Low-Complexity Interference Cancellation Algorithms for Detection in Media-based Modulated Uplink Massive-MIMO Systems
NNoname manuscript No. (will be inserted by the editor)
Low-Complexity Interference CancellationAlgorithms for Detection in Media-based ModulatedUplink Massive-MIMO Systems
Manish Mandloi* · Devendra Singh Gurjar
Received: date / Accepted: date
Abstract
Media-based modulation (MBM) is a novel modulation techniquethat can improve the spectral efficiency of the existing wireless systems. InMBM, multiple radio frequency (RF) mirrors are placed near the transmitantenna(s) and are switched ON/OFF to create different channel fade re-alizations. In such systems, additional information is conveyed through theON/OFF status of RF mirrors along with conventional modulation symbols.A challenging task at the receiver is to detect the transmitted informationsymbols and extract the additional information from the channel fade real-ization used for transmission. In this paper, we consider a massive MIMO(mMIMO) system where each user relies on MBM for transmitting informa-tion to the base station, and investigate the problem of symbol detection at thebase station. First, we propose a mirror activation pattern (MAP) selectionbased modified iterative sequential detection algorithm. With the proposedalgorithm, the most favorable MAP is selected, followed by the detection ofsymbol corresponding to the selected MAP. Each solution is subjected to thereliability check before getting the update. Next, we introduce a K favorableMAP search based iterative interference cancellation (KMAP-IIC) algorithm.In particular, a selection rule is introduced in KMAP-IIC for deciding the setof favorable MAPs over which iterative interference cancellation is performed,followed by a greedy update scheme for detecting the MBM symbols corre-sponding to each user. Simulation results show that the proposed detection M. Mandloi (Corresponding author)Department of Electronics and Telecommunication Engineering, SVKM’s NMIMS (Deemedto be University) Shirpur campus, Maharashtra 425405, IndiaTel.: +91-9584858734E-mail: [email protected]. S. GurjarDepartment of Electronics and Communication Engineering, National Institute of Technol-ogy Silchar, Assam 788010, IndiaE-mail: [email protected] a r X i v : . [ c s . I T ] J a n Manish Mandloi*, Devendra Singh Gurjar algorithms exhibit superior performance-complexity trade-off over the exist-ing detection techniques in MBM-mMIMO systems.
Keywords
Media-based modulation · RF mirrors · massive MIMO · iterativeinterference cancellation · structured sparsity · channel hardening The explosive increase in the number of subscribers, the use of data thirstyapplications, and ubiquitous computing for low latency applications pose aserious challenge towards the design of advanced communication techniquesfor 5G and beyond wireless systems. Over the last decade, different wire-less techniques, such as non-orthogonal multiple access (NOMA), millimeter-wave (mm-Wave) communications, vehicle-to-everything (V2X) communica-tions, massive MIMO (mMIMO), and machine learning in wireless networks,have been proposed in the literature to enhance the performance of the exist-ing wireless systems [1,2,3]. Amongst these, mMIMO has been considered asa promising technique to satisfy the requirement of high data rate for 5G andbeyond wireless systems [4,5,6]. In mMIMO, a large number of base station(BS) antennas are used to serve comparatively small number of single/multipleantenna users [4,5]. One of the key advantages of such systems is that simplelinear precoders and decoders can achieve near-optimal bit error rate (BER)performance which includes zero-forcing and minimum mean squared errordetectors [4,7]. Some of the key challenges in practical implementation ofmultiple antenna systems include inter channel interference, requirement ofdedicated RF chains at each transmit antenna, and constraints on the totalnumber of receive antennas. Index modulation is one such digital modulationtechnique proposed to overcome these challenging issues in MIMO systems [8,9]. Through index modulation, extra information bits are embedded in theswitching pattern of the building blocks along with the transmission of sym-bols selected from the conventional constellation set such as M-ary PSK or M-QAM [10]. This results in relaxing the need of all the available resources at thetransmitter such as transmit antennas and RF chains [11,12]. In conventionalindex modulation (IM) schemes, such as spatial modulation (SM) and gener-alised SM (GSM), only one or fewer antenna elements and RF chains are activewhich reduces the energy consumption as well as the inter channel interferencethereby providing a higher energy efficiency. IM together with mMIMO [13,14] has been evolved as an emerging technique for fulfilling the higher spectralas well as the higher energy efficiency requirements in beyond fifth-generation(B5G) wireless systems [10]. One of the key challenges in IM-mMIMO systemsis the detection of the transmitted information which requires detection of theselected switching pattern and the detection of transmitted symbol.Recently, media-based modulation (MBM) has been introduced as a po-tential IM scheme to enhance the spectral efficiency and the energy efficiencyof the existing wireless systems by embedding information in different channelfade realizations [15], [16]. These different end-to-end channel fade realizations itle Suppressed Due to Excessive Length 3 are created by modifying the radio frequency (RF) properties of the propaga-tion medium near to the transmitter [15]. To achieve these variations, multipleRF mirrors (parasitic elements), which are digitally controlled by the inputdata bits, are placed near the transmit antenna(s) [16]–[18] . The ON/OFF sta-tus of RF mirrors also referred to as mirror activation pattern (MAP) conveyadditional information bits along with the conventional modulation symbol.Through n rf RF mirrors, 2 n rf MAPs can be generated. Each MAP resultsin an independent channel fade realization which is used by the transmittingantenna to transmit information symbol