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Millimeter wave communications require multibeam beamforming in order to utilize wireless channels that suffer from obstructions, path loss, and multi-path effects. Digital multibeam beamforming has maximum degrees of freedom compared to analog phased arrays. However, circuit complexity and power consumption are important constraints for digital multibeam systems. A low-complexity digital computing architecture is proposed for a multiplication-free 32-point linear transform that approximates multiple simultaneous RF beams similar to a discrete Fourier transform (DFT). Arithmetic complexity due to multiplication is reduced from the FFT complexity of $\mathcal{O}(N\: \log N)$ for DFT realizations, down to zero, thus yielding a 46% and 55% reduction in chip area and dynamic power consumption, respectively, for the $N=32$ case considered. The paper describes the proposed 32-point DFT approximation targeting a 1024-beams using a 2D array, and shows the multiplierless approximation and its mapping to a 32-beam sub-system consisting of 5.8 GHz antennas that can be used for generating 1024 digital beams without multiplications. Real-time beam computation is achieved using a Xilinx FPGA at 120 MHz bandwidth per beam. Theoretical beam performance is compared with measured RF patterns from both a fixed-point FFT as well as the proposed multiplier-free algorithm and are in good agreement.