A Novel 4-Point Discrete Fourier Transforms Circuit based on Product of Rademacher Functions

This paper presents a new circuit design for implementing 4-point DFT algorithm based on product of Rademacher functions. The circuit has been derived from the similarity of how Fourier transforms and Walsh transforms are implemented. Walsh matrices contain numbers either +1 or -1 except for first row. Similarly, the 4-point DFT matrix contain numbers either positive or negative except for first row. This similarity has been taken into the case of how to implement the DFT circuit based on how Walsh transforms is generated. Since Walsh transforms is derived based on product of Rademacher functions, the proposed 4-point DFT circuit is designed according to product Rademacher functions. The circuit consist of negative circuit, multiplexers, accumulator (real and imaginary), buffers, and control circuit. The control circuit is designed to produce Rademacher functions for controlling and managing data flow. The 4-point DFT circuit has been successfully designed and implemented to FPGA platform. Among the selected chips, Artix 7 is the fastest one.