selected from a constellation set. Thepractical advantages of MBM systems have been demonstrated in [16] with14 RF mirrors placed near a dipole transmit antenna. Moreover, the antennadesign, radiation pattern, and potential advantages of MBM are discussed in[16], [17]. The use of MBM in mMIMO systems, which is referred as MBM-mMIMO, has been manifested to achieve significant improvements in terms ofBER performance, and spectral efficiency [19,20] over the existing modulationschemes such as SM and GSM [21,22,23,24,25,26]. It is worth noting that,in MBM the throughput enhancement is linear with respect to the number ofRF mirrors, whereas, in SM and GSM, the increase in throughput is logarith-mic [19]. Therefore, MBM is being considered as an emerging spectral efficientIM scheme in B5G wireless systems. However, one of the major challenges inMBM-mMIMO systems is to detect the transmitted information symbol, andto extract the additional information conveyed through the selection of chan-nel fade realization by each user, reliably with low computational complexity[19,27].Optimal detection rule suggests an exhaustive search over the set of allthe possible MBM symbol vector [19,20,27], which is termed as maximum-likelihood (ML) detection. However, due to exponentially high computationalcomplexity, ML detection is impractical in MBM-mMIMO systems [19]. It isdue to the fact that the set of all the possible symbol vectors for each useris generated by using all the possible combinations of MAPs and constella-tion points which ultimately results in an exponential increase in the size ofthe search space for ML detection with respect to the number of transmitantennas and n rf . Though, ZF/MMSE achieves near-optimal performance inmMIMO systems [4], [28,29,30,31], their performance in MBM-mMIMO sys-tems is sub-optimal which makes ZF/MMSE less selective for MBM-mMIMOsystems [11,12]. The sub-optimal BER performance of ZF/MMSE and ex-ponentially high computational complexity of ML detection motivates for thedesign of low-complexity detection algorithms capable of achieving better BERperformance in MBM-mMIMO systems.Recently, inclusion-exclusion subspace pursuit (IESP) [19] and iterativeinterference cancellation (IIC) [20] algorithms have been proposed to achievebetter BER performance over the conventional detection techniques such asMMSE and successive interference cancellation (SIC) [16,20,32] in MBM-mMIMO systems. However, the IESP algorithm [19] requires computation ofpseudo inverse at multiple stages which is computationally expensive in MBM-mMIMO system due to large dimensional channel matrix. On the other hand, Manish Mandloi*, Devendra Singh Gurjar
IIC algorithm [20] outperforms MMSE, SIC and IESP algorithms in terms ofboth the BER performance and the computational complexity. In IIC [20],symbol vector transmitted by each user is detected by canceling interferencefrom all the other users followed by an exhaustive search over the set of all thepossible MAPs in an iterative manner. Although the computational complex-ity of IIC is less compared with MMSE, SIC and IESP algorithms, there isscope to reduce the computational complexity of IIC algorithm further by se-lecting a list of favorable MAPs for performing the search. In order to solve thedetection problem in MBM-mMIMO systems, we propose two low-complexitydetection schemes in this paper. First, we propose a MAP selection based it-erative sequential detection (MAP-ISD) algorithm where the most favorableMAP is selected using a selection metric. The symbol corresponding to theselected MAP is detected sequentially for each user while nullifying the inter-ference from all the other users. Next, we propose a K favorable MAP searchbased iterative interference cancellation (KMAP-IIC) algorithm. In particular, K most favorable MAPs are selected over which the search is performed fordetecting the symbol corresponding to each user. The selection metric is ob-tained by exploiting the channel hardening which occurs in mMIMO systems[33].Our key contributions in this article are; – A technique for selection of favorable mirror activation pattern (MAP) foreach user is proposed by utilizing channel hardening in mMIMO. – A low-complexity MAP-ISD algorithm is proposed for symbol detection inthe MBM-mMIMO systems which utilizes the concept of favorable MAPand low-complexity ML search in sparse vectors. – The concept of reliability check of the solution in each iteration and stop-ping rule are utilized in the MAP-ISD algorithm to obtain better BERwith low computational complexity. – Selection of K favorable MAPs and low-complexity ML search are alsointegrated with IIC and proposed KMAP-IIC algorithm. – Simulation results on BER and computational complexity of the proposedalgorithms are shown to validate superiority of the proposed algorithmsover the existing algorithms.Simulation results show that the proposed detection algorithms exhibit su-perior performance-complexity trade-off over the existing detection techniquessuch as MMSE, IESP, and IIC [19,20] in MBM-mMIMO systems. It is alsoobserved through simulations that the selection of a detection scheme betweenMAP-ISD and KMAP-IIC depends on the number of RF mirrors and the num-ber of users in the system.
Notation:
Boldface upper-case and lower-case letters denote matrices and col-umn vectors, respectively. ( · ) H and ( · ) − represent matrix Hermitian andmatrix-inversion, respectively. x i is the i th element of x . I U refers to U × U identity matrix, a i,j denotes the element in i th row and j th column of A , and a j denotes j th column of A . Q [ · ] denotes the quantization operation whichmaps the soft values to the nearest constellation point. itle Suppressed Due to Excessive Length 5 In this section, we discuss the mathematical model of MBM-mMIMO system,and introduce the ML detection rule for optimal detection of symbols in suchsystems. We also shed light on the prohibitive computational complexity ofML detection in MBM-mMIMO systems. Finally, we provide the detailed de-scription of ISD and IIC algorithms for detection in mMIMO systems andMBM-mMIMO systems, respectively.
