Mixed-radix discrete cosine transform

Presents two new fast discrete cosine transform computation algorithms: a radix-3 and a radix-6 algorithm. These two new algorithms are superior to the conventional radix-3 algorithm as they (i) require less computational complexity in terms of the number of multiplications per point, (ii) provide a wider choice of the sequence length for which the DCT can be realized and, (iii) support the prime factor-decomposed computation algorithm to realize the 2^m3ⁿ-point DCT. Furthermore, a mixed-radix algorithm is also proposed such that an optimal performance can be achieved by applying the proposed radix-3 and radix-6 and the well-developed radix-2 decomposition techniques in a proper sequence     

Automatic generation of prime length FFT programs

Describes a set of programs for circular convolution and prime length fast Fourier transforms (FFTs) that are relatively short, possess great structure, share many computational procedures, and cover a large variety of lengths. The programs make clear the structure of the algorithms and clearly enumerate independent computational branches that can be performed in parallel. Moreover, each of these independent operations is made up of a sequence of suboperations that can be implemented as vector/parallel operations. This is in contrast with previously existing programs for prime length FFTs: They consist of straight line code, no code is shared between them, and they cannot be easily adapted for vector/parallel implementations. The authors have also developed a program that automatically generates these programs for prime length FTTs. This code-generating program requires information only about a set of modules for computing cyclotomic convolutions     

Asymptotic average redundancy of Huffman (and other) block codes

We study asymptotically the redundancy of Huffman (and other) codes. It has been known from the inception of the Huffman (1952) code that in the worst case its redundancy-defined as the excess of the code length over the optimal (ideal) code length-is not more than one. However, to the best of our knowledge no precise asymptotic results have been reported in literature thus far. We consider here a memoryless binary source generating a sequence of length n distributed as binomial (n, p) with p being the probability of emitting 0. Based on the results of Stubley (1994), we prove that for p<1/2 the average redundancy R¯_n^H of the Huffman code becomes as n→∞: R¯_n^H={(3/2-(1/ln2+o(1))=0.057304…, α irrational); (3/2-(1/M)(〈βMn〉-½)); (-(1/M(1-2^-1M/))2^-〈nβM〉M/); (+O(ρⁿ), α=N/M rational); where α=log₂ (1-p)/p and β=-log₂(1-p), ρ<1, M, N are integers such that gcd (N, M)=1, and 〈x〉=x-[x] is the fractional part of x. The appearance of the fractal-like function 〈βMn〉 explains the erratic behavior of the Huffman redundancy, and its “resistance” to succumb to a precise analysis. As a side result, we prove that the average redundancy of the Shannon block code is as n→∞: R¯_n^S{(½+o(1), α irrational); (½-1/M (〈Mnβ〉-½)); (+O(ρⁿ), α=N/M rational); where ρ<1. Finally, we derive the redundancy of the Golomb (1966) code (for the geometric distribution) which can be viewed as a special case of the Huffman and Shannon codes, Golomb's code redundancy has only oscillating behavior (i.e., there is not convergent mode). These findings are obtained by analytic methods such as theory of distribution of sequences modulo 1 and Fourier series     

基于CORDIC的精确快速幅相解算方法

针对传统CORDIC算法进行高精度幅度相位解算时迭代次数过多、时延较长、相位收敛较慢等局限,提出了一种基于最佳一致逼近方法的幅度与相位补偿算法,即利用传统CORDIC算法迭代一定次数后得到的向量信息,采用最佳一致逼近方法对幅度和相位分区间进行一阶多项式补偿,有效提高了计算精度.仿真及实测结果表明,对传统CORDIC算法4次迭代后的结果进行补偿,幅度相对误差可达到10^-5量级、相位绝对误差可达到10^-5度量级,最大输出时延不大于100ns.在使用部分专用乘法器的条件下,寄存器消耗降低了42.5%,查找表消耗降低了15.5%.采用该补偿算法,每多一次CORDIC迭代其相位精度可提高约一个数量级.因此,本文提出的补偿CORDIC算法在迭代次数、计算精度等方面优于传统CORDIC算法,适合于高精度计算的场合.     

