مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

Sequential circuits comprising 1) modulo- $p$ ( $p$ = prime) summers, 2) amplifiers whose gains are integers $< p$ , and 3) unit delays are considered in this paper which constitutes an extension of earlier work by Huffman. Such circuits are characterized in terms of the modular field $GF(p)$ and vectors and matrices defined thereover. A summary of the properties of $GF(p)$ is given. A linear sequential circuit is defined in terms of $\\overrightarrow{y}(n) = C \\overrightarrow{s}(n) + D \\overrightarrow{x}(n)$ $\\vec{s}(n + 1) = A \\vec{s}(n) + B \\vec{x}(n)$ where $A, B, C,$ and $D$ are $k \\times k$ matrices defined over $GF(p)$ . The latter equations constitute a canonical representation of any circuit comprising the above listed components. It is shown that circuits of this type meet the usual additivity criterion of linear systems. The behavior of the circuit is described in a finite state space of $k$ dimensions and $p^k$ states. The autonomous circuit $(A, B, C, D =$ constant and $\\overrightarrow{x}(n) \\equiv \\overrightarrow{O}$ , all $n$ ) is characterized by the matrix $A$ . If $A$ is nonsingular all initial states are either finite equilibrium points or lie in periodic sequences of length $T_{\\tau } \\leq T_{\\max } = p^{k} - 1$ . If the minimum polynomial of $A_{\\tau }$ has distinct roots, $T_{\\tau }$ , divides $r(p - 1)$ . If $A$ is singular, there are some singular initial states to which the circuit cannot return in the absence of excitation. The use of $Z$ transforms for linear modular sequential circuits is demonstrated. Inputs and outputs are represented by their "transforms" and the circuit by its "transfer function." The transform of the output is the product of the transfer function and the transform of the input. Several illustrative examples are included.