Table 1
List of Notations N r Number of BS antennas U Number of users n rf Number of RF mirrors M = 2 n rf Number of mirror activation pattern (MAP) A Constellation set S MBM
MBM signal set for a single user S UMBM
MBM signal set for U user system L Number of iterations
MIMO Receiver
Detector
Index Detection+Symbol Detection Index DemappingSymbol Demapping
Output bits
Channel Estimation
User N r n rf RF Mirror control SwitchRF Chain
User i User 2 User U CHA N NEL
Fig. 1
System model for MBM-mMIMO system.
We consider an N r × U mMIMO system with N r BS antennas and U singleantenna users ( N r >> U for e.g., N r = 128 , U = 16 ). Each user employsMBM for transmission of information to the BS as depicted in Fig. 1. We Manish Mandloi*, Devendra Singh Gurjar also consider that each user is having n rf RF mirrors placed near the an-tenna. Using MBM, the information is transmitted in two parts: 1. the mirrorON/OFF status, and 2. the conventional modulation symbol using the con-stellation set 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 arich scattering environment, each MAP corresponds to an independent channelfade realization [15,16]. After selection of an MAP, log | A | additional bits aretransmitted through the antenna by selecting one symbol from the modulationalphabet A . Therefore, the spectral efficiency of a multi-user MBM system interms of bits per channel use (bpcu) can be mathematically given by η MBM = U ( n rf + log | A | ) . (1)Let, the channel state vector between the j th MAP selected by k -th user andthe base station is represented by h jk . Each h jk = [ h j ,k , h j ,k , · · · , h jN r ,k ] TN r × ,for all j = 1 , , · · · , M and k = 1 , , · · · , U , where h ji,k is assumed to be inde-pendent and identically distributed (i.i.d.) complex Gaussian random variablewith zero mean and unit variance i.e. ∼ CN (0 , k -th user is H k = [ h k , h k , · · · , h Mk ]. The MBM signal set for a single usercan be defined as the set of all the possible transmit symbol vectors as S MBM = { s j,i : j = 1 , , · · · , M, i = 1 , , · · · , | A |} , s.t. s j,i = [0 , · · · , , q i (cid:124)(cid:123)(cid:122)(cid:125) j th coordinate , , · · · , T , q i ∈ A , (2)where s j,i is an M × q i ∈ A correspondingto the j th channel fade realization. For the k -th user, let us denote the transmitsymbol vector is denoted by x k ∈ S MBM . Thus, the received symbol vectorat the BS after performing matched filtering and sampling operations can bewritten as y = U (cid:88) k =1 H k x k + n , (3)where n is an N r × n ∼ CN (0 , σ I N r ). Further, for simplicity, we can rewrite the received symbolvector as y = Hx + n , (4)where H = [ H H · · · H U ] is an N r × U M
MBM-mMIMO channel matrixand x = [ x T x T · · · x TU ] T is U M × itle Suppressed Due to Excessive Length 7 x which minimize theML cost, where ML cost associated with the symbol vector x is given by C ( x ) = (cid:107) y − Hx (cid:107) , (5)= (cid:107) y − U (cid:88) k =1 H k x k (cid:107) . (6)Therefore, the ML detection problem given the received vector y and thechannel state information H can be formulated asˆ x = arg min x ∈ S UMBM (cid:107) y − Hx (cid:107) . (7)where S UMBM is the set of all the possible combinations of x . In MBM-mMIMOsystems, for a given value of U , M and | A | the dimensions of x equals M × U and the set S UMBM contains ( M × | A | ) U possible combinations of x for e.g.,if U = 16 and M = 2 n rf = 8 then for a 4-QAM modulated MBM-mMIMOsystems the dimensions of x is 128 ( M × U = 16 × | S UMBM | = 32 ≈ . × . Clearly, ML detection is unreasonable insuch systems. Therefore, the design of low-complexity detection techniques isa challenging and a crucial problem for practical realization of MBM-mMIMOin the B5G wireless systems.2.3 Iterative Sequential Detection for mMIMO SystemsIn this section, we discuss the ISD algorithm proposed in [30] for detectingthe transmitted information symbols in mMIMO systems. In ISD, symbolscorresponding to each user are detected in a sequential manner. In each iter-ation of ISD, symbols corresponding to all the users are detected which arerefined in next iterations. An initial solution is used to initialize the algorithm,which could be an all zero solution, i.e., ˆ x j = 0 for all j = 1 , , · · · , 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 r ( t ) k = y − k − (cid:88) i =1 h i ˆ x ( t ) i − U (cid:88) i = k +1 h i ˆ x ( t − i , (8)where ˆ x t − i and ˆ x ti is the symbol corresponding to the i -th user in the ( t − t )-th iterations, respectively. The residual error vector r ( t ) k is then Manish Mandloi*, Devendra Singh Gurjar passed through a matched filter for detecting ˆ x ti asˆ x ti = Q (cid:20) h Hi (cid:107) h i (cid:107) r ( t ) k (cid:21) , (9)where Q [ · ] is the quantization operation which maps the soft values to thenearest constellation points. These steps are repeated for multiple iterations toobtain a better solution. It is worth noting that the detection of MBM symbolsusing the ISD algorithm directly is an inefficient way of detection due to thepresence of MAPs. Moreover, selecting a reliable solution for each user anda proper stopping rule are required to avoid error propagation and terminatethe algorithm early, respectively. Therefore, the selection of a reliable MAPand solution for each user is crucial for detecting the transmitted symbolin MBM-mMIMO systems. Also, the stopping rule will help in terminatingthe algorithm early, thereby saving the required computations. This motivatesfurther modifications in ISD to detect symbols in MBM-mMIMO systems withlow-complexity. Algorithm 1 presents the pseudo code of ISD algorithm formMIMO detection. Algorithm 1
ISD Algorithm Input: y , H , N r , U, L Initialization: x (0) = , r (0) = y , t = 0 (Iteration index)3: repeat
4: ˆ x ( t ) = ˆ x ( t − , r ( t ) = r ( t − , t = t + 15: for ( j = 1 , + + j, j ≤ U ) do
6: Compute : r ( t ) k = y − (cid:80) k − i =1 h i ˆ x ( t ) i − (cid:80) Ui = k +1 h i ˆ x ( t − i
7: Compute : z i = h Hi (cid:107) h i (cid:107) r ( t ) k
8: Update the symbol for i th user as : ˆ x ti = Q [ z i ]9: end for until t = L (terminate the iterative processing)11: Output: ˆ x L x (0) i = 0 ∀ i = 1 , , · · · , U . In the first step,the residual signal is obtained as r ( t ) = y − U (cid:88) i =1 H i ˆ x ( t ) i , (10) y ( t ) j = r ( t ) + H j ˆ x ( t ) j (11)where ˆ x ( t − i is the estimated MBM symbol vector of the i -th user in ( t − r ( t ) is the residual signal. y ( t ) j is the interference cancellation itle Suppressed Due to Excessive Length 9 vector for the j -th user in the t -th iteration. Next, a search is performed overthe set of all the possible MBM symbol vectors for detecting the symbol vectorcorresponding to the j th user asˆ s ( t ) j = arg min s j ∈ S MBM (cid:107) y ( t ) j − H j s j (cid:107) . (12)IIC algorithm is initialised with an all-zero solution. After performing thesetwo steps for all the users, a greedy search is performed multiple times toupdate the solution for each user by using the selection metric given as u = arg min k =1 , ··· ,U (cid:107) r ( t ) + H j (ˆ s ( t ) j − ˆ x ( t ) i ) (cid:107) . (13)These steps are performed for multiple iterations so that the algorithm con-verge to a better solution. The pseudo code of IIC algorithm is described inAlgorithm 2. Algorithm 2
IIC Algorithm Input: y , H , N r , U, n rf , L Initialization: x (0) = (Symbol estimation) , r (0) = y (Residual signal) , W i , ∀ i =1 , , · · · , U, t = 0 (Iteration index)3: repeat
4: ˆ x ( t ) = ˆ x ( t − , r ( t ) = r ( t − , t = t + 15: for ( j = 1 , + + j, j ≤ U ) do
6: Apply interference cancellation for the j -th user as: y ( t ) j = r ( t ) + H j ˆ x ( t ) j
7: Perform ML search for the j -th user as: ˆ s ( t ) j = arg min s j ∈ S MBM (cid:107) y ( t ) j − H j s j (cid:107) end for for ( j = 1 , + + j, j ≤ U ) do
10: Select the first element of u = arg min k =1 , ··· ,U (cid:107) r ( t ) + H j (ˆ s ( t ) j − ˆ x ( t ) i ) (cid:107) if (cid:107) r ( t ) (cid:107) > (cid:107) r ( t ) + H u (ˆ x ( t ) u − ˆ s ( t ) u ) (cid:107) then
12: Update the residual error vector as r ( t ) = r ( t ) + H u (ˆ x ( t ) u − ˆ s ( t ) u )13: Update the symbol vector for v th user as ˆ x ( t ) u = ˆ s ( t ) u else
15: break16: end if end for until ˆ x ( t ) = ˆ x ( t − or t = L (terminate the iterative processing or if no update in thesolution)19: ˆ x = ˆ x ( t ) Output: ˆ x The size of search space S MBM for each user in MBM-mMIMO is ( M × | A | )which grows exponentially with the number of RF mirrors and linearly with thesize of constellation set used. To perform ML search for each user in a single it-eration, it requires IIC to search over M ×| A | possible solutions. Therefore, thetotal search operations in performing ML search is ( M × | A | ) U L . This makesthe algorithm computationally expensive and motivates for further researchin the algorithm to improve the practical feasibility of IIC in MBM-mMIMOsystems.
