A fast algorithm for determining the linear complexity of asequence with period pn over GF(q)

A fast algorithm is presented for determining the linear complexity of a sequence with period pⁿ over GF (q), where p is an odd prime, and where q is a prime and a primitive root (mod p²)     

Avalanche multiplication noise characteristics in thin GaAs p+-i-n+ diodes

Avalanche noise measurements have been performed on a range of homojunction GaAs p⁺-i-n⁺ and n⁺-i-p ⁺ diodes with “i” region widths, ω from 2.61 to 0.05 μm. The results show that for ω⩽1 μm the dependence of excess noise factor F on multiplication does not follow the well-established continuous noise theory of McIntyre [1966]. Instead, a decreasing noise factor is observed as ω decreases for a constant multiplication. This reduction in F occurs for both electron and hole initiated multiplication in the thinner ω structures even though the ionization coefficient ratio is close to unity. The dead-space, the minimum distance a carrier must travel to gain the ionization threshold energy, becomes increasingly important in these thinner structures and largely accounts for the reduction in noise     

Fast and precise Fourier transforms

Many applications of fast Fourier transforms (FFTs), such as computer tomography, geophysical signal processing, high-resolution imaging radars, and prediction filters, require high-precision output. An error analysis reveals that the usual method of fixed-point computation of FFTs of vectors of length 2^l leads to an average loss of l/2 bits of precision. This phenomenon, often referred to as computational noise, causes major problems for arithmetic units with limited precision which are often used for real-time applications. Several researchers have noted that calculation of FFTs with algebraic integers avoids computational noise entirely. We combine a new algorithm for approximating complex numbers by cyclotomic integers with Chinese remaindering strategies to give an efficient algorithm to compute b-bit precision FFTs of length L. More precisely, we approximate complex numbers by cyclotomic integers in Z[e(2πi/2ⁿ)] whose coefficients, when expressed as polynomials in e(2πi/2ⁿ), are bounded in absolute value by some integer M. For fixed n our algorithm runs in time O(log(M)), and produces an approximation with worst case error of O(1/M(2^n-2-1)). We prove that this algorithm has optimal worst case error by proving a corresponding lower bound on the worst case error of any approximation algorithm for this task. The main tool for designing the algorithms is the use of the cyclotomic units, a subgroup of finite index in the unit group of the cyclotomic field. First implementations of our algorithms indicate that they are fast enough to be used for the design of low-cost high-speed/high-precision FFT chips     

Three space-economical algorithms for calculatingminimum-redundancy prefix codes

The minimum-redundancy prefix code problem is to determine, for a given list W=[ω₁,..., ω_n] of n positive symbol weights, a list L=[l₁,...,l_n] of n corresponding integer codeword lengths such that Σ_i=1 ⁿ 2^-li⩽1 and Σ_i=1ⁿ ω_il_i is minimized. Let us consider the case where W is already sorted. In this case, the output list L can be represented by a list M=[m₁,..., m_H], where m_l, for l=1,...,H, denotes the multiplicity of the codeword length l in L and H is the length of the greatest codeword. Fortunately, H is proved to be O(min(log(1/p₁),n)), where p₁ is the smallest symbol probability, given by ω₁/Σ _i=1ⁿ ω_i. We present the Fast LazyHuff (F-LazyHuff), the Economical LazyHuff (E-LazyHuff), and the Best LazyHuff (B-LazyHuff) algorithms. F-LazyHuff runs in O(n) time but requires O(min(H², n)) additional space. On the other hand, E-LazyHuff runs in O(n+nlog(n/H)) time, requiring only O(H) additional space. Finally, B-LazyHuff asymptotically overcomes, the previous algorithms, requiring only O(n) time and O(H) additional space. Moreover, our three algorithms have the advantage of not writing over the input buffer during code calculation, a feature that is very useful in some applications     

A New Family of Ternary Almost Perfect Nonlinear Mappings

A mapping f(x) from GF(pⁿ) to GF(pⁿ) is differentially k-uniform if k is the maximum number of solutions x isin GF(pⁿ) of f(x+a) - f(x) = b, where a, b isin GF(pⁿ) and a ne 0. A 2-uniform mapping is called almost perfect nonlinear (APN). This correspondence describes new families of ternary APN mappings over GF(3ⁿ), n>3 odd, of the form f(x) = ux^d + x^{d 2} where d₁ = (3^n-1)/2 - 1 and d₂ = 3ⁿ - 2.     

Perfect Codes for Metrics Induced by Circulant Graphs

An algebraic methodology for defining new metrics over two-dimensional signal spaces is presented in this work. We have mainly considered quadrature amplitude modulation (QAM) constellations which have previously been modeled by quotient rings of Gaussian integers. The metric over these constellations, based on the distance concept in circulant graphs, is one of the main contributions of this work. A detailed analysis of some degree-four circulant graphs has allowed us to detail the weight distribution for these signal spaces. A new family of perfect codes over Gaussian integers will be defined and characterized by providing a solution to the perfect t-dominating set problem over the circulant graphs presented. Finally, we will show how this new metric can be extended to other signal sets by considering hexagonal constellations and circulant graphs of degree six.     

