Modeling and analysis of communication circuit performance using Markov chains and efficient graph representations

Abstract

In high-speed data networks, the bit-error-rate specification on the system can be very stringent, i.e., 10¢ 14. At such error rates, it is not feasible to evaluate the performance of a design using straightforward, simulation based, approaches. Nevertheless performance prediction before actual hardware is built is essential for the design process. This work introduces a stochastic model and an analysis-based, non-Monte-Carlo method for performance evaluation of digital data communication circuits. The analyzed circuit is modeled by a number of interacting finite state machines with inputs described as functions on a Markov chain state-space. The composition of these elements results in a typically very large Markov chain. System performance measures, such as probability of bit errors and rate of synchronization loss, can be evaluated by solving linear problems involving the large Markov chain's transition probability matrix. This paper first describes a dedicated multi-grid method used to solve these very large linear problems. The principal bottleneck in such an approach is the size of the Markov chain state-space, which grows exponentially with system complexity. The second part of this paper introduces a novel, graph based, data structure capable of efficiently storing and manipulating transition probability matrices for several million state Markov chains. The methods are illustrated on a real industrial clock-recovery circuit design. 1 Introduction High-speed communication systems have extremely tight bit-error-rate (BER) specifications. For SONET/SDH applications it is not uncommon to have BER requirements in the order of 10£ 14. Such specifications are practically impossible to verify through straightforward simulation because of the extremely long sequence that would need to be simulated in order to get meaningful error statistics. In the absence of a performance analysis tool, designers rely on the experience of previous designs, intuition, and good luck. This environment discourages innovative solutions and non-incremental applications. On the other hand, the design process of communication systems would benefit significantly from the existence of a reliable design performance evaluation capability. Such a capability would permit the evaluation of a number of alternative algorithms, architectures, circuit techniques, and technologies in a short time and without the commitment of expensive resources. A situation that illustrates the need for a reliable evaluation capability of the BER occurred in the design of a SONET-type application at a well-known microelectronics company. The specification for a mul-tiplexer chip required a BER of 10£ 14. The prototype implementation, based on the modification of an existing design delivered performance that was more than an order of magnitude bellow the specification. The designers suspected that the main cause for the errors is the interference noise in the PLL-based clock recovery circuit, induced by the rest of the chip's circuitry. A number of circuit, technol...

Stochastic steady-state and AC analyses of mixed-signal systems

“…With continuous state variables in mixed-signal systems, the number of states in the Markov chain model can grow quickly with the number of variables. A related work by Demir and Feldman [16] also recognized this problem and used a decompositional graph representation in order to efficiently store and compute the large transition probability matrices with millions of states. As an orthogonal approach, we try to reduce the number of states in the Markov chain itself.…”

“…The steady-state response of such a system is statistical in nature, representing an ensemble of waveforms rather than a single one; the probability density function (PDF) and the power spectral density (PSD) are the examples. While computing the steady-state distribution of a Markov chain model is a well-established art [14,15], the main challenge lies in constructing a finite Markov chain model that has a feasible number of states [16]. We describe three methods to limit the number of states: a state discretization scheme that uses Gaussian decomposition to represent the probability distributions across continuous state space, a state exploration algorithm that includes only the states that are part of the steady-state response, and a state truncation algorithm that eliminates the states whose stationary probabilities are negligible.…”