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International audience; Use of unmanned aerial vehicles (UAVs) to provide on-demand communications has been receiving growing interest, especially for use in remote and hard-to-reach areas. Also, the use of light-emitting diode-based lighting in UAVs has opened opportunities for data transmission through visible-light communications. To manage multiuser interference while avoiding complex handover procedures, we consider the use of zero forcing (ZF) pre-coding. Since the performance of ZF pre-coding depends on the correlation between channel gains of users, we propose in this paper to reduce it by means of location optimization of UAVs. More specifically, we use particle swarm optimization with the objective of maximizing the overall achievable network throughput. Furthermore, to relax the optimization requirements at UAVs, we investigate the case when the optimization is performed at a specific rate under different mobility conditions.

A leak is a vertex that is not allowed to perform a force during the zero forcing process. Leaky forcing was recently introduced as a new variation of zero forcing in order to analyze how leaks in a network disrupt the zero forcing process. The l -leaky forcing number of a graph is the size of the smallest zero forcing set that can force a graph despite l leaks. A graph G is l -resilient if its zero forcing number is the same as its l -leaky forcing number. In this paper, we analyze l -leaky forcing and show that if an ( l − 1 ) -leaky forcing set B is robust enough, then B is an l -leaky forcing set. This provides the framework for characterizing l -leaky forcing sets. Furthermore, we consider structural implications of l -resilient graphs. We apply these results to bound the l -leaky forcing number of several graph families including trees, supertriangles, and grid graphs. In particular, we resolve a question posed by Dillman and Kenter concerning the upper bound on the 1-leaky forcing number of grid graphs.

Given a finite graph G, the maximum length of a sequence $$(v_1,\ldots ,v_k)$$ of vertices in G such that each $$v_i$$ dominates a vertex that is not dominated by any vertex in $$\{v_1,\ldots ,v_{i-1}\}$$ is called the Grundy domination number, $$\gamma _\mathrm{gr}(G)$$ , of G. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that $$\gamma _\mathrm{gr}(G) \ge \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}$$ holds for every connected k-regular graph of order n different from $$K_{k+1}$$ and $$\overline{2C_4}$$ . The bound in the case $$k=3$$ reduces to $$\gamma _\mathrm{gr}(G)\ge \frac{n}{2}$$ , and we characterize the connected cubic graphs with $$\gamma _\mathrm{gr}(G)=\frac{n}{2}$$ . If G is different from $$K_4$$ and $$K_{3,3}$$ , then $$\frac{n}{2}$$ is also an upper bound for the zero forcing number of a connected cubic graph, and we characterize the connected cubic graphs attaining this bound.

Pada pengiriman data, sinyal yang diterima oleh antena penerima pasti tidak sama dengan sinyal yang dikirim oleh antena pengirim. Pada penelitian ini, teknik K-Means Clustering diaplikasikan untuk proses pengembalian data bit sebagai fungsi demodulasi pada konstelasi M-Ary yang sebelumnya telah terganggu oleh noise dan fading di jaringan pita lebar. Modulasi digital yang digunakan dalam penelitian ini yaitu 16 QAM. Simulasi dilakukan pada kanal ideal (AWGN) menggunakan teknik multicarrier Filter Bank Multicarrier Offset QAM (FBMC OQAM) dengan equalizer Zero Forcing (ZF) dan tanpa ZF. Kinerja dari demapper k-means dievaluasi menggunakan Silhouette Coefficient menghasilkan sistem FBMC OQAM dengan penggunaan ZF memiliki kinerja lebih baik daripada sistem FBMC OQAM tanpa penggunaan equalizer . Seperti BER FBMC OQAM ZF pada SNR 0 dB hingga SNR 20 dB mengalami penurunan sebesar 0,3849, sedangkan pada FBMC OQAM memiliki BER yang relatif stabil. Selain mengalami penurunan pada parameter BER, sistem FBMC OQAM ZF mengalami peningkatan pada kapasitas kanal yang dihasilkan.

