Erdem Koyuncu

Distributed beamforming in wireless relay networks with quantized feedback

This paper is on quantized beamforming in wireless amplify-and-forward (AF) relay networks. We use the generalized Lloyd algorithm (GLA) to design the quantizer of the feedback information and specifically to optimize the bit error rate (BER) performance of the system. Achievable bounds for different performance measures are derived. First, we analytically show that a simple feedback scheme based on relay selection can achieve full diversity. Unlike the previous diversity analysis on the relay selection scheme, our analysis is not aided by any approximations or modified forwarding schemes. Then, for highrate feedback, we find an upper bound on the average signalto- noise ratio (SNR) loss. Using this result, we demonstrate that both the average SNR loss and the capacity loss decay at least exponentially with the number of feedback bits. In addition, we provide approximate upper and lower bounds on the BER, which can be calculated numerically.We observe that our designs can achieve both full diversity as well as high array gain with only a moderate number of feedback bits. Simulations also show that our approximate BER is a reliable estimation on the actual BER. We also generalize our analytical results to asynchronous networks, where perfect carrier level synchronization is not available among the relays.

Distributed Beamforming in Wireless Multiuser Relay-Interference Networks With Quantized Feedback

We study fixed data rate communication schemes for wireless relay-interference networks with any number of transmitters, relays, and receivers. The transmitters and the relays have individual short-term power constraints. We analyze both amplify- and-forward (AF) and decode-and-forward (DF) relaying strategies with a two channel use quantized network beamforming protocol. We design the quantizer of the channel state information to minimize the probability that at least one receiver incorrectly decodes its desired symbol(s). Correspondingly, we introduce a generalized diversity measure that encapsulates the conventional one as the first-order diversity. Additionally, it incorporates the second-order diversity, which is concerned with the transmitter power dependent logarithmic terms that appear in the error rate expression. We first show that for AF relays, the maximal achievable diversity in the presence of interference is strictly less than the transmit diversity bound in terms of the second-order diversity. We then prove that it is possible to achieve the transmit diversity bound using DF relays as if there is no interference and as if coding over an arbitrary number of channel uses is allowed. Relay selection provides the best possible diversity gain for both relaying strategies. Finally, we show that all the aforementioned diversity gains can be achieved using distributed decision making with asymptotically zero feedback rate per receiver. Such a performance is made possible by a special distributed quantizer design method we have called localization.

Variable-Length Limited Feedback Beamforming in Multiple-Antenna Fading Channels

We study a multiple-input single-output fading channel, where we would like to minimize the channel outage probability or symbol error rate (SER) by employing beamforming via quantized channel state information at the transmitter (CSIT). We consider a variable-length limited feedback scheme where the quantized CSIT is acquired through feedback binary codewords of possibly different lengths. We design and analyze the performance of the associated variable-length quantizers (VLQs) and compare their performance with the previously studied fixed-length quantizers (FLQs). For the outage probability performance measure, we construct VLQs that can achieve the full-CSIT performance with finite rate. Moreover, as the signal-to-noise ratio P tends to infinity, we show that VLQs can achieve the full-CSIT outage probability performance with asymptotically zero feedback rate. For the SER performance measure, we show that while the SER with full-CSIT is not achievable at any finite feedback rate, the diversity and array gains with full-CSIT can be achieved using VLQs with asymptotically zero feedback rate as P → ∞. Our results show that VLQs can significantly improve upon the traditional FLQs that require infinite feedback rate to achieve the outage probability or the diversity and array gains with full-CSIT.

Multicast Networks With Variable-Length Limited Feedback

We investigate the channel quantization problem for two-user multicast networks where the transmitter is equipped with multiple antennas and either receiver is equipped with only a single antenna. Our goal is to design a global quantizer to minimize the outage probability. It is known that any fixed-length quantizer with a finite-cardinality codebook cannot obtain the same minimum outage probability as the case where all nodes in the network know perfect channel state information (CSI). To achieve the minimum outage probability, we propose a variable-length global quantizer that knows perfect CSI and sends quantized CSI to the transmitter and receivers. With a random infinite-cardinality codebook, we prove that the proposed quantizer is able to achieve the minimum outage probability with a low average feedback rate. We also extend the proposed quantizer to the multicast networks with more than two users. Numerical simulations validate our theoretical analysis.

A Systematic Distributed Quantizer Design Method with an Application to MIMO Broadcast Channels

We introduce a systematic distributed quantizer design method, called {\it{localization}}, in which, out of an existing centralized (global) quantizer, one synthesizes the distributed (local) quantizer using high-rate scalar quantization combined with entropy coding. The general localization procedure is presented, along with a practical application to a quantized beamforming problem for multiple-input multiple-output broadcast channels. For our particular application, not only localization provides high performance distributed quantizers with very low feedback rates, but also reveals an interesting property of finite rate feedback schemes that might be of theoretical interest: For single-user multiple-input single-output systems, one can achieve the performance of almost any quantized beamforming scheme with an arbitrarily low feedback rate, when the transmitter power is sufficiently large.

