Computing Arithmetic Functions Using Stochastic Logic by Series Expansion
Computing Arithmetic Functions Using Stochastic Logic by Series Expansion
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DOI：
10.1109/tetc.2016.2618750
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发表时间：
2019-01-01
影响因子：
5.9
通讯作者：
Liu, Yin
中科院分区：
文献类型：
--
作者：
Parhi, Keshab K.;Liu, Yin
复杂算术函数（例如三角函数、指数函数和 S 型函数）的随机逻辑实现是基于其麦克劳林级数展开式的截断版本而导出的。本文做出了三点贡献。首先，表明如果系数交替正负且其大小单调递减，则可以根据荷马规则使用多级与非门来实现多项式。算术函数的截断麦克劳林级数展开式用于生成满足这些约束的多项式。这些函数中的输入和输出由单极表示法表示。使用该方法可以实现包括正弦、余弦、正切双曲、对数和指数在内的函数。其次，对于不满足这些约束的多项式，如果多项式的每个因子都满足这些约束，那么仍然可以根据荷马规则来实现。结果表明，诸如 sin pi x/pi、e (ax)、tanh ax 和 sigmoid(ax3)（对于 a > 1 的值）等函数可以使用因式分解结合荷马法则的随机逻辑来实现。第三，对于输入和输出以不同格式表示的算术函数提出了格式转换，例如给定 x 是 [0, 1] 的元素的 cos pi x 和给定 x 是 [-1, 1] 的元素的 sigmoid(x)。多项式被转换为自然利用格式转换的等价形式。所提出的随机逻辑电路优于众所周知的基于伯恩斯坦多项式和基于有限状态机（FSM）的实现。此外，在大多数情况下，所提出的实现的硬件复杂性和关键路径低于众所周知的基于 Bernstein 多项式和基于 FSM 的实现。
Stochastic logic implementations of complex arithmetic functions, such as trigonometric, exponential, and sigmoid, are derived based on truncated versions of their Maclaurin series expansions. This paper makes three contributions. First, it is shown that a polynomial can be implemented using multiple levels of NAND gates based on Homer's rule, if the coefficients are alternately positive and negative and their magnitudes are monotonically decreasing. Truncated Maclaurin series expansions of arithmetic functions are used to generate polynomials which satisfy these constraints. The input and output in these functions are represented by unipolar representation. Functions including sine, cosine, tangent hyperbolic, logarithm and exponential can be implemented using this method. Second, for a polynomial that does not satisfy these constraints, it still can be implemented based on Homer's rule if each factor of the polynomial satisfies these constraints. It is shown that functions such as sin pi x/pi, e (ax), tanh ax and sigmoid(ax3) (for values of a > 1) can be implemented using stochastic logic using factorization in combination with Homer's rule. Third, format conversion is proposed for arithmetic functions with input and output represented in different formats, such as cos pi x given x is an element of [0, 1] and sigmoid(x) given x is an element of [-1, 1]. Polynomials are transformed to equivalent forms that naturally exploit format conversions. The proposed stochastic logic circuits outperform the well-known Bernstein polynomial based and finite-state-machine (FSM) based implementations. Furthermore, the hardware complexity and the critical path of the proposed implementations are less than the well-known Bernstein polynomial based and FSM based implementations for most cases.