Let G be a graph such that the color of its vertices is white or black. A dynamic vertex coloring for G is defined as follows. One starts with a certain set of black vertices. Then, at each time step, a black vertex with exactly one white neighbor forces its white neighbor to become black. The initial set of black vertices is called a zero forcing set if by iterating this process, all of the vertices of G become black. The zero forcing number of G (denoted by Z(G)) is the minimum cardinality of a zero forcing set in G. In this paper, we study the zero forcing number of trees. Let T be a tree with at least two vertices. We show that $$\Delta (T)-1\le Z(T)\le r(T)-1$$ , where $$\Delta (T)$$ and r(T) are the maximum degree and the number of pendant vertices of T, respectively. As a consequence, we obtain that $$Z(L(T))\ge Z(T)$$ , where L(T) is the line graph of T. We characterize all trees T such that $$Z(T)=\Delta (T)-1$$ . Finally, we study trees T with $$Z(T)= r(T)-1$$ .

Zero-Forcing (ZF) detector is used in Space-Division Multiplexing (SDM) receiver to remove interference among the received symbols. Previous works showed that the power of channel noise is enhanced in the output of the ZF detector. They recommend using the ZF detector when the received Signal-to-Noise Ratio (SNR) is high. This work proves that the performance of the ZF detector depends on the eigenvalues of the channel correlation matrix. The paper shows that if the sum of the eigenvalues of this correlation matrix is equal to the rank of the channel matrix, the ZF detector will not enhance noise power at its outputs. Moreover, if the sum of the eigenvalues is smaller than the rank of the channel matrix, the ZF detector will reduce noise power at its outputs. In this work, a theorem, which demonstrates the performance of the ZF detector in SDM receiver, is introduced and proved. The proposed work uses smart antennas in the transmitter and receiver to control the elements and eigenvalues of the channel matrix. The introduced theorem and a complete SDM receiver with ZF detector are simulated and evaluated at different conditions with different criteria. A real-time SDM receiver with ZF detector is also implemented and evaluated. The simulation and implementation results are shown at the end of this study. The results of the proposed systems show that a ZF detector can be used to remove interference in the SDM system without enhancing the channel noise.

The work is devoted to the study of the zero forcing number of some families of graphs. The concept of zero forcing is a relatively new research topic in discrete mathematics, which already has some practical applications, in particular, is used in studies of the minimum rank of the matrices of adjacent graphs. The zero forcing process is an example of the spreading process on graphs. Such processes are interesting not only in terms of mathematical and computer research, but also interesting and are used to model technical or social processes in other areas: statistical mechanics, physics, analysis of social networks, and so on. Let the vertices of the graph G be considered white, except for a certain set of S black vertices. We will repaint the vertices of the graph from white to black, using a certain rule.Colour change rule: A white vertex turns black if it is the only white vertex adjacent to the black vertex.[5] The zero forcing number Z(G) of the graph G is the minimum cardinality of the set of black vertices S required to convert all vertices of the graph G to black in a finite number of steps using the ”colour change rule”.It is known [10] that for any graph G, its zero forcing number cannot be less than the minimum degree of its vertices. Such and other already known facts became the basis for finding the zero forcing number for two given below families of graphs:A gear graph, denoted W2,n is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, W2,n has 2n + 1 vertices and 3n edges.A prism graph, denoted Yn, or in general case Ym,n, and sometimes also called a circular ladder graph, is a graph corresponding to the skeleton of an n-prism.A wheel graph, denoted Wn is a graph formed by connecting a single universal vertex to all vertices of a cycle of length n.In this article some known results are reviewed, there is also a definition, proof and some examples of the zero forcing number and the zero forcing process of gear graphs and prism graphs.

Zero forcing is a process of coloring in a graph in time steps known as propagation time. These graph-theoretic parameters have diverse applications in computer science, electrical engineering and mathematics itself. The problem of evaluating these parameters for a network is known to be NP-hard. Therefore, it is interesting to study these parameters for special families of networks. Perila et al. (2017) studied properties of these parameters for some basic oriented graph families such as cycles, stars and caterpillar networks. In this paper, we extend their study to more non-trivial structures such as oriented wheel graphs, fan graphs, friendship graphs, helm graphs and generalized comb graphs. We also investigate the change in propagation time when the orientation of one edge is flipped.