The Correlation Distribution of Quaternary Sequences of Period $2(2^n-1)$

Family A is a family of sequences of period 2ⁿ - 1 over Zi, the ring of integers modulo 4. This family has optimal correlation properties and its correlation distribution is well known. Two related families of quaternary sequences are the families B and C. These are families of sequences over Z₄ of period 2(2ⁿ - 1). In recent years, new families of quaternary sequences of period 2(2ⁿ - 1) have been constructed by modifying the sequence families B and C in a nonlinear way. This has resulted in a new family D of sequences of period 2(2ⁿ - 1) which has optimal correlation properties, but until now the correlation distribution of this family has not been known. In this paper, we completely determine the correlation distribution of family D by making use of properties of exponential sums.     

Cyclotomic FFTs With Reduced Additive Complexities Based on a Novel Common Subexpression Elimination Algorithm

In this paper, we first propose a novel common subexpression elimination (CSE) algorithm for matrix-vector multiplications over characteristic-2 fields. As opposed to previously proposed CSE algorithms, which usually focus on complexity savings due to recurrences of subexpressions, our CSE algorithm achieves two types of complexity reductions, differential savings and recurrence savings, by taking advantage of the cancelation property of characteristic-2 fields. Using our CSE algorithm, we reduce the additive complexities of cyclotomic fast Fourier transforms (CFFTs). Using a weighted sum of the numbers of multiplications and additions as a metric, our CFFTs achieve smaller total complexities than previously proposed CFFTs and other FFTs, requiring both fewer multiplications and fewer additions in many cases.      

2n modified prime codes for use in fibre optic CDMAnetworks

2ⁿ modified prime codes are designed for all-optical code-division multiple access (CDMA) networks using very simple encoders and decoders. The proposed code is obtained from an original 2ⁿ prime code of prime number P. By padding P-1 zeros in each `subsequence' of codewords in the corresponding 2ⁿ prime code. The cross-correlation constraint of the resulting 2ⁿ modified prime code is equal to one, as opposed to two for a 2ⁿ prime code. For a given bit error rate (BER), the proposed code can thus be used to support a larger number of active users in the fibre optic CDMA network than a 2ⁿ prime code. Moreover, using the former can also reduce code length and weight compared with employing the latter to achieve the same BER     

Trace-Orthonormal Full-Diversity Cyclotomic Space–Time Codes

In this paper, we consider the design of full-diversity space-time codes for a coherent multiple-input multiple-output (MIMO) communication system. Starting from both the information theoretic and detection error viewpoints, we first establish that a desirable property for general linear dispersion (LD) codes is to have an interunitary as well as an intraunitary structure-a structure we call trace-orthonormality. By imposing the trace-orthonormal structure on an LD code and applying cyclotomic number theory, we establish, for an arbitrary number of transmitter and receiver antennas, a systematic and simple method to jointly design a unitary cyclotomic matrix, the Diophantine number, and the corresponding constellation for an LD code. As a result, this enables us to construct full-diversity rectangular cyclotomic LD codes with any symbol transmission rate less than or equal to the number of transmitter antennas. In addition, for the case when the number of transmitter antennas is greater than the number of receiver antennas, by taking advantage of the delay, we also arrive at the design of a special trace-orthonormal full-diversity cyclotomic space-time block code which, for the number of transmitter antenna being equal to 2^m, can be proved to minimize the worst case pairwise error probability of a maximum-likelihood (ML) detector for a q-ary quadrature amplitude modulation (QAM) signal constellation and, therefore, achieves optimal coding gain. Computer simulations show that these codes have bit-error performance advantages over currently available codes     

RNS-to-Binary Converters for Two Four-Moduli Sets {2n−1,2n,2n+1,2n+1−1} and {2n−1,2n,2n+1,2n+1+1}

