Effect of Bit-Level Correlation in Stochastic Computing

Impact of bit-level correlation in stochastic computing and its implications on system efficiency and performance. This study delves into the theoretical and simulated results, highlighting the properties and applications of stochastic computing. The research also analyzes previous works and aims to generate correlated bit streams for multi-sensor processing systems.

• Stochastic Computing
• Correlation Analysis
• System Efficiency
• Multi-Sensor Processing
• Bit-Level Simulation

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1. Effect of Bit-Level Correlation in Stochastic Computing MEGHA MEGHA PARHI, MARC D. RIEDEL, PARHI, MARC D. RIEDEL, KESHAB KESHAB K. PARHI K. PARHI DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING UNIVERSITY OF MINNESOTA, MINNEAPOLIS MN, USA

2. Outline Introduction Objective Theoretical Results Simulated Results Conclusions and Future Work

3. Stochastic Computing Stochastic number can be represented in two formats, where each bit has the same weight. Unipolar: ? = ? ? = 1 = ?(?) and ? [0,1] Bipolar: ? = 2? ? = 1 1 = 2? ? 1 and ? [ 1,1]

4. Properties of Stochastic Computing Stochastic Computing: A number is represented by a string of 1 s and 0 s. The percent of 1 s in the number represents the value of the number represented as a probability. It was proposed in 1967 by Gaines as an alternative to binary computing. Stochastic logic gates compute an approximation of the output as opposed to an exact value. Applications: These are well suited in low-speed area-constrained applications such as biomedical applications, and cyber-physical systems operating at low rates. Advantages: Low complexity in computing, small in size, low power Fault-tolerance due to redundancy Disadvantages Long computation time (if bit stream is long) Low Accuracy (if bit stream is short) Multiplying by 2 and checking sign in bipolar are expensive operations

5. Example of Stochastic Multiplication 100110 0.500 0.333 000010 0.166 010010

6. Outline Introduction Objective Theoretical Results Simulated Results Conclusions and Future Work

7. Previous Work Parker and McCluskey discuss how to treat probability in a logic gate without using stochastic bit streams where multiple bit streams are uncorrelated at bit-level (1975). Qian et al present approaches to synthesize a certain probability assuming that the bit streams are independent (2009, 2011). Alaghi and Hayes use an approach that uses Stochastic Correlation and have proposed a method to generate correlated bit streams using probabilistic transfer matrices (2013). Objective-1: Analyze output when multiple bit streams are correlated at the bit-level. Objective-2: Generate correlated bit streams.

8. Multi-Sensor Processing System ?1 ?1 MIMO System ?2 ?2 ?3

9. Bit-Level Correlation This work presents a method to analyze effect of bit-level correlation and generate correlated bit streams using Pearson correlation for unipolar [0,1] ? ?? ???? ???? ???= Each bit is a Bernouli random variable. Sum of Bernouli is a Binomial RV, For long bit stream, binomial approximates a Gaussian RV

10. Outline Introduction Previous Work Objective Theoretical Results Simulated Results Conclusions and Future Work

11. Closed Form Expressions for Single Logic Gates Gate Type Independent Correlated AND ?1?2 ?1?2+ ?1?2 AND (?1 inverted) ?2(1 ?1) ?2 ?1?2 ?1?2 NAND 1 ?1?2 1 ?1?2 ?1?2 OR ?1+?2 ?1?2 ?1+?2 ?1?2 ?1?2 OR (?1 inverted) ?2+(1 ?1) ?2(1 ?1) 1 ?1 ?1?2 ?1?2 NOR (1 ?1)(1 ?2) 1 ?1 ?2+ ?1?2+ ?1?2 XOR ?1+?2 2?1?2 ?1+?2 2?1?2 2 ?1?2 XNOR 1 ?1+?2+2?1?2 1 ?1+?2+2?1?2+ 2 ?1?2 MUX (?3 select signal) ?1 ?1?3+ ?2?3 ?1 ?1?3+ ?2?3

12. Error Analysis ? Deviation = Error) Gate Type Error AND NAND OR NOR XOR XNOR ?1?2 ?1?2 ?1?2 ?1?2 2 ?1?2 2 ?1?2

13. Synthesis Correlated Bit Streams from Uncorrelated Bit Streams (Unipolar) ?1= 0 ?1= 1 Marginal ?2= 0 ?2= 1 Marginal 1 ?2 ?1+ ? ?1 ? 1 ?2 ?2 ? ? ?2 1 ?1 ?1 1 Let ?(?1= 1,?2= 1) = ? Calculate: ? = ??1?2+ ?1?2

14. Synthesis of Two Correlated Stochastic Bit Streams Input: ?1, ?2 and ?. Output: ?1 and ?2.

15. Synthesized Circuit using LFSR, MUX p1[9:0] X1 Comparator a/p1[9:0] M U X X2 (p2-a)/(1-p1)[9:0] Comparator

16. Range MINIMUM CORRELATION COEFFICIENT MAXIMUM CORRELATION COEFFICIENT

17. Constraints for Correlation ? ? max( ?1?2 ? min(?1 ?1?2 ?1?2, ?1?2+?1+?2 1 ,?2 ?1?2 ) ?1?2 ) ?1?2 ?1?2

18. Synthesis of Three Correlated Stochastic Bit Streams Input: ?1, ?2 and ?3; ?12, ?13, ?23 and ?123. Output: ?1, ?2 and ?3.

19. Circuit Diagram of Synthesized Circuit

20. Outline Introduction Previous Work Objective Theoretical Results Simulated Results Conclusions and Future Work

21. Simulated Results ? = 0.2 ? = ?1?2+ ?1?2 0.4 0.5 0.5 0.6 100 = 0.2 + 0.2 ? Deviation = Error = ? ? = 0.2489 0.2 = 0.0489 = 0.2489 0.4 0.5 0.2489

22. Simulation Results of Stochastic Logic Given Correlated inputs

23. Example using Logic Gates X1 Y1 X1 X3 Y1 X2 Z Z X2 X3 Y2 ? ?1 = ?1?3+ ? ?1?31 ?1 1 ?3 2?3+ ??1?2?1?2 2?31 ?1?2 1 ?2?3 ? ? = ? ?1?2 = ?1?2 ? ? = ? ?2?1 = ?2 ?1?3+ ? ?1?31 ?1 1 ?3 = ?2 ?1?2?3+ ??1?2?1?31 ?1?2 1 ?2?3 ??1?2=?1?31 ?2+? ?1?3(1 ?1)(1 ?3) ?1?3(1 ?1?2)(1 ?2?3)

24. Conclusion Presented an approach to analyze effect of bit-level correlation Presented synthesis of correlated bit streams Simulation results confirm results predicted from theory