Mark S. K. Lau

Energy-aware probabilistic multiplier: design and analysis

Probabilistic CMOS is considered to be a promising technology for substantial energy savings for computing devices, such as DSPs and graphics chips. The basic principle is to relax the energy requirement by allowing possibly incorrect computation results. For devices with probabilistic components, energy should be assigned to each component wisely, in order to achieve a good trade-off between energy consumption and correctness of the outputs. Recently, a few schemes have been proposed for energy assignment of ripple-carry adders, which are often based on intuitive arguments. In the present paper, we extend the idea of energy assignment to probabilistic multipliers. We focus on a fundamental type of multipliers, known as array multipliers. We derive some analytical results. Guided by these results, we devise an energy assignment scheme. We also find that energy assignment for array multipliers and ripple-carry adders can be quite different, due to differences in their structures. To our best knowledge, our work here is the first attempt in the literature to consider energy assignment for multipliers. Some examples, including digital image enhancement, are presented to demonstrate the effectiveness of the proposed scheme.

A More Precise Model of Noise Based PCMOS Errors

In this paper we present a new model for characterization of probabilistic gates. While still not mainstream, probabilistic CMOS has the potential to dramatically reduce energy consumption by trading off with error rates on individual bits, e.g., least significant bits of an adder. Our contribution helps account for the filtering effect seen in noise based PCMOS in a novel way. The characterization proposed here can enable accurate multi-bit models based on fast mathematical extrapolation instead of expensive and slow HSPICE simulations.

Modeling of Probabilistic Ripple-Carry Adders

This paper proposes a mathematical model for probabilistic ripple-carry adders. The model gives explicit expressions for calculating error probabilities of sum and carry bits. The expressions show how errors propagate through the carry, which accumulate and eventually influence the correctness of a ripple-carry adders outputs. The proposed model is flexible since it only requires mild assumptions on the probability distribution of noise. Hence, in addition to Gaussian, it is applicable to a wide class of distributions. We validate the model through HSPICE simulation. The model is able to predict error-rates of a simulated probabilistic ripple-carry adder with reasonable accuracy.

A general mathematical model of probabilistic ripple-carry adders

Probabilistic CMOS is considered a promising technology for future generations of computing devices. By embracing possibly incorrect calculations, the technology makes it possible to trade correctness of circuit operations for potentially significant energy saving. For systematic design of probabilistic circuits, accurate mathematical models are indispensable. To this end, we propose a model of probabilistic ripplecarry adders. Compared to existing models, ours is applicable under a wide range of noise assumptions, including the popular additive-noise assumption. Our model provides recursive equations that can accurately capture propagation of carry errors. The proposed model is validated by HSPICE simulation, and we find that the model is able to predict multi-bit error-rates of a simulated probabilistic ripple-carry adder with reasonable accuracy.

A Linear Zero-One Formulation of Optimal Power Allocation of IDMA Systems

Transmitted-power allocation (TPA) is vital for minimizing power consumption and managing interference in interleave-division multiple access (IDMA) systems. Conventionally, optimization problems arising from TPA assume that the transmitted powers can take any values in a pre-defined interval. Hence, a practical limitation is not taken into consideration: the transmitted powers of most mobile devices are discrete nowadays. In this paper, we consider an optimization problem in which transmitted powers can only take a finite number of levels. This problem is non-convex, but we show that it can be converted into a linear zero-one programming (LZOP) problem. Numerical examples demonstrate that the problem in the LZOP form can be solved efficiently using existing solvers.

On a Power Allocation Method for IDMA Systems

A method was proposed in this journal (vol. 6, no. 1, pp. 192-201, 2007), which tackles a power allocation problem for interleave-division multiple-access (IDMA) systems over fading channels. In this paper, we show that the method may not provide valid solutions in some situations. We illustrate this with many numerical examples. Our findings indicate that the method should be applied carefully, and further investigation is required to find efficient methods for more general situations.

Optimality and Feasibility of Equal Power Allocation of IDMA Systems

Interleave-division multiple access (IDMA) is a recently developed multiple access technique for wireless communications. A power allocation problem of IDMA is to find a power vector (a vector of the received powers of the signals transmitted by all users) with the minimum total power. The feasible set of this global optimization problem contains all power vectors satisfying a performance requirement. Because the problem is non-convex, finding an optimal solution unavoidably requires a lots of computation and a sophisticated solver. As we observe in the literature, it is sometimes preferable to simply require the received powers for all users to be equal. This approach is not necessarily optimal, but optimality is sacrificed in exchange for significantly cheaper computation and a simpler implementation. However, we are not aware of any systematic study on the optimality and feasibility of this approach. In this paper, we derive conditions that can be used to determine the optimality and feasibility of the equal power allocation for IDMA. The conditions suggest that the equal power allocation is not only feasible, but also optimal if the number of users is small enough.

Equal power allocation of IDMA systems: Feasibility, optimality, and throughput

Equal power allocation (EPA) has been truly successful when applied in IS-95 CDMA. It is therefore worthwhile to see whether EPA is also applicable in IDMA systems, which can be considered as a special case of the direct-sequence CDMA. However, the applicability of EPA in IDMA systems must be considered in light of two fundamental problems. First, EPA is not efficient in suppressing multi-user interference and hence often results in severe performance degradation (infeasibility). Second, EPA schemes often require higher received power than other schemes that allow unequal received powers (sub-optimality). In this paper, we derive the necessary and sufficient conditions for the feasibility and optimality of IDMA systems having equal received powers. The conditions show how feasibility and optimality are related to system parameters such as number of users, code rate and bit-error rate target. The conditions also suggest an upper limit on the throughput of IDMA systems with equal received powers. The results obtained in this paper may provide a theoretical foundation for designing future EPA schemes for IDMA.

A Branch-and-Bound Method for Power Minimization of IDMA

This paper tackles a power minimization problem of interleave-division multiple-access (IDMA) systems over a fading multiple-access channel. The problem is minimizing the total power received by the receiver while keeping the bit error rates (BERs) of all users below a predefined value. The original formulation of the problem has highly nonlinear and implicitly defined functions, which render most existing optimization methods incapable. A new formulation is proposed in this paper, whose solution can effectively be obtained by a branch-and-bound (B&B) technique. An algorithm is devised based on B&B, and its effectiveness is also demonstrated by numerical experiments of systems with a moderate numbers of users.

A Method to Compute All Fixed Points for the Iterative MUD of CDMA Systems with Chip-Level Interleavers

This paper proposes a simple method to compute all fixed points for the iterative MUD of CDMA systems with chip-level interleavers. The task of finding all fixed points is difficult because a set of nonlinear equations with multiple variables has to be solved. It is shown that this set of equations can be reduced to an equivalent equation with just one variable. As a result, all fixed points can be obtained by simply locating the roots of this univariate equation. An application of the proposed method in optimal power allocation with a numerical example is presented as well