We explore the relationship between propagation tree decompositions of a graph of a given size and the OS-sets of the same size, showing that each can generate the other. We give a short constructive proof that the OS-sets are in bijective correspondence with a subset of the propagation tree decompositions.

We introduce and study forcing number for fuzzy graphs. Also, we compute zero forcing numbers for some classes of graphs and extend this concept to fuzzy graphs. In this regard we obtain upper bounds for zero forcing of some classes of fuzzy graphs. We will proceed to obtain a new algorithm to computing zero forcing set and finding a formula for zero forcing number, and by some examples we illustrate these notions. Finally, we introduce some applications of fuzzy zero forcing in medical treatments.

In a zero forcing process, an initial vertex coloring of a graph is updated iteratively according to the following conversion rule: an uncolored vertex becomes colored if it is the only uncolored neighbor of some colored vertex. In this process, a zero forcing set is an initial set of colored vertices such that all the remaining vertices will ultimately change from uncolored to colored, and the zero forcing number is the minimum cardinality of a zero forcing set. We investigate two related concepts: a zero blocking set is the complement of a set which is not a zero forcing set, and the zero blocking number is the minimum cardinality of a zero blocking set. We provide a tight upper bound for the zero blocking number of Cartesian products of two arbitrary graphs. Exact values that are smaller than this general upper bound are also established when one of the graphs in the product is a cycle and the other is a path with at least as many vertices as the cycle.

A dynamic coloring of a graph G starts with an initial subset F ⊆ V ( G ) of colored vertices, while all the remaining vertices are non-colored. At each time step, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set F is called a zero forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The zero forcing number of G , denoted by F ( G ) , is the cardinality of a minimum zero forcing set of G . The maximum nullity of G , denoted by M ( G ) , is the largest possible nullity over all | V ( G ) | by | V ( G ) | real symmetric matrices A whose non-diagonal entries are non-zero if the corresponding vertices are adjacent in G and with no restriction for its diagonal entries. In this paper, we characterize all graphs G of order n , maximum degree at most three, and F ( G ) = 3 . Also we classify these graphs with their maximum nullity.

In a zero forcing process, an initial vertex coloring of a graph is updated iteratively according to the following conversion rule: an uncolored vertex becomes colored if it is the only uncolored neighbor of some colored vertex. In this process, a zero forcing set is an initial set of colored vertices such that all the remaining vertices will ultimately change from uncolored to colored, and the zero forcing number is the minimum cardinality of a zero forcing set. We investigate two related concepts: a zero blocking set is the complement of a set which is not a zero forcing set, and the zero blocking number is the minimum cardinality of a zero blocking set. We provide upper and lower bounds for the zero blocking number of rectangular grids and discuss conditions under which these bounds coincide. We go on to use the same machinery to provide similar results for certain cylindrical and Mobius grids.

The concept of zero forcing is extended from graphs to uniform hypergraphs in analogy with the way zero forcing was defined as an upper bound for the maximum nullity of the family of symmetric matrices whose nonzero pattern of entries is described by a given graph: A family of symmetric hypermatrices is associated with a uniform hypergraph and zeros are forced in a null vector. The value of the hypergraph zero forcing number and maximum nullity are determined for various families of uniform hypergraphs and the effects of several graph operations on the hypergraph zero forcing number are determined. The hypergraph zero forcing number is compared to the infection number of a hypergraph and the iteration process in hypergraph power domination.

Zero forcing is a combinatorial game played on a graph where the goal is to start with all vertices unfilled and to change them to filled at minimal cost. In the original variation of the game there were two options. Namely, to fill any one single vertex at the cost of a single token; or if any currently filled vertex has a unique non-filled neighbor, then the neighbor is filled for free. This paper investigates a q-analogue of zero forcing which introduces a third option involving an oracle. Basic properties of this game are established including determining all graphs which have minimal cost 1 or 2 for all possible q, and finding the zero forcing number for all trees when $$q=1$$ .

