Study Of Range Analysis For Multiplication And Precision Analysis For System With Several Sub-systems In Word-length Optimization

In the design of VLSI, it 50% of the design time contribute to the word-length optimization and floating-point to fixed-point transformation generally. To shorten the design cycle and the time to market of product, it is important to improve the efficiency of the word-length optimization. The range analysis of multiplication operation play a key role in the range analysis of word-length optimization, and the precision analysis of the hierarchically defined system is useful since the system of VLSI is usually divided into sub-systems in practical.Based on the detailed discussion of the disadvantage of range analysis of multiplication operation and precision analysis of the hierarchically defined system, Approximation Affine based on Space Extreme Estimation(AASEE) for range analysis of multiplication operation and Fractional Word-Length Dividing Precision Analysis(FWLDPA) for the hierarchically defined system are proposed in this dissertation, respectively. At the meantime, these proposed methods are applied in the hardware acceleration design of Helical Cone-Beam CT(CBCT) image reconstruction, which is based on FPGA, and the efficiencies and advantages of the proposed methods are demonstrated. The concrete works of this dissertation as follows.1、In the existing methods of the range analysis of multiplication operation, high computational accuracy and low computational complexity cannot be reached simultaneously. To solve the problem above, a new method which was named that Approximation Affine based on Space Extreme Estimation(AASEE) was proposed. In AASEE, the affine form of multiplication operation was divided into two parts, the one was the approximate affine form and the other was the equivalent affine form. The approximate affine form was used to describe the first order of noise in the result of multiplication operation, while the equivalent one was used to describe the second order. Based on the extreme value theory of multi-variable functions, the approximate maximum and minimum of the second order expression of noise were calculated, it means, the equivalent affine form was calculated. Compared with the existing approximation affine methods, approximation affine of Chebyshev approximation(AACHA) and approximation affine of trivial range estimation(AATRE), both the low computational complexity and the high computational accuracy were reached simultaneously in AASEE.2、To improve the efficiency of the precision analysis of the hierarchically defined system, a new static precision analysis method which was called that Fractional Wordlength Dividing Precision Analysis(FWLDPA) was proposed in this paper. The aim of the precision analysis was to optimize the fractional word-lengths while the error constraint of the system was satisfied and the system requirement of the primary output accuracy was met. In the proposed precision analysis method, the fractional word-length of each signal was divided into two parts, the uniform fractional word-length(UF) and the rectificatory fractional word-length(RF), and the procedure of precision analysis were executed in system level and sub-system level. In the sub-system level, the UF of the precision analysis was a constant, and the RFs were optimized using the optimization algorithm. In the system level, the optimized RFs in the sub-system level were used and UFs are optimized using the optimization algorithm. Because only a UF was used for a sub-system, it was obviously that the total computational complexity was decreased.3、To solve the problem of word-length optimization on the hardware for the nonlinear operation in COordinate Rotation DIgital Computer(CORDIC) algorithm, the word-length optimization method which was based on the affine approximation and the FWLDPA was proposed. This new method provided a solution of the hardware system which was include the CORDIC algorithm. Affine approximation was proposed for each non-linear operation in CORDIC algorithm for range analysis by AA. The error propagation models, area models and FWLDPA for CORDIC were proposed either.4、In the floating-point to fixed-point transformation of the FPGA based design of CBCT image reconstruction, the proposed methods were used and verified. In the design flow, the data flow graphs of the sub-systems were described, and the proposed methods were used to optimize the integer and fractional word-length, respectively. Compared with the existing method, it was more efficient to compute the word-length using the proposed methods, it means, to reach the optimized system performance while the accuracy the primary output signal was met.