Generalized Triangular Decomposition Research Articles

Optimal hybrid precoder design for millimeter-wave massive MIMO systems

Millimeter-waves (mmWave) have emerged as a promising candidate for next-generation communication systems. However, they have severe degradation issues that can be overcome by multiple-input multiple-output (MIMO) technology. Still, high-power consumption and budget issues are associated with multiple antennas. Therefore, hybrid precoders are exploited to combat these inevitable issues. The prime objective of this manuscript is to develop a hybrid precoder for massive MIMO systems that can provide better performance than the existing ones in terms of bit error rate (BER) versus signal-to-noise ratio (SNR). In this context, we exploit iterative geometric mean decomposition (IGMD) method that presents asymptotically optimum performance not only in traditional MIMO systems but also in massive MIMO systems. Later, the outcomes of the proposed scheme are compared with a recent design exploiting generalized triangular decomposition (GTD) scheme. The simulation studies show that the proposed method provides more than 7 dB improvement according to other designs.

Iterative geometric mean decomposition based secure hybrid precoder design for mmWave massive MIMO communication systems

Physical layer security (PLS) is one of the major concerns in modern mobile communication systems and has got special attention in millimeter-wave (mmWave) communication systems. The principal objective of this study is to construct a hybrid precoder that can secure the mmWave communication systems by incorporating the finest techniques to obtain a secure precoder and combiner for the legitimate receiver and to ensure minimum information leakage towards illegal-user. In this context, firstly, a traditionally coordinated precoder and combiner design algorithm is exploited in the analog domain to protect the legitimate receiver, and then, iterative geometric mean decompositions (IGMD) scheme is applied to obtain a digital precoder and combiner. Moreover, artificial noise (AN) is also transmitted towards illegal-user that further alleviates the privacy of the system. Finally, we compared our results with our latest proposed algorithm for a secure hybrid precoder system based on the generalized triangular decomposition (GTD) method and a recent algorithm presented in the literature. The simulation studies confirmed that the IGMD based approach considerably degrades the eavesdropping capabilities of the undesired user and provides significantly better secrecy rate performance as much as 3.5 bps/Hz than its counterparts.

Simultaneous Wireless Information and Power Transfer Based on Generalized Triangular Decomposition

In this paper, a new approach, based on the generalized triangular decomposition (GTD), is proposed for simultaneous wireless information and power transfer (SWIPT) in the spatial domain for a point-to-point multiple-input multiple-output (MIMO) system. The proposed approach takes advantage of the GTD structure to allow the transmitter to use the strongest eigenchannel jointly for energy harvesting and information exchange while these transmissions can be separated at the receiver. The optimal structure of the GTD that maximizes the total information rate constrained by a given power allocation and a required amount of energy harvesting is derived. An algorithm is developed that minimizes the total transmitted power for given information rate and energy harvesting constraints with a limited total power at the transmitter. Both theoretical and simulation results show that our proposed GTD based SWIPT outperforms singular value decomposition (SVD) based SWIPT. This is due to the flexibility introduced by the GTD to increase the energy harvested via interstream interference.

A Systolic Array Based GTD Processor With a Parallel Algorithm

Generalized triangular decomposition (GTD) has been found to be useful in the field of signal processing, but the feasibility of the related hardware has not yet been established. This paper presents (for the first time) a GTD processor architecture with a parallel algorithm. The proposed parallel GTD algorithm achieves an increase in speed of up to 1.66 times, compared to the speed of its conventional sequential counterpart for an 8 ×8 matrix. For hardware implementation, the proposed reconfigurable architecture is capable of computing singular value decomposition (SVD), geometric mean decomposition (GMD), and GTD for matrix sizes from 1 ×1 to 8 ×8. The proposed GTD processor is composed of 16 processing cores in a heterogeneous systolic array. Computations are distributed over area-efficient coordinate rotation digital computers (CORDICs) to achieve a high throughput. To establish the validity of the concept, a GTD processor was designed and implemented. The latency constraint of 16 μs specified in the 802.11ac standard is adopted for the hardware realization. The proposed design achieves a maximum throughput of 83.3k matrices/s for an 8 ×8 matrix at 112.4 MHz. The estimated power and core area are 172.7 mW and 1.96 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , respectively, based on standard 90 nm CMOS technology.

FIR Approximation of GTD Filter Banks and their Multiresolution Optimality

The theory and design of signal adapted filter banks in the coding gain objective as well as the multiresolution objective are of great interest in many signal processing applications. The role of the generalized triangular decomposition (GTD) filter banks in optimizing perfect reconstruction filter banks has been proposed recently by Ching-Chih Weng et al. They have proposed the GTD filter bank as a subband coder for optimizing the theoretical coding gain. In this paper, we show that the design of the GTD filter bank via the singular value decomposition (SVD) will be reduced to the principal component filter bank (PCFB) and it gives optimal performance in the multiresolution objective. The FIR approximation of the optimal GTD filter banks is also discussed in this paper. This is done by using the iterative greedy algorithm.

