of single-user massive multiple input multiple output (SU-MMIMO) over multi-user (MU) MMIMO systems. The bit-error-rate (BER) performance of SU-MMIMO using serially concatenated turbo codes (SCTC) over uncorrelated channel is presented. A semi-analytic approach to estimating the BER of a turbo code is derived. A detailed signal-to-interference-plus-noise ratio analysis for SU-MMIMO over correlated channel is presented. The BER performance of SU-MMIMO with parallel concatenated turbo code (PCTC) over correlated channel is studied. Future work could involve estimating the MMIMO channel, since the present work assumes perfect knowledge of the channel.