In this section, we discuss the proposed algorithms, namely, MAP-ISD al-gorithm and KMAP-IIC algorithm for MBM symbol detection in mMIMOsystems.3.1 MAP Selection based ISDIn this algorithm, we propose to modify the ISD algorithm [30], which wasoriginally proposed to detect the symbols in mMIMO systems. Here, we im-provise the ISD algorithm for detecting MBM symbols corresponding to eachuser in MBM-mMIMO systems. In particular, we introduce a rule for selectingthe most favorable MAP corresponding to each user from the list of all thepossible MAPs in MBM-mMIMO system. For this, first, we utilize the con-cept of channel hardening , which occurs in mMIMO systems, and then findthe pseudo-inverse of the individual channel matrices H k . The key idea forcomputing the pseudo-inverse is to find the highly erroneous locations by ap-plying low-complexity zero-forcing over the residual vector (as discussed laterin this section). The MAP corresponding to these erroneous locations are thenexplored for detecting the symbol transmitted by a particular user. Upon de-tection, the residual vector is updated accordingly. In mMIMO systems, due to a large number of receive antennas as comparedto the number of users, the Gram matrix, G = H H H , becomes diagonaldominant which is a consequence of channel hardening [4,33]. Mathematically,it can be expressed as h Ti h j N r → , ∀ i (cid:54) = j, i, j ∈ { , , . . . , U M } , (14)where h j is the j th column of channel gain matrix H . Due to diagonal dominance, we can approximate the inverse of matrix G byusing matrix D [22] as G − ≈ D − , (15)where D is the diagonal matrix containing only the diagonal entries of G .Therefore, the problem of finding pseudo-inverse in mMIMO systems [22,23,24] can be approximated by using a low-complexity approximation as( H H H ) − H H = G − H H ≈ D − H H . (16)It is worth noting that, the approximate inverse of matrix G is computed onlyonce for each user and requires the computational complexity of O ( M N r ). itle Suppressed Due to Excessive Length 11 Next, we use the approximate pseudo-inverse obtained in Eq. (16) over theresidual vector defined in Eq. 10 to obtain the favorable MAP as e ( t ) j = W j y − U (cid:88) i =1 ,i (cid:54) = j H i ˆ x ( t − i . (17)where W j = D − j H Hj and D j is the diagonal of the Gram matrix H Hj H j .From e ( t ) j , we select the element having the largest magnitude value, i.e., | e ( t ) j,i | for all i = 1 , , · · · , M and select k as the index of the largest value whichis nothing but the index of the most favorable MAP. The magnitude valueof | e ( t ) j,i | resembles the non-zero location in the transmitted symbol vector x j which can be mathematically analysed using Eq. (17). The vector e ( t ) j can besimplified as e ( t ) j = W j H j x j + U (cid:88) i =1 ,i (cid:54) = j H i x i + n − U (cid:88) i =1 ,i (cid:54) = j H i ˆ x ( t − i = W j H j x j + U (cid:88) i =1 ,i (cid:54) = j W j H i (cid:16) x i − ˆ x ( t − i (cid:17) + W j n . (18)After incorporating the value of W j in Eq. (18) and further solving we get= D − j G j x j + U (cid:88) i =1 ,i (cid:54) = j D − j G ji (cid:16) x i − ˆ x ( t − i (cid:17) + (cid:101) n , (19)where (cid:101) n = D − j H Hj n and G ji = H Hj H i . Due to channel hardening D − j G j ≈ I and D − j G ji ≈ O . Clearly, the vector e ( t ) j reduces to a combination thetransmitted symbol x j and the noise vector (cid:101) n , i.e., e ( t ) j = x j + (cid:101) n . Obviously,the vector x j has only one non-zero location, and therefore, the magnitude | e ( t ) j,i | for all i = 1 , , · · · , M have the maximum value corresponding to thatnon-zero location.Finally, we decide the MBM symbol corresponding to the j th user asˆ s ( t ) j = [0 , · · · , , β (cid:124)(cid:123)(cid:122)(cid:125) k th coordinate , , · · · , T (20)where β = Q [ e j,k ] such that β ∈ A . The symbol for the j th user is updatedonly if it results in the reduction of the norm of residual vector, i.e., ˆ x ( t ) j = ˆ s ( t ) j if (cid:107) r ( t ) (cid:107) > (cid:107) r ( t ) + H j (ˆ x ( t − j − ˆ s ( t ) j ) (cid:107) . Similarly, the symbol corresponding toeach user is detected in a single iteration. The algorithm is run for multipleiterations say L for refining the detected symbols and obtaining the minimum norm of residual vector. Algorithm 3 presents the pseudo code of the MAP-ISDalgorithm for symbol detection in MBM-mMIMO systems. Algorithm 3
MAP-ISD Algorithm Input: y , H , N r , U, n rf , L Initialization: x (0) = , r (0) = y , W i , ∀ i = 1 , , · · · , U, t = 0 (Iteration index)3: repeat
4: ˆ x ( t ) = ˆ x ( t − , r ( t ) = r ( t − , t = t + 15: for ( j = 1 , + + j, j ≤ U ) do
6: Compute : e j = W j ( r ( t ) + H j ˆ x ( t − j )7: Find the element having largest | e ( t ) j,i | for all i = 1 , , · · · , M
8: Find the MBM symbol for the j th user as: ˆ s ( t ) j =[0 , · · · , , β (cid:124)(cid:123)(cid:122)(cid:125) k th coordinate , , · · · , T where β = Q [ e j,k ]9: if (cid:107) r ( t ) (cid:107) > (cid:107) r ( t ) + H j (ˆ x ( t − j − ˆ s ( t ) j ) (cid:107) then