Let $G$ be a simple undirected graph with each vertex colored either white or black, $ u $ be a black vertex of $ G, $ and exactly one neighbor $ v $ of $ u $ be white. Then change the color of $ v $ to black. When this rule is applied, we say $ u $ forces $ v, $ and write $ u \rightarrow v $. A $zero\ forcing\ set$ of a graph $ G$ is a subset $Z$ of vertices such that if initially the vertices in $ Z $ are colored black and remaining vertices are colored white, the entire graph $ G $ may be colored black by repeatedly applying the color-change rule. The zero forcing number of $ G$, denoted $Z(G), $ is the minimum size of a zero forcing set.\\ In this paper, we investigate the zero forcing number for the generalized Petersen graphs (It is denoted by $P(n,k)$). We obtain upper and lower bounds for the zero forcing number for $P(n,k)$. We show that $Z(P(n,2))=6$ for $n\geq 10$, $Z(P(n,3))=8$ for $n\geq 12$ and $Z(P(2k+1,k))=6$ for $k\geq 5$.

Generalised Frequency Division Multiplexing (GFDM) is utilized in the fifth era cell organize as opposed to the Orthogonal Frequency Division Multiplexing (OFDM) since GFDM has numerous appealing highlights like heartiness towards recurrence particular blurring, simple execution and sensible, and so forth. GFDM has just a single cyclic prefix (CP) for a lot of images in a period OFDM it uses a solitary CP per image. Right now, research the effects of nonlinear mutilation on the different information numerous yield (MIMO) - GFDM framework when the sign is passed the HPA, which is displayed with sufficiency and stage contortion. The execution of the proposed strategy is resolved as far as range investigation, PAPR examination, and bit blunder rate (BER) investigation. The recreation results show that the proposed plan prevails with regards to making up for nonlinear contortions brought about by the HPA for huge contribution back-off (IBO) values.

The notions of critical ideals and characteristic ideals of graphs are introduced by Corrales and Valencia to study properties of graphs, including clique number, zero forcing number, minimum rank and critical group. In this paper, we provide methods to compute critical ideals of complete multipartite graphs and obtain complete answers for the characteristic ideals of complete multipartite graphs.

In this paper, we investigate the fuzzification of zero forcing process. For this, first we introduce a new embedding of a graph [Formula: see text] by considering a minimal zero forcing set of [Formula: see text] and an arbitrary list of maximal forcing chains of this zero forcing set. Then we get a comparison between zero forcing sets of a graph by using fuzzy concepts. Finally, we give an application for this procedure.

This study investigates the equalisation techniques for the diffusive molecular communication (DMC) systems with binary molecular-shift keying (BMoSK) modulation, referred to as the BMoSK-DMC systems, in order to mitigate the effect of inter-symbol interference (ISI). The authors first show that in terms of receiving equalisation, a BMoSK-DMC system is in fact equivalent to a conventional binary phase-shift keying modulated radio communication system encountering ISI. Hence, a wealth of equalisation techniques developed with the conventional radio communication systems may be introduced for equalisation of BMoSK-DMC signals. However, considering the limited capability of molecular transceivers on computation and storage, in this study, the authors investigate only the linear equalisers (LEQ) in the principles of matched-filtering, zero-forcing (ZF) and minimum mean-square error (MMSE). They characterise the effects of the different aspects related to DMC signalling and propagation on the achievable performance of the BMoSK-DMC systems with these LEQ. Their studies and performance results demonstrate that both the ZF- and MMSE-LEQ are capable of efficiently suppressing the ISI and attaining promising performance, while still have low-complexity to facilitate implementation.

The zero forcing number of a graph is the minimum size of a zero forcing set. This parameter bounds the path cover number which is the minimum number of vertex-disjoint-induced paths that cover all the vertices of graph. In this paper, we investigate these two parameters and present an infinite number of graphs with the large difference between these two parameters and an infinite number of graphs with no difference between these parameters. Also, we prove a conjecture about the relationship between these parameters for some graphs.

