Nice Error Bases and Sylow Subgroups

Nice error bases have various applications in quantum computing such as in the teleportation of quantum states and in the theory of noiseless systems. Associated in a natural way to any nice error basis there is a so-called index group and also a group of central type. In this paper, we will extend the number of known index groups for nice error bases by exploiting the connections between groups of central type and their Sylow subgroups. Our methods will allow us find all index groups of degree up to $44$ with two exceptions.      

On the monomiality of nice error bases

Unitary error bases generalize the Pauli matrices to higher dimensional systems. Two basic constructions of unitary error bases are known: An algebraic construction by Knill that yields nice error bases, and a combinatorial construction by Werner that yields shift-and-multiply bases. An open problem posed by Schlingemann and Werner relates these two constructions and asks whether each nice error basis is equivalent to a shift-and-multiply basis. We solve this problem and show that the answer is negative. However, we find that nice error bases have more structure than one can anticipate from their definition. In particular, we show that nice error bases can be written in a form in which at least half of the matrix entries are 0.     

Cortical patch basis model for spatially extended neural activity

A new source model for representing spatially distributed neural activity is presented. The signal of interest is modeled as originating from a patch of cortex and is represented using a set of basis functions. Each cortical patch has its own set of bases, which allows representation of arbitrary source activity within the patch. This is in contrast to previously proposed cortical patch models which assume a specific distribution of activity within the patch. We present a procedure for designing bases that minimize the normalized mean squared representation error, averaged over different activity distributions within the patch. Extension of existing algorithms to the basis function framework is straightforward and is illustrated using linearly constrained minimum variance (LCMV) spatial filtering and maximum-likelihood signal estimation/generalized likelihood ratio test (ML/GLRT). The number of bases chosen for each patch determines a tradeoff between representation accuracy and the ability to differentiate between distinct patches. We propose choosing the minimum number of bases that satisfy a constraint on the normalized mean squared representation accuracy. A mismatch analysis for LCMV and ML/GLRT is presented to show that this is an appropriate strategy for choosing the number of bases. The effectiveness of the patch basis model is demonstrated using real and simulated evoked response data. We show that significant changes in performance occur as the number of basis functions varies, and that very good results are obtained by allowing modest representation error.     

On denoising and best signal representation

We propose a best basis algorithm for signal enhancement in white Gaussian noise. The best basis search is performed in families of orthonormal bases constructed with wavelet packets or local cosine bases. We base our search for the “best” basis on a criterion of minimal reconstruction error of the underlying signal. This approach is intuitively appealing, because the enhanced or estimated signal has an associated measure of performance, namely, the resulting mean-square error. Previous approaches in this framework have focused on obtaining the most “compact” signal representations, which consequently contribute to effective denoising. These approaches, however, do not possess the inherent measure of performance which our algorithm provides. We first propose an estimator of the mean-square error, based on a heuristic argument and subsequently compare the reconstruction performance based upon it to that based on the Stein (1981) unbiased risk estimator. We compare the two proposed estimators by providing both qualitative and quantitative analyses of the bias term. Having two estimators of the mean-square error, we incorporate these cost functions into the search for the “best” basis, and subsequently provide a substantiating example to demonstrate their performance     

Shear madness: new orthonormal bases and frames using chirpfunctions

The proportional-bandwidth and constant-bandwidth time-frequency signal decompositions of the wavelet, Gabor, and Wilson orthonormal bases have attracted substantial interest for representing nonstationary signals. However, these representations are limited in that they are based on rectangular tessellations of the time-frequency plane. While much effort has gone into methods for designing nice wavelet and window functions for these frameworks, little consideration has been given to methods for constructing orthonormal bases employing nonrectangular time-frequency tilings. The authors take a first step in this direction by deriving two new families of orthonormal bases and frames employing elements that shear, or chirp, in the time-frequency plane, in addition to translate and scale. The new scale-shear fan bases and shift-shear chevron bases are obtained by operating on an existing: wavelet, Gabor (1946), or Wilson basis set with two special unitary warping transformations. In addition to the theoretical benefit of broadening the class of valid time-frequency plane tilings, these new bases could possibly also be useful for representing certain types of signals, such as chirping and dispersed signals     

Enumerative source encoding

LetSbe a given subset of binary n-sequences. We provide an explicit scheme for calculating the index of any sequence inSaccording to its position in the lexicographic ordering ofS. A simple inverse algorithm is also given. Particularly nice formulas arise whenSis the set of alln-sequences of weightkand also whenSis the set of all sequences having a given empirical Markov property. Schalkwijk and Lynch have investigated the former case. The envisioned use of this indexing scheme is to transmit or store the index rather than the sequence, thus resulting in a data compression of(logmidSmid)/n.     