Binary sequence sets with favorable correlations from differencesets and MDS codes

We propose new families of pseudorandom binary sequences based on Hadamard difference sets and MDS codes. We obtain, for p=4k-1 prime and t an integer with 1⩽t⩽(p-1)/2, a set of p^t binary sequences of period p² whose peak correlation is bounded by 1+2t(p+1). The sequences are balanced, have high linear complexity, and are easily generated     

Nonmatrix Cramer-Rao bound expressions for high-resolutionfrequency estimators

Analytical expressions are derived for the Cramer-Rao (CR) lower bound on the variance of frequency estimates for the two-signal time-series data models consisting of either one real sinusoid or two complex sinusoids in white Gaussian noise. The expressions give the bound in terms of the signal-to-noise ratio (SNR), the number N of data samples, and a function dependent on the frequency separation and the initial phase difference between the two signal components of each model. The bounds are examined as the phase difference is varied, and the largest and smallest bound expressions and the corresponding critical values of the phase difference are obtained. The exact expressions are analyzed for the case of small frequency separations δω. It is found that the largest bound is proportional to (N·δω)^-4/N³·SNR and that the smallest bound is proportional to (N·δω)^-2 N³·SNR for small δω. Examples indicate that the small δω results closely approximate the exact ones whenever the frequency separation is smaller than the Fourier resolution limit. Based on the asymptotic results, it is found that the threshold SNR at which an unbiased estimator can resolve the two signal frequencies is at least proportional to (N·δω)^-6 /N for the worst phase difference case and to (N·δω)^-4/N for the best phase difference case for small δω. The results are applicable to the general case of sampling where the samples are taken at arbitrary instants     

The Cramer-Rao bound on frequency estimates of signals closelyspaced in frequency

This paper examines the Cramer-Rao (CR) lower bound on the variance of frequency estimates for the problem of n signals closely spaced in frequency. The main results presented are simple analytic expressions for the CR bound in terms of the maximum frequency separation, δω, SNR, and the number of data vectors, N, that are valid for small δω. The results are applicable to the conditional (deterministic) signal model. The results show that the CR bound on frequency estimates is proportional to (δω)^-2(n-1)/N×SNR. Therefore, the bound increases rapidly as the signal separation is reduced. Examples indicate that the expressions closely approximate the exact CR bounds whenever the signal separation is smaller than one resolution cell. Based upon the results, it is argued that the threshold SNR at which an unbiased estimator can resolve n closely spaced signals is at least proportional to (δω)^-2n/N. The results are quite general and apply to many different types of temporal and spatial sampling grids     

A self-aligned retrograde twin-well structure with buried p+-layer

A self-aligned retrograde twin-well structure with a buried p⁺-layer surrounding the n-well is presented. The retrograde twin well and buried p⁺-layer are fabricated by a single lithographic step using high-energy ion implantation. The retrograde n-well is self-aligned to the retrograde p-well regions, and the channel stop processes are eliminated by using tight spatial distributions of retrograde n- and p-wells. This simple process is compatible with both local oxidation of silicon (LOCOS) and trench isolation processes and allows a scalable CMOS structure for very tight n⁺-to-p⁺ spacing. The present CMOS structure provides high latchup immunity at 1.5-μm n⁺-to-p⁺ spacing and good isolation characteristics without additional n- and p-channel stop dopings     

Characterization of Zak Space Support of a Discrete Chirp

General conditions are derived for an (N=KL²)-point discrete chirp with chirp rate a and carrier frequency b to have minimal support on the LtimesKL Zak transform lattice. Earlier, it has been shown that when the normalized chirp parameters amacr= aK, amacr= aK ², and 2bmacr = 2bK are integers, the last two of the same parity, then the discrete chirp is supported at KL points. Here, this condition is relaxed, by allowing amacr to be a rational number, i.e., amacr = n/d, n, disin Z, (n,d)=1, and requiring only that amacr and bmacrL be integers of arbitrary parity. It is shown that the support of the Zak space chirp satisfying the new condition then increases to dKL points. The results provide foundations for future constructions of sophisticated radar and communications signal processing algorithms. Examples of direct applications of the Zak space conditions in chirp parameter estimation, chirp detection, and chirp de-noising are included     

Extended One-Dimensional Analysis to Effectively Derive Quantum Efficiency of Various CMOS Photodiodes

This paper proposes an extended 1-D analysis to derive quantum efficiency of various commonly used CMOS photodiodes. The theoretical model of the CMOS photodiode with the n-/p-epitaxial/p + substrate (n-/p-epi/p + sub) structure is established from steady-state continuity equations, where most existing boundary conditions are applied. In particular, the minority carrier and current densities are continuous across the interface between two layers with the same dopant type. Models of the other commonly used CMOS photodiodes are also examined. Three CMOS photodiodes with n-/p-substrate (n-/p-sub), p⁺/n-/p-substrate (p⁺/n-/p-sub), and n-/p-epi/p + sub structures are fabricated and characterized to validate the proposed model. Additionally, the surface recombination velocity is adequately determined by fitting the simulated quantum efficiency to the measured value. The simulated quantum efficiency of the proposed model for these three photodiodes is quite consistent with the measured values, revealing the feasibility and effectiveness of the proposed model in characterizing various CMOS photodiodes.     