In this paper, reverse converters for two recently proposed four-moduli sets {2ⁿ - 1,2ⁿ,2ⁿ + 1,2ⁿ+1 - 1} and {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ+1 + 1} are described. The reverse conversion in the three-moduli set {2ⁿ - 1,2ⁿ,2ⁿ + 1} has been optimized in literature. Hence, the proposed converters are based on two new moduli sets {(2ⁿ(2²ⁿ-1)),2ⁿ⁺¹-1} and {(2ⁿ(2²ⁿ-1)), 2ⁿ⁺¹+1} and use mixed radix conversion. The resulting designs do not require any ROM. Both are similar in their architecture except that the converter for the moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ+1 + 1} is slightly complicated due to the difficulty in performing reduction modulo (2ⁿ⁺¹+1) as compared with modulo (2ⁿ⁺¹-1). The proposed conversion techniques are compared with earlier realizations described in literature with regard to conversion time as well as area requirements.     

Adaptive H∞ channel equalization for wirelesspersonal communications

In this paper, a new adaptive H_∞ filtering algorithm is developed to recursively update the tap-coefficient vector of a decision feedback equalizer (DFE) in order to adaptively equalize the time-variant dispersive fading channel of a high-rate indoor wireless personal communication system. Different from conventional L ₂ (such as the recursive least squares (RLS)) filtering algorithms which minimize the squared equalization error, the adaptive H _∞ filtering algorithm is a worst case optimization. It minimizes the effect of the worst disturbances (including input noise and modeling error) on the equalization error. Hence, the DFE with the adaptive H_∞ filtering algorithm is more robust to the disturbances than that with the RLS algorithm. Computer simulation demonstrates that better transmission performance can be achieved using the adaptive H_∞ algorithm when the received signal-to-noise ratio (SNR) is larger than 20 dB     

Groups of algebraic integers used for coding QAM signals

Linear block codes over Gaussian integers and Eisenstein integers were used for coding over two-dimensional signal space. A group of Gaussian integers with 2²ⁿ elements was constructed to code quadrature amplitude modulation (QAM) signals such that a differentially coherent method can be applied to demodulate the QAM signals. This paper shows that one subgroup of the multiplicative group of units in the algebraic integer ring of any quadratic number field with unique factorization, modulo the ideal (Pⁿ), can be used to obtain a QAM signal space of 2p^2n-2 points, where p is any given odd prime number. Furthermore, one subgroup of the multiplicative group of units in the quotient ring Z[ω]/(pⁿ) can be used to obtain a QAM signal space of 6p^2n-2 points; one subgroup of the multiplicative group of units in the quotient ring Z[i](pⁿ) can be used to obtain a QAM signal space of 4p^2n-2 points which is symmetrical over the quadrants of the complex plane and useful for differentially coherent detection of QAM signals; the multiplicative group of units in the quotient ring Z[ω]/(2ⁿ) can be used to obtain a QAM signal space of 3·2^2n-2 points, where i=√-1, ω=(-1+√-3)/2=(-1+i√3)/2, p is any given odd prime number, Z[i] and Z[ω] are, respectively, the Gaussian integer ring and the Eisenstein integer ring. These multiplicative groups can also be used to construct block codes over Gaussian integers or Eisenstein integers which are able to correct some error patterns     

Parallel-decomposition algorithm for discrete computationalproblems and its application in developing an efficient discreteconvolution algorithm

The authors present a parallel-decomposition algorithm for discrete computational problems. An efficient convolution algorithm is developed under the proposed decomposition algorithm. The decomposition operation is based on integer modular arithmetic. Congruent sets are generated from integer modular operation over the index of the problem and constitute a partition. The partition is used to decompose the problem into several small subproblems. The partition under the integer modular operation is unique. Complete reconstruction of the problem is guaranteed by the uniqueness of the partition. Since the algorithm is established on the foundation of all algebraic systems, i.e. number and set theories, it is highly problem-independent, and provides an uniform approach to parallel decomposition of general problems on the discrete domain. Because the subproblems are highly regular, it is suitable for VLSI hardware implementation. The computational complexity of the developed convolution algorithm is reduced by a factor of p², and (p²)ⁿ for n-dimensional problems (p can be any common factor of the input sequences). Compared to the popular FFT methods, where round-off error is introduced, the convolution result from the algorithm is exact. In addition, it does not have restrictions on the length of the input sequences, as in the well known block convolution algorithm. As a result, the algorithm is highly efficient for convolution of large input sequences or correlation operations     

