Asymptotically optimal classification for multiple tests withempirically observed statistics

The decision problem of testing M hypotheses when the source is Kth-order Markov and there are M (or fewer) training sequences of length N and a single test sequence of length n is considered. K, M, n, N are all given. It is shown what the requirements are on M , n, N to achieve vanishing (exponential) error probabilities and how to determine or bound the exponent. A likelihood ratio test that is allowed to produce a no-match decision is shown to provide asymptotically optimal error probabilities and minimum no-match decisions. As an important serial case, the binary hypotheses problem without rejection is discussed. It is shown that, for this configuration, only one training sequence is needed to achieve an asymptotically optimal test     

On the Hamming distance properties of group codes

Under certain mild conditions, the minimum Hamming distance D of an (N, K, D) group code C over a non-abelian group G is bounded by D⩽N -2K+2 if K⩽N/2, and is equal to 1 if K>N/2. Consequently, there exists no (N, K, N-K+1) group code C over an non-abelian group G if 1<K<N. Moreover, any normal code C with a non-abelian output space has minimum Hamming distance equal to D=1. These results follow from the fact that non-abelian groups have nontrivial commutator subgroups. Finally, if C is an (N, K, D) group code over an abelian group G that is not elementary abelian, then there exists an (N, K, D) group code over a smaller elementary abelian group G'. Thus, a group code over a general group G cannot have better parameters than a conventional linear code over a field of the same size as G     

Improved sphere bounds on the covering radius of codes

The sphere bound is a trivial lower bound on K(n,R), the minimal cardinality of any binary code of length n and with covering radius R. By simple arguments it is considerably improved, to K(n,1)⩾2 ⁿ/n for n even. A table of lower and upper bounds on K(n,R) for n⩽33, R ⩽10 is included     

On a very tight truncation error bound for stationary stochasticprocesses

A very tight truncation error upper bound is established for bandlimited weakly stationary stochastic processes if the sampling interval is closed. In particular, the magnitude of the upper bound is O(N^-2q ln² N) for a symmetric sampling reconstruction from 2N+1 sampled values, where q is an arbitrary positive integer. The results are derived with the help of the Bernstein bound on the remainder of a symmetric complex Fourier series of the function exp (iλ t). Convergence rates are given for mean square and almost sure sampling reconstructions     

Optimization of signal sets for partial-response channels. I.Numerical techniques

Given a linear, time-invariant, discrete-time channel, the problem of constructing N input signals of finite length K that maximize minimum l₂ distance between pairs of outputs is considered. Two constraints on the input signals are considered: a power constraint on each of the N inputs (hard constraint) and an average power constraint over the entire set of inputs (soft constraint). The hard constraint, problem is equivalent to packing N points in an ellipsoid in min(K,N-1) dimensions to maximize the minimum Euclidean distance between pairs of points. Gradient-based numerical algorithms and a constructive technique based on dense lattices are used to find locally optimal solutions to the preceding signal design problems. Two numerical examples are shown for which the average spectrum of an optimized signal set resembles the water pouring spectrum that achieves Shannon capacity, assuming additive white Gaussian noise     

A linear bound for sliding-block decoder window size

The author gives an upper bound on the necessary length of a sliding-block decoder window for finite-state codes from arbitrary n -ary data into any constrained system Σ with capacity at least log(n) presented by a graph G with memory m and anticipation a. Specifically, it is shown that the ACH code construction algorithm can be used to construct a code with a sliding-block decoder at rate t:t and with window length m+a+2t, where t is upper-bounded by a linear function of the number of states of G. It is demonstrated that this is the best one can do in the sense that any general upper bound on the decoder window length for finite-state codes into systems Σ with finite memory must grow at least linearly with the number of states of the graph G presenting Σ     

Modulatable orthogonal sequences and their application to SSMAsystems

A set of N-1 orthogonal sequences of period N ² is proposed, where N is a natural number. Each orthogonal sequence proposed can be modulated by N complex numbers of absolute value 1, so the modulated sequence is also orthogonal. When N is an odd prime number, the absolute value of the cross-correlation function between any two of the N-1 orthogonal sequences is constant and satisfies the mathematical lower bound. This property of the cross-correlation function is not changed when each of the two orthogonal sequences is modulated by N complex numbers of absolute value 1. Two spread-spectrum multiple-access (SSMA) systems using these sequences are proposed. One system is an asynchronous SSMA system, using the proposed sequences unmodulated. The cochannel interference peak between any two channels in this system realizes the mathematical lower bound for an asynchronous SSMA system using a set of orthogonal sequences. The other system is a synchronous SSMA system without cochannel interference which uses the modulated form of the proposed sequences     

