Sarah Spence Adams

The Theory of Quaternion Orthogonal Designs

Over the past several years, there has been a renewed interest in complex orthogonal designs for their application in space-time block coding. Motivated by the success of this application, this paper generalizes the definition of complex orthogonal designs by introducing orthogonal designs over the quaternion domain. This paper builds a theory of these novel quaternion orthogonal designs, offers examples, and provides several construction techniques. These theoretical results, along with the results of preliminary simulations, lay the foundation for developing applications of these designs as orthogonal space-time-polarization block codes.

The Minimum Decoding Delay of Maximum Rate Complex Orthogonal Space–Time Block Codes

The growing demand for efficient wireless transmissions over fading channels motivated the development of space-time block codes. Space-time block codes built from generalized complex orthogonal designs are particularly attractive because the orthogonality permits a simple decoupled maximum-likelihood decoding algorithm while achieving full transmit diversity. The two main research problems for these complex orthogonal space-time block codes (COSTBCs) have been to determine for any number of antennas the maximum rate and the minimum decoding delay for a maximum rate code. The maximum rate for COSTBCs was determined by Liang in 2003. This paper addresses the second fundamental problem by providing a tight lower bound on the decoding delay for maximum rate codes. It is shown that for a maximum rate COSTBC for 2m - 1 or 2m antennas, a tight lower bound on decoding delay is r = (m-1 2m) . This lower bound on decoding delay is achievable when the number of antennas is congruent to 0, 1, or 3 modulo 4. This paper also derives a tight lower bound on the number of variables required to construct a maximum rate COSTBC for any given number of antennas. Furthermore, it is shown that if a maximum rate COSTBC has a decoding delay of r where r < r les 2r, then r=2r. This is used to provide evidence that when the number of antennas is congruent to 2 modulo 4, the best achievable decoding delay is 2(m-1 2m_).

Modeling the spread of fault in majority-based network systems: Dynamic monopolies in triangular grids

In a graph theoretical model of the spread of fault in distributed computing and communication networks, each element in the network is represented by a vertex of a graph where edges connect pairs of communicating elements, and each colored vertex corresponds to a faulty element at discrete time periods. Majority-based systems have been used to model the spread of fault to a certain vertex by checking for faults within a majority of its neighbors. Our focus is on irreversible majority processes wherein a vertex becomes permanently colored in a certain time period if at least half of its neighbors were in the colored state in the previous time period. We study such processes on planar, cylindrical, and toroidal triangular grid graphs. More specifically, we provide bounds for the minimum number of vertices in a dynamic monopoly defined as a set of vertices that, if initially colored, will result in the entire graph becoming colored in a finite number of time periods.

Dynamic monopolies and feedback vertex sets in hexagonal grids

In a majority conversion process, the vertices of a graph can be in one of the two states, colored or uncolored, and these states are dynamically updated so that a vertex becomes colored at a certain time period if at least half of its neighbors were in the colored state in the previous time period. A dynamic monopoly is a set of vertices in a graph that when initially colored will eventually cause all vertices in the graph to become colored. This paper establishes a connection between dynamic monopolies and the well-known feedback vertex sets which are sets of vertices whose removal results in an acyclic graph. More specifically, we show that dynamic monopolies and feedback vertex sets are equivalent in graphs wherein all vertices have degree 2 or 3. We use this equivalence to provide exact values for the minimum size of dynamic monopolies of planar hexagonal grids, as well as upper and lower bounds on the minimum size of dynamic monopolies of cylindrical and toroidal hexagonal grids. For these last two topologies, the respective upper and lower bounds differ by at most one.

The Final Case of the Decoding Delay Problem for Maximum Rate Complex Orthogonal Designs

Complex orthogonal space-time block codes (COSTBCs) based on generalized complex orthogonal designs (CODs) have been successfully implemented in wireless systems with multiple transmit antennas and single or multiple receive antennas. It has been shown that for a maximum rate COD with 2m-1 or 2m columns, a lower bound on decoding delay is (m-1 2m) and this delay is achievable when the number of columns is congruent to 0, 1 , or 3 modulo 4. In this paper, the final case is addressed, and it is shown that when the number of columns is congruent to 2 modulo 4, the lower bound on decoding delay cannot be achieved. In this case, the shortest decoding delay a maximum rate COD can achieve is twice the lower bound. New techniques for analyzing CODs are introduced with connections to binary vector spaces.