Adaptive estimation of HMM transition probabilities

This paper presents new schemes for recursive estimation of the state transition probabilities for hidden Markov models (HMMs) via extended least squares (ELS) and recursive state prediction error (RSPE) methods. Local convergence analysis for the proposed RSPE algorithm is shown using the ordinary differential equation (ODE) approach developed for the more familiar recursive output prediction error (RPE) methods. The presented scheme converges and is relatively well conditioned compared with the previously proposed RPE scheme for estimating the transition probabilities that perform poorly in low noise. The ELS algorithm presented is computationally of order N², which is less than the computational effort of order N⁴ required to implement the RSPE (and previous RPE) scheme, where N is the number of Markov states. Building on earlier work, an algorithm for simultaneous estimation of the state output mappings and the state transition probabilities that requires less computational effort than earlier schemes is also presented and discussed. Implementation aspects of the proposed algorithms are discussed, and simulation studies are presented to illustrate the convergence and convergence rates     

RNS-To-Binary Converter for a New Three-Moduli Set {2n+1−1,2n,2n−1}

In this brief, the design of residue number system (RNS) to binary converters for a new powers-of-two related three-moduli set {2ⁿ⁺¹ - 1, 2ⁿ, 2ⁿ - 1} is considered. This moduli set uses moduli of uniform word length (n to n + 1 bits). It is derived from a previously investigated four-moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2^{n +1} - 1}. Three RNS-to-binary converters are proposed for this moduli set: one using mixed radix conversion and the other two using Chinese remainder theorem. Detailed architectures of the three converters as well as comparison with some earlier proposed converters for three-moduli sets with uniform word length and the four-moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ⁺¹ - 1} are presented.     

A Truncation method for computing slant transforms withapplications to image processing

A truncation method for computing the slant transform is presented. The slant transform truncation (STT) algorithm uses the divide and conquer principle of hierarchical data structures to factorize coherent image data into sparse subregions. In one dimension with a data array of size N=2ⁿ, the truncation method takes a time between O(N) and O(Nlog₂N), degenerating to the performance of the fast slant transform (FST) method in its worst case. In two dimensions, for a data array of size N×N, the one-dimensional truncation method is applied to each row, then to each column of the array, to compute the transform in a time between O(N²) and O(N²log₂N). Coherence is a fundamental characteristic of digital images and so the truncation method is superior to the FST method when computing slant transforms of digital images. Experimental results are presented to justify this assertion     

Error-trellises for convolutional codes .II. Decoding methods

For pt. I see ibid. vol.46, p.1592-1601 (1998). Soft-decision maximum-likelihood decoding of convolutional codes over GF(q) can be accomplished via searching through an error-trellis for the least weighing error sequence. The error-trellis is obtained by a syndrome-based construction. Its structure lends itself particularly well to the application of expedited search procedures. The method to carry out such error-trellis-based decoding is formulated by four algorithms. Three of these algorithms are aimed at reducing the worst case computational complexity, whereas by applying the fourth algorithm, the average computational complexity is reduced under low to moderate channel wise level. The syndrome decoder achieves substantial worst case and average computational gains in comparison with the conventional maximum-likelihood decoder, namely the Viterbi decoder, which searches for the most likely codeword directly within the code     

On the computation of two-dimensional DCT

A new fast algorithm for the type-II two-dimensional (2-D) discrete cosine transform (DCT) is presented. It shows that the 2-D DCT can be decomposed into cosine-cosine, cosine-sine, sine-cosine, and sine-sine sequences that can be further decomposed into a number of similar sequences. Compared with other reported algorithms, the proposed one achieves savings on the number of arithmetic operations and has a recursive computational structure that leads to a simplification of the input/output indexing process. Furthermore, the new algorithm supports transform sizes (p₁*2^m1)×(p₂*2 ^m2), where p₁ and p₂ are arbitrarily odd integers, which provides a wider range of choices on transform sizes for various applications     

