The discrete Fourier transform (DFT) is widely employed for multi-beam digital beamforming. The DFT can be efficiently implemented through the use of fast Fourier transform (FFT) algorithms, thus reducing chip area, power consumption, processing time, and consumption of other hardware resources. This paper proposes three new hybrid DFT 1024-point DFT approximations and their respective fast algorithms. These approximate DFT (ADFT) algorithms have significantly reduced circuit complexity and power consumption compared to traditional FFT approaches while trading off a subtle loss in computational precision which is acceptable for digital beamforming applications in RF antenna implementations. ADFT algorithms have not been introduced for beamforming beyond N = 32, but this paper anticipates the need for massively large adaptive arrays for future 5G and 6G systems. Digital CMOS circuit designs for the ADFTs show the resulting improvements in both circuit complexity and power consumption metrics. Simulation results show similar or lower critical path delay with up to 48.5% lower chip area compared to a standard Cooley-Tukey FFT. The time-area and dynamic power metrics are reduced up to 66.0%. The 1024-point ADFT beamformers produce SNR gains between 29.2–30.1 dB, which is a loss of ≤ 0.9 dB SNR gain compared to exact 1024-point DFT beamformers (worst case) realizable at using an FFT.

In this paper, we introduce low-complexity multidimensional discrete cosine transform (DCT) approximations. 3D DCT approximations are formalized in terms of high-order tensor theory. The formulation is extended to higher dimensions with arbitrary lengths. Several multiplierless $8\times 8 \times 8$ approximate methods are proposed and the computational complexity is discussed for the general multidimensional case. The proposed methods complexity cost was assessed, presenting considerably lower arithmetic operations when compared with the exact 3D DCT. The proposed approximations were embedded into 3D DCT-based video coding scheme and a modified quantization step was introduced. The simulation results showed that the approximate 3D DCT coding methods offer almost identical output visual quality when compared with exact 3D DCT scheme. The proposed 3D approximations were also employed as a tool for visual tracking. The approximate 3D DCT-based proposed system performs similarly to the original exact 3D DCT-based method. In general, the suggested methods showed competitive performance at a considerably lower computational cost.

A low-complexity approximation for the 16-point DFT and its respective multiplierless fast algorithm is proposed. A receive mode multi-beam phased-array experiment was realized at 2.4 GHz employing a 16-element IQ receiver array that uses the proposed approximate spatial DFT in real-time in order to achieve multi-beam digital beamforming. The 16- beam digital receiver experiment uses a ROACH-2 based Xilinx Virtex-6 FPGA platform for both digital beam computation as well as to perform the multi-receiver analog-to-digital conversion. Receive mode RF beams were measured and compared to the exact DFT (realized with fixed-point multipliers with 8-bit twiddle factors). The measured approximate DFT closely followed the measured beams resulting from the fixed-point conventional DFT implementation. The approximate DFT achieves RF beam performance (mainlobe gain, sidelobes) similar to the DFT at the cost of a small error which would be tolerable for the majority of multi-beam phased-array receivers. The 16-point approximate DFT provides a hardware reduction of ∼70% with respect to FFTs, setting up a low size, weight and power (SWaP) system. The maximum magnitude error of the filter bank response is 0.106 (≍ -20 dB).

The discrete cosine transform (DCT) is a relevant tool in signal processing applications, mainly known for its good decorrelation properties. Current image and video coding standards—such as JPEG and HEVC—adopt the DCT as a fundamental building block for compression. Recent works have introduced low-complexity approximations for the DCT, which become paramount in applications demanding real-time computation and low-power consumption. The design of DCT approximations involves a trade-off between computational complexity and performance. This paper introduces a new multiparametric transform class encompassing the round-off DCT (RDCT) and the modified RDCT (MRDCT), two relevant multiplierless 8-point approximate DCTs. The associated fast algorithm is provided. Four novel orthogonal low-complexity 8-point DCT approximations are obtained by solving a multicriteria optimization problem. The optimal 8-point transforms are scaled to lengths 16 and 32 while keeping the arithmetic complexity low. The proposed methods are assessed by proximity and coding measures with respect to the exact DCT. Image and video coding experiments and hardware realization are performed. The novel transforms perform close to or outperform the current state-of-the-art DCT approximations.