Rejoinder: Best Subset, Forward Stepwise or Lasso? Analysis and Recommendations Based on Extensive Comparisons

Cell-free massive multiple-input multiple-output (MIMO) is considered as a promising technology for satisfying higher rate requirement of users in beyond-5G networks. This paper investigates the impact of phase noise on the performance of cell-free MIMO systems which employ a large number of multi-antenna access points (APs). For both downlink and uplink transmissions, we derive the tractable closed-form achievable rate expressions when signal detection with channel state information (CSI) and channel distribution information (CDI), respectively, by utilizing the extended deterministic equivalence method in random matrix theory. The closed-form achievable rate expressions provide efficient evaluation for the impact of phase noise on system performance. The analytical results indicate that the phase noise influences the power of the desired signal, the beamforming uncertainty gain, and the inter-user interference increment caused by the pilot contamination. In addition, the special cases that only APs have phase noise and only users have phase noise are discussed. It is found that the impact of phase noise at users is more severe than that at APs. Besides, we discuss the system performance at low and high SNR regimes. It demonstrates that the phase noise is the major limitation of the system performance in high SNR regime while it can be neglected in low SNR regime. Furthermore, we formulate the max-min power control problem for both downlink and uplink transmissions to guarantee the users' fairness and prove that the downlink max-min problem is a quasi-concave problem while the uplink max-min problem is a geometric programming (GP) problem. Then, the phase noise-aware max-min power control schemes for both downlink and uplink transmissions are proposed to ensure the users' fairness. Finally, numerical results are provided to validate the analytical results and the proposed phase noise-aware power control scheme.

Compressed sensing (CS) enables measurement reconstruction by using sampling rates below the Nyquist rate, as long as the amplitude vector of interest is sparse. In this paper, we first derive and analyze the Bayesian Cramer–Rao bound (BCRB) for the amplitude vector when the set of indices (the support) of its nonzero entries is known. We consider the following context: i) The dictionary is nonstochastic but randomly generated; ii) the number of measurements and the support cardinality grow to infinity in a controlled manner, i.e., the ratio of these quantities converges to a constant; iii) the support is random; and iv) the vector of nonzero amplitudes follow a multidimensional generalized normal distribution. Using results from random matrix theory, we obtain closed-form approximations of the BCRB. These approximations can be formulated in a very compact form in low and high SNR regimes. Second, we provide a statistical analysis of the variance and the statistical efficiency of the oracle linear mean-square-error (LMMSE) estimator. Finally, we present results from numerical investigations in the context of non-bandlimited finite-rate-of-innovation (FRI) signal sampling. We show that the performance of Bayesian mean-square error (BMSE) estimators that are aware of the cardinality of the support, such as OMP and CoSaMP, are in good agreement with the developed lower bounds in the high SNR regime. Conversely, sparse estimators exploiting only the knowledge of the parameter vector and the noise variance in form of a priori distributions of these parameters, like LASSO and BPDN, are not efficient at high SNR. However, at low SNR, their BMSE is lower than that of the former estimators and may be close to the BCRB.

The design of the precoder the maximizes the mutual information in linear vector Gaussian channels with an arbitrary input distribution is studied. Precisely, the precoder optimal left singular vectors and singular values are derived. The characterization of the right singular vectors is left, in general, as an open problem whose computational complexity is then studied in three cases: Gaussian signaling, low SNR, and high SNR. For the Gaussian signaling case and the low SNR regime, the dependence of the mutual information on the right singular vectors vanishes, making the optimal precoder design problem easy to solve. In the high SNR regime, however, the dependence on the right singular vectors cannot be avoided and we show the difficulty of computing the optimal precoder through an NPhardness analysis. © 2009 IEEE.

