Fast evaluation of an integral occurring in digital filteringapplications

A fast evaluation procedure for the integral I_m,n,p=1/2πj∯_|z|=1H_m,n(z)H _m,n(z^-1)z^p-1dz for arbitrary nonnegative integer-valued m, n, and p, is presented, where H_m,n (z)=Σ_k=0^mb_m,kz^-k /Σ_l=0ⁿa_n,lz^-1,a _n,0≠0 is the transfer function of an arbitrary digital filter. Evaluation of this integral frequently appears in control, communication, and digital filtering. A notable result is the one-term recursion on p, for arbitrary but fixed nonnegative integers m and n. The computational complexity is analyzed, and two illustrative examples demonstrate some of the advantages of this approach     

On ternary complementary sequences

A pair of real-valued sequences A=(a₁,a₂,...,a_N) and B=(b₁,b ₂,...,b_N) is called complementary if the sum R(·) of their autocorrelation functions R_A(·) and R_B(·) satisfies R(τ)=R_A(τ)+R _B(τ)=Σ_i=1^N$ -^τa_ia_i+τ+Σ_j=1 ^N-τb_jb_j+τ=0, ∀τ≠0. In this paper we introduce a new family of complementary pairs of sequences over the alphabet α₃=+{1,-1,0}. The inclusion of zero in the alphabet, which may correspond to a pause in transmission, leads both to a better understanding of the conventional binary case, where the alphabet is α₂={+1,-1}, and to new nontrivial constructions over the ternary alphabet α₃. For every length N, we derive restrictions on the location of the zero elements and on the form of the member sequences of the pair. We also derive a bound on the minimum number of zeros necessary for the existence of a complementary pair of length N over α₃. The bound is tight, as it is met by some of the proposed constructions, for infinitely many lengths     

Theoretical analysis of word-level switching activity in thepresence of glitching and correlation

This paper presents a novel analytical approach to compute the switching activity in digital circuits at the word level in the presence of glitching and correlation. The proposed approach makes use of signal statistics such as mean, variance, and autocorrelation. It is shown that the switching activity α_f at the output node f of any arbitrary circuit in the presence of glitching and correlation is computed as α_f=Σ_i=1^S-1α(f _i,i+1)=Σ_i=1^{S- 1}p(f_i+1)(1-p(f_i))(1-ρ(f_i,i+1 )) (1) where ρ(f_i,i+1)=ρ(f_i,i+1)=(E[f_i(Sn)f _i+1(Sn)]- p(f_i)p(f_i+1))/(√(p(f _i)-p(f_i)²)(p(f_i+1)- p(f_i+1²))) (2). S number of time slots in a cycle; ρ(f_i,+1) time-slot autocorrelation coefficient; E[x]=expected value of x; p_x=probability of the signal x being “one”. The switching activity analysis of a signal at the word level is computed by summing the activities of all the individual bits constituting the signal. It is also shown that if the correlation coefficient of the higher order bits of a normally distributed signal x is ρ(x_c), then the bit P₀ where the correlation begins and the correlation coefficient is related hy ρ(x_c)=erfc{(2(P₀-1)-1)/(√2σ_x )} where erfc(x)=complementary error function; σ_x=variance of x. The proposed approach can estimate the switching activity in less than a second which is orders of magnitude faster than simulation-based approaches. Simulation results show that the errors using the proposed approach are about 6.1% on an average and that the approach is well suited even for highly correlated speech and music signals     

Eigenvalues of (↓2)H and convergence of the cascade algorithm

This paper is about the eigenvalues and eigenvectors of (↓2)H. The ordinary FIR filter H is a convolution with a vector h=(h(O),...,h(N)), which is the impulse response. The operator (↓2) downsamples the output y=h*x, keeping the even-numbered components y(2n). Where H is represented by a constant-diagonal matrix, this is a Toeplitz matrix with h(k) on its kth diagonal, the odd-numbered rows are removed in (↓2)H. The result is a double shift between rows, yielding a block Toeplitz matrix with 1×2 blocks. Iteration of the filter is governed by the eigenvalues. If the transfer function H(z)=Σh(k)z^-k has a zero of order p at z=-1, corresponding to ω=π, then (↓2)H has p special eigenvalues ½,¼...,(½)^p. We show how each additional “zero at π” divides all eigenvalues by 2 and creates a new eigenvector for λ=½. This eigenvector solves the dilation equation φ(t)=2Σh(k)φ(2t-k) at the integers t=n. The left eigenvectors show how 1,t,...,t^p-1 can be produced as combinations of φ(t-k). The dilation equation is solved by the cascade algorithm, which is an infinite iteration of M=(↓2)2H. Convergence in L² is governed by the eigenvalues of T=(↓2)2HH^T corresponding to the response 2H(z)H(z^-1 ). We find a simple proof of the necessary and sufficient condition for convergence     

The inductance matrix of a multiconductor transmission line inmultiple magnetic media