Stochastic gradient adaptation under general error criteria

Examines a family of adaptive filter algorithms of the form W_k+1=W_k+μf(d_k-W_k^t X_k)X_k in which f(·) is a memoryless odd-symmetric nonlinearity acting upon the error. Such algorithms are a generalization of the least-mean-square (LMS) adaptive filtering algorithm for even-symmetric error criteria. For this algorithm family, the authors derive general expressions for the mean and mean-square convergence of the filter coefficients For both arbitrary stochastic input data and Gaussian input data. They then provide methods for optimizing the nonlinearity to minimize the algorithm misadjustment for a given convergence rate. Using the calculus of variations, it is shown that the optimum nonlinearity to minimize misadjustment near convergence under slow adaptation conditions is independent of the statistics of the input data and can be expressed as -p'(x)/p(x), where p(x) is the probability density function of the uncorrelated plant noise. For faster adaptation under the white Gaussian input and noise assumptions, the nonlinearity is shown to be x/{1+μλx²/σ_k ²}, where λ is the input signal power and σ_k² is the conditional error power. Thus, the optimum stochastic gradient error criterion for Gaussian noise is not mean-square. It is shown that the equations governing the convergence of the nonlinear algorithm are exactly those which describe the behavior of the optimum scalar data nonlinear adaptive algorithm for white Gaussian input. Simulations verify the results for a host of noise interferences and indicate the improvement using non-mean-square error criteria     

Low complexity detection algorithm based on optimized Neumann series for massive MIMO system

An optimized Neumann series ( NS ) approximation isdescribed based on Frobenius matrix decomposition, this method aims to reduce the high complexity, which caused by the large matrix inversion of detection algorithm in the massive multiple input multiple output (MIMO) system. The large matrix in the inversion is decomposed into the sum of the hollow matrix and a Frobenius matrix, and the Frobenius matrix has the diagonal elements and the first column of the large matrix. In order to ensure the detection performance approach to minimum mean square error (MMSE) algorithm, the first three terms of the series approximation are needed, which results in high complexity as O(K³), where K is the number of users. This paper further optimize the third term of the series approximation to reduce the computational complexity from O(K³) to O(K²). The computational complexity analysis and simulation results show that the performance of proposed algorithm can approach to MMSE algorithm with low complexity O(K²).     

PPM performance with BWT complexity: a fast and effective datacompression algorithm

This paper introduces a new data compression algorithm. The goal underlying this new code design is to achieve a single lossless compression algorithm with the excellent compression ratios of the prediction by partial mapping (PPM) algorithms and the low complexity of codes based on the Burrows Wheeler Transform (BWT). Like the BWT-based codes, the proposed algorithm requires worst case O(n) computational complexity and memory; in contrast, the unbounded-context PPM algorithm, called PPM*, requires worst case O(n²) computational complexity. Like PPM*, the proposed algorithm allows the use of unbounded contexts. Using standard data sets for comparison, the proposed algorithm achieves compression performance better than that of the BWT-based codes and comparable to that of PPM*. In particular, the proposed algorithm yields an average rate of 2.29 bits per character (bpc) on the Calgary corpus; this result compares favorably with the 2.33 and 2.34 bpc of PPM5 and PPM* (PPM algorithms), the 2.43 bpc of BW94 (the original BWT-based code), and the 3.64 and 2.69 bpc of compress and gzip (popular Unix compression algorithms based on Lempel-Ziv (LZ) coding techniques) on the same data set. The given code does not, however, match the best reported compression performance-2.12 bpc with PPMZ9-listed on the Calgary corpus results web page at the time of this publication. Results on the Canterbury corpus give a similar relative standing. The proposed algorithm gives an average rate of 2.15 bpc on the Canterbury corpus, while the Canterbury corpus web page gives average rates of 1.99 bpc for PPMZ9, 2.11 bpc for PPM5, 2.15 bpc for PPM7, 2.23 bpc for BZIP2 (a popular BWT-based code), and 3.31 and 2.53 bpc for compress and gzip, respectively