Systems and Methods for Shaped Single Carrier Orthogonal Frequency Division Multiplexing with Low Peak to Average Power Ratio

Systems and methods for shaped single carrier orthogonal frequency division multiplexing with low peak to average power ratio. The system receives an input signal and modulates the input signal to form Dirichlet kernels in a time domain to generate an offset Dirichlet kernel output time array where each Dirichlet kernel has a main lobe and a plurality of side lobes. The system modulates the input signal by receiving the input signal by an N-point time input array and transforming the N-point time input array to the frequency domain by a discrete Fourier transform to generate an N-point input frequency array. The system replicates the N-point input frequency array to generate an M-point input frequency array where M is greater than N and utilizes a filter to generate a shaped M-point output filtered frequency array by multiplying the M-point input frequency array and the filter. The system transforms the shaped M-point output filtered frequency array by an inverse discrete Fourier transform to generate an M-point offset Dirichlet kernel output time array. The system generates a cyclic prefix time array by replicating duration points of an end of the M-point offset Dirichlet kernel output time array, and appends the cyclic prefix time array to a beginning of the M-point offset Dirichlet kernel output time array to generate an M-point and duration point output time array.