Simple systolic arrays for discrete cosine transform

in this paper, simple 1-D and 2-D systolic array for realizing the discrete cosine transform (DCT) based on the discrete Fourier transform (DFT) fo an input sequence are presented. The proposed arrays are obtained by a simple modified DFT (MDFT) and an inverse DFT (IDFT) version of the Goertzel algorithm combined with Kung's approach. The 1-D array requiresN cells, one multiplier and takesN clock cycles to produce a completeN-point DCT. The 2-D array takes N clock cycles, faster than the 1-D array, but the area complexity is larger. A continuous flow of input data is allowed and no idle time is required between the input sequences.     

Orthogonalization of circular stationary vector sequences and itsapplication to the Gabor decomposition

Certain vector sequences in Hermitian or in Hilbert spaces, can be orthogonalized by a Fourier transform. In the finite-dimensional case, the discrete Fourier transform (DFT) accomplishes the orthogonalization. The property of a vector sequence which allows the orthogonalization of the sequence by the DFT, called circular stationarity (CS), is discussed in this paper. Applying the DFT to a given CS vector sequence results in an orthogonal vector sequence, which has the same span as the original one. In order to obtain coefficients of the decomposition of a vector upon a particular nonorthogonal CS vector sequence, the decomposition is first found upon the equivalent DFT-orthogonalized one and then the required coefficients are found through the DFT. It is shown that the sequence of discrete Gabor (1946) basis functions with periodic kernel and with a certain inner product on the space of N-periodic discrete functions, satisfies the CS condition. The theory of decomposition upon CS vector sequences is then applied to the Gabor basis functions to produce a fast algorithm for calculation of the Gabor coefficients     

共轭对称数据的DFT及其FFT算法

该文对共轭对称复数序列的离散傅里叶交换(DFT)及其快速傅里叶变换(FFT)算法进行了研究,获得共轭对称序列的DFT具有虚部为零的性质,并开发出适用于共轭对称数据的FFT算法。该算法与传统FFT算法相比减少了一半的计算量和存储单元,运算速度提高了一倍。     

On the Peak Factors of BPSK and QPSK Modulated MC-CDMA Signals Employing WH Sequences and WH-Based Complementary Sequences

In this paper, the inverse discrete Fourier transform (IDFT) results of oversampled Walsh Hadamard (WH) sequences and WH-based complementary (CP) sequences are derived, based on which, the peak factors (PF) of MC-CDMA signals are investigated for both the two sequence sets. The upper bounds of PF for both BPSK and QPSK modulation are presented in this paper and the PF bounds presented in B.J. Choi and L. Hanzo (2003) can be regarded as special cases of the bounds derived here. Compared with accurate computing which requires prohibitive computations, the upper bounds obtained provide a more effective way to estimate the PF of BPSK and QPSK modulated MC-CDMA signals employing WH or WH-based CP sequences. In addition, based on the IDFT properties of WH sequences, a simplified search for optimum WH subset in the context of BPSK modulation is derived.     

Low complexity implementation of Alamouti space-time coded OFDM transmitters

Space-time and space-frequency coded orthogonal frequency division multiplexing systems employing the Alamouti scheme were proposed by Lee and Williams. Straightforward implementation of these systems requires separate inverse discrete Fourier transform (IDFT) processing blocks for each of the two transmit antennas. In this letter, DFT/IDFT symmetry properties are utilized to approximately halve the computational load of these systems at the transmitter.     

Simple FFT and DCT algorithms with reduced number of operations

A simple algorithm for the evaluation of discrete Fourier transforms (DFT) and discrete cosine transforms (DCT) is presented. This approach, based on the divide and conquer technique, achieves a substantial decrease in the number of additions when compared to currently used FFT algorithms (30% for a DFT on real data, 15% for a DFT on complex data and 25% for a DCT) and keeps the same number of multiplications as the best known FFT algorithms. The simple structure of the algorithm and the fact that it is best suited for real data (one does not have to take a transform of two real sequences simultaneously anymore) should lead to efficient implementations and to a wide range of applications.     

Discrete multiple tone modulation with coset coding for thespectrally shaped channel

A discrete approach to multiple tone modulation is developed for digital communication channels with arbitrary intersymbol interference (ISI) and additive Gaussian noise. Multiple tone modulation is achieved through the concatenation of a finite block length modulator based on discrete Fourier transform (DFT) code vectors, and high gain coset or trellis codes. Symbol blocks from an inverse DFT (IDFT) are cyclically extended to generate ISI-free channel-output symbols that decompose the channel into a group of orthogonal and independent parallel subchannels. Asymptotic performance of this system is derived, and examples of asymptotic and finite block length coding gain performance for several channels are evaluated at different values of bits per sample. This discrete multiple tone technique is linear in both the modulation and the demodulation, and is free from the effects of error propagation that often afflict systems employing bandwidth-optimized decision feedback plus coset codes     

Efficient Recursive IDFT Scheme for Complex-valued Signals in Tap-selective Maximum-likelihood Channel Estimation

This paper presents several efficient, recursive inverse discrete Fourier transform (IDFT) schemes for complex-valued input data in tap-selective maximum-likelihood channel estimation; the results of their implementation are also presented. The proposed schemes employ only real-valued arithmetic, which reduces the number of required real multiplication operations in comparison with conventional IDFT approaches; however, the number of real additions increases significantly due to the sliding window scheme. The results show that the schemes can reduce the computational complexity and enhance flexibility when only several subsets of the IDFT output bins are required.     