Reduction of correlated noise using a library of orthonormal bases

We study the application of a library of orthonormal bases to the reduction of correlated Gaussian noise. A joint condition on the library and the noise covariance is derived which ensures that simple thresholding in an adaptively chosen basis yields an estimation error within a logarithmic factor of the ideal risk. In the model example of a wavelet packet library and stationary noise the condition can be translated into a reverse Holder inequality on the power spectrum     

A generalized Forney formula for algebraic-geometric codes

This correspondence contains a straightforward generalization of decoding of BCH codes to the decoding of algebraic-geometric codes, couched in terms of varieties, ideals, and Grobner bases. This consists of 1) a Berlekamp-Massey-type lattice-shifting row-reduction algorithm with majority voting similar to algorithms in the current literature, 2) a realization that it produces a minimal Grobner basis B for the error-locator ideal I(V) relative to a particular weighted total degree monomial ordering, 3) a factoring of that basis into several minimal PLEX bases, that facilitates finding the variety V of error positions, and 4) a direct generalization of Forney's formula to calculate error magnitudes using functions σp, which are by-products of this factoring     

Construction of sparse signal representations with adaptive multiscale orthogonal bases

We propose a new approach to construct adaptive multiscale orthonormal (AMO) bases of R^N that provide highly sparse signal representations. Our new multilayer AMO basis design produces a high proportion of small scale vectors. The basis vectors are built from small scale to large scales, layer by layer. For each layer, the basis vector maximizes a p-norm measure of sparsity. We compare the sparsity ratios SR (i.e. the percentage of negligibly small coefficients) obtained with AMO and Daubechies wavelet bases for seven families of piecewise smooth signals with randomly located discontinuities. The signals are composed of polynomial, sinusoidal and exponential pieces. In all cases, AMO bases produce a SR increase ranging from 6% to 37%. AMO bases have three main advantages over wavelets. First, they are found automatically by solving a sequence of optimization problems, which eliminates the problem of selecting a wavelet for a given signal. Second, they can provide a significantly sparser representation. Finally, they have the ability to produce zero coefficients for a larger family of piecewise smooth signals. The drawbacks of AMO bases are computational: the basis computation is more expensive, the basis vectors require storage space and no fast transform is known.     

广播电视工艺用房建筑声学指标与控制

探讨广播电视工艺用房所涉及的建筑声学技术指标与噪声控制问题,为了获得令人满意的音质效果,工艺用房良好的建声环境是基础,具体阐述录音与配音室、演播室和音控室的技术需求,列举出与工艺用房相关的噪声与振动控制、噪声参考限值,指出从工艺用房的设计规划与内装设计入手,遵循相关的建筑声学标准,才能达到良好的录播效果。     

Penalized discriminant analysis of [15O]-water PET brain images with prediction error selection of smoothness and regularization hyperparameters

We propose a flexible, comprehensive approach for analysis of [15O]-water positron emission tomography (PET) brain images using a penalized version of linear discriminant analysis (PDA). We applied it to scans from 20 subjects (eight scans/subject) performing a finger movement task and analyzed: 1) two classes to obtain a covariance-normalized baseline-activation image, and 2) eight classes for the mean within subject temporal structure which contained baseline-activation and time-dependent changes in a two-dimensional canonical subspace. We imposed spatial smoothness on the resulting image(s) by expanding it in five tensor-product B-spline (TPS) bases of varying smoothness, and further regularized with a ridge-type penalty on the noise covariance matrix. The discrimination approach of PDA provides a probabilistic framework within which prediction error (PE) estimates are derived. We used these to optimize over TPS bases and a ridge hyperparameter (expressed as equivalent degrees of freedom, EDF). We obtained unbiased, low variance PE estimates using modern resampling tools (.632+ Bootstrap and cross validation), and compared PDA of 1) TPS-projected, mean-normalized and unnormalized scans and 2) mean-normalized scans with and without additional presmoothing. By examining the tradeoffs between PE and EDF, as a function of basis selection and image smoothing we demonstrate the utility of PDA, the PE framework, and the relationship between singular value decomposition and smooth TPS bases in the analysis of functional neuroimages.     

鲁棒总体均方最小自适应滤波:算法与分析

本文研究了在输入输出观测数据均含有噪声的情况下如何有效地进行鲁棒自适应滤波的问题.以总体均方误差(TMSE)最小为准则,基于最速下降原理,通过对总体均方误差梯度进行修正,提出了一种鲁棒的总体均方最小自适应滤波算法.通过与已有算法的对比分析表明,该算法能够有效地降低权向量的每步调整量对噪声的敏感程度.仿真实验的结果进一步表明,该算法的鲁棒抗噪性能和稳态收敛精度明显地高于其它同类方法,而且可以使用较大的学习因子,在高噪声环境下仍然保持良好的收敛性.     

Sparse geometric image representations with bandelets.

This paper introduces a new class of bases, called bandelet bases, which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates directions in which the image gray levels have regular variations. The image decomposition in a bandelet basis is implemented with a fast subband-filtering algorithm. Bandelet bases lead to optimal approximation rates for geometrically regular images. For image compression and noise removal applications, the geometric flow is optimized with fast algorithms so that the resulting bandelet basis produces minimum distortion. Comparisons are made with wavelet image compression and noise-removal algorithms.     