Very Low-Rate Variable-Length Channel Quantization for Minimum Outage Probability

We identify a practical vector quantizer design problem where any fixed-length quantizer (FLQ) yields non-zero distortion at any finite rate, while there is a variable-length quantizer (VLQ) that can achieve zero distortion with arbitrarily low rate. The problem arises in a t × 1 multiple-antenna fading channel where we would like to minimize the channel outage probability by employing beam forming via quantized channel state information at the transmitter (CSIT). It is well-known that in such a scenario, finite-rate FLQs cannot achieve the full-CSIT (zero distortion) outage performance. We construct VLQs that can achieve the full-CSIT performance with finite rate. In particular, with P denoting the power constraint of the transmitter, we show that the necessary and sufficient VLQ rate that guarantees the full-CSIT performance is Θ(1/P). We also discuss several extensions (e.g. to precoding) of this result.

Cooperative Quantization for Two-UserInterference Channels

We introduce cooperative quantizers for two-user interference channels where interference signals are treated as noise. Compared with the conventional quantizers where each receiver quantizes its own channel independently, the proposed cooperative quantizers allow multiple rounds of feedback communication in the form of conferencing between receivers. For both time-sharing and concurrent transmission strategies, we propose different cooperative quantizers to achieve the full-channel-state-information (full-CSI) network outage probability of sum rate and the full-CSI network outage probability of minimum rate, respectively. Our proposed quantizers only require finite average feedback rates, whereas the conventional quantizers require infinite rate to achieve the full-CSI performance. For the minimum rate, we also design cooperative quantizers for a joint time-sharing and concurrent transmission strategy that can approach the previously established optimal network outage probability with a negligible gap. Numerical simulations confirm that our cooperative quantizers based on conferencing outperform the conventional quantizers.

On the Minimum Distortion of Quantizers with Heterogeneous Reproduction Points

In quantization theory, one typically works with a unique distortion function (e.g. the squared-error distortion function) that quantifies the cost of quantizing a given source sample to any given reproduction point of the quantizer. Many applications, however, induce quantization problems where different distortion functions should be associated with different reproduction points. In this paper, we consider the case where the distortion of a given reproduction point is the squared distance to the source sample weighted by a factor that varies from one reproduction point to another. For a uniform distribution of source samples, we determine the corresponding optimal scalar quantizers and their distortions. We also find upper and lower bounds on the distortion of optimal vector quantizers. For non-uniform distributions, we provide a high resolution analysis of the minimum possible distortion. As a byproduct of our analysis, we show that for certain distributions of weights, a tessellation of non-congruent quantization cells can outperform tessellations of congruent polytopes. This suggests that Gershos conjecture cannot be extended to the case of squared-error distortion functions with weighted reproduction points.

Delay-Limited and Ergodic Capacities of MIMO Channels With Limited Feedback

We consider a fixed data rate slow-fading MIMO channel with a long-term power constraint P at the transmitter. A relevant performance limit is the delay-limited capacity, which is the largest data rate at which the outage probability is zero. It is well known that if both the transmitter and the receiver have full channel state information (CSI) and if either of them has multiple antennas, the delay-limited capacity is non-zero and grows logarithmically with P. Achieving even a positive delay-limited capacity, however, becomes a difficult task when the CSI at the transmitter (CSIT) is imperfect. In this context, the standard partial CSIT model where the transmitter has a fixed finite bit of quantized CSI feedback for each channel state results in zero delay-limited capacity. We show that by using a variable-length feedback scheme that utilizes a different number of feedback bits for different channel states, a non-zero delaylimited capacity can be achieved if the feedback rate is greater than 1 bit per channel state. Moreover, we show that the delaylimited capacity loss due to finite-rate feedback decays at least inverse linearly with respect to the feedback rate. We also discuss the applications to ergodic MIMO channels.

Energy efficiency in two-tiered wireless sensor networks

We study a two-tiered wireless sensor network (WSN) consisting of N access points (APs) and M base stations (BSs). The sensing data, which is distributed on the sensing field according to a density function f, is first transmitted to the APs and then forwarded to the BSs. Our goal is to find an optimal deployment of APs and BSs to minimize the average weighted total, or Lagrangian, of sensor and AP powers. For M = 1, we show that the optimal deployment of APs is simply a linear transformation of the optimal N-level quantizer for density f, and the sole BS should be located at the geometric centroid of the sensing field. Also, for a one-dimensional network and uniform f, we determine the optimal deployment of APs and BSs for any N and M. Moreover, to numerically optimize node deployment for general scenarios, we propose one-and two-tiered Lloyd algorithms and analyze their convergence properties. Simulation results show that, when compared to random deployment, our algorithms can save up to 79% of the power on average.