Based on the high probability that primary user is idle in cognitive radio networks, we propose an optimal Bayesian detector structure for spectrum sensing. Although the optimal detector by Neyman-Pearson theorem maximizes the detection probability for a given false alarm probability, Bayesian detector can achieve a higher overall spectrum utilization and SU throughput and at the same time the primary user is well protected from secondary user's interference. For BPSK modulated primary signals we show that the optimal Bayesian detector can be reduced to an energy detector in lower SNR regime, and it can be approximated to a detector employing the sum of received signal magnitudes in high SNR regime to detect primary signals. We give the analysis for optimal Bayesian detector and the corresponding suboptimal detector structure in both low and high SNR regimes, and verify the performance of the detector with simulation results.

Communication systems with low-resolution analog-to-digital-converters (ADCs) can exploit channel state information at the transmitter and receiver. This paper presents codebook designs and performance analyses for limited feedback MIMO systems with finite-bit ADCs. A point-to-point single-user channel is firstly considered. When the received signal is sliced by 1-bit ADCs, the absolute phase at the receiver is important to align the phase of the received signals. A new codebook design for beamforming, which separately quantizes the channel direction and the residual phase, is therefore proposed. For the multi-bit case where the optimal transmission method is unknown, suboptimal Gaussian signaling and eigenvector beamforming is assumed to obtain a lower bound of the achievable rate. It is found that to limit the rate loss, more feedback bits are needed in the medium SNR regime than the low and high SNR regimes, which is quite different from the conventional infinite-bit ADC case. Second, a multi-user system where a multiple-antenna transmitter sends signals to multiple single-antenna receivers with finite-bit ADCs is considered. Based on the derived performance loss due to finite-bit ADCs and finite-bit CSI feedback, the number of bits per feedback should increase linearly with the ADC resolution in order to restrict the rate loss.

A two-way relay non-orthogonal multiple access (TWR-NOMA) system is investigated, where two groups of NOMA users exchange messages with the aid of one half-duplex (HD) decode-and-forward (DF) relay. Since the signal-plus-interference-to-noise ratios (SINRs) of NOMA signals mainly depend on effective successive interference cancellation (SIC) schemes, imperfect SIC (ipSIC) and perfect SIC (pSIC) are taken into account. In order to characterize the performance of TWR-NOMA systems, we first derive closed-form expressions for both exact and asymptotic outage probabilities of NOMA users' signals with ipSIC/pSIC. Based on the derived results, the diversity order and throughput of the system are examined. Then we study the ergodic rates of users' signals by providing the asymptotic analysis in high SNR regimes. Lastly, numerical simulations are provided to verify the analytical results and show that: 1) TWR-NOMA is superior to TWR-OMA in terms of outage probability in low SNR regimes; 2) Due to the impact of interference signal (IS) at the relay, error floors and throughput ceilings exist in outage probabilities and ergodic rates for TWR-NOMA, respectively; and 3) In delay-limited transmission mode, TWR-NOMA with ipSIC and pSIC have almost the same energy efficiency. However, in delay-tolerant transmission mode, TWR-NOMA with pSIC is capable of achieving larger energy efficiency compared to TWR-NOMA with ipSIC.

In this paper, we study the performance of the two-way relay network (TWRN) with NOMA-based transmission, where the information is exchanged during three time slots. To provide a fair comparison, we also examine the performance of a three-time-slot (3T) OMA-based transmission scheme. In the following, these schemes are referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3T-NOMA-TWRN</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3T-OMA-TWRN</i> . In the 3T-NOMA-TWRN scheme, each transceiver simultaneously transmits two symbols with different powers to exploit the multiplexing gain, implemented through the power-domain NOMA technique. The performance of both schemes is analyzed in terms of the outage probability, the ergodic sum-rate, and their asymptotic behaviors. The outage and rate performances of the schemes are enhanced by optimizing the power allocation coefficients, especially in the NOMA-based scheme. The proposed power allocation algorithm imposes negligible overhead because of considering the closed-form expressions of the outage probability and the ergodic sum-rate as the objective function. For the NOMA-based scheme, we propose sub-optimal power allocation strategies that closely approximate the optimal solution. The validity of the analytical results is confirmed by extensive numerical examples. The results show that the 3T-NOMA-TWRN scheme outperforms its OMA-based counterpart in terms of the outage probability and the ergodic sum-rate, especially in the high-rate and high-SNR regimes. Additionally, the 3T-NOMA-TWRN scheme significantly outperforms the existing two-time-slot (2T)-NOMA-TWRN scheme in the high-SNR regime. However, in the low-SNR regime, only in some cases, the situation changes in favor of the 2T-NOMA-TWRN scheme. Surprisingly, the performance of the 3T-OMA-TWRN scheme can significantly surpass that of the 2T-NOMA-TWRN scheme in certain situations, depending on the rate and SNR regimes and the network geometry. Furthermore, the 3T-NOMA-TWRN scheme significantly reduces the TWRN performance sensitivity to the location of relay node, which is a critical concern in both the 3T-OMA-TWRN and 2T-NOMA-TWRN schemes, especially in the latter case.