Full Newton and constraint methods for semilinear signal modeling

The authors derive an expression for the Hessian matrix of the variable projection functional (VPF) and implement the Hessian using QR factorization. This is incorporated into a full Newton variable projection (FNVP) algorithm for estimating parameters in semilinear signals. They introduce a deflation technique for constraining the VPF to contain known basis vectors. For modeling exponential signals, the computational cost of the FNVP algorithm is shown to vary linearly with N, the size of the data vector, while other algorithms vary as N² or N log ₂ N     

An efficient solution for the response of electrical well loggingtools in a complex environment

A symmetrical form of the solution for an electrical source in a multibed well-logging environment is derived. The method uses local reflection and transmission operators of a single-bed boundary and a general recursive algorithm to derive generalized reflection and transmission operators. Using this method, the computation time scales linearly as N, where N is the number of beds in the environment. A computer program was developed to implement the solution. The program is robust and generates accurate results from 20 kHz to 25 MHz     

Bounds on maximum throughput for digital communications withfinite-precision and amplitude constraints

The problem of finding the maximum achievable data rate over a linear time-invariant channel is considered under constraints different from those typically assumed. The limiting factor is taken to be the accuracy with which the receiver can measure the channel output. More precisely, the following problem is considered. Given a channel with known impulse response h(t), a transmitter with an output amplitude constraint, and a receiver that can distinguish between two signals only if they are separated in amplitude at some time t ₀ by at least some small positive constant d, what is the maximum number of messages, N_max, that can be transmitted in a given time interval [0,T]? Lower bounds on N_max can be easily computed by constructing a particular set of inputs to the channel. The main result is an upper bound on N_max for arbitrary h(t). The upper bound depends on the spread of h(t), which is the maximum range of values the channel output may take at some time t₀>0 given that the output takes on a particular value α at time t=0. Numerical results are shown for different impulse responses, including two simulated telephone subscriber loop impulse responses     

Estimating the information content of symbol sequences andefficient codes

Several variants of an algorithm for estimating Shannon entropies of symbol sequences are presented. They are all related to the Lempel-Ziv algorithm (1976, 1977) and to recent algorithms for estimating Hausdorff dimensions. The average storage and running times increase as N and Nlog N, respectively, with the sequence length N. These algorithms proceed basically by constructing efficient codes. They seem to be the optimal algorithms for sequences with strong long-range correlations, e.g. natural languages. An application to written English illustrates their use     

Message delay and queue-size analysis for circuit-switched TDMAsystems

A multiple-access communications channel which is shared among network stations using a circuit-switched time-division multiple-access (CS-TDMA) scheme is examined. Each station is allocated a fixed number of slots (N) during each frame. The authors carry out queue-size message delay analysis for CS-TDMA systems. They derive the generating function of the queue size and of the waiting time distribution for a discrete-time Geom^[x]/Geom/N queuing system. This result is used to obtain the generating function of the system size for the CS-TDMA scheme. The associated computation requires the solution of (N+1)2^N linear equations. To derive a more computationally effective procedure, tight lower and upper bounds are obtained, requiring the solution of at most 3N linear equations. The authors prove that a slot allocation scheme which distributes station slots uniformly over the frame yields a message delay lower bound. The application of the results to the analysis of demand-assigned CS-TDMA systems is also discussed     

K-band quasi-planar tapped combline filter and diplexer

Quasi-planar realizations of a combline bandpass filter and diplexer using multiple coupled suspended substrate striplines (MCSSSs) have demonstrated good performance at K-band without any tuning. The N MCSSSs excite N zero-cutoff-frequency quasi-TEM modes. A computer-aided filter design approach employing a rigorous spectral domain approach and 2N-port microwave circuit theory accounts for the effects of the N quasi-TEM modes, the couplings through nonadjacent MCSSSs, and cover height. Two 19.5-20.5 GHz MCSSS combline filters with different cover heights have been built and tested to compare their filter characteristics. The reduction in cover height has been found to decrease the amount of nonadjacent coupling through MCSSSs and to result in better filter stopband performance. An 18.5-19 GHz and 20-20.5 GHz MCSSS diplexer is also presented. All the measured results for the combline filters and diplexers agree well with the theoretic calculations     

