Low-Complexity Interference Cancellation Algorithms for Detection in Media-based Modulated Uplink Massive-MIMO Systems

We consider an $N_{r}\times U$ mMIMO system with $N_{r}$ BS antennas and $U$ single antenna users ($N_{r}>>U$ for e.g., $N_{r}=128,U=16$ ). Each user employs MBM for transmission of information to the BS as depicted in Fig. 1. We also consider that each user is having $n_{rf}$ RF mirrors placed near the antenna. Using MBM, the information is transmitted in two parts: 1. the mirror ON/OFF status, and 2. the conventional modulation symbol using the constellation set $\mathbb{A}$ (e.g., 4-QAM, QPSK, 16-QAM). To switch mirrors ON/OFF, each user requires $n_{rf}$ bits of incoming information, and therefore, there are $M=2^{n_{rf}}$ ON/OFF combinations possible which are termed as MAPs. In a rich scattering environment, each MAP corresponds to an independent channel fade realization r11 ; r12 . After selection of an MAP, $\log_{2}{|{\mathbb{A}}|}$ additional bits are transmitted through the antenna by selecting one symbol from the modulation alphabet $\mathbb{A}$. Therefore, the spectral efficiency of a multi-user MBM system in terms of bits per channel use (bpcu) can be mathematically given by

Let, the channel state vector between the $j$th MAP selected by $k$-th user and the base station is represented by $\mathbf{h}^{j}_{k}$. Each $\mathbf{h}^{j}_{k}=[h^{j}_{1,k},h^{j}_{2,k},\cdots,h^{j}_{N_{r},k}]_{N_{r}\times 1}^{T}$, for all $j=1,2,\cdots,M$ and $k=1,2,\cdots,U$, where $h^{j}_{i,k}$ is assumed to be independent and identically distributed (i.i.d.) complex Gaussian random variable with zero mean and unit variance i.e. $\sim\mathcal{CN}(0,1)$. Therefore, the channel matrix comprising of channel vectors corresponding to all the possible MAPs for the $k$-th user is $\mathbf{H}_{k}=[\mathbf{h}^{1}_{k},\mathbf{h}^{2}_{k},\cdots,\mathbf{h}^{M}_{k}]$. The MBM signal set for a single user can be defined as the set of all the possible transmit symbol vectors as

where $\mathbf{s}_{j,i}$ is an $M\times 1$ vector with only one non-zero entry $q_{i}\in\mathbb{A}$ corresponding to the $j$th channel fade realization. For the $k$-th user, let us denote the transmit symbol vector is denoted by $\mathbf{x}_{k}\in\mathbb{S}_{MBM}$. Thus, the received symbol vector at the BS after performing matched filtering and sampling operations can be written as

where $\mathbf{n}$ is an $N_{r}\times 1$ additive white Gaussian noise (AWGN) vector with $\mathbf{n}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{N_{r}})$. Further, for simplicity, we can rewrite the received symbol vector as

where $\mathbf{H}=[\mathbf{H}_{1}\ \ \mathbf{H}_{2}\ \ \cdots\ \ \mathbf{H}_{U}]$ is an $N_{r}\times UM$ MBM-mMIMO channel matrix and $\mathbf{x}=[\mathbf{x}_{1}^{T}\ \ \mathbf{x}_{2}^{T}\ \ \cdots\ \ \mathbf{x}_{U}^{T}]^{T}$ is $UM\times 1$ MBM-mMIMO transmit symbol vector comprising of transmit symbol vectors of all the users.

The objective at the receiver is to extract the information embedded in the selection of MAP as well as detect the transmitted information symbol from the constellation set for realizing the potential benefits of MBM. An optimal detection rule termed as ML detection suggests an exhaustive search over all the possible combinations of the transmit symbol vector $\mathbf{x}$ which minimize the ML cost, where ML cost associated with the symbol vector $\mathbf{x}$ is given by

