Research On Optimization Of Remainder System And Its Application In FIR Filter Design

In modern digital signal processing applications,the existence of digital filters is indispensable.Finite Impulse Response?FIR?filters have become the focus of people's research due to their linear phase,stability,and simple design characteristics.As the data to be processed becomes larger and larger,the processing speed becomes faster and faster,the FIR filter using the traditional two's complement?TCS?cannot smeet real-time requirements while processing large-scale data.The residue number system?RNS?has the characteristics of parallelism,weightlessness,and no carry propagation between parallel channels.It is a non-weighted number system that can be well applied in the design of FIR filters with a large number of multiplications and additions.Firstly,the thesis introduces the basic principle and structure of the Nested Residue number system.Unlike the residue number system,the nested residue number system requires multiple forward and backward conversions,and its modular operation is based on the nested moduli.The forward conversion of the inner layer can be realized directly by look-up table.The modular adder and the modular multiplier use the form of a two-level look-up table.Compared with the general modular adder and multiplier can save more resources.In the inner backward conversion,the operation which is converted to binary value in the mixed radix system is replaced by the modulo operation of the corresponding moduli basis to obtain the module value directly instead of the need to obtain the binary value to calculate the module again.Secondly,the principle of probability calculation is briefly introduced,and then the error-free probability calculation is designed and implemented to be combined with the residue number system.In the accurate stochastic calculation with input data x,the length of the generated accurate stochastic sequence is2^2×[log2x],and it takes2^2×[log2x]cycles to realize the accurate stochastic multiplication.In the system combining RNS and the accurate stochastic calculation,the input data x is split into{x ₁,x ₂,…,x _L}.The length of the generated accurate stochastic sequence is{m₁×m₁,m₂×m₂,…,m_L×m_L}is much smaller than2^2×[log2x].However,the result of each calculation needs to be converted to the corresponding modulus value and then the probability sequence is regenerated for calculation in the system that combines the residue number and error-free probability calculation.Compare the resource consumption of modules with different remainder base modulo multipliers and probability multipliers.When the moduli is large,the multiplication unit combining the remainder and the error-free probability calculation can save more resources.Finally,a series of FIR filters with different dynamic ranges are implemented in the FPGA based on the traditional method,the remainder system,the nested remainder system,and the combination of the remainder and error-free probability calculation to compare resource consumption and power consumption.The results show that the architecture of the FIR filter with a large dynamic range using the remainder system and error-free probability calculation consumes the least resources and power.When the input data and filter coefficients are 24 bits wide,compared with the FIR filter directly generated by Maltab,the LUTs is reduced by 42.32%,and the power consumption is reduced by 7.14%.