Conjugate Interleaved Partitioning PTS Scheme for PAPR Reduction of OFDM Signals

The interleaved partitioning partial transmit sequence (IP-PTS) scheme is an attractive technique for peak-to-average power ratio (PAPR) reduction in orthogonal frequency division multiplexing (OFDM) systems. But the PAPR performance of IP-PTS is inferior to that of the adjacent partitioning PTS (AP-PTS) scheme because the candidates generated in IP-PTS are not fully independent. This paper analyzes the independence of candidates in IP-PTS in detail and finds the effective phase factor vectors. In order to improve the PAPR performance of IP-PTS, a conjugate IP-PTS (C-IP-PTS) scheme is proposed. By performing the conjugate operations on some sub-blocks, the number of candidates is increased. Because of the conjugate property of the discrete Fourier transform (DFT), the additional inverse DFT can be avoided. By optimizing the conjugate sequence, the complexity can be further lowered. Simulation results show that C-IP-PTS can obtain better PAPR performance compared with AP-PTS; moreover, the computational complexity of C-IP-PTS is not high.     

关于核苷酸序列频谱分析方法的探讨

由A、T、C、G四种字符组成的核苷酸序列本质上是生物分子的一种符号表示。在计算生物学中,常采用离散傅立叶变换进行频谱分析。根据核苷酸序列频率分布的特点和离散傅立叶变换所固有的“栅栏效应”,本文提出采用调频Z变换的分析方法,定义了相应的表达式,同时构造了短时调频Z变换用于识别基因区域中的外显子部分和内含子部分。与已有的基因区域识别算法相比,该方法不需要依赖于训练样本序列。测试结果表明了该方法的有效性。     

多维DFT的多维多项式变换与离散W变换算法

本文首先通过引进一种序列的重排技术将m(m2) 维离散Fourier变换 (m-D DFT)转化为一系列的一维广义离散Fourier变换(GDFT)的多重和.然后引入一维离散W变换(DWT)以及多维多项式变换(MD-PT)计算该多重和以减少冗余的算术运算,从而得到了高效的多维DFT算法,该算法与常用的行-列DFT算法相比,乘法仅约为行-列法的1/2m,而加法仅约为行-列法的(2m+1)/4m.对于2维DFT的计算,本文方法同单纯的多项式变换方法相比,乘法与加法分别减少50%与40%左右.另外,本文算法计算结构简单,易于编程实现,通过数值实验验证了本文算法的高效性.     

Efficient computation of DFT of Zadoff-Chu sequences

An important property of a Zadoff-Chu (ZC) sequence is derived, namely that the discrete Fourier transform (DFT) of a ZC sequence is a time-scaled conjugate of the ZC sequence, multiplied by a constant factor. This result has many practical applications. For example, it can be used to generate 3GPP LTE access preambles more efficiently than the standard suggests as it allows the DFT of a ZC sequence of prime length P to be computed with P instead of PlogP arithmetic operations.     

Efficient VLSI architectures for fast computation of the discreteFourier transform and its inverse

In this paper, we propose two new VLSI architectures for computing the N-point discrete Fourier transform (DFT) and its inverse (IDFT) based on a radix-2 fast algorithm, where N is a power of two. The first part of this work presents a linear systolic array that requires log₂ N complex multipliers and is able to provide a throughput of one transform sample per clock cycle. Compared with other related systolic designs based on direct computation or a radix-2 fast algorithm, the proposed one has the same throughput performance but involves less hardware complexity. This design is suitable for high-speed real-time applications, but it would not be easily realized in a single chip when N gets large. To balance the chip area and the processing speed, we further present a new reduced-complexity design for the DFT/IDFT computation. The alternative design is a memory-based architecture that consists of one complex multiplier, two complex adders, and some special memory units. The new design has the capability of computing one transform sample every log₂ N+1 clock cycles on average. In comparison with the first design, the second design reaches a lower throughput with less hardware complexity. As N=512, the chip area required for the memory-based design is about 5742×5222 μm², and the corresponding throughput can attain a rate as high as 4M transform samples per second under 0.6 μm CMOS technology. Such area-time performance makes this design very competitive for use in long-length DFT applications, such as asymmetric digital subscriber lines (ADSL) and orthogonal frequency-division multiplexing (OFDM) systems     