Therefore, the ML detection problem given the received vector $\mathbf{y}$ and the channel state information $\mathbf{H}$ can be formulated as

where $\mathbb{S}^{U}_{MBM}$ is the set of all the possible combinations of $\mathbf{x}$. In MBM-mMIMO systems, for a given value of $U$, $M$ and $|{\mathbb{A}}|$ the dimensions of $\mathbf{x}$ equals $M\times U$ and the set $\mathbb{S}^{U}_{MBM}$ contains $\left(M\times|{\mathbb{A}}|\right)^{U}$ possible combinations of $\mathbf{x}$ for e.g., if $U=16$ and $M=2^{n_{rf}}=8$ then for a 4-QAM modulated MBM-mMIMO systems the dimensions of $\mathbf{x}$ is 128 ($M\times U=16\times 8=128$) and the size of $|\mathbb{S}^{U}_{MBM}|=32^{16}\approx 1.2\times 10^{24}$. Clearly, ML detection is unreasonable in such systems. Therefore, the design of low-complexity detection techniques is a challenging and a crucial problem for practical realization of MBM-mMIMO in the B5G wireless systems.

In this section, we discuss the ISD algorithm proposed in r24 for detecting the transmitted information symbols in mMIMO systems. In ISD, symbols corresponding to each user are detected in a sequential manner. In each iteration of ISD, symbols corresponding to all the users are detected which are refined in next iterations. An initial solution is used to initialize the algorithm, which could be an all zero solution, i.e., $\hat{\mathbf{x}}_{j}^{0}=0$ for all $j=1,2,\cdots,U$. Next, for detection of symbol corresponding to the $k$-th user in the $t$-th iteration, interference from all the other users is cancelled as

where $\hat{\mathbf{x}}_{i}^{t-1}$ and $\hat{\mathbf{x}}_{i}^{t}$ is the symbol corresponding to the $i$-th user in the $(t-1)$-th and the $(t)$-th iterations, respectively. The residual error vector $\mathbf{r}_{k}^{(t)}$ is then passed through a matched filter for detecting $\hat{\mathbf{x}}_{i}^{t}$ as

where $\mathcal{Q}[\cdot]$ is the quantization operation which maps the soft values to the nearest constellation points. These steps are repeated for multiple iterations to obtain a better solution. It is worth noting that the detection of MBM symbols using the ISD algorithm directly is an inefficient way of detection due to the presence of MAPs. Moreover, selecting a reliable solution for each user and a proper stopping rule are required to avoid error propagation and terminate the algorithm early, respectively. Therefore, the selection of a reliable MAP and solution for each user is crucial for detecting the transmitted symbol in MBM-mMIMO systems. Also, the stopping rule will help in terminating the algorithm early, thereby saving the required computations. This motivates further modifications in ISD to detect symbols in MBM-mMIMO systems with low-complexity. Algorithm 1 presents the pseudo code of ISD algorithm for mMIMO detection.

In this section, we discuss the IIC algorithm proposed in r14 . In IIC, the symbol detection corresponding to each user is a two step process. The algorithm starts with an all zero solution, i.e., $\hat{\mathbf{x}}_{i}^{(0)}=0$ $\forall i=1,2,\cdots,U$. In the first step, the residual signal is obtained as

where $\hat{\mathbf{x}}_{i}^{(t-1)}$ is the estimated MBM symbol vector of the $i$-th user in $(t-1)$-th iteration, and $\mathbf{r}^{(t)}$ is the residual signal. $\mathbf{y}_{j}^{(t)}$ is the interference cancellation vector for the $j$-th user in the $t$-th iteration. Next, a search is performed over the set of all the possible MBM symbol vectors for detecting the symbol vector corresponding to the $j$th user as

IIC algorithm is initialised with an all-zero solution. After performing these two steps for all the users, a greedy search is performed multiple times to update the solution for each user by using the selection metric given as

These steps are performed for multiple iterations so that the algorithm converge to a better solution. The pseudo code of IIC algorithm is described in Algorithm 2.

The size of search space $\mathbb{S}_{MBM}$ for each user in MBM-mMIMO is $\left(M\times|\mathbb{A}|\right)$ which grows exponentially with the number of RF mirrors and linearly with the size of constellation set used. To perform ML search for each user in a single iteration, it requires IIC to search over $M\times|\mathbb{A}|$ possible solutions. Therefore, the total search operations in performing ML search is $\left(M\times|\mathbb{A}|\right)UL$. This makes the algorithm computationally expensive and motivates for further research in the algorithm to improve the practical feasibility of IIC in MBM-mMIMO systems.

