arXiv subjects

Youjian Liu

Publications and source records attributed to Youjian Liu.

14 recordsLinked to original sources

arXiv2025

Maximum a Posteriori Probability (MAP) Joint Carrier Frequency Offset (CFO) and Channel Estimation for MIMO Channels with Spatial and Temporal Correlations

We consider time varying MIMO fading channels with known spatial and temporal correlation and solve the problem of joint carrier frequency offset (CFO) and channel estimation with prior distributions. The maximum a posteriori probability (MAP) joint estimation is proved to be equivalent to a separate MAP estimation of the CFO followed by minimum mean square error (MMSE) estimation of the channel while treating the estimated CFO as true. The MAP solution is useful to take advantage of the estimates from the previous data packet. A low complexity universal CFO estimation algorithm is extended from the time invariant case to the time varying case. Unlike past algorithms, the universal algorithm does not need phase unwrapping to take advantage of the full range of symbol correlation and achieves the derived Bayesian Cramér-Rao lower bound (BCRLB) in almost all SNR range. We provide insight on the the relation among the temporal correlation coefficient of the fading, the CFO estimation performance, and the pilot signal structure. An unexpected observation is that the BCRLB is not a monotone function of the temporal correlation and is strongly influenced by the pilot signal structures. A simple rearrangement of the 0's and 1's in the pilot signal matrix will render the BCRLB from being non-monotone to being monotone in certain temporal correlation ranges. Since the BCRLB is shown to be achieved by the proposed algorithm, it provides a guideline for pilot signal design.

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arXiv2013

Duality and Optimization for Generalized Multi-hop MIMO Amplify-and-Forward Relay Networks with Linear Constraints

We consider a generalized multi-hop MIMO amplify-and-forward (AF) relay network with multiple sources/destinations and arbitrarily number of relays. We establish two dualities and the corresponding dual transformations between such a network and its dual, respectively under single network linear constraint and per-hop linear constraint. The result is a generalization of the previous dualities under different special cases and is proved using new techniques which reveal more insight on the duality structure that can be exploited to optimize MIMO precoders. A unified optimization framework is proposed to find a stationary point for an important class of non-convex optimization problems of AF relay networks based on a local Lagrange dual method, where the primal algorithm only finds a stationary point for the inner loop problem of maximizing the Lagrangian w.r.t. the primal variables. The input covariance matrices are shown to satisfy a polite water-filling structure at a stationary point of the inner loop problem. The duality and polite water-filling are exploited to design fast primal algorithms. Compared to the existing algorithms, the proposed optimization framework with duality-based primal algorithms can be used to solve more general problems with lower computation cost.

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arXiv2012

Downlink Performance Analysis for a Generalized Shotgun Cellular System

In this paper, we analyze the signal-to-interference-plus-noise ratio (SINR) performance at a mobile station (MS) in a random cellular network. The cellular network is formed by base-stations (BSs) placed in a one, two or three dimensional space according to a possibly non-homogeneous Poisson point process, which is a generalization of the so-called shotgun cellular system. We develop a sequence of equivalence relations for the SCSs and use them to derive semi-analytical expressions for the coverage probability at the MS when the transmissions from each BS may be affected by random fading with arbitrary distributions as well as attenuation following arbitrary path-loss models. For homogeneous Poisson point processes in the interference-limited case with power-law path-loss model, we show that the SINR distribution is the same for all fading distributions and is not a function of the base station density. In addition, the influence of random transmission powers, power control, multiple channel reuse groups on the downlink performance are also discussed. The techniques developed for the analysis of SINR have applications beyond cellular networks and can be used in similar studies for cognitive radio networks, femtocell networks and other heterogeneous and multi-tier networks.

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arXiv2009

Sum-capacity of Interference Channels with a Local View: Impact of Distributed Decisions

Due to the large size of wireless networks, it is often impractical for nodes to track changes in the complete network state. As a result, nodes have to make distributed decisions about their transmission and reception parameters based on their local view of the network. In this paper, we characterize the impact of distributed decisions on the global network performance in terms of achievable sum-rates. We first formalize the concept of local view by proposing a protocol abstraction using the concept of local message passing. In the proposed protocol, nodes forward information about the network state to other neighboring nodes, thereby allowing network state information to trickle to all the nodes. The protocol proceeds in rounds, where all transmitters send a message followed by a message by all receivers. The number of rounds then provides a natural metric to quantify the extent of local information at each node. We next study three network connectivities, Z-channel, a three-user double Z-channel and a reduced-parametrization $K$-user stacked Z-channel. In each case, we characterize achievable sum-rate with partial message passing leading to three main results. First, in many cases, nodes can make distributed decisions with only local information about the network and can still achieve the same sum-capacity as can be attained with global information irrespective of the actual channel gains. Second, for the case of three-user double Z-channel, we show that universal optimality is not achievable if the per node information is below a threshold. Third, using reduced parametrization $K$-user channel, we show that very few protocol rounds are needed for the case of very weak or very strong interference.

