Number Of Real Multiplications Research Articles

Pre-Equalized DMT Signal Transmission Utilizing Low-Resolution DAC With Channel Response Dependent Noise Shaping Technique

In this paper, a channel response dependent noise shaping (CRD-NS) technique is proposed and experimentally demonstrated for the pre-equalized discrete multi-tone (DMT) signal transmission in intensity modulation and direct detection (IM/DD) system with a low-resolution digital-to-analog converter (DAC). Compared with traditional NS technique, the proposed CRD-NS technique can bring signal-to-quantization noise ratio (SQNR) improvement and solve the unevenly distributed SQNR problem in pre-equalized DMT signal induced by residual quantization noise without introducing additional computational complexity. Transmission and reception of 100 Gb/s 16 quadrature amplitude modulation (QAM)-DMT signal generated by a 3-bit DAC are experimentally demonstrated to verify the effectiveness of the proposed CRD-NS technique. The experimental results show that the bit-error ratio (BER) performance of 100 Gb/s 16 QAM-DMT signal with CRD-NS technique outperforms the corresponding signal with traditional NS technique. Compared with digital resolution enhancer (DRE) technique, in which channel response is also considered, the CRD-NS technique shows a similar noise shaping capability with DRE in pre-equalized DMT signal transmission, while the computational complexity can be significantly reduced, the number of real multiplications per sample is reduced from 81 to 7.

Realization of Second-Order Structure of Recursive Algorithm for Discrete Cosine Transform

A computational and hardware-efficient second-order infinite impulse response filter structure is proposed in this paper. It can compute discrete cosine transform (DCT) with improved processing speed and is valid for N = 2r, where N is the length of the input sequence and r > 1. A new algorithm is also proposed in this paper which is an improvement over previously reported algorithms in the literature. The proposed algorithm reduces the total number of real multiplications and additions in comparison with the existing algorithms. Using the suggested algorithm, computational cycles required to compute a DCT coefficient are less which further reduces the truncation error while processing a long length of input data.

Структура алгоритму швидкої двомірної згортки за допомогою ізоморфних гіперкомплексних числових систем

In the mathematical modeling of linear systems, it is necessary to repeatedly perform a linear convolution of discrete signals. The complexity of calculating the linear convolution rapidly increases with the length of the convoluted arrays and their dimension, thus the methods of «fast» convolution calculations are used. One of the most common methods is convolution using Fast Fourier Transform (FFT) algorithms, which are based on decomposition of the original large-dimensional problem into a large number of low-dimensional problems. Thus, it is very important to develop such methods for solving problems for small dimension, which use, possibly, a smaller number of real operations. There are a number of methods for the rapid calculation of linear convolution: the methods of Cook-Toom, Vine, Fast Fourier Transform (FFT), Cooley-Tuke, Good-Thomas, and others. The algorithms for performing convolution based on the transition to hypercomplex spaces are considered. The basis of this approach has been developed by the authors. Convoluted numerical sequences are considered as components of hypercomplex numbers belonging to some HNS. The product of these numbers will contain paired products of components of convolutional numerical sequences. Nevertheless, they will be combined in amounts not in the same composition as necessary for organizing convolution components. In addition, the number of real multiplications with multiplication of hypercomplex numbers in the general case is the same as in the direct calculation of convolution, that’s why there is no profit. In this way there are two problems: the first one is a reduction in the number of real operations when multiplying hypercomplex numbers; the second one is the organization of the choice of paired products of convolution components. The solution of these two problems allows synthesize such convolution algorithms, which by the number of operations are more efficient than direct calculation algorithms for convolution.

Datapath-regular implementation and scaled technique for N=3×2m DFTs

Discrete Fourier transform (DFT) is used widely in almost all fields of science and engineering, and is generally calculated using the fast Fourier transform (FFT) algorithm. In this paper, we present a fast algorithm for efficiently computing a DFT of size 3×2m. The proposed algorithm decomposes the DFT, obtaining one length-2m unscaled sub-DFT and two length-2m sub-DFTs scaled by constant real numbers. For efficiently computing the scaled sub-DFTs, the constant real factors are attached to twiddle factors, combining them into new twiddle factors. By using this approach, the number of real multiplications is reduced compared with existing algorithms. To obtain regular datapath, a novel implementation method is presented aiming at the implementation of the proposed algorithm and making its datapath regular like the radix-2 FFT algorithm. The method can be applied to other algorithms with L-shape butterfly. Experimental result shows that, the proposed algorithm consumes less processing time than the existing algorithms for all scale DFTs, and than FFTW, a C subroutine library of FFTs, just for small scale DFTs.