In this algorithm, we propose to modify the ISD algorithm r24 , which was originally proposed to detect the symbols in mMIMO systems. Here, we improvise the ISD algorithm for detecting MBM symbols corresponding to each user in MBM-mMIMO systems. In particular, we introduce a rule for selecting the most favorable MAP corresponding to each user from the list of all the possible MAPs in MBM-mMIMO system. For this, first, we utilize the concept of channel hardening, which occurs in mMIMO systems, and then find the pseudo-inverse of the individual channel matrices $\mathbf{H}_{k}$. The key idea for computing the pseudo-inverse is to find the highly erroneous locations by applying low-complexity zero-forcing over the residual vector (as discussed later in this section). The MAP corresponding to these erroneous locations are then explored for detecting the symbol transmitted by a particular user. Upon detection, the residual vector is updated accordingly.

In this section, we discuss the proposed KMAP-IIC algorithm for detecting symbols in MBM-mMIMO systems. Through selection of multiple favorable MAPs, the computational complexity of the existing IIC can be reduced significantly without compromising the BER performance (as discussed in Section 4). In the proposed algorithm, multiple favorable MAPs are selected using Eq. (17) and a list $\mathbb{L}_{j}^{(t)},\ \forall j=1,2,\cdots,U$, of $K$-favorable MAPs is generated. From $\mathbf{e}_{j}^{(t)}$, we select $K$ elements having largest magnitude value to update $\mathbb{L}_{j}^{(t)}$ i.e. we sort $|{e}_{j,i}^{(t)}|$ for all $i=1,2,\cdots,M$ in descending order, and select the indices of first $K$ element.
Next, we define the reduced size set $\tilde{\mathbb{S}}_{MBM}^{j,(t)}$ of favorable MBM symbol vectors for the $j$th user in the $t$th iteration as

After finding the list of favorable MAPs and the reduced size set of MBM symbol vectors, a search is performed over the set $\tilde{\mathbb{S}}_{MBM}^{j,(t)}$ for all the users in order to determine the best candidate solution corresponding to each user. The solution in the set $\tilde{\mathbb{S}}_{MBM}^{j,(t)}$ which minimizes the norm of residual vector is selected as the best candidate solution for the $j$th user defined as

Moreover, for a square- and rectangular-QAM constellation, a low-complexity search, proposed in r22 , is utilized in the proposed algorithm to reduce the computational complexity further. The real and imaginary parts of the estimated solution for square- and rectangular-QAM constellations can be expressed mathematically as

where $m_{1}=\Re({z}_{l})$, $m_{2}=\Im({z}_{l})$ and $z_{l}=\frac{\mathbf{h}_{j,l}^{H}\mathbf{r}_{j}^{(t)}}{\|\mathbf{h}_{j,l}\|^{2}}$, and $N_{1},N_{2}$ are the sizes two PAMs on the real and imaginary axis of the QAM constellation, respectively. Thus the best possible solution corresponding to the $l$th MAP of the $j$th user is $\hat{\mathbf{s}}_{l}=\Re(\hat{\mathbf{s}}_{l})+i\Im(\hat{\mathbf{s}}_{l})$ r22 . This makes the search independent of the number of points in the constellation set, and therefore, reduces the computational complexity incurred during the detection.
The search for the best possible solution is followed by symbol update stage where symbol vectors corresponding to the selected user is updated. A greedy update strategy proposed in r8 is used which requires $U$ iterations to update the solution corresponding to several users in order to minimize the ML cost. In each iteration, the user for which the ML cost is selected by using

and the solution corresponding to the selected user is updated. This update strategy is initiated only after completion of search over reduced size set for all the users. Algorithm 4 presents the pseudo code of the KMAP-IIC algorithm.

In Figs. 2 and 3, we present the BER performance of MAP-ISD with different iterations $L=1,2,4,6$ and $8$ for $128\times 20$ mMIMO system, $n_{rf}=3$ and $128\times 16$ mMIMO system, $n_{rf}=4$, respectively. It is observed that the BER performance of MAP-ISD improves with increase in the number of iterations and converges after $L=6$. Significant improvement in BER performance is observed for increase in the value of $L$ from 1 to 2 and from 2 to 4 whereas marginal improvement is observed for further increase in the value of $L$.

In Fig. 4, we simulate the BER performance of KMAP-IIC algorithm with different iterations $L=1,2,4$ and $6$ for $128\times 20$ mMIMO systems with 4-QAM, $K=M/2$ and $n_{rf}=3$ MBM. Similar observations can be drawn which suggests that the BER performance of the KMAP-IIC algorithm converges for $L=6$. Therefore, for comparison of BER of the proposed algorithms, i.e. MAP-ISD and KMAP-IIC, with other detection techniques (discussed later in the section) we use $L=6$ iterations.

