In this work, we present some novel results for recovering the frequency components (frequencies and their complex coefficients) of ann-dimensional complex signal/vector from noiseless measurements of its frequency spectrum. In the first part of the work, the frequency vectors are the general Vandermonde vectors (with an amplitude varying exponentially across the vector components) and the spectrum is a “Vandermonde” spectrum (a vector’s Vandermonde transform). It is shown that if then-dimensional vector is composed ofsunknown frequency components, exact recovery of the frequency components is possible with$2s$uniform measurements of the spectrum and$O(\mathrm {poly}(s))$complexity when$n \geq 2s$. Furthermore, exact recovery with$O(\mathrm {poly}(s))$complexity is also possible with arbitrary$3s$non-uniform measurements of the spectrum. In the second part of the work, the frequency vectors are Fourier vectors and we consider the challenging problem of recovering the frequency components fromphase-lessFourier spectrum measurements. One main result shows that the frequencies of the vector can be recovered with$4s - 1$uniform spectrum measurements and$O(\mathrm {poly}(s))$complexity if$n \geq 4s - 1$, but not their complex coefficients (there being in general$2^{s-1}$solutions for the coefficients). Another main result shows the surprising fact that if then-dimensional vector is composed ofsunknown frequency components with at least one having anon-harmonicfrequency,$n \geq 4s - 1$and we take$8s-3$non-uniform (phase-less) spectrum measurements, the frequencies can be recovered exactly while, remarkably, there areonly twopossible coefficient vectors (up to a phase) and they are easily obtainable from each other. The frequencies and the two coefficient vectors can be found by an algorithm with only$O(\mathrm {poly}(s))$complexity. Based on the results from the first part, a new class of compressed sensing measurement matrices is presented from which it is possible to recovers-sparsen-dimensional vectors for$n \geq 2s$with as few as$2s$measurements and with a recovery algorithm of$O(\mathrm {poly}(s))$complexity. Similarly, from the results in the second part, we provide a class of measurement matrices from which it is possible to recover almost alls-sparsen-dimensional vectors (up to a phase) from$8s - 2$phase-less measurements and$O(\mathrm {poly}(s))$recovery complexity when$n \geq 4s - 1$.

In this paper, we study channel estimation at a uniform linear array (ULA) with$N$antennas, where the channel at the ULA is composed of$L$paths with different angles of arrival (AoAs). It is assumed that Discrete-Time Fourier Transform (DTFT) beams (also known as analog beams with DFT beams as special cases) are applied at the ULA to project the incoming signal onto a single (or multiple) RF chain(s), after which the signal is sampled and measured in baseband domain; the underlying signal is assumed to be constant during the projections. This measurement procedure arises in various communication systems, such as the receive beam sweeping phase in 5G NR, where DTFT beams are used due to their simple implementation as linear phase shifts on analog antennas. A fundamental question about this procedure is the number of DTFT measurements$K$needed to recover the$L$AoAs. Previous work on this problem showed (by applying compressed sensing theory) that$K \approx L\mathcal {O}(\log (N/L))$measurements are sufficient for recovering the AoAs, which grows with$N$. First, we show that necessary conditions for recovery are$N \geq 2L$and$K \geq 2L$. Second, by using properties of DTFT beam projections, we are able to show that if$N \geq 2L$then$K \geq 3L$arbitrary DTFT measurements suffice; hence, dependency on$N$is completely removed. Furthermore, if the DTFT beams are chosen to equal DFT beams with period$N$, then$K \geq 2L$beam measurements are enough, achieving sufficiency of the necessary conditions. With these results, an AoA estimation algorithm is formulated which has enormous complexity savings compared to$L$-dimensional AoA search such as maximum likelihood (ML) estimation. Numerical simulations demonstrate the algorithm’s improved performance over conventional algorithms such as beamspace ESPRIT and compressed sensing.