An extension of the channel-assignment problem: L(2, 1)-labelings of generalized Petersen graphs

The channel-assignment problem involves assigning frequencies represented by nonnegative integers to radio transmitters such that transmitters in close proximity receive frequencies that are sufficiently far apart to avoid interference. In one of its variations, the problem is commonly quantified as follows: transmitters separated by the smallest unit distance must be assigned frequencies that are at least two apart and transmitters separated by twice the smallest unit distance must be assigned frequencies that are at least one apart. Naturally, this channel-assignment problem can be modeled with vertex labelings of graphs. An L(2, 1)-labeling of a graph G is a function f from the vertex set V(G) to the nonnegative integers such that |f(x)-f(y)|/spl ges/2 if d(x,y)=1 and |f(x)-f(y)|/spl ges/1 if d(x,y)=2. The /spl lambda/-number of G, denoted /spl lambda/(G), is the smallest number k such that G has an L(2, 1)-labeling using integers from {0,1,...,k}. A long-standing conjecture by Griggs and Yeh stating that /spl lambda/(G) can not exceed the square of the maximum degree of vertices in G has motivated the study of the /spl lambda/-numbers of particular classes of graphs. This paper provides upper bounds for the /spl lambda/-numbers of generalized Petersen graphs of orders 6, 7, and 8. The results for orders 7 and 8 establish two cases in a conjecture by Georges and Mauro, while the result for order 6 improves the best known upper bound. Furthermore, this paper provides exact values for the /spl lambda/-numbers of all generalized Petersen graphs of order 6.

Novel Constructions of Improved Square Complex Orthogonal Designs for Eight Transmit Antennas

Constructions of square, maximum rate complex orthogonal space-time block codes (CO STBCs) are well known, however codes constructed via the known methods include numerous zeros, which impede their practical implementation. By modifying the Williamson and Wallis-Whiteman arrays to apply to complex matrices, we propose two methods of construction of square, order-4n CO STBCs from square, order-n codes which satisfy certain properties. Applying the proposed methods, we construct square, maximum rate, order-8 CO STBCs with no zeros, such that the transmitted symbols are equally dispersed through transmit antennas. Those codes, referred to as the improved square CO STBCs, have the advantages that the power is equally transmitted via each transmit antenna during every symbol time slot and that a lower peak-to-average power ratio (PAPR) is required to achieve the same bit error rates as the conventional CO STBCs with zeros.

On Transceiver Signal Linearization and the Decoding Delay of Maximum Rate Complex Orthogonal Space-Time Block Codes

Complex orthogonal designs (CODs) have been successfully implemented in wireless systems as complex orthogonal space-time block codes (COSTBCs). Certain properties of the underlying CODs affect the performance of the codes. In addition to the main properties of a CODs rate and decoding delay, a third consideration is whether the COD can achieve transceiver signal linearization, a property that facilitates practical implementation by, for example, significantly simplifying the receiver structure for iterative decoding. It has been shown that a COD can achieve this transceiver signal linearization if the nonzero entries in any given row of the matrix are either all conjugated or all nonconjugated. This paper determines the conditions under which maximum rate CODs can achieve this desirable property. For an odd number of transmit antennas, it is shown that maximum rate CODs that achieve the lower bound on decoding delay can also achieve transceiver signal linearization. In contrast, for an even number of transmit antennas, maximum rate CODs that achieve the lower bound on delay cannot achieve this linearization. In this latter case, linearization is possible only if the COD achieves at least twice the lower bound on delay. This work highlights the tradeoffs among these three important properties.

On an Orthogonal Space-Time-Polarization Block Code

Over the past several years, diversity methods such as space, time, and polarization diversity have been successfully implemented in wireless communications systems. Orthogonal space-time block codes efficiently combine space and time diversity, and they have been studied in detail. Polarization diversity has also been studied, however it is usually considered in a simple concatenation with other coding methods. In this paper, an efficient method for incorporating polarization diversity with space and time diversity is studied. The simple yet highly efficient technique is based on extending orthogonal space-time block codes into the quaternion domain and utilizing a description of the dual-polarized signal by means of quaternions. The resulting orthogonal space-timepolarization block codes have given promising results in simulations. In the example described in this paper, the achievable performance gain for two transmit and one receive antennas is approximately 6 dB at a bit error rate of 10-4 when compared with the Alamouti code.

Novel Classes of Minimal Delay and Low PAPR Rate

Complex orthogonal designs (CODs) of rate 1/2 have been considered recently for use in analog transmissions and as an alternative to maximum rate CODs due to the savings in decoding delay as the number of antennas increases. While algorithms have been developed to show that an upper bound on the minimum decoding delay for rate 1/2 CODs with n=2m-1 or n=2m columns is ν(n) = 2m-1 or ν(n) = 2m, depending on the parity of n modulo 8, it remains open to determine the exact minimum delay. This paper shows that this bound ν(n) is also a lower bound on minimum decoding delay for a major class of rate 1/2 CODs, named balanced complex orthogonal designs (BCODs), and that this is the exact minimum decoding delay for most BCODs. These rate 1/2 codes are conjugation-separated and thus permit a linearized description of the transceiver signal. BCODs also display other combinatorial properties that are expected to be useful in implementation, such as having no linear processing. An elegant construction is provided for a class of rate 1/2 CODs that have no zero entries, effectively no irrational coefficients, no linear processing, and have each variable appearing exactly twice per column. The resulting codes meet the aforementioned bound on decoding delay in most cases. This class of CODs will be useful in practice due to their low peak-to-average power ratio (PAPR) and other desirable properties.