Ahmet H. Kayran

On Feasibility of Interference Alignment in MIMO Interference Networks

We explore the feasibility of interference alignment in signal vector space-based only on beamforming-for K-user MIMO interference channels. Our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry. It is well known, e.g., from Bezouts theorem, that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables. Following this intuition, we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables. Rigorous connections between feasible and proper systems are made through Bernshteins theorem for the case where each transmitter uses only one beamforming vector. The multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems. In this case, we show that the connection between feasible and proper systems can be further strengthened (since the equivalency between feasible and proper systems does not always hold) by including standard information theoretic outer bounds in the feasibility analysis.

Feasibility Conditions for Interference Alignment

The degrees of freedom (DoF) of K-user MIMO interference networks with constant channel coefficients are not known in general. Determining the feasibility of a linear interference alignment is a key step toward solving this open problem. Our approach in this paper is to view the alignment problem for interference networks as a multivariate polynomial system and determine its solvability by comparing the number of equations and the number of variables. Consequently, we divide the interference networks into two classes - proper and improper, where interference alignment is and is not achievable, respectively. An interference network is called proper if the cardinality of every subset of equations in the corresponding polynomial system is less than or equal to the number of variables involved in that subset of equations. Otherwise, it is called improper. Our intuition in this paper is that for general channel matrices, proper systems are almost surely feasible and improper systems are almost surely infeasible. We prove the direct link between proper (improper) and feasible (infeasible) systems for some important cases, thus significantly strengthening our intuition. Numerical simulation results also support our intuition.

Design of recursive and nonrecursive fan filters with complex transformations

This paper develops a technique for designing recursive and nonrecursive two-dimensional digital filters by the application of a complex transformation to a one-dimensional low-pass filter. A set of transformed filters is presented. The appropriate combination of these filters produces zero-phase fan filters. The advantage of such an approach is that the resulting fan filters are inherently stable and no optimization is needed. Several examples are presented for both finite impulse response (FIR) and infinite impulse response (IIR) fan filters and quadrant fan filters. The paper also includes implementation techniques using the properties of finite area arrays which reduce both computational and storage requirements compared with conventional implementations.

Two-dimensional orthogonal lattice structures for autoregressive modeling of random fields

Two-dimensional orthogonal lattice filters are developed as a natural extension of the 1-D lattice parameter theory. The method offers a complete solution for the Levinson-type algorithm to compute the prediction error filter coefficients using lattice parameters from the given 2-D augmented normal equations. The proposed theory can be used for the quarter-plane and asymmetric half-plane models. Depending on the indexing scheme in the prediction region, it is shown that the final order backward prediction error may correspond to different quarter-plane models. In addition to developing the basic theory, the article includes several properties of this lattice model. Conditions for lattice model stability and an efficient method for factoring the 2-D correlation matrix are given. It is shown that the unended forward and backward prediction errors form orthogonal bases. A simple procedure for reduced complexity 2-D orthogonal lattice filters is presented. The proposed 2-D lattice method is compared with other alternative structures both in terms of conceptual background and complexity. Examples are considered for the given covariance case.

ARMA model parameter estimation based on the equivalent MA approach

The paper investigates the relation between the parameters of an autoregressive moving average (ARMA) model and its equivalent moving average (EMA) model. On the basis of this relation, a new method is proposed for determining the ARMA model parameters from the coefficients of a finite-order EMA model. This method is a three-step approach: in the first step, a simple recursion relating the EMA model parameters and the cepstral coefficients of an ARMA process is derived to estimate the EMA model parameters; in the second step, the AR parameters are estimated by solving the linear equation set composed of EMA parameters; then, the MA parameters are obtained via simple computations using the estimated EMA and AR parameters. Simulations including both low- and high-order ARMA processes are given to demonstrate the performance of the new method. The end results are compared with the existing method in the literature over some performance criteria. It is observed from the simulations that our new algorithm produces the satisfactory and acceptable results.