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arXiv2008

A Simple Converse Proof and a Unified Capacity Formula for Channels with Input Constraints

Given the single-letter capacity formula and the converse proof of a channel without constraints, we provide a simple approach to extend the results for the same channel but with constraints. The resulting capacity formula is the minimum of a Lagrange dual function. It gives an unified formula in the sense that it works regardless whether the problem is convex. If the problem is non-convex, we show that the capacity can be larger than the formula obtained by the naive approach of imposing constraints on the maximization in the capacity formula of the case without the constraints. The extension on the converse proof is simply by adding a term involving the Lagrange multiplier and the constraints. The rest of the proof does not need to be changed. We name the proof method the Lagrangian Converse Proof. In contrast, traditional approaches need to construct a better input distribution for convex problems or need to introduce a time sharing variable for non-convex problems. We illustrate the Lagrangian Converse Proof for three channels, the classic discrete time memoryless channel, the channel with non-causal channel-state information at the transmitter, the channel with limited channel-state feedback. The extension to the rate distortion theory is also provided.

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arXiv2008

Joint Beamforming for Multiaccess MIMO Systems with Finite Rate Feedback

This paper considers multiaccess multiple-input multiple-output (MIMO) systems with finite rate feedback. The goal is to understand how to efficiently employ the given finite feedback resource to maximize the sum rate by characterizing the performance analytically. Towards this, we propose a joint quantization and feedback strategy: the base station selects the strongest users, jointly quantizes their strongest eigen-channel vectors and broadcasts a common feedback to all the users. This joint strategy is different from an individual strategy, in which quantization and feedback are performed across users independently, and it improves upon the individual strategy in the same way that vector quantization improves upon scalar quantization. In our proposed strategy, the effect of user selection is analyzed by extreme order statistics, while the effect of joint quantization is quantified by what we term ``the composite Grassmann manifold''. The achievable sum rate is then estimated by random matrix theory. Due to its simple implementation and solid performance analysis, the proposed scheme provides a benchmark for multiaccess MIMO systems with finite rate feedback.

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arXiv2007

Multi-Access MIMO Systems with Finite Rate Channel State Feedback

This paper characterizes the effect of finite rate channel state feedback on the sum rate of a multi-access multiple-input multiple-output (MIMO) system. We propose to control the users jointly, specifically, we first choose the users jointly and then select the corresponding beamforming vectors jointly. To quantify the sum rate, this paper introduces the composite Grassmann manifold and the composite Grassmann matrix. By characterizing the distortion rate function on the composite Grassmann manifold and calculating the logdet function of a random composite Grassmann matrix, a good sum rate approximation is derived. According to the distortion rate function on the composite Grassmann manifold, the loss due to finite beamforming decreases exponentially as the feedback bits on beamforming increases.

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arXiv2007

Quantization Bounds on Grassmann Manifolds of Arbitrary Dimensions and MIMO Communications with Feedback

This paper considers the quantization problem on the Grassmann manifold with dimension n and p. The unique contribution is the derivation of a closed-form formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary dimension n and p, while previous results are only valid for either p=1 or a fixed p with asymptotically large n. Based on the volume formula, the Gilbert-Varshamov and Hamming bounds for sphere packings are obtained. Assuming a uniformly distributed source and a distortion metric based on the squared chordal distance, tight lower and upper bounds are established for the distortion rate tradeoff. Simulation results match the derived results. As an application of the derived quantization bounds, the information rate of a Multiple-Input Multiple-Output (MIMO) system with finite-rate channel-state feedback is accurately quantified for arbitrary finite number of antennas, while previous results are only valid for either Multiple-Input Single-Output (MISO) systems or those with asymptotically large number of transmit antennas but fixed number of receive antennas.

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arXiv2007

On the Information Rate of MIMO Systems with Finite Rate Channel State Feedback and Power On/Off Strategy

This paper quantifies the information rate of multiple-input multiple-output (MIMO) systems with finite rate channel state feedback and power on/off strategy. In power on/off strategy, a beamforming vector (beam) is either turned on (denoted by on-beam) with a constant power or turned off. We prove that the ratio of the optimal number of on-beams and the number of antennas converges to a constant for a given signal-to-noise ratio (SNR) when the number of transmit and receive antennas approaches infinity simultaneously and when beamforming is perfect. Based on this result, a near optimal strategy, i.e., power on/off strategy with a constant number of on-beams, is discussed. For such a strategy, we propose the power efficiency factor to quantify the effect of imperfect beamforming. A formula is proposed to compute the maximum power efficiency factor achievable given a feedback rate. The information rate of the overall MIMO system can be approximated by combining the asymptotic results and the formula for power efficiency factor. Simulations show that this approximation is accurate for all SNR regimes.