Recently, as the demand for tremendous spectral efficiency has increased, the massive multiple-input multiple-output (MIMO) system has attracted attention in the wireless communication system. In massive MIMO, the zero forcing (ZF) precoder provides optimal performance. However, the complexity for process of matrix inversion is burden in terms of practical implementation. Therefore, many researches for approximate inversion of channel matrix have been performed in order to reduce the complexity. The typical linear precoder based on approximate matrix inversion is the Gauss Seidel (GS) precoder. The GS precoder provides the similar precoded signals to ZF precoder with low complexity. However, the GS precoder does not adopt parallel implementation because of inner structure. Consequently, precoder for the GS iterative method spends a lot of times in order to estimate precoded signal. Therefore, this problem makes the GS precoder impractical. In this article, the punctured GS (PGS) is proposed in order to mitigate the problem of parallel operation by modifying inner structure for the GS precoder. However, the performance for the PGS precoder is degraded due to modified inner structure. Therefore, the ordering PGS precoder which performance degradation due to modified inner structure is mitigated is additionally introduced. As a result, although the delay when precoded signal for the PGS precoder is obtained decreases than the GS precoder, the BER performance for the PGS precoder is degraded than the GS precoder. In contrast, the ordering PGS precoder provides improved BER performance with decrease of delay compared with the GS precoder.

We give a brief survey about the existing results on the power domination of the Cartesian product of graphs, and improve two of the results by determining the exact power domination numbers of two families of graphs, namely, the cylinder Pn□Cmand the tori Cn□Cm. We also establish the power domination number for Kn□K1,m, the Cartesian product of a complete graph and a star. Keywords: Electric power monitoring, Power domination, Zero forcing, Cartesian product

Given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Zero forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper, we investigate the largest size of a set S that does not force all of the vertices in a graph to be in S. This quantity is known as the failed zero forcing number of a graph and will be denoted by F(G). We present new results involving this parameter. In particular, we completely characterize all graphs G where F(G)=2, solving a problem posed in 2015 by Fetcie, Jacob, and Saavedra.

We are testing the proposed approach in several scenarios generated using Quadriga - an open source software for generating realistic radio channel impulse response.

In massive MIMO systems, inter-user interference has effect on the transmitted signals, so linear precoding techniques are employed for multiuser transmission channels to remove this inter-user int...

Given a simple graph G, let $$Z(G),\, p(G),\, \Phi (G), ex(G)$$ and M(G), respectively, be the zero forcing number, the number of pendant vertices, the cyclomatic number, the number of exterior major vertices and the maximum nullity of G. Wang et al. (Linear Multilinear Algebra, 2018. https://doi.org/10.1080/03081087.2018.1545829) established upper and lower bounds on Z(G) with respect to $$p(G),\, ex(G)$$ and $$\Phi (G)$$: $$p(G)-ex(G)-1\leqslant Z(G)\leqslant p(G)+2\Phi (G)+1$$. Hence, it is interesting to study the distribution of the zero forcing number Z(G) in the interval $$[p(G)-ex(G)-1,\, p(G)+2\Phi (G)+1]$$. Wang et al. (2018) determined all the connected graphs G with $$Z(G)=p(G)-ex(G)$$ and $$Z(G)=p(G)+2\Phi (G)-1.$$ In this paper we identify all the connected graphs G with $$Z(G)=p(G)-ex(G)+1$$ and $$Z(G)=p(G)+2\Phi (G)-2.$$ On the other hand, ‘AIM Minimum Rank-Special Graphs Work Group’ (Linear Algebra Appl 428(7):1628–1648, 2008) established the inequality $$Z(G)\geqslant M(G)$$. The authors posted an attractive question: What is the class of graphs G for which $$Z(G)=M(G)$$? In this paper, we show that the equality holds for threshold graphs.