The Role of GTD in Optimizing Perfect Reconstruction Filter Banks

Filter bank optimization for specific input statistics has been of great interest in both theory and practice in many signal processing applications. In this paper we propose GTD (generalized triangular decomposition) filter banks as a subband coder for optimizing the theoretical coding gain. We focus on perfect reconstruction orthonormal GTD filter banks and biorthogonal GTD filter banks. We show that in both cases there are two fundamental properties in the optimal solutions, namely, total decorrelation and spectrum equalization. The optimal solutions can be obtained by performing the frequency dependent GTD on the Cholesky factor of the input power spectrum density matrices. We also show that in both theory and numerical simulations, the optimal GTD subband coders have superior performance than optimal traditional subband coders. In addition, the uniform bit loading scheme can, with no loss of optimality, be used in the optimal biorthogonal GTD coders, which solves the granularity problem in the conventional optimum bit loading formula. We then extend the use of GTD filter banks to wireless communication systems, where linear precoding and zero-forcing decision feedback equalization is used in frequency selective channels. We consider the quality of service (QoS) problem of minimizing the transmitted power subject to the bit error rate and total bit rate constraints. Optimal systems with orthonormal precoder and unconstrained precoder are both derived and shown to be related to the frequency dependent GTD of the channel frequency response.

MIMO Transceivers With Decision Feedback and Bit Loading: Theory and Optimization

This paper considers MIMO transceivers with linear precoders and decision feedback equalizers (DFEs), with bit allocation at the transmitter. Zero-forcing (ZF) is assumed. Considered first is the minimization of transmitted power, for a given total bit rate and a specified set of error probabilities for the symbol streams. The precoder and DFE matrices are optimized jointly with bit allocation. It is shown that the generalized triangular decomposition (GTD) introduced by Jiang, Li, and Hager offers an optimal family of solutions. The optimal linear transceiver (which has a linear equalizer rather than a DFE) with optimal bit allocation is a member of this family. This shows formally that, under optimal bit allocation, linear and DFE transceivers achieve the same minimum power. The DFE transceiver using the geometric mean decomposition (GMD) is another member of this optimal family, and is such that optimal bit allocation yields identical bits for all symbol streams-no bit allocation is necessary-when the specified error probabilities are identical for all streams. The QR-based system used in VBLAST is yet another member of the optimal family and is particularly well-suited when limited feedback is allowed from receiver to transmitter. Two other optimization problems are then considered: (a) minimization of power for specified set of bit rates and error probabilities (the QoS problem), and (b) maximization of bit rate for fixed set of error probabilities and power. It is shown in both cases that the GTD yields an optimal family of solutions.

Generalized Triangular Decomposition in Transform Coding

A general family of optimal transform coders (TCs) is introduced here based on the generalized triangular decomposition (GTD) developed by Jiang This family includes the Karhunen-Loeve transform (KLT) and the generalized version of the prediction-based lower triangular transform (PLT) introduced by Phoong and Lin as special cases. The coding gain of the entire family, with optimal bit allocation, is equal to that of the KLT and the PLT. Even though the original PLT introduced by Phoong is not applicable for vectors that are not blocked versions of scalar wide sense stationary processes, the GTD-based family includes members that are natural extensions of the PLT, and therefore also enjoy the so-called MINLAB structure of the PLT, which has the unit noise-gain property. Other special cases of the GTD-TC are the geometric mean decomposition (GMD) and the bidiagonal decomposition (BID) transform coders. The GMD-TC in particular has the property that the optimum bit allocation is a uniform allocation; this is because all its transform domain coefficients have the same variance, implying thereby that the dynamic ranges of the coefficients to be quantized are identical.

The generalized triangular decomposition

Given a complex matrix H, we consider the decomposition H = QRP*, where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal. We show that any diagonal for R can be achieved that satisfies Weyl's multiplicative majorization conditions: k k K K Π i=1 |r i | ≤ Π i=1 σ ι, 1 ≤ k < K, K, Π i=1 |r i | = Π i=1 σ i , where K is the rank of H, σ i is the i-th largest singular value of H, and r i is the i-th largest (in magnitude) diagonal element of R. Given a vector r which satisfies Weyl's conditions, we call the decomposition H = QRP*, where R is upper triangular with prescribed diagonal r, the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

Tunable Channel Decomposition for MIMO Communications Using Channel State Information

We consider jointly designing transceivers for multiple-input multiple-output (MIMO) communications. Assuming the availability of the channel state information (CSI) at the transmitter (CSIT) and receiver (CSIR), we propose a scheme that can decompose a MIMO channel, in a capacity lossless manner, into multiple subchannels with prescribed capacities, or equivalently, signal-to-interference-and-noise ratios (SINRs). We refer to this scheme as the tunable channel decomposition (TCD), which is based on the recently developed generalized triangular decomposition (GTD) algorithm and the closed-form representation of minimum mean-squared-error VBLAST (MMSE-VBLAST) equalizer. The TCD scheme is particularly relevant to the applications where independent data streams with different qualities-of-service (QoS) share the same MIMO channel. The TCD scheme has two implementation forms. One is the combination of a linear precoder and a minimum mean-squared-error VBLAST (MMSE-VBLAST) equalizer, which is referred to as TCD-VBLAST, and the other includes a dirty paper (DP) precoder and a linear equalizer followed by a DP decoder, which we refer to as TCD-DP. We also include the optimal code-division multiple-access (CDMA) sequence design as a special case in the framework of MIMO transceiver designs. Hence, our scheme can be directly applied to optimal CDMA sequence design, both in the uplink and downlink scenarios.