Due to the fast deployment and commercial use of fifth generation (5G) communication systems, there is an increasing demand for higher uplink rates, and thus the deployment of more transmit antennas at user equipment (UE) becomes even more urgent. Nowadays, to better balance the uplink user experience and terminal cost, a feasible way to deploy larger number of transmit antennas at UE is to patch together multiple smaller radio frequency integrated circuits (RFICs), e.g., two RFICs of 2 transmit antennas (2T) will be used to implement 4T. However, this setup will induce a random phase difference between the two RFICs that is unknown at the transmitter. In this paper, we investigate the optimal uplink digital precoder design under such random phase imperfections. In terms of maximizing the average channel capacity, we find that the optimal precoder will be the eigen-vectors of an adjusted transmit correlation matrix. Block-diagonal precoders are also shown to be robust to random phase errors at the expense of some loss in degrees of freedom and channel capacity. Numerical simulations are provided to verify the effectiveness of the proposed uplink precoders under random phase impacts.

We consider a downlink resource allocation problem that maximizes the downlink rate, where several users can be allocated to the same resource by employing a larger modulation alphabet. Moreover, each user is constrained to receive across one resource only. The latter constraint is especially suitable for computational and power constrained users. These requirements give rise to a different resource allocation problem compared to previous studies. The throughput of a user is not measured by conventional logarithm formulae for throughput (which implicitly assume Gaussian alphabets). Instead, we let UEs calculate effective signal-to-noise-ratios for each resource and modulation alphabet, which corresponds to the achievable data rate on a resource with the specific modulation. These values are reported to the transmitter, which then uses them to find the optimal allocation. The underlying optimization problem is discrete, and it is made linear through our formulation of the problem. We solve the problem exactly, and formulate a simple heuristic allocation procedure that is close to the optimal allocation in simulated scenarios.

The need for accurate and relevant CSI feedback is one of the pillars for the efficiency of cellular systems. However, one problem facing CSI reporting is the effect of CSI aging such that the reported CSI information is obsolete when data is scheduled based on it. CSI may instead be predicted in order to mitigate some of the aging loss. In doing so two things need to be considered. First, the CSI prediction needs to have low bias also for cases where prediction is hard, such as short channel coherence time or infrequent CSI reporting. Second, it is desirable not to predict the interference part of the channel quality, as the interference is in the control of the network and hence there is no need for the UE to predict it. This paper addresses the CSI prediction problem taking these two aspects into consideration and proposes a method based on squared channel matrix prediction.

This work considers a two-user uplink MIMO channel, where the receiving node applies a zero forcing equalizer to jointly estimate the transmitted symbols. The goal is to design linear precoders at the two users in order to improve the performance of the equalizer. In one part of this work, assuming local channel knowledge at each user, we quantify the minimum amount of feedback from the receiving node necessary for construction of optimal precoders at the users. In contrast to common feedback procedures, where the precoders are fed back to the users, we show that the amount of complex numbers to feedback is at most half when compared to feeding back the precoders. Furthermore, we investigate construction of precoders when limited feedback is available from the receiving node. Numerical results are presented that quantify the loss due to absence of optimal feedback, as well as the gain obtained from limited feedback with local channel knowledge compared to precoding without feedback and channel knowledge at the users.

This paper investigates linear precoding over nonsingular linear channels with additive white Gaussian noise, with lattice-type inputs. The aim is to maximize the minimum distance of the received lattice points, where the precoder is subject to an energy constraint. It is shown that the optimal precoder only produces a finite number of different lattices, namely perfect lattices, at the receiver. The well-known densest lattice packings are instances of perfect lattices, but are not always the solution. This is a counter-intuitive result as previous work in the area showed a tight connection between densest lattices and minimum distance. Since there are only finite many different perfect lattices, they can theoretically be enumerated offline. A new upper bound on the optimal minimum distance is derived, which significantly improves upon a previously reported bound, and is useful when actually constructing the precoders.