四维信号星座图改进及相应OFDM系统模型设计

针对当前三维OFDM系统存在的频谱效率较低问题,提出了一种具有规则分布的四维信号星座图改进设计方法,并建立了相应的OFDM系统模型.在所提出的四维OFDM系统中,输入的比特信息通过设计的四维信号星座图映射到OFDM信号的子载波,再利用二维离散傅里叶反变换把OFDM信号从频域调制到时域.所设计的8点和16点四维星座图信号点分布弥补了以往高维星座图信号点无规则的缺陷,具有进一步降低误码率的潜能.通过对提出系统在AWGN和频率选择性衰落信道环境下的性能仿真验证,所提出的基于四维8点和16点OFDM系统比传统三维OFDM具有更优的误码率性能,且系统的频谱效率提高了三分之一.     

Joint detection with low computational complexity for hybridTD-CDMA systems

In the hybrid multiple access technique time division-code division multiple access (TD-CDMA), both multiple-access interference and intersymbol interference (ISI) arise. In order to combat the overall interference, we propose four efficient joint detection schemes based on zero-forcing and minimum-mean square error criteria. By exploiting the Toeplitz-block structure of matrices and the asymptotic equivalence between finite-order Toeplitz matrices and circulant matrices, most computations can be carried out very efficiently through extensive use of discrete Fourier transform (DFT) and independent DFT (IDFT) transforms. Performance results based on the UMTS scenario are presented     

Algorithmes de transformations de Fourier et en cosinus mono et bidimensionnels

This paper presents a fast algorithm for the computation of the discrete Fourier and cosine transform, and this for transform lengths which are powers of 2. This approach achieves the lowest known number of operations (multiplications and additions) for the discrete Fourier transform of real, complex, symmetrical and antisymmetrical sequences, for the odd discrete Fourier transform and for the discrete cosine transform. The extension to the two-dimensional Fourier and cosine transform is presented as well.     

Computing the inverse DFT with the in-place, in-order prime factorFFT algorithm

We present a method for computing the inverse discrete Fourier transform (IDFT) by the in-place, in-order prime factor FFT algorithm (PFA). This is achieved by modifying the input and the output index mapping equations. This approach does not result in any additional cost in terms of program length and computational time     

The integer transforms analogous to discrete trigonometric transforms

The integer transform (such as the Walsh transform) is the discrete transform that all the entries of the transform matrix are integer. It is much easier to implement because the real number multiplication operations can be avoided, but the performance is usually worse. On the other hand, the noninteger transform, such as the DFT and DCT, has a good performance, but real number multiplication is required. W derive the integer transforms analogous to some popular noninteger transforms. These integer transforms retain most of the performance quality of the original transform, but the implementation is much simpler. Especially, for the two-dimensional (2-D) block transform in image/video, the saving can be huge using integer operations. In 1989, Cham had derived the integer cosine transform. Here, we will derive the integer sine, Hartley, and Fourier transforms. We also introduce the general method to derive the integer transform from some noninteger transform. Besides, the integer transform derived by Cham still requires real number multiplication for the inverse transform. We modify the integer transform introduced by Cham and introduce the complete integer transform. It requires no real number multiplication operation, no matter what the forward or inverse transform. The integer transform we derive would be more efficient than the original transform. For example, for the 8-point DFT and IDFT, there are in total four real numbers and eight fixed-point multiplication operations required, but for the forward and inverse 8-point complete integer Fourier transforms, there are totally 20 fixed-point multiplication operations required. However, for the integer transform, the implementation is simpler, and many of the properties of the original transform are kept.     

Closed-Form Orthogonal DFT Eigenvectors Generated by Complete Generalized Legendre Sequence

In this paper, we propose a new method for deriving the closed-form orthogonal discrete Fourier transform (DFT) eigenvectors of arbitrary length using the complete generalized Legendre sequence (CGLS). From the eigenvectors, we then develop a novel method for computing the DFT. By taking a specific eigendecomposition to the DFT matrix, after proper arrangement, we can derive a new fast DFT algorithm with systematic construction of an arbitrary length that reduces the number of multiplications needed as compared with the existing fast algorithm. Moreover, we can also use the proposed CGLS-like DFT eigenvectors to define a new type of the discrete fractional Fourier transform, which is efficient in implementation and effective for encryption and OFDM.      

Conjugate pair fast Fourier transform

A new algorithm for the fast computation of the discrete Fourier transform is introduced. The algorithm, called the conjugate pair FFT (CPFFT), is used to compute a length-2/sup n/ DFT. The number of multiplications and additions required by the CPFFT is less than that required by the SRFFT algorithm.<>