Weikang Qian

1. 4/8 5/8 ( is integer coefficient) Suppose 3/8 Abstract The Synthesis of Stochastic Logic to Perform Multivariate Polynomial Arithmetic As the feature size of integrated circuits continues scaling down, maintaining the paradigm of deterministic Boolean computation is increasingly challenging. Indeed, mounting concerns over noise and uncertainty in signal values motivate a new approach: the design of stochastic logic, that is to say, digital circuitry that processes signals probabilistically, and so can cope with errors and uncertainty. In this work, we present a general methodology for synthesizing stochastic logic for the computation of multivariate polynomial, a category that is important for applications such as digital signal processing. The method is based on converting polynomials into a particular mathematical form – multivariate Bernstein polynomials -- and then implementing the computation with stochastic logic. The resulting logic processes serial or parallel streams that are random at the bit level. In the aggregate, the computation becomes accurate, since the results depend only on the precision of the statistics. Experiments show that our method produces circuits that are highly tolerant of errors in the input stream, while the area-delay product of the circuit is comparable to that of deterministic implementations. Marc D. Riedel Weikang Qian Assistant Professor, University of Minnesota Ph.D. Student,University of Minnesota Mathematical Model Example: Multiplexer Motivation Stochastic Bit Streams probabilistic interpretation bit stream X: each bit with prob. x to be 1 • The traditional IC design is based on deterministic sequence of zeroes and ones. • Pros: precise • Cons: complex design to handle variability/noise X1 Independent Random Boolean Variable Also a Random Boolean Variable! combinationallogic real value x A FunctionF X2 Y combinationallogic Xn 0,1,0,1,1,0,1,0 1,1,0,1,0,0,1,1 How about using stochastic input bit streams to do computation? 1,0,0,0,0,1,0,1 Implement multivariate Bernstein polynomial F is integer-coefficient multivariate polynomial, with bounded degree! with all coefficients in the unit interval Bernstein Polynomial Problem Stochastic Logic Implementing Bernstein Polynomial Decoding Block (8-Input Example ) Can we implement arbitrary multivariate polynomial? Derive Bernstein Coefficient from Power-Basis Coefficient (d=2) Univariate Bernstein Basis Polynomial Possible! Consider Multivariate Bernstein Basis Polynomial Degree Elevation (d=2) Set , , , Multivariate Bernstein Polynomial Then, we have Probability Assignment is Bernstein Coefficient Experimental Result: Performance with Noisy Input Data Experimental Result: Hardware Comparison Synthesize Stochastic Logic to Compute Multivariate Power-Form Polynomial Ratio of Area-Time Product of Stoch. Impl. Over Deter. Impl. Implement Bivariate Polynomial Butterworth Polynomial, used in signal processing, is Theorem: If a multivariate polynomial for any then, it can be converted into a multivariate Bernstein polynomial with all coefficients in the unit interval. We implement and , for n = 2, 3, 4 and 5. Synthesizing Step: Perform linear transform on the original polynomial, so that the polynomial satisfies the condition in the above theorem. Compute Bernstein coefficients from power-basis coefficients. While there exists one Bernstein coefficient that is not in the unit interval, perform degree elevation to obtain next set of Bernstein coefficients. Build stochastic logic to implement the Bernstein polynomial with all coefficients in the unit interval. • Deterministic Implementation: binary radix encoding with M bits. Need one multiplier and adder and a iterations. • Stochastic Implementation: bit stream of length 2M. Implemented either in serial or in parallel.