10: Update the residual vector as r ( t ) = r ( t ) + H j (ˆ x ( t − j − ˆ s ( t ) j )11: Update the symbol vector for j th user as ˆ x ( t ) j = ˆ s ( t ) j end if end for until ˆ x ( t ) = ˆ x ( t − or t = L (terminate the iterative processing or if no update in thesolution)15: Output: ˆ x t K -favorable MAP based IICIn this section, we discuss the proposed KMAP-IIC algorithm for detectingsymbols in MBM-mMIMO systems. Through selection of multiple favorableMAPs, the computational complexity of the existing IIC can be reduced signif-icantly without compromising the BER performance (as discussed in Section4). In the proposed algorithm, multiple favorable MAPs are selected using Eq.(17) and a list L ( t ) j , ∀ j = 1 , , · · · , U , of K -favorable MAPs is generated. From e ( t ) j , we select K elements having largest magnitude value to update L ( t ) j i.e.we sort | e ( t ) j,i | for all i = 1 , , · · · , M in descending order, and select the indicesof first K element.Next, we define the reduced size set ˜ S j, ( t ) MBM of favorable MBM symbol vectorsfor the j th user in the t th iteration as˜ S j, ( t ) MBM = { s k,i ∈ A : k = L ( t ) j (1) , L ( t ) j (2) , · · · , L ( t ) j ( K ) ,i = 1 , , · · · , | A |} s.t. s k,i = [0 , · · · , , α i (cid:124)(cid:123)(cid:122)(cid:125) k th coordinate , , · · · , T , α i ∈ A . (21)After finding the list of favorable MAPs and the reduced size set of MBMsymbol vectors, a search is performed over the set ˜ S j, ( t ) MBM for all the users in itle Suppressed Due to Excessive Length 13 order to determine the best candidate solution corresponding to each user.The solution in the set ˜ S j, ( t ) MBM which minimizes the norm of residual vector isselected as the best candidate solution for the j th user defined asˆ s ( t ) j = arg min ˜ s q ∈ ˜ S j, ( t ) MBM (cid:107) r ( t ) j − H j ˜ s q (cid:107) . (22)Moreover, for a square- and rectangular-QAM constellation, a low-complexitysearch, proposed in [28], is utilized in the proposed algorithm to reduce thecomputational complexity further. The real and imaginary parts of the es-timated solution for square- and rectangular-QAM constellations can be ex-pressed mathematically as (cid:60) (ˆ s l ) = min (cid:20) max (cid:18) (cid:98) m + 12 (cid:101) − , − N + 1 (cid:19) , N − (cid:21) , (cid:61) (ˆ s l ) = min (cid:20) max (cid:18) (cid:98) m + 12 (cid:101) − , − N + 1 (cid:19) , N − (cid:21) , (23)where m = (cid:60) ( z l ), m = (cid:61) ( z l ) and z l = h Hj,l r ( t ) j (cid:107) h j,l (cid:107) , and N , N are the sizes twoPAMs 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 useris ˆ s l = (cid:60) (ˆ s l ) + i (cid:61) (ˆ s l ) [28]. This makes the search independent of the numberof points in the constellation set, and therefore, reduces the computationalcomplexity incurred during the detection.The search for the best possible solution is followed by symbol update stagewhere symbol vectors corresponding to the selected user is updated. A greedyupdate strategy proposed in [12] is used which requires U iterations to updatethe 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 v = arg min i =1 , , ··· ,U (cid:107) r ( t ) + H i (ˆ x ( t ) i − ˆ s ( t ) i ) (cid:107) , (24)and the solution corresponding to the selected user is updated. This updatestrategy is initiated only after completion of search over reduced size set for allthe users. Algorithm 4 presents the pseudo code of the KMAP-IIC algorithm. Algorithm 4
KMAP-IIC Algorithm Input: y , H , N r , U, n rf , L, K Initialization: x (0) = , r (0) = y , W i , ∀ i = 1 , , · · · , U, t = 0 (Iteration index)3: repeat
4: ˆ x ( t ) = ˆ x ( t − , r ( t ) = r ( t − , t = t + 15: for ( j = 1 , + + j, j ≤ U ) do
6: Compute : e j = W j ( r ( t ) + H j ˆ x ( t − j )7: Sort the elements in Descending Order and update L ( t ) j i.e. select indices of first K elements from arg sort ( | e ( t ) j, | , | e ( t ) j, | , . . . , | e ( t ) j,M | , descend )8: Generate the Reduced Search Space ˜ S j, ( t ) MBM
9: Perform search for the j th user as: ˆ s ( t ) j = arg min ˜ s q ∈ ˜ S j, ( t ) MBM (cid:107) r ( t ) j − H j ˜ s q (cid:107) end for for ( j = 1 , + + j, j ≤ U ) do
12: Select the first element of v = arg min i =1 , , ··· ,U (cid:107) r ( t ) + H i (ˆ x ( t ) i − ˆ s ( t ) i ) (cid:107) if (cid:107) r ( t ) (cid:107) > (cid:107) r ( t ) + H v (ˆ x ( t ) v − ˆ s ( t ) v ) (cid:107) then
14: Update the residual vector as r ( t ) = r ( t ) + H v (ˆ x ( t ) v − ˆ s ( t ) v )15: Update the symbol vector for v th user as ˆ x ( t ) v = ˆ s ( t ) v else
17: break18: end if if ˆ x ( t ) = ˆ x ( t − then
20: break21: end if end for until ˆ x ( t ) = ˆ x ( t − or t = L (terminate the iterative processing or if no update in thesolution)24: Output: ˆ x t In this section, we present the simulation results of the proposed detectionalgorithms in terms of BER performance and the computational complexityfor MBM-mMIMO systems. We also compare the BER performance of theproposed algorithms with the performance of MMSE detector [11], IESP algo-rithm [11] and IIC algorithm [12]. We consider 128 ×
16 and 128 ×
20 MBM-mMIMO systems with n rf = 3, n rf = 4, n rf = 5 and n rf = 6, respectively.In summary, Table 2 shows list of the parameters used in simulation of BERperformance and computational complexity. itle Suppressed Due to Excessive Length 15 Table 2
List of parameters used in simulation.Figure MBM-mMIMO System Parameters UsedFig. 2 N r = 128, U = 20, n rf = 3 4-QAM, L = 1 , , , , N r = 128, U = 16, n rf = 4 4-QAM, L = 1 , , , , N r = 128, U = 20, n rf = 3 4-QAM, L = 1 , , , K = M/ N r = 128, U = 20, n rf = 3 4-QAM, L = 6, K = 1 , M/ , M/ N r = 128, U = 20, n rf = 4 4-QAM, L = 6, K = 1 , M/ , M/ N r = 128, U = 16, n rf = 6 4-QAM, L = 6, K = 1 , M/ , M/ N r = 128, U = 20, n rf = 3 16-QAM, L = 6, K = 1 , M/ , M/ N r = 128, U = 20, n rf = 5 16-QAM, L = 6, K = 1 , M/ , M/ U = 20, n rf = 4 4-QAM, L = 6, K = 1 , M/ , M/ N r = 128, U = 16 , n rf = 3 , L = 6, K = 1 , M/ , M/