Reduced-Complexity (Quasi-)Maximum-Likelihood Detectors with No Performance Degradation for S-QAM Modulated MIMO Systems

This paper proposes a rotation and scaling based transformation method to simplify the (quasi-)maximum-likelihood (ML) detectors (such as QRD-M, sphere decoding (SD), etc.) for the square quadrature amplitude modulation (S-QAM) modulated multiple-input and multiple-output (MIMO) systems with no performance degradation. The key idea of the simplification is to utilize the symmetric shape of the S-QAM constellation so that after rotation and scaling, real multiplication and addition with certain constellation points in symbol replica generation simply transform into bit shifting or even vanish. As an illustrative example, the QRD-M detector is employed in this work. By using the proposed method, the number of real multiplications of the symbol replica generation required for the QRD-M detector is reduced by 50% and 25% for 4-QAM and 16-QAM, respectively. The numbers of real addition reduction is twice the amount of multiplication. The underlying principle of the proposed rotation and scaling transformation can be straightforwardly extended to amplitude and phase-shift keying (APSK) modulated MIMO systems employing the QRD-M, SD, list sequential sphere decoder (LISS), list sphere decoding (LSD), and the full ML detectors.

A Discrete Newton's Method for Gain Based Predistorter

Gain based predistorter (PD) is a highly effective and simple digital baseband predistorter which compensates for the nonlinear distortion of PAs. Lookup table (LUT) is the core of the gain based PD. This paper presents a discrete Newton’s method based adaptive technique to modify LUT. We simplify and convert the hardship of adaptive updating LUT to the roots finding problem for a system of two element real equations on athematics. And we deduce discrete Newton’s method based adaptive iterative formula used for updating LUT. The iterative formula of the proposed method is in real number field, but secant method previously published is in complex number field. So the proposed method reduces the number of real multiplications and is implemented with ease by hardware. Furthermore, computer simulation results verify gain based PD using discrete Newton’s method could rectify nonlinear distortion and improve system performance. Also, the simulation results reveal the proposed method reaches to the stable statement in fewer iteration times and less runtime than secant method.

ML Approaching MIMO Detection Based on Orthogonal Projection

A detection algorithm for spatially multiplexed multiple input multiple output (MIMO) systems is proposed. The receiver first estimates the MIMO channel and rearranges the layers according to the measured SNRs. To determine the candidate vectors, an orthogonal projection combined with the M-algorithm is used. Without performing the exhaustive full search of the maximum likelihood (ML) method, the proposed algorithm can reach the performance which is closely akin to the ML method. The computational complexity markedly decreases: 0.66% over the ML method in terms of the number of real multiplications.

Multiplicative complexity of discrete cosine and sine transform

By use of the closure operation for an odd integer set, the minimal number of real multiplications needed to compute length-2m four-type discrete cosine and sine transforms over the field Q, of rational numbers has been demonstrated.

Algorithms meeting the lower bounds on the multiplicative complexity of length-2/sup n/ DFTs and their connection with practical algorithms

In a previous work (see Electron. Lett., vol.20, no.17, p.690, 1984), the author described an algorithm that computes a length-2/sup n/ discrete Fourier transform using 2/sup 2+1/-2n/sup 2/+4n-8 nontrivial (i.e. not=+or-j=+or- square root -1) complex multiplications. In the present work, it is first shown that this algorithm actually provides the attainable lower bound on the number of complex multiplications. A slight modification of the last step of this algorithm is also shown to provide the attainable lower bound on the number of real multiplications. A connection with the split-radix FFT algorithm (SRFFT) is then explained, showing that SRFFT is another variation of these optimal algorithms, where the last step is computed recursively from shorter FFTs in a suboptimal manner. Finally, once the connection between the minimal complexity and SRFFT (which is the best known practical algorithm) is understood, it provides useful information on the possibility of further improvements of the SRFFT.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Comment: Conjugate pair fast Fourier transform

A recently introduced algorithm for the fast computation of the discrete Fourier transform ,called conjugate pair fast Fourier transform (CPFFT),seemed to require a smaller number of real multiplications and additions than that required for the split radix (SR) FFT algorithm.Here it is shown that the CPFFT is the same as the SRFFT algorithm from the arithmetic complexity point of view.

Multiplicative complexity of the two-dimensional discrete Fourier transform DFT (2n; 2)

We show that the two-dimensional discrete Fourier transform DFT (2n; 2) can be transformed into a series of distinct second versions of the discrete cosine transform DCT. As a result of the minimal number of real multiplications over the field Q to compute the length 2n second version DCT, we derive the minimum number of real multiplications over the field Q necessary to compute a two-dimensional discrete Fourier transform for 2n × 2n real input data.