We address the Least Quantile of Squares (LQS) (and in particular the Least Median of Squares) regression problem using modern optimization methods. We propose a Mixed Integer Optimization (MIO) formulation of the LQS problem which allows us to find a provably global optimal solution for the LQS problem. Our MIO framework has the appealing characteristic that if we terminate the algorithm early, we obtain a solution with a guarantee on its sub-optimality. We also propose continuous optimization methods based on first-order subdifferential methods, sequential linear optimization and hybrid combinations of them to obtain near optimal solutions to the LQS problem. The MIO algorithm is found to benefit significantly from high quality solutions delivered by our continuous optimization based methods. We further show that the MIO approach leads to (a) an optimal solution for any dataset, where the data-points $(y_{i},\\mathbf{x}_{i})$’s are not necessarily in general position, (b) a simple proof of the breakdown point of the LQS objective value that holds for any dataset and (c) an extension to situations where there are polyhedral constraints on the regression coefficient vector. We report computational results with both synthetic and real-world datasets showing that the MIO algorithm with warm starts from the continuous optimization methods solve small ($n=100$) and medium ($n=500$) size problems to provable optimality in under two hours, and outperform all publicly available methods for large-scale ($n=10,000$) LQS problems.

This paper addresses the following question, which is of interest in the design of a multiuser decentralized network. Given a total system bandwidth of W Hz and a fixed data rate constraint of R bps for each transmission, how many frequency slots N of size W/N should the band be partitioned into in order to maximize the number of simultaneous links in the network? Dividing the available spectrum results in two competing effects. On the positive side, a larger N allows for more parallel, noninterfering communications to take place in the same area. On the negative side, a larger N increases the SINR requirement for each link because the same information rate must be achieved over less bandwidth. Exploring this tradeoff and determining the optimum value of N in terms of the system parameters is the focus of the paper. Using stochastic geometry, the optimal SINR threshold - which directly corresponds to the optimal spectral efficiency - is derived for both the low SNR (power-limited) and high SNR (interference-limited) regimes. This leads to the optimum choice of the number of frequency bands N in terms of the path loss exponent, power and noise spectral density, desired rate, and total bandwidth.

Channel state information (CSI) is important for achieving large rates in MIMO channels. However, in time-varying MIMO channels, there is a tradeoff between the time/energy spent acquiring channel state information (CSI) and the time/energy remaining for data transmission. This tradeoff is accentuated in the MIMO multiple access channel (MAC), since the number of channel vectors to be estimated increases with the number of users. Furthermore, the problem of acquiring CSI is tightly coupled with the problem of exploiting CSI through multiuser scheduling. This paper considers a block-fading MAC with coherence time <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> , n uncoordinated users-each with one transmit antenna and the same average power constraint, and a base station with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> receive antennas and no a priori CSI. For this scenario, a training-based communication scheme is proposed and the training and multiuser-scheduling aspects of the scheme are jointly optimized. In the high-SNR regime, the sum capacity of the non-coherent SIMO MAC is characterized and used to establish the SNR-scaling-law optimality of the proposed scheme. In the low-SNR regime, the sum-rate of the proposed scheme is found to decay linearly with vanishing SNR when flash signaling is incorporated. Furthermore, this linear decay is shown to be order-optimal through comparison to the low-SNR sum capacity of the non-coherent SIMO MAC. A by product of these SNR-asymptotic analyses is the observation that non-trivial scheduling (i.e., scheduling a strict subset of trained users) is advantageous at low SNR, but not at high SNR. The sum-rate and per-user throughput are also explored in the large- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> and large- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> regimes. Non-coherent capacity, training, multiple access channel, multiuser scheduling, opportunistic scheduling.