Upper bounds on capacity for a constrained Gaussian channel

A low-pass and a bandpass additive white Gaussian noise channel with a peak-power constraint imposed on otherwise arbitrary input signals are considered. Upper bounds on the capacity of such channels are derived. They are strictly less than the capacity of the channel when the peak-power constrain is removed and replaced by the average-power constraint, for which the Gaussian inputs are optimum. This provides the answer to an often-posed question: peak-power limiting in the case of bandlimited channels does reduce capacity, whereas in infinite bandwidth channels it does not, as is well known. For an ideal low-pass filter of bandwidth B, the upper bound is Blog 0.934P/(N₀B) for P/( N₀B)≫1, where P is the peak power of the input signal and N₀/2 is the double-sided power spectral density of the additive white Gaussian noise     

Fast K-dimensional tree algorithms for nearest neighborsearch with application to vector quantization encoding

Fast search algorithms are proposed and studied for vector quantization encoding using the K-dimensional (K-d) tree structure. Here, the emphasis is on the optimal design of the K -d tree for efficient nearest neighbor search in multidimensional space under a bucket-Voronoi intersection search framework. Efficient optimization criteria and procedures are proposed for designing the K-d tree, for the case when the test data distribution is available (as in vector quantization application in the form of training data) as well as for the case when the test data distribution is not available and only the Voronoi intersection information is to be used. The criteria and bucket-Voronoi intersection search procedure are studied in the context of vector quantization encoding of speech waveform. They are empirically observed to achieve constant search complexity for O(log N) tree depths and are found to be more efficient in reducing the search complexity. A geometric interpretation is given for the maximum product criterion, explaining reasons for its inefficiency with respect to the optimization criteria     

Fast cosine transform of Toeplitz matrices, algorithm andapplications

A fast algorithm for the discrete cosine transform (DCT) of a Toeplitz matrix of order N is derived. Only O(N log N)+O(M) time is needed for the computation of M elements. The storage requirement is O(N). The method carries over to other transforms (DFT, DST) and to Hankel or circulant matrices. Some applications of the algorithm are discussed     

Electromagnetic scattering of finite strip array on a dielectricslab

A feast recursive algorithm is used to compute the scattering properties of a finite array of strip gratings on a dielectric slab. this algorithm has a computational complexity of O(N log² N) for one incident angle and O(N ² log N) for N incident angles. It uses plane wave basis for expanding the incident wave and the scattered wave. The scattered wave is expanded in terms of a Sommerfeld-type integral with spectral distribution along a vertical branch cut, rendering its expansion very efficient. To validate the scattering solution obtained using the recursive algorithm, comparisons with the method of moments are illustrated. The current distributions on the strips and scattering patterns are both presented. Since this algorithm has reduced computational complexity and is fast compared to other conventional methods, it can be used to analyze very large strip arrays. Scattering solution of a 50-wavelength wide strip is illustrated     

A convolutional single-parity-check concatenated coding scheme forhigh-data-rate applications

The coding scheme uses a set of n convolutional codes multiplexed into an inner code and a (n,n-1) single-parity-check code serving as the outer code. Each of the inner convolutional codes is decoded independently, with maximum-likelihood decoding being achieved using n parallel implementations of the Viterbi algorithm. The Viterbi decoding is followed by additional outer soft-decision single-parity-check decoding. Considering n=12 and the set of short constraint length K=3, rate 1/2 convolutional codes, it is shown that the performance of the concatenated scheme is comparable to the performance of the constraint length K=7, rate 1/2 convolutional code with standard soft-decision Viterbi decoding. Simulation results are presented for the K=3, rate 1/2 as well as for the punctured K=3, rate 2/3 and rate 3/4 inner convolutional codes. The performance of the proposed concatenated scheme using a set of K=7, rate 1/2 inner convolutional codes is given     

In the optimum design of the block adaptive FIR digital filter

A general optimum block adaptive (GOBA) algorithm for adaptive FIR (finite impulse response) filtering is presented. In this algorithm, the correction terms for the filter coefficients in each block, instead of the convergence factors, are optimized in a least squares sense. There are no constraints on the block length L and the filter tap number N. It is shown that the GOBA algorithm is reduced to the normalized LMS algorithm when L⩾N. The convergence of the GOBA algorithm can be assured if the correlation matrix of the input signal is positive definite. Computer simulations based on an efficient computing procedure confirm that the GOBA algorithm achieves faster convergence with slightly degraded convergence accuracy in stationary environments and better weight tracking capability in nonstationary environments as compared to existing block adaptive algorithms with no constraints on L and N