Distributed storage systems have been receiving increasing attention lately due to the developments in cloud and grid computing. Furthermore, a major part of the stored information comprises of multimedia, whose content can be communicated even with a lossy (non-perfect) reconstruction. In this context, Multiple Description Lattice Quantizers (MDLQ) can be employed to encode such sources for distributed storage and store them across distributed nodes. Their inherent properties yield that having access to all nodes gives perfect reconstruction of the source, while the reconstruction quality decreases gracefully with fewer available nodes. If a set of nodes fails, lossy repair techniques could be applied to reconstruct the failed nodes from the available ones. This problem has mostly been studied with the lossless (perfect) reconstruction assumption. In this work, a general model, Multiple Description Lattice Quantizer with Repairs (MDLQR), is introduced that encompasses the lossy repair problem for distributed storage applications. New performance measures and repair techniques are introduced for MDLQR, and a non-trivial identity is derived, which is related to other results in the literature. This enables us to find the optimal encoder for a certain repair technique used in the MDLQR. Furthermore, simulation results are used to evaluate the performance of the different repair techniques.

We consider physical layer security of massive MIMO systems in TDD mode. We show that with massive MIMO a passive eavesdropper is not very dangerous and must therefore be active and attack the training phase. An attack on the training phase is potentially very harmful to the physical layer security, and we therefore investigate three different schemes for detecting the presence of an active eavesdropper. The three schemes differ in the location where the detection is done (base station, intended user, or jointly), and also in the level of system parameters that are assumed known to the base station and/or intended user.

Channel estimation attacks can degrade the performance of the legitimate system and facilitate eavesdropping. It is known that pilot contamination can alter the legitimate transmit precoder design and strengthen the quality of the received signal at the eavesdropper, without being detected. In this paper, we devise a technique which employs random pilots chosen from a known set of phase-shift keying (PSK) symbols to detect pilot contamination. The scheme only requires two training periods without any prior channel knowledge. Our analysis demonstrates that using the proposed technique in a massive MIMO system, the detection probability of pilot contamination attacks can be made arbitrarily close to 1. Simulation results reveal that the proposed technique can significantly increase the detection probability and is robust to noise power as well as the eavesdropper's power.

This work addresses scheduling in a cognitive radio scenario where a minimum throughput for the downlink primary network (PN) is guaranteed to each user with an associated violation probability (probability of not obtaining the guaranteed throughput). The primary network is surrounded by multiple downlink secondary networks, each aiming to maximize its network throughput. Scheduling in PN is performed independent of the secondary networks. Some information about the PN is available at the central scheduler that is responsible for scheduling the secondary networks. The contribution of this work is to apply a novel scheduler to the PN which is more robust to QoS degradations resulting from the secondary networks than other state of the art schedulers. This is validated by numerical simulations of the cognitive radio network.

Consider the communication system model y = HFx + n, where H and F are the channel and precoder matrices, x is a vector of data symbols drawn from some lattice-type constellation, such as M-QAM, n is an additive white Gaussian noise vector and y is the received vector. It is assumed that both the transmitter and the receiver have perfect knowledge of the channel matrix H and that the transmitted signal Fx is subject to an average energy constraint. The columns of the matrix HF can be viewed as the basis vectors that span a lattice, and we are interested in the precoder F that maximizes the minimum distance of this lattice. This particular problem remains open within the theory of lattices and the communication theory. This paper provides the complete solution for any nonsingular M × 2 channel matrix H. For real-valued matrices and vectors, the solution is that HF spans the hexagonal lattice. For complex-valued matrices and vectors, the solution is that HF, when viewed in four-dimensional real-valued space, spans the Schlafli lattice D4.

We consider the effect of signaling rate (baud rate) on the information rate of single carrier linear transmission systems with Gaussian inputs. Several different communication scenarios are investigated: correlated or uncorrelated symbols, a fixed modulation pulse or a modulation pulse varying with the signaling rate and frequency selective or flat channels. For uncorrelated symbols, we show that the information rate grows monotonically with signaling rate for some cases while it can in fact decrease in other cases. Sufficient conditions on the modulation pulse and the channel impulse response are derived so that the information rate is increasing with increased signaling rate. Especially, these conditions give criterias for when non-orthogonal signaling is beneficial compared to orthogonal signaling in the case of flat fading. For modulation pulses varying with the signaling rate, it is shown that there are pulses for which the information rate is non-decreasing with increasing signaling rate. When correlation between symbols is allowed, we show that one can guarantee increasing information rate with increased signaling rate, no matter the pulse-channel shape (except for some hypothetical special cases), by signaling with an SNR above a certain finite threshold.