An improved 2-D lattice filter and its entropy relations

Abstract In this paper, an improved lattice filter structure to model two-dimensional (2-D) autoregressive (AR) fields is presented. This work is the generalization of the three-parameter lattice filter developed by Parker and Kayran. The proposed structure generates four prediction error fields (one forward and three backward prediction error fields) at the first stage. After the first stage, two additional prediction error fields are generated using two of the backward prediction error fields at the output of the first stage. This leads to six prediction error fields whose linear combination defines the successive lattice stages and the reflection coefficients. A recursive relationship between the reflection coefficients of the lattice filter and the AR coefficients is derived. In addition, the new structure and the three-parameter lattice filter are compared from information-theoretic point of view. The entropy calculations are carried out for Gaussian distributed data. It is concluded that the new structure approximates the maximum entropy more closely compared to the three-parameter structure. The increase in entropy naturally leads to a more reliable and better modelling of AR data fields.

Adaptive Volterra filtering with complete lattice orthogonalization

The article presents a new recursive least squares (RLS) adaptive nonlinear filter, based on the Volterra series expansion. The main approach is to transform the nonlinear filtering problem into an equivalent multichannel, but linear, filtering problem. Then, the multichannel input signal is completely orthogonalized using sequential processing multichannel lattice stages. With the complete orthogonalization of the input signal, only scalar operations are required, instability problems due to matrix inversion are avoided and good numerical properties are achieved. The avoidance of matrix inversion and vector operations reduce the complexity considerably, making the filter simple, highly modular and suitable for VLSI implementations. Several experiments demonstrating the fast convergence properties of the filter are also included.

Estimation of 2-D ARMA model parameters by using equivalent AR approach

Abstract In this paper, the problem of estimating the parameters of a two-dimensional autoregressive moving-average (2-D ARMA) model driven by an unobservable input noise is addressed. In order to solve this problem, the relation between the parameters of a 2-D ARMA model and their 2-D equivalent autoregressive (EAR) model parameters is investigated. Based on this relation, a new algorithm is proposed for determining the 2-D ARMA model parameters from the coefficients of the 2-D EAR model. This algorithm is a three-step approach. In the first step, the parameters of the 2-D EAR model that is approximately equivalent to the 2-D ARMA model are estimated by applying 2-D modified Yule–Walker (MYW) equation to the process under consideration. Then, the moving-average parameters of the 2-D ARMA model are obtained solving the linear equation set constituted by using the EAR coefficients acquired in the first step. Finally, the autoregressive parameters of the 2-D ARMA model are found by exploiting the values obtained in first and second steps. The performance of the proposed algorithm is compared with other 2-D ARMA parameter and spectral estimation algorithms available in the technical literature by means of three different criteria. As a result of this comparison, it is shown that the parameters and the corresponding power spectrums estimated by using the proposed algorithm are converged to the original parameters and the original power spectrums, respectively.

Optimum quarter-plane autoregressive modeling of 2-D fields using four-field lattice approach

A new orthogonal four-field two-dimensional (2-D) quarter-plane lattice structure with a complete set of reflection coefficients is developed by employing appropriately defined auxiliary prediction errors. This work is the generalization of the three-parameter lattice filter proposed by Parker and Kayran (1984). After the first stage, four auxiliary forward and four auxiliary backward prediction errors are generated in order to obtain a growing number of 2-D reflection coefficients at successive stages. The theory has been proven by using a geometrical formulation based on the mathematical concepts of vector space, orthogonal projection, and subspace decomposition. It is shown that all four quarter-plane filters are orthogonal and thus optimum for all stages. In addition to developing the basic theory, a set of orthogonal backward prediction error fields for successive lattice parameter model stages is presented.

Optimum asymmetric half-plane autoregressive lattice parameter modeling of 2-D fields

In this paper, we present a new optimum asymmetric half-plane (ASHP) autoregressive lattice parameter modeling of two-dimensional (2-D) random fields. This structure introduces 4N points into the prediction support region when the order of the model increases from (N-1) to N. Starting with a given data field, a set of four auxiliary prediction errors are generated in order to obtain the growing number of 2-D ASHP reflection coefficients at successive stages. The theory has been applied to the high-resolution radar imaging problem and has also been proven using the concepts of vector space, orthogonal projection, and subspace decomposition. It is shown that the proposed ASHP structure generates the orthogonal realization subspaces for different recurse directions. In addition to developing the basic theory, the presentation includes a comparison between the proposed theory and other alternative structures, both in terms of conceptual background and complexity. While the recently developed reduced-complexity ASHP lattice modeling structure requires O(4N/sup 3/) lattice sections with N equal to the order of the error filter, the proposed configuration requires only O(2N/sup 2/) lattice sections.