Motivated by the demand for energy-efficient communication solutions in the next generation cellular network, a mixed-ADC receiver architecture for massive multiple input multiple output (MIMO) systems is proposed, which differs from previous works in that herein one-bit analog-to-digital converters (ADCs) partially replace the conventionally assumed high-resolution ADCs. The information-theoretic tool of generalized mutual information (GMI) is exploited to analyze the achievable data rates of the proposed system architecture and an array of analytical results of engineering interest are obtained. For deterministic single input multiple output (SIMO) channels, a closed-form expression of the GMI is derived, based on which the linear combiner is optimized. Then, the asymptotic behaviors of the GMI in both low and high SNR regimes are explored, and the analytical results suggest a plausible ADC assignment scheme. Finally, the analytical framework is applied to the multi-user access scenario, and the corresponding numerical results demonstrate that the mixed system architecture with a relatively small number of high-resolution ADCs is able to achieve a large fraction of the channel capacity without output quantization.

The novel advancement in the wireless communication system demands Quality-of-service (QoS) awareness to support low latency applications, such as the internet of things, multimedia communication, and autonomous vehicles under delay constraints. To explore the performance of wireless networks supporting such applications, effective capacity (EC) is introduced as a significant performance parameter. This paper proposes a closed-form expression of the EC over gamma shadowed &#x03B1;-&#x03BA;-&#x03BC; composite fading channel, and using this expression performance analysis of the EC over the presented channel is conducted. Furthermore, asymptotic analysis under low and high SNR regimes is performed to get more insight into the system.

In this work, we consider a K-user Gaussian wiretap multiple-access channel (GW-MAC) in which each transmitter has an independent confidential message for the receiver. There is also an external eavesdropper who intercepts the communications. The goal is to transmit the messages reliably while keeping them confidential from the eavesdropper. To accomplish this goal, two different approaches have been proposed in prior works, namely, i.i.d. Gaussian random coding and real alignment. However, the former approach fails at moderate and high SNR regimes as its achievable result does not grow with SNR. On the other hand, while the latter approach gives a promising result at the infinite SNR regime, its extension to the finite-SNR regime is a challenging task. To fill the gap between the performance of the existing approaches, in this work, we establish a new scheme in which, at the receiver's side, it utilizes an extension of the compute-and-forward decoding strategy and at the transmitters' side it exploits lattice alignment, cooperative jamming, and i.i.d. random codes. For the proposed scheme, we derive a new achievable bound on sum secure rate which scales with log(SNR) and hence it outperforms the i.i.d. Gaussian codes in moderate and high SNR regimes. We evaluate the performance of our scheme, both theoretically and numerically. Furthermore, we show that our sum secure rate achieves the optimal sum secure degrees of freedom in the infinite-SNR regime.

This paper studies the low-SNR regime performance of a scalar complex K-user interference channel with the Gaussian noise. The finite bandwidth case is considered, where the low-SNR regime is approached by letting the input power go to zero, while the bandwidth is small and fixed. We show that for all δ > 0, there exists a set of channel coefficients with non-zero measure (probability), in which the wideband slope per user satisfies S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <; 2/K +δ. This is quite contrary to the large bandwidth case, where a slope of 1 per user is achievable with probability 1. We also develop an interference alignment scheme for the finite bandwidth case that shows some gain in wideband slope.

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