This paper considers polarization shift keying (PolSK) modulation in wireless communication systems. The PolSK has previously been analyzed in additive white Gaussian noise (AWGN) for optical fiber and wireless communications. In our work, analytical results additionally focus on Rayleigh and Rician fading channels. It is shown that the dual-polarized signaling enjoys an ergodic capacity advantage over co-polarized signaling for a sufficiently large Rician $K$ factor and signal-to-noise ratio. Specific PolSK constellations are formulated and analyzed in AWGN and Rayleigh fading channels. We examine the effect of the channel’s polarization parameters on symbol error rates and achievable information. Also, considered are detection candidates to exploit statistical and instantaneous channel state information at the receiver. Over-the-air experiments were conducted to validate the principles. At higher modulation orders, uncoded PolSK shows an error rate advantage over unit symbol power constellations such as phase shift keying (PSK) and differential PSK with/without adverse carrier frequency offsets. The PolSK with dual-polarized antennas also outperforms the same modulation technique implemented with spatially separated co-polarized antenna elements in a line-of-sight (LoS) scenario. The proposed maximum likelihood PolSK detectors exploiting non-Gaussian statistics of the noise on the Poincare sphere are validated to be superior to zero forcing and minimum mean square error equalization for a non-LoS scenario.

Zero forcing is an iterative process on a graph used to bound the maximum nullity. The process begins with select vertices as colored, and the remaining vertices can become colored under a specific color change rule. The goal is to find a minimum set of vertices such that after iteratively applying the rule, all of the vertices become colored (i.e., a minimum zero forcing set). Of particular interest is the propagation time of a chosen set which is the number of steps the rule must be applied in order to color all the vertices of a graph. We give a purely linear algebraic view of zero forcing: Find a set of vertices S such that for any weighted adjacency matrix A, whenever Ax = 0 , the entirety of x can be recovered using only x S , the entries corresponding to S. The key here is that S must be chosen before A. In this light, we are able to give a linear algebraic interpretation of the propagation time: Any error in x S effects the error of x exponentially in the propagation time. This error can be quantitatively measured using newly defined zero forcing-related parameters, the error polynomial vector and the variance polynomial vector. In this sense, the quality of two zero forcing sets can objectively be compared even if the sets are the same size and their propagation times are the same. Examples and constructions are given.

The positive semidefinite zero forcing number is a variant of the zero forcing number, which is an important parameter in the study of minimum rank/maximum nullity problems. In this paper, we consider two classes of graphs: matching-chain graphs and claw-free graphs. We first introduce the propagation decomposition of graphs; then we use this decomposition to prove a lower bound for the positive semidefinite zero forcing number of a graph. We apply this lower bound to find the positive semidefinite zero forcing number of matching-chain graphs. We show that the positive semidefinite zero forcing number and the zero forcing number agree for matching-chain graphs. As a consequence, we prove the conjecture about the positive semidefinite zero forcing number of the Cartesian product of two paths, and partially prove the conjecture about the positive semidefinite zero forcing number of the Cartesian product of a cycle and a path. For claw-free graphs, we establish a relationship between the positive semidefinite zero forcing number and the zero forcing number. We also prove that it is NP-complete to find the positive semidefinite zero forcing number of line graphs.

A set S of vertices in a graph G is a total dominating set of G if every vertex has a neighbor in S. The total domination number, γt(G), is the minimum cardinality of a total dominating set of G. A total forcing set in a graph G is a forcing set (zero forcing set) in G which induces a subgraph without isolated vertices. The total forcing number of G, denoted Ft(G), is the minimum cardinality of a total forcing set in G. Our main contribution is to show that the total forcing number and the total domination number of a cubic graph are related. More precisely, we prove that if G is a connected cubic graph different from K3,3, then F t ( G ) ≤ 3 2 γ t ( G ) .

In this letter, we propose a downlink precoder for massive multi-input multi-output (MIMO) system in the presence of I/Q imbalance (IQI) at both base station and user terminals (UTs). We formulate a minimum mean-square error (MMSE) precoding problem with multiple regularization parameters for different channel qualities and channel correlations of UTs with different IQIs. Then we obtain the asymptotically optimal precoding matrix for the massive MIMO system. Moreover, we derive the asymptotic sum rate of the proposed regularized zero-forcing (RZF) precoder. We show that the proposed RZF precoder achieves significantly higher sum rate than conventional RZF precoder. In addition, the proposed precoder shows robust performances to severe IQI conditions.

