Formation of optimal Boolean functions for analog-digital conversion

[EN] This paper describes a situation in which the function obtained from an analog-to-digital conversion is considered a partially-certain function. In this case it is possible to neglect the least significant bits and therefore redefine them arbitrarily. A realization of proposed method simplifies the further processing of the received signal, and saves hardware consumption at designing and usage of hardware and software systems that include analog-to-digital converters

Combinational logic synthesis based on the dual form of Reed-Muller representation

In certain applications, AND/XOR (Reed-Muller), and ORlXNOR (Dual form of Reed-Muller) logic have shown some attractive advantages over the standard Sum of Products (SOP) and Product of Sums (POS). Bidirectional conversion algorithms between SOP and AND/XOR also between POS and ORlXNOR based on Sparse and partitioning techniques are presented for multiple output Boolean functions. The developed programs are tested for some benchmarks with up to 20 inputs and 40 outputs. A new direct method is presented to calculate the coefficients of the Fixed Polarity Dual Reed-Muller (FPDRM) from the truth vector of the POS. Any Boolean function can be expressed by FPDRM forms. There are 211 polarities for an n-variable function and the number of sum terms depends on these polarities. Finding the best polarity is costly interims of CPU time, in order to search for the best polarity which will lead to the minimum number of sums for a particular function. Therefore, an algorithm is developed to compute all the coefficients of the Fixed Polarity Dual Reed-Muller (FPDRM) with polarity p from any polarity q. This technique is used to find the best polarity of FPDRM among the 211 fixed polarities. The algorithm is based on the Dual- polarity property and the Gray code strategy. Therefore, there is no need to start from POS form to find FPDRM coefficients for all the polarities. The proposed methods are efficient in terms of memory size and CPU time. A fast algorithm is developed and implemented in C language which can convert between POSs and FPDRMs. The program was tested for up to 23 variables. A modified version of the same program was used to find the best polarity. For up to 13 variables the CPU time was less than 42 seconds. To search for the optimal polarity for large number of variables and to reduce the se arch time 0 ffinding the 0 ptimal polarity 0 fthe function, two new algorithms are developed and presented in this thesis. The first one is used to convert between P OS and Positive Polarity Dual Reed-Muller (PPDRM) forms. The second algorithm will find the optimal fixed polarity for the FPDRM among the 211 different polarities for large n-variable functions. The most popular minimization criterion of the FPDRM form is obtained by the exhaustive search of the entire polarity vector. A non-exhaustive method for FPDRM expansions is presented. The new algorithms are based on separation of the truth vector (T) of POSs around each variable Xi into two groups. Instead of generating all of the polarity sets and searching for the best polarity, this algorithm will find the optimal polarity using the separation and sparse techniques, which will lead to optimal polarity. Time efficiency and computing speed are thus achieved in this technique. The algorithms don't require a large size of memory and don't require a long CPU time. The two algorithms are implemented in C language and tested for some benchmark. The proposed methods are fast and efficient as shown in the experimental results and can be used for large number of variables.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Combinational logic synthesis based on the dual form of Reed-Muller representation.

In certain applications, AND/XOR (Reed-Muller), and ORlXNOR (Dualform of Reed-Muller) logic have shown some attractive advantages over thestandard Sum of Products (SOP) and Product of Sums (POS). Bidirectionalconversion algorithms between SOP and AND/XOR also between POS andORlXNOR based on Sparse and partitioning techniques are presented for multipleoutput Boolean functions. The developed programs are tested for somebenchmarks with up to 20 inputs and 40 outputs.A new direct method is presented to calculate the coefficients of the FixedPolarity Dual Reed-Muller (FPDRM) from the truth vector of the POS. AnyBoolean function can be expressed by FPDRM forms. There are 211 polarities foran n-variable function and the number of sum terms depends on these polarities.Finding the best polarity is costly interims of CPU time, in order to search for thebest polarity which will lead to the minimum number of sums for a particularfunction. Therefore, an algorithm is developed to compute all the coefficients ofthe Fixed Polarity Dual Reed-Muller (FPDRM) with polarity p from any polarity q.This technique is used to find the best polarity of FPDRM among the 211 fixedpolarities. The algorithm is based on the Dual- polarity property and the Gray codestrategy. Therefore, there is no need to start from POS form to find FPDRMcoefficients for all the polarities. The proposed methods are efficient in terms ofmemory size and CPU time. A fast algorithm is developed and implemented in Clanguage which can convert between POSs and FPDRMs. The program was testedfor up to 23 variables. A modified version of the same program was used to findthe best polarity. For up to 13 variables the CPU time was less than 42 seconds.To search for the optimal polarity for large number of variables and toreduce the se arch time 0 ffinding the 0 ptimal polarity 0 fthe function, two newalgorithms are developed and presented in this thesis. The first one is used toconvert between P OS and Positive Polarity Dual Reed-Muller (PPDRM) forms.The second algorithm will find the optimal fixed polarity for the FPDRM amongthe 211 different polarities for large n-variable functions. The most popularminimization criterion of the FPDRM form is obtained by the exhaustive search ofthe entire polarity vector. A non-exhaustive method for FPDRM expansions ispresented. The new algorithms are based on separation of the truth vector (T) ofPOSs around each variable Xi into two groups. Instead of generating all of thepolarity sets and searching for the best polarity, this algorithm will find the optimalpolarity using the separation and sparse techniques, which will lead to optimalpolarity. Time efficiency and computing speed are thus achieved in this technique.The algorithms don't require a large size of memory and don't require a long CPUtime. The two algorithms are implemented in C language and tested for somebenchmark. The proposed methods are fast and efficient as shown in theexperimental results and can be used for large number of variables