2, SNR = 5 dBFig. 12 U = 16, n rf = 4 4-QAM, L = 6, K = 1 , M/ , M/
2, SNR = 5 dB L = 1 , , , ×
20 mMIMO system, n rf = 3 and128 ×
16 mMIMO system, n rf = 4, respectively. It is observed that the BERperformance of MAP-ISD improves with increase in the number of iterationsand converges after L = 6. Significant improvement in BER performance isobserved for increase in the value of L from 1 to 2 and from 2 to 4 whereasmarginal improvement is observed for further increase in the value of L . -5 -4 -3 -2 -1 -4 Fig. 2
BER performance of MAP-ISD for 128 × n rf = 3 MBM-mMIMO system with4-QAM modulation.6 Manish Mandloi*, Devendra Singh Gurjar -5 -4 -3 -2 -1 -4 Fig. 3
BER performance of MAP-ISD for 128 × n rf = 4 MBM-mMIMO system with4-QAM modulation. -6 -5 -4 -3 -2 -1 -4 Fig. 4
BER performance of KMAP-IIC with K = M/ × n rf = 4 MBM-mMIMOsystem with 4-QAM modulation.itle Suppressed Due to Excessive Length 17 In Fig. 4, we simulate the BER performance of KMAP-IIC algorithm withdifferent iterations L = 1 , , ×
20 mMIMO systems with 4-QAM, K = M/ n rf = 3 MBM. Similar observations can be drawn whichsuggests that the BER performance of the KMAP-IIC algorithm convergesfor L = 6. Therefore, for comparison of BER of the proposed algorithms, i.e.MAP-ISD and KMAP-IIC, with other detection techniques (discussed later inthe section) we use L = 6 iterations. -6 -5 -4 -3 -2 -1 -5 -4 Fig. 5
BER performance comparison for 128 × n rf = 3 MBM-mMIMO system with4-QAM modulation. In Fig. 5, we compare the BER performance of the proposed algorithmswith other algorithms such as IIC, IESP and MMSE for 128 ×
20 mMIMOwith 4-QAM, n rf = 3 MBM system. For comparison of KMAP-IIC, we con-sider three different values for K , i.e. K = 1, K = M/ K = M/
2. It isobserved that, MAP-ISD and KMAP-IIC algorithms outperform the MMSEand the IESP algorithms. Moreover, the KMAP-IIC achieves BER perfor-mance close to within 0 . K = M/ − , MAP-ISD achieves performance within 0.5 dB ofthe IIC algorithm. Therefore, due to significant low-computational complexityof MAP-ISD (discussed later in Section 4.2), it would be a better choice overother detection techniques with marginal loss in performance. Fig. 6 showsthe comparison of BER for 128 ×
20 mMIMO with 4-QAM, n rf = 4 MBMsystems. It is observed that in contrast to Fig. 5, the BER performance gapbetween MAP-ISD and KMAP-IIC increases with increase in n rf from 3 to
4. The key reason behind such performance degradation is the error propa-gation in interference cancellation of MAP-ISD which increases with increasein n rf . It is interesting to note that, KMAP-IIC with K = 1 achieves BERperformance close to within 0.1 dB to that of IIC, KMAP-IIC with K = M/ K = M/
2. Therefore, in MBM-mMIMO systems withmoderate values of n rf , KMAP-IIC with K = 1 could be a better choice. -5 -4 -3 -2 -1 -4 Fig. 6
BER performance comparison for 128 × n rf = 4 MBM-mMIMO system with4-QAM modulation. Fig. 7 presents the BER performance comparison for 128 ×
20 mMIMOwith 4-QAM, n rf = 6 MBM system. It is observed that the performance ofKMAP-IIC with K = M/ K = M/ K = 1 is close to within 0 . . n rf . However, in suchscenarios KMAP-IIC still performs up to mark. Therefore, from the afore-mentioned BER performance results, it can be concluded that the choice oflow-complexity (discussed in next subsection) reasonable detection techniquebetween MAP-ISD and KMAP-IIC depends significantly on the value of n rf RF mirrors and the number of uplink users. itle Suppressed Due to Excessive Length 19 -5 -4 -3 -2 -1 Fig. 7
BER performance comparison for 128 × n rf = 6 MBM-mMIMO system with4-QAM modulation. -5 -4 -3 -2 -1 -4 Fig. 8
BER performance comparison for 128 × n rf = 3 MBM-mMIMO system with16-QAM modulation.0 Manish Mandloi*, Devendra Singh Gurjar -6 -5 -4 -3 -2 -1
11 11.2 11.4 11.6 11.8 12 12.210 -4 Fig. 9
BER performance comparison for 128 × n rf = 5 MBM-mMIMO system with16-QAM modulation. In Figs. 8 and 9, the BER performance comparison is performed for 128 × n rf = 3 and n rf = 5 with 16-QAM modulation. Obser-vations reveal that the proposed algorithms perform superior over the IESPalgorithm. However, the performance of MAP-ISD and KMAP-IIC with K = 1degrades with an increase in n rf as compared to KMAP-IIC with K = M/ K = M/ K = M/ K = M/ N r = 80 to N r = 140 for U = 20, n rf = 4 and 4-QAMMBM-mMIMO system. It is observed that the BER performance improveswith an increase in the BS antennas. Also, the performance of KMAP-IICwith different values of K , i.e., K = 1 , M/ , M/
2, is close to that of the IICalgorithm. On the other hand, the performance of MAP-ISD is far from IIC.For example, for a target BER of 5 × − , required BS antennas for IIC andKMAP-IIC are around 125, whereas MAP-ISD requires around 140 BS anten-nas. Moreover, channel-hardening is not observed to degrade the performancecompared to the IIC algorithm. itle Suppressed Due to Excessive Length 21
80 90 100 110 120 130 14010 -4 -3 -2 -1 Fig. 10
BER performance comparison for U = 20 mMIMO with n rf = 4 and 4-QAMmodulation for different values of N r . Table 3
Approximate number of computations in each iteration of IIC algorithmSteps involved in IIC Computational complexityInterference cancellation 2 N r Performing ML search (4 N r − MU | A | Finding the solution using greedy search (6 N r − U In Table 4 and 5, we present the computations required in different stepsof the MAP-ISD and KMAP-IIC algorithms, respectively.
