This paper presents the lowest multiplication complexity, self-recursive, radix-2 DCT II/III algorithms for any n = 2(t) (t >= 1) with implementations in signal-flow graphs and image compression. These algorithms a...
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This paper presents the lowest multiplication complexity, self-recursive, radix-2 DCT II/III algorithms for any n = 2(t) (t >= 1) with implementations in signal-flow graphs and image compression. These algorithms are derived mainly using a matrix factorization technique to factor DCT II/III matrices into sparse and scaled orthogonal matrices. Although there are small length DCT II algorithms available only for n = 8 and n = 16, there is no existing self-recursive fast radix-2 DCT II/III algorithms for any n. To fill this gap, this paper presents the lowest multiplication complexity radix-2 self-recursive DCT II/III algorithms. We also attain the lowest theoretical multiplication bound for n = 8 with the new DCT II algorithm and establish new lowest bounds for DCT III for any n and for DCT II for any n >= 32. The paper also establishes a novel relationship between DCT II an DCT IV having sparse factors. This enables one to see the connection of most traditional factorization of DCT II/III matrices with the proposed DCT II/III matrices.
This paper presents a self-contained factorization for the delay Vandermonde matrix (DVM), which is the super class of the discrete Fourier transform, using sparse and companion matrices. An efficient DVM algorithm is...
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This paper presents a self-contained factorization for the delay Vandermonde matrix (DVM), which is the super class of the discrete Fourier transform, using sparse and companion matrices. An efficient DVM algorithm is proposed to reduce the complexity of radio-frequency (RF) N-beam analog beamforming systems. There exist applications for wideband multi-beam beamformers in wireless communication networks such as 5G/6G systems, system capacity can be improved by exploiting the improvement of the signal to noise ratio (SNR) using coherent summation of propagating waves based on their directions of propagation. The presence of a multitude of RF beams allows multiple independent wireless links to be established at high SNR, or used in conjunction with multiple-input multiple-output (MIMO) wireless systems, with the overall goal of improving system SNR and therefore capacity. To realize such multi-beam beamformers at acceptable analog circuit complexities, we use sparse factorization of the DVM in order to derive a low arithmetic complexity DVM algorithm. The paper also establishes an error bound and stability analysis of the proposed DVM algorithm. The proposed efficient DVM algorithm is aimed at implementation using analog realizations. For purposes of evaluation, the algorithm can be realized using both digital hardware as well as software defined radio platforms.
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