量子密码分配过程中的测量基问题

推导了利用单量子通信系统中Alice发送基与Bob接收基方向间有偏差时双方建立的Sifted-key的误码率公式.在对公式进行考查的基础上,我们建议设立一个万有方向来统一通信双方使用的基的方向.     

Magnetic field integral equation at very low frequencies

It is known that there is a low-frequency breakdown problem when the method of moments (MOM) with Rao-Wilton-Glisson (RWG) basis is used in the electric field integral equation (EFIE); it can be solved through the loop and tree basis decomposition. The behavior of the magnetic field integral equation (MFIE) at very low frequencies is investigated using MOM, where two approaches are presented based on the RWG basis and loop and tree bases. The study shows that MFIE can be solved by the conventional MOM with the RWG basis at arbitrarily low frequencies, but there exists an accuracy problem in the real part of the electric current. Although the error in the current distribution is small, it results in a large error in the far-field computation. This is because a big cancellation occurs during the far field computation. The source of error in the current distribution is easily detected through the MOM analysis using the loop and tree basis decomposition. To eliminate the error, a perturbation method is proposed, from which a very accurate real part of the tree current has been obtained. Using the perturbation method, the error in the far-field computation is also removed. Numerical examples show that both the current distribution and the far field can be accurately computed at extremely low frequencies by the proposed method.     

量子通信中的测量基方向问题

利用单量子的通信系统中Alice发送基与Bob接收基方向间有偏差时双方建立的Sifted -key的误码率公式。在对公式进行考查的基础上 ,我们建议设立一个万有方向来统一通信双方使用的基的方向。     

A Systematic Approach to Multistage Detectors in Multipath Fading Channels

We consider linear multistage detectors with universal (large system) weighting for synchronous code-division multiple access (CDMA) in multipath fading channels with many users. A convenient choice of the basis of the projection subspace allows a joint projection of all users. Taking advantage of this property, the complexity per bit of multistage detectors with universal weights scales linearly with the number of users on the uplink CDMA channel, while other known multistage detectors with universal weights and different bases of the projection subspace keep the same quadratic complexity order per bit as the linear minimum mean-square error (LMMSE) detector. We focus on the design of two kinds of detectors with linear complexity. The detector of Type I is obtained as an asymptotic approximation of the polynomial expansion detector proposed by Moshavi The detector of Type II has the same performance as the multistage Wiener filter (MSWF) in large systems. Additionally, general performance expressions for large systems, applicable to any multistage detector with the same basis of the projection subspace (e.g., linear parallel interference canceling detectors), are derived. As a by-product, the performance analysis disproves the widespread belief that the MSWF and the polynomial expansion detector are equivalent. We show that, in general, the MSWF outperforms the latter one and they are equivalent only asymptotically in the case of equal received powers.     

Multidimensional multichannel FIR deconvolution using Gr?bner bases.

We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using Gr?bner bases. Previous work formulates the problem of multichannel FIR deconvolution as the construction of a left inverse of the convolution matrix, which is solved by numerical linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry and Gr?bner bases, we find necessary and sufficient conditions for the existence of exact deconvolution FIR filters and propose simple algorithms to find these deconvolution filters. The main contribution of our work is to extend the previous Gr?bner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general FIR filters. The proposed algorithms obtain a set of FIR deconvolution filters with a small number of nonzero coefficients (a desirable feature in the impulsive noise environment) and do not require the prior information of the support. Moreover, we provide a complete characterization of all exact deconvolution FIR filters, from which good FIR deconvolution filters under the additive white noise environment are found. Simulation results show that our approaches achieve good results under different noise settings.     

飞行仿真中气动特性数据拟合方法研究

飞机气动特性数据是飞行仿真的基础,其精确性直接影响着飞行仿真效果.提出一种基于遗传算法的B样条曲线拟合方法,建立了飞机气动特性数据的数学模型.B样条曲线可对复杂曲线进行拟合,但数据点的参数选择影响着拟合的精度.参数选择通常有两种方法,一种是固定B样条基的节点选择参数,另一种是选定参数然后变化B样条基的节点.采用遗传算法将二者统一考虑,同时变化节点与参数使得拟合误差最小.通过气动特性数据拟合验证了该方法的有效性、精确性和鲁棒性.     

排列模式索引调制正交频分复用系统

多模索引调制正交频分复用系统(MM-OFDM-IM)在索引调制正交频分复用系统的基础上采用不同星座集对系统中的全部子载波进行索引调制,能有效地提高系统的子载波利用率和频谱效率。但全部子载波的利用影响了系统的子载波间抗干扰能力,导致误码率性能下降。针对这一问题,该文提出排列模式索引调制正交频分复用系统(PM-OFDM-IM)。该系统在MM-OFDM-IM的基础上重新引入静默子载波,既能保证系统较高的频谱效率,又能提高系统的误码率性能。同时该文提出一种基于幅值相移键控的分类映射模式,即按半径大小排列的星座集分类模式(PCC-R),该模式能够良好结合系统传输的额外信息。最后仿真结果验证,该系统能够更优地均衡系统的频谱效率和误码率性能,且所提分类映射方案可以达到更优的系统性能。