Paper aims to enhance the performance of bit error rate (BER) in wireless communication based on the multiple-input multiple-output (MIMO) system of vertical Bell laboratories layered space-time (VBLAST) algorithm. The VBLAST algorithm uses zero-forcing (ZF) and the minimum mean square error (MMSE) to evaluate the BER of wireless communication. MIMO VBLAST techniques function as an adaptive filter and can minimize the interference and multipath fading in the received signal of the channel. Physical layer network coding (PNC) is a new technique used to exploit the spatial diversity of the MIMO VBLAST system to improve the throughput and performance of wireless communication. The bit-error-rate (BER) of proposed VBLAST MIMO with PNC with binary phase-shift keying (BPSK) and quadrature phase-shift keying (QPSK) modulation over the additive white Gaussian noise and Rayleigh fading channel are analyzed. The performance of both BPSK and QPSK modulation in two and four antennas are compared. From the simulation results, it was found that the proposed scheme MIMO VBLAST PNC has a 45.2 % higher BER performance compared to the traditional MIMO scheme with an increase in the BER using MMSE and ZF respectively in both two and four antennas.

In massive multiple-input multiple-output (MIMO) systems when the number of base station antennas is much higher than the number of users, linear detectors, such as zero forcing (ZF) and minimum mean-square error (MMSE), are able to achieve the near-optimal performance due to the favorable massive MIMO channel propagation. But, these detectors employ, in general, exact matrix inversion which is computationally complex for such systems. In this paper, we affirm that computing the exact matrix inversion by direct methods is not necessary to find ZF or MMSE solution. An iterative matrix inversion procedure would yield similar performance. Thus, an efficient iterative matrix inversion based on the hyper-power (HP) method is proposed for massive MIMO detection. The computing efficiency of the iterative matrix inversion is further improved by optimizing the number of terms from the infinite series used in the HP method. Analytical results show that the optimum choice for the number of terms is three from the HP method. Simulation results show that the HP method with the optimum number of terms achieves the near-optimal ZF performance in a small number of iterations. Finally, the Coppersmith–Winograd algorithm for matrix multiplication is employed in order to reduce the computational complexity from $$O(K^{3})$$ to $$O(K^{2.373})$$ , where K represents the number of users.

In diffusion-based communication, for molecular systems, the achievable data rate depends on the stochastic nature of diffusion, which exhibits a severe inter-symbol-interference (ISI). Multiple-input multiple-output (MIMO) multiplexing improves the data rate at the expense of an inter-link interference (ILI). This paper investigates training-based channel estimation schemes for diffusive MIMO (D-MIMO) systems and corresponding equalization methods. Maximum likelihood and least-squares estimators of mean channel are derived, and the training sequence is designed to minimize the mean square error (MSE). The numerical validations in terms of MSE are compared with Cramer–Rao bound derived herein. Equalization is based on decision feedback equalizer (DFE) structure as this is effective in mitigating diffusive ISI/ILI. Zero-forcing, minimum MSE, and least-squares criteria have been paired to DFE, and their performances are evaluated in terms of bit error probability. D-MIMO time interleaving is exploited as an additional countermeasure to mitigate the ILI with remarkable performance improvements. The configuration of nano-transceivers is not static but affected by a Brownian motion. A block-type communication is proposed for D-MIMO channel estimation and equalization, and the corresponding time-varying D-MIMO MC system is numerically evaluated.

The Zero Forcing (ZF) equalizer suffers from noise enhancement and high complexity due to direct matrix inversion. On the other hand, the Minimum Mean Square Error (MMSE) equalizer suffers from high complexity, and requires Signal-to-Noise Ratio (SNR) estimation to work properly. In this paper, we use the Discrete Sine Transform (DST) for Multiple-Input–Multiple-Output (MIMO) Orthogonal Frequency Division Multiplexing (OFDM) instead of the Discrete Fourier Transform (DFT). Moreover, we present a Joint Low-Complexity Regularized ZF (JLRZF) equalizer to perform both equalization and Carrier Frequency Offset (CFO) compensation, jointly, in Underwater Acoustic (UWA)-OFDM systems. This equalizer mitigates the noise enhancement problem by using a constant regularization parameter. It has low complexity as it is based on banded-matrix approximation. The whole proposed system is compared with that based on the DFT. Simulation results prove the superiority of the proposed system compared to the traditional one.