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arXiv2007

Unequal dimensional small balls and quantization on Grassmann Manifolds

The Grassmann manifold G_{n,p}(L) is the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space L^{n}, where L is either R or C. This paper considers an unequal dimensional quantization in which a source in G_{n,p}(L) is quantized through a code in G_{n,q}(L), where p and q are not necessarily the same. It is different from most works in literature where p\equiv q. The analysis for unequal dimensional quantization is based on the volume of a metric ball in G_{n,p}(L) whose center is in G_{n,q}(L). Our chief result is a closed-form formula for the volume of a metric ball when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary n, p, q and L, while previous results pertained only to some special cases. Based on this volume formula, several bounds are derived for the rate distortion tradeoff assuming the quantization rate is sufficiently high. The lower and upper bounds on the distortion rate function are asymptotically identical, and so precisely quantify the asymptotic rate distortion tradeoff. We also show that random codes are asymptotically optimal in the sense that they achieve the minimum achievable distortion with probability one as n and the code rate approach infinity linearly. Finally, we discuss some applications of the derived results to communication theory. A geometric interpretation in the Grassmann manifold is developed for capacity calculation of additive white Gaussian noise channel. Further, the derived distortion rate function is beneficial to characterizing the effect of beamforming matrix selection in multi-antenna communications.

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arXiv2007

Effect of Finite Rate Feedback on CDMA Signature Optimization and MIMO Beamforming Vector Selection

We analyze the effect of finite rate feedback on CDMA (code-division multiple access) signature optimization and MIMO (multi-input-multi-output) beamforming vector selection. In CDMA signature optimization, for a particular user, the receiver selects a signature vector from a codebook to best avoid interference from other users, and then feeds the corresponding index back to the specified user. For MIMO beamforming vector selection, the receiver chooses a beamforming vector from a given codebook to maximize throughput, and feeds back the corresponding index to the transmitter. These two problems are dual: both can be modeled as selecting a unit norm vector from a finite size codebook to "match" a randomly generated Gaussian matrix. In signature optimization, the least match is required while the maximum match is preferred for beamforming selection. Assuming that the feedback link is rate limited, our main result is an exact asymptotic performance formulae where the length of the signature/beamforming vector, the dimensions of interference/channel matrix, and the feedback rate approach infinity with constant ratios. The proof rests on a large deviation principle over a random matrix ensemble. Further, we show that random codebooks generated from the isotropic distritution are asymptotically optimal not only on average, but also with probability one.

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arXiv2006

Quantization Bounds on Grassmann Manifolds and Applications to MIMO Communications

This paper considers the quantization problem on the Grassmann manifold \mathcal{G}_{n,p}, the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space. The chief result is a closed-form formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary dimension n and p, while previous results pertained only to p=1, or a fixed p with asymptotically large n. Based on this result, several quantization bounds are derived for sphere packing and rate distortion tradeoff. We establish asymptotically equivalent lower and upper bounds for the rate distortion tradeoff. Since the upper bound is derived by constructing random codes, this result implies that the random codes are asymptotically optimal. The above results are also extended to the more general case, in which \mathcal{G}_{n,q} is quantized through a code in \mathcal{G}_{n,p}, where p and q are not necessarily the same. Finally, we discuss some applications of the derived results to multi-antenna communication systems.

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arXiv2006

Performance Analysis of CDMA Signature Optimization with Finite Rate Feedback

We analyze the performance of CDMA signature optimization with finite rate feedback. For a particular user, the receiver selects a signature vector from a signature codebook to avoid the interference from other users, and feeds the corresponding index back to this user through a finite rate and error-free feedback link. We assume the codebook is randomly constructed where the entries are independent and isotropically distributed. It has been shown that the randomly constructed codebook is asymptotically optimal. In this paper, we consider two types of signature selection criteria. One is to select the signature vector that minimizes the interference from other users. The other one is to select the signature vector to match the weakest interference directions. By letting the processing gain, number of users and feedback bits approach infinity with fixed ratios, we derive the exact asymptotic formulas to calculate the average interference for both criteria. Our simulations demonstrate the theoretical formulas. The analysis can be extended to evaluate the signal-to-interference plus noise ratio performance for both match filter and linear minimum mean-square error receivers.

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arXiv2006

On the Information Rate of MIMO Systems with Finite Rate Channel State Feedback Using Beamforming and Power On/Off Strategy

It is well known that Multiple-Input Multiple-Output (MIMO) systems have high spectral efficiency, especially when channel state information at the transmitter (CSIT) is available. When CSIT is obtained by feedback, it is practical to assume that the channel state feedback rate is finite and the CSIT is not perfect. For such a system, we consider beamforming and power on/off strategy for its simplicity and near optimality, where power on/off means that a beamforming vector (beam) is either turned on with a constant power or turned off. The main contribution of this paper is to accurately evaluate the information rate as a function of the channel state feedback rate. Name a beam turned on as an on-beam and the minimum number of the transmit and receive antennas as the dimension of a MIMO system. We prove that the ratio of the optimal number of on-beams and the system dimension converges to a constant for a given signal-to-noise ratio (SNR) when the numbers of transmit and receive antennas approach infinity simultaneously and when beamforming is perfect. Asymptotic formulas are derived to evaluate this ratio and the corresponding information rate per dimension. The asymptotic results can be accurately applied to finite dimensional systems and suggest a power on/off strategy with a constant number of on-beams. For this suboptimal strategy, we take a novel approach to introduce power efficiency factor, which is a function of the feedback rate, to quantify the effect of imperfect beamforming. By combining power efficiency factor and the asymptotic formulas for perfect beamforming case, the information rate of the power on/off strategy with a constant number of on-beams is accurately characterized.

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