Algebraic construction of cyclic codes over Z8 with agood Euclidean minimum distance

Let S(8) denote the set of the eight admissible signals of an 8PSK communication system. The alphabet S(8) is endowed with the structure of Z₈, the set of integers taken modulo 8, and codes are defined to be Z₈-submodules of Z₈ⁿ. Three cyclic codes over Z₈ are then constructed. Their length is equal to 6, 8, and 7, and they, respectively, contain 64, 64, and 512 codewords. The square of their Euclidean minimum distance is equal to 8, 16-4√2 and 10-2√2, respectively. The size of the codes of length 6 and 7 can be doubled while the Euclidean minimum distance remains the same     

Upper bounds on trellis complexity of lattices

Unlike block codes, n-dimensional lattices can have minimal trellis diagrams with an arbitrarily large number of states, branches, and paths. In particular, we show by a counterexample that there is no f(n), a function of n, such that all rational lattices of dimension n have a trellis with less than f(n) states. Nevertheless, using a theorem due to Hermite, we prove that every integral lattice Λ of dimension n has a trellis T, such that the total number of paths in T is upper-bounded by P(T)⩽n!(2/√3)^n2(n-1/2)V(Λ) ^n-1 where V(n) is the volume of Λ. Furthermore, the number of states at time i in T is upper-bounded by |S_i|⩽(2/√3)^i2(n-1)V(Λ)²ⁱ² n/. Although these bounds are seldom tight, these are the first known general upper bounds on trellis complexity of lattices     

Eigenvalues and eigenvectors of covariance matrices for signalsclosely spaced in frequency

The eigenstructures of common covariance matrices are identified for the general case of M closely spaced signals. It is shown that the largest signal-space eigenvalue is relatively insensitive to signal separation. By contrast, the ith largest eigenvalue is proportional to δω^2(i-1) or δω^4(i-1), where δω is a measure of signal separation. Therefore, matrix conditioning degrades rapidly as signal separation is reduced. It is also shown that the limiting eigenvectors have remarkably simple structures. The results are very general, and apply to planar far-field direction-finding problems involving almost arbitrary scenarios, and also to time-series analysis of sinusoids, exponentials, and other signals     

New families of almost perfect nonlinear power mappings

A power mapping f(x)=x^d over GF(pⁿ) is said to be differentially k-uniform if k is the maximum number of solutions x∈GF(pⁿ) of f(x+a)-f(x)=b where a, b∈GF(pⁿ ) and a≠0. A 2-uniform mapping is called almost perfect nonlinear (APN). We construct several new infinite families of nonbinary APN power mappings     

Polarity dependent gate tunneling currents in dual-gate CMOSFETs

Polarity dependence of the gate tunneling current in dual-gate CMOSFETs is studied over a gate oxide range of 2-6 nm. It is shown that, when measured in accumulation, the I_g versus V_g characteristics for the p⁺/pMOSFET are essentially identical to those for the n⁺/nMOSFET; however, when measured in inversion, the p⁺/pMOSFET exhibits much lower gate current for the same |V_g|. This polarity dependence is explained by the difference in the supply of the tunneling electrons. The carrier transport processes in p⁺/pMOSFET biased in inversion are discussed in detail. Three tunneling processes are considered: (1) valence band hole tunneling from the Si substrate; (2) valence band electron tunneling from the p⁺-polysilicon gate; and (3) conduction band electron tunneling from the p⁺-polysilicon gate. The results indicate that all three contribute to the gate tunneling current in an inverted p⁺/pMOSFET, with one of them dominating in a certain voltage range     

Nickel ohmic contacts to p and n-type 4H-SiC

The first demonstration of Ni ohmic contacts to both p⁺ and n⁺ 4H-SiC formed by ion implantation is reported. Sample preparation conditions are described and experimental results presented. Specific contact resistances in the range of 10^-4 Ω cm ² and 10^-6 Ω cm² for p⁺ and n⁺ 4H-SiC, respectively, have been determined by the transfer length method     

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     

An approach to Ungerboeck coding for rectangular signal sets (Corresp.)

A method of finding good Ungerboeck codes for large rectangular [quadrature amplitude modulation (QAM)] signal sets is described. Using the concept of Euclidean weights due to Ungerboeck, we prove that a2^{n}point basic constellation may be employed to determine exactly the free distance for an Ungerboeck-coded rectangular2^{m}point set, whenm-n-1bits are uncoded and the remaining bits pass through a rate(n-1)/nconvolutionai encoder. It is shown that rate2/3encoders may be used to achieve most of the theoretically possible coding gain in the proposed scheme where the effect of the error coefficient on the coding gain has been considered.