In Figs. 8 and 9, the BER performance comparison is performed for $128\times 20$ mMIMO system with $n_{rf}=3$ and $n_{rf}=5$ with 16-QAM modulation. Observations reveal that the proposed algorithms perform superior over the IESP algorithm. However, the performance of MAP-ISD and KMAP-IIC with $K=1$ degrades with an increase in $n_{rf}$ as compared to KMAP-IIC with $K=M/4$, KMAP-IIC with $K=M/2$ and IIC. It is also observed that KMAP-IIC with $K=M/4$ achieves performance close to KMAP-IIC with $K=M/2$ and IIC algorithms.
In Fig. 10, we present the BER performance with variation in the number of BS antennas from $N_{r}=80$ to $N_{r}=140$ for $U=20$, $n_{rf}=4$ and 4-QAM MBM-mMIMO system. It is observed that the BER performance improves with an increase in the BS antennas. Also, the performance of KMAP-IIC with different values of $K$, i.e., $K=1,M/4,M/2$, is close to that of the IIC algorithm. On the other hand, the performance of MAP-ISD is far from IIC. For example, for a target BER of $5\times 10^{-4}$, required BS antennas for IIC and KMAP-IIC are around $125$, whereas MAP-ISD requires around $140$ BS antennas. Moreover, channel-hardening is not observed to degrade the performance compared to the IIC algorithm.

This section discusses the computational complexity of the proposed algorithms and compares them with the IIC algorithm r8 . First, we discuss the approximate computations involved in different steps of the IIC algorithm in Table 3 in terms of floating-point operations (FLOPs) r23 ,r24 . It is to note that the algorithms’ initialization does not require any computation due to all zero initial solutions.

In Table 4 and 5, we present the computations required in different steps of the MAP-ISD and KMAP-IIC algorithms, respectively.

It is observed from Tables 3, 4, and 5 that the computations in MAP-ISD are significantly less than that of IIC and KMAP-IIC algorithm, which is due to the complex search mechanism in IIC and KMAP-IIC algorithm. The computation of finding a low-complexity matrix inversion in MAP-ISD and KMAP-IIC is required only once. Therefore, the reduction in search space due to favorable MAP selection and low-complexity search are the key factors for the overall reduction in the proposed algorithms’ computational complexity. Fig. 11 presents the comparison of average number of FLOPs for different MBM-mMIMO systems at SNR= 5 dB. In Fig. 11, System 1 refers to $128\times 20$ mMIMO systems with $n_{rf}=3$, system 2 refers to $128\times 20$ mMIMO systems with $n_{rf}=4$ and system 3 refers to $128\times 16$ mMIMO systems with $n_{rf}=6$. Observations reveal that KMAP-IIC with $K=M/2$ achieves significant computational gain of upto $70\%$ and $63\%$ for $128\times 16$, $n_{rf}=6$ and $128\times 20$, $n_{rf}=4$ systems, respectively. This gain increases further to $76\%$ and $80\%$ when MAP-ISD algorithm is used for $128\times 16$, $n_{rf}=6$ and $128\times 20$, $n_{rf}=4$ systems, respectively. It is clear from the figure that the computational complexity of MAP-ISD is significantly less as compared to the KMAP-IIC with $K=1$, $K=M/4$ and $K=M/2$. Therefore, in case of MBM-mMIMO systems with less value of $n_{rf}$, MAP-ISD is suitable over KMAP-IIC with $K=1$ and other algorithms. On the other hand for moderate values of $n={rf}$ KMAP-IIC with $K=1$ and $K=M/4$ can be used to achieve a reasonable BER performance.

In Fig. 12, we present the comparison on the average number of floating point operations (FLOPs) of the IIC algorithm with MAP-ISD and KMAP-IIC algorithms, respectively, with respect to the number of receive antennas for MBM-mMIMO system having $16$ users and $n_{rf}=4$ at each user. Observation reveals that the computational complexity increase linearly with increase in the number of receive antennas at the station. It is also observed that the computational complexity increases with increase in $K$. Furthermore, the complexity of MAP-ISD is significantly less compared with the KMAP-IIC algorithm and the IIC algorithm r8 .