#### Date of Award

Spring 1-1-2012

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Electrical, Computer & Energy Engineering

#### First Advisor

Mahesh K. Varanasi

#### Second Advisor

Youjian Liu

#### Third Advisor

Brian C. Rider

#### Abstract

In this thesis, the information theoretic performance limits of two important building blocks of the general multi-user wireless network, namely, the * interference channel* and the *relay channel*, are characterized. We consider both time-invariant and time-varying or fading channel. In the first part, we focus on the 2-user interference channel with time-invariant channel coefficients. First, we characterize the capacity region of a class of MIMO IC called *strong in partial order* ICs. It turns out that for this class of channels decoding both the messages at both the receivers is optimal, i.e., the capacity region is identical to that of the compound multiple access channel (MAC). The defining constraints on the channel coefficients for the class of *strong in partial order * ICs enable us to derive a novel tight upper bound to the sum rate of the channel --- a problem that is very difficult for general channel coefficients. To avoid this difficulty for the general IC, we next derive upper and lower bounds which are not identical but are within a constant number of bits to each other which characterizes the capacity region of the 2-user multi-input multi-output (MIMO) Gaussian interference channel (IC) with an arbitrary number of antennas at each node to within a constant gap that is independent of the signal-to-noise ratio (SNR) and all channel parameters. In contrast to an earlier result in [Telatar and Tse, ISIT, 2007], where both the achievable rate region and upper bounds to the capacity region of a general class of interference channels was specified as the union over all possible input distributions here we provide, a simple and an explicit achievable coding scheme for the achievable region and an explicit outer bound. We also illustrate an interesting connection of the simple achievable coding scheme to MMSE estimators at the receivers. A reciprocity result is also proved which is that the capacity of the reciprocal MIMO IC is within a constant gap of the capacity region of the forward MIMO IC.

We also analyze the channel's performance in the high SNR regime, which is obtained from the explicit expressions of the approximate capacity region and the resulting asymptotic rate region is known as the generalized degrees of freedom (GDoF) region. A close examination of the super position coding scheme which is both GDoF and approximate capacity optimal reveals that joint signal-space and signal-level interference alignment is necessary to achieve the GDoF region of the channel. The admissible DoF-splits between the private and common messages of the HK scheme are also specified. A study of the GDoF region reveals various insights through the joint dependence of optimal interference management techniques (at high SNR) on the SNR exponents and the numbers of antennas at the four terminals. For instance, it reveals that, unlike in the scalar IC, treating interference as noise is not always GDoF-optimal even in the very weak interference regime. Moreover, while the DoF-optimal strategy that relies just on transmit/receive zero-forcing beamforming and time-sharing is not GDoF optimal (and thus has an unbounded gap to capacity), the precise characterization of the very strong interference regime - where single-user DoF performance can be achieved simultaneously for both users- depends on the relative numbers of antennas at the four terminals and thus deviates from what it is in the SISO case. For asymmetric numbers of antennas at the four nodes the shape of the symmetric GDoF curve can be a "distorted W" curve to the extent that for certain MIMO ICs it is a "V" curve.

In the second part of the thesis, we concentrate on time varying fading channels. We first characterize the fundamental diversity-multiplexing tradeoff (DMT) of the quasi-static fading MIMO Z interference channel (ZIC) with channel state information at the transmitters (CSIT) and arbitrary number of antennas at each node. A short-term average power constraint is assumed. It is shown that a variant of the superposition coding scheme described above, where the 2nd transmitter's signal depends on the channel matrix to the first receiver and the 1st user's transmit signal is independent of CSIT, can achieve the full CSIT DMT of the ZIC. We also characterize the achievable DMT of a transmission scheme, which does not utilize any CSIT and show that for some range of multiplexing gains, the full CSIT DMT of the ZIC can be achieved by it. The size of this range of multiplexing gains depends on the system parameters such as the number of antennas at the four nodes (referred to hereafter as “antenna configuration”), signal-to-noise ratios (SNR) and interference-to-noise ratio (INR) of the direct links and cross link, respectively. Interestingly, for certain special cases such as when the interfered receiver has a relatively larger number of antennas than that at the other nodes or when the INR is stronger than the SNRs, the No-CSIT scheme can achieve the F-CSIT DMT for all multiplexing gains. Thus, under these circumstances, the optimal DMT of the MIMO ZIC with F-CSIT is same as the DMT of the corresponding ZIC with No-CSIT. For other channel configurations, the DMT achievable by the No-CSIT scheme serves as a lower bound to the fundamental No-CSIT DMT of the MIMO ZIC.

We also characterize the fundamental diversity-multiplexing tradeoff of the three-node, multi-input, multi-output (MIMO), quasi-static, Rayleigh faded, half-duplex relay channel for an arbitrary number of antennas at each node and in which opportunistic scheduling (or dynamic operation) of the relay is allowed, i.e., the relay can switch between receive and transmit modes at a channel dependent time. In this most general case, the diversity-multiplexing tradeoff is characterized as a solution to a simple, two-variable optimization problem. This problem is then solved in closed form for special classes of channels defined by certain restrictions on the numbers of antennas at the three nodes. The key mathematical tool developed here that enables the explicit characterization of the diversity-multiplexing tradeoff is the joint eigenvalue distribution of three mutually correlated random Wishart matrices. Besides being relevant here, this distribution result is interesting in its own right. Previously, without actually characterizing the diversity-multiplexing tradeoff, the optimality in this tradeoff metric of the dynamic compress-and-forward (DCF) protocol based on the classical compress-and-forward scheme of Cover and El Gamal was shown by Yuksel and Erkip. However, this scheme requires global channel state information (CSI) at the relay. In this work, the so-called quantize-map and forward (QMF) coding scheme is adopted as the achievability scheme with the added benefit that it achieves optimal tradeoff with only the knowledge of the (channel dependent) switching time at the relay node. Moreover, in special classes of the MIMO half-duplex relay channel, the optimal tradeoff is shown to be attainable even without this knowledge. Such a result was previously known only for the half-duplex relay channel with a single antenna at each node, also via the QMF scheme. More generally, the explicit characterization of the tradeoff curve in this work enables the in-depth comparisons herein of full-duplex versus half-duplex relaying as well as static versus dynamic relaying, both as a function of the numbers of antennas at the three nodes.

#### Recommended Citation

Karmakar, Sanjay, "Information Theoretic Limits of MIMO Interference and Relay Networks" (2012). *Electrical, Computer & Energy Engineering Graduate Theses & Dissertations*. 43.

https://scholar.colorado.edu/ecen_gradetds/43