Consider the communication model ȳ = HF x̄ + n̄, where H; F are real-valued matrices, x̄ is a data vector drawn from some real-valued lattice (e.g. M-PAM), n̄ is additive white Gaussian noise and ȳ is the received vector. It is assumed that the transmitter and the receiver have perfect knowledge of the channel matrix H (perfect CSI) and that the transmitted signal F x̄ is subject to an average energy constraint. The columns of the matrix HF can be viewed as basis vectors that span a lattice, and we are interested in the minimum distance of this lattice. More precisely, for a given H, which F under an average energy constraint will maximize the minimum distance of the lattice HF? This particular question remains open within the theory of lattices. This work provides the solution for 2×2 matrices H; F. The answer is an F such that HF is a hexagonal lattice.

Consider the transmission of complex-valued symbols over $N$ parallell channels in additive white Gaussian noise. It is well known that linear precoding of the complex-valued data improves system performance (e.g. symbol error rate, information rate, MMSE, etc.) at a cost of increased decoding complexity at the receiver. This work constructs precoders that are constrained to have a decoding complexity which equals that of no precoding, while still improving the system performance significantly compared with the no precoding case. This is achieved by designing the precoder so that it precodes the complex data streams separately, by utilizing the latest result from optimal real-valued precoding, and transmitting the real and complex parts of one symbol over different antennas.

This work studies the design of linear precoder codebooks for NxM MIMO channels with MMSE detection at the receiver. A natural split of precoder-design is unitary precoding and non-unitary precoding. Unitary precoding is only performing rotation of the data in a way beneficial for the channel. Non-unitary precoding additionally also uses power-loading to further improve the performance. Somewhat surprisingly, unitary precoding facilitates a performance boosting by a re-enumeration of the antenna elements at the receiver side that can not be accomodated in the non-unitary precoding setting. This operation leads to substantial performance gains. The question investigated in this paper is whether this re-enumeration can compensate for the lack of power-loading. The outcome is that for small precoder codebooks, unitary precoding performs as good as non-unitary, while for larger codebooks non-unitary precoding outperforms unitary precoding.

We consider the problem of constructing linear precoders for quasi-stationary multiple-input multiple-output channels. Maximum-likelihood detection is assumed and the objective of the precoding is to maximize the minimum Euclidean distance of the signaling. Since the channel remains constant for some time, the precoding is performed spatially as well as across time. As will be shown, the precoder design is tightly connected to the theory of partial response signaling and precoders can be designed by usage of existing methods. The decoding complexity will be controlled and can be maintained small.

In this work we study the problem of constructing precoders for spatially multiplexed multiple-input multiple output (MIMO) channels with close to optimal minimum Euclidean distance. In order to exploit the full potential of such designs, an ML detector must be used. Our design takes the decoding complexity into account and constrains it to a reasonable level. For our simplest case, the ML detector can be implemented by a Viterbi algorithm operating on a state space of size equal to the size of the modulation alphabet. The design problem will be relaxed by using precoders F such that F*H*HF is a cyclic Toeplitz matrix. Within this class of precoders, the optimal precoder can be found via linear programming. Of uttermost practical importance is the discovery that there only exist very few different effective channels HF even for large MIMO setups; thus, the optimization at the transmitter side reduces into choosing the best precoder from a small list. Receiver tests verify that our method improves upon the currently best precoder designs.

We consider the effect of symbol rate on the constrained capacity of linear modulation with a fixed spectral density. We show that constrained capacity grows with the symbol rate for some modulation pulses but shrinks with others. Sufficient conditions on the pulse are derived for the constrained capacity to be monotonically increasing with faster symbol rate. Most standard pulses fulfill these.

In this paper we consider the least time-frequency product necessary to transmit a small finite symbol packet such that the symbols can be independently detected. The system model assumed is offset QAM-OFDM, based on a finite duration pulse shape. The outcome is that the optimal pulse shape is of very short duration and that the optimal symbol allocation strategy is often to use as many subcarriers as there are symbols to transmit. Symbol packets up to 150 symbols are considered.