Table 4
Approximate number of computations in each iteration of MAP-ISDSteps involved in MAP-ISD Computational complexityFinding low-complexity matrix inverse (required only once) (2 N r − MU Finding the most favorable MAP 2 MN r U Finding reliability of the solution (4 N r − U Table 5
Approximate number of computations in each iteration of KMAP-IICSteps involved in KMAP-IIC Computational complexityFinding low-complexity matrix inverse (required only once) (2 N r − MU Finding favorable MAPs 2 MN r U Performing low-complexity search (2 N r + 4) KU Finding the solution using greedy search (6 N r − U 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 isdue to the complex search mechanism in IIC and KMAP-IIC algorithm. Thecomputation of finding a low-complexity matrix inversion in MAP-ISD andKMAP-IIC is required only once. Therefore, the reduction in search space dueto favorable MAP selection and low-complexity search are the key factors forthe overall reduction in the proposed algorithms’ computational complexity.Fig. 11 presents the comparison of average number of FLOPs for differentMBM-mMIMO systems at SNR= 5 dB. In Fig. 11, System 1 refers to 128 × n rf = 3, system 2 refers to 128 ×
20 mMIMO systemswith n rf = 4 and system 3 refers to 128 ×
16 mMIMO systems with n rf =6. Observations reveal that KMAP-IIC with K = M/ × n rf = 6 and 128 × n rf = 4 systems, respectively. This gain increases further to 76% and 80%when MAP-ISD algorithm is used for 128 × n rf = 6 and 128 × n rf =4 systems, respectively. It is clear from the figure that the computationalcomplexity of MAP-ISD is significantly less as compared to the KMAP-IICwith K = 1, K = M/ K = M/
2. Therefore, in case of MBM-mMIMOsystems 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/ itle Suppressed Due to Excessive Length 23 Fig. 11
Computational complexity comparison of different detection techniques for differentMBM-mMIMO systems.
80 90 100 110 120 130 14010 Fig. 12
Computational complexity comparison of different detection techniques versus thenumber of receive antennas.4 Manish Mandloi*, Devendra Singh Gurjar
In Fig. 12, we present the comparison on the average number of floatingpoint operations (FLOPs) of the IIC algorithm with MAP-ISD and KMAP-IIC algorithms, respectively, with respect to the number of receive antennasfor MBM-mMIMO system having 16 users and n rf = 4 at each user. Observa-tion reveals that the computational complexity increase linearly with increasein the number of receive antennas at the station. It is also observed thatthe computational complexity increases with increase in K . Furthermore, thecomplexity of MAP-ISD is significantly less compared with the KMAP-IICalgorithm and the IIC algorithm [12]. We proposed low complexity interference cancellation based algorithms forsymbol detection in MBM-mMIMO systems. First, we proposed a MAP selec-tion based ISD algorithm which detects symbols in a sequential manner whilenullifying interference from all other users. Then, we presented an approach toreduce the search space in the IIC algorithm and devised the KMAP-IIC algo-rithm. The key idea was to introduce a metric and a selection rule for selectingthe list of the favorable MAPs in MBM corresponding to each user. The pro-posed algorithms achieve superior performance-complexity trade-off over theexisting detection techniques in MBM-mMIMO systems. Observations revealthat when the number of users and the RF mirrors are comparatively lessMAP-ISD algorithm achieves similar performance with almost 75% and 87%savings in computational complexity as compared to KMAP-IIC and IIC algo-rithm, respectively. However, with an increase in the number of users and theRF mirrors, KMAP-IIC delivers better BER performance over MAP-ISD withalmost 65% to 70% savings in computational complexity for different value of K over the IIC algorithm. Acknowledgements
This work is supported by the Start-up Research Grant (file no.SRG/2019/000654) scheme of Science and Engineering Research Board, Department of Sci-ence and Technology, Government of India.
Conflict of interest
The authors declare that they have no conflict of interest.
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