A simple relationship between the inductance matrix and the auxiliary capacitance matrix is given. For a multiconductor transmission line consisting of N_c conducting cylinders in inhomogeneous media consisting of N_d homogeneous regions with permeabilities μ_i and permittivities ϵ _i, the inductance matrix [L] for the line is obtained by solving the magnetostatic problem of N_c conductors in N_d regions with permeabilities μ _i. The capacitance matrix [C] for the line is obtained by solving the electrostatic problem of N_c conductors in N_d regions with permittivities ϵ _i. It is shown that [L]=μ₀ϵ₀[C^'] ^-1, where [C^'] is the capacitance matrix of an auxiliary electrostatic problem of N_c conductors in N_d regions with relative permittivities set equal to the reciprocals of the relative permeabilities of the magnetostatic problem, i.e. ϵ^' _i/ϵ₀=μ₀/μ_i     

The length of a typical Huffman codeword

If p_i(i=1,···, N) is the probability of the ith letter of a memoryless source, the length l_i of the corresponding binary Huffman codeword can be very different from the value -log p_i. For a typical letter, however, l_i≈-logp_i. More precisely, P_m ^-=Σ/sub j∈{i|l<-logpj-m}/p_j<2^-m and P_m ⁺=Σ/sub j∈{i|li>-logpi+m/}p_j<2^-c(m-2)+2, where c≈2.27     

Nondegenerate diffraction on transient gratings in InSb

Nondegenerate diffraction in InSb has been investigated. In experiments, the grating is excited by two CO laser beams with the frequency ω₁, and a third beam with a different frequency ω₂ is used to detect the grating. The diffraction efficiency can be represented by the nonlinear susceptibility χ⁽³⁾(ω₂, -ω₂ , ω₁), which can be obtained with the Bloch equation. The frequency dependence given by theory is in agreement with experiments, implying that the nonlinear index is caused by the saturation     

The first-order asymptotic of waiting times with distortion betweenstationary processes

Let X and Y be two independent stationary processes on general metric spaces, with distributions P and Q, respectively. The first-order asymptotic of the waiting time W_n(D) between X and Y, allowing distortion, is established in the presence of one-sided ψ-mixing conditions for Y. With probability one, n^-1log W _n(D) has the same limit as -n^-1logQ(B(X₁ⁿ, D)), where Q(B(X₁ ⁿ, D)) is the Q-measure of the D-ball around (X₁ ,...,X_n), with respect to a given distortion measure. Large deviations techniques are used to get the convergence of -n^-1 log Q(B(X₁ⁿ, D)). First, a sequence of functions R_n in terms of the marginal distributions of X₁ⁿ and Y₁ⁿ as well as D are constructed and demonstrated to converge to a function R(P, Q, D). The functions R_n and R(P, Q, D) are different from rate distortion functions. Then -n^-1logQ(B(X₁ⁿ , D)) is shown to converge to R(P, Q, D) with probability one     

Constellations matched to the Rayleigh fading channel

We introduce a new technique for designing signal sets matched to the Rayleigh fading channel, In particular, we look for n-dimensional (n⩾2) lattices whose structure provides nth-order diversity. Our approach is based on a geometric formulation of the design problem which in turn can be solved by using a number-geometric approach. Specifically, a suitable upper bound on the pairwise error probability makes the design problem tantamount to the determination of what is called a critical lattice of the body S={x=(x₁, ···, x_n)∈Rⁿ, |Π_i=1ⁿx_i|⩽1}. The lattices among which we search for an optimal solution are the standard embeddings in R ⁿ of the number ring of some totally real number field of degree n over Q. Simulation results confirm that this approach yields lattices with considerable coding gains     

Statistical characterization of sample fourth-order cumulants of anoisy complex sinusoidal process

The paper deals with the statistical characterization of sample estimates of the fourth-order cumulants of a random process consisting of multiple complex sinusoids and additive colored Gaussian noise. In particular, it presents necessary and sufficient conditions for strong consistency of the sample cumulants of arbitrary orders, and derives expressions for the asymptotic covariance of the sample estimates of the fourth-order cumulants. It is shown that the fourth-order cumulant C_4y (τ₁,...,τ₄) can be written as a function of a single argument τ=τ₃+τ₄ -τ₁-τ₂, which implies large flexibility in estimating the cumulant. It is recommended that the estimate be based upon lags such that τ₁ is distant from τ₂ and τ₃ is distant from τ₄, and/or as a linear combination of such terms. The asymptotic variance of a cumulant-based frequency estimator is shown to have the form c₂·SNR^-2+c₃·SNR^-3 +c₄·SNR^-4, where the coefficient c ₂ may possibly vanish. The theory is illustrated via numerical examples. The results of this paper will be useful in analyzing the performance of various cumulant-based frequency estimation algorithms     

An acoustooptic phased array antenna beamformer with independentphase and carrier control using single sideband signals

An acoustooptic phase-array antenna beamformer is experimentally demonstrated using single sideband signals with frequencies ω_c+ω₀ and ω_c-ω ₀ to drive the two acoustooptic devices, respectively. Varying ω_c controls the antenna carrier (2ω_c ), while changing ω₀ only effects the phase. Phase control of 0-2π is achieved using 0-340 kHz change in ω ₀, with the carrier fixed at 120 MHz using ω_c=60 MHz. A dynamic range of 66.6 dB at 2 MHz and carrier-to-noise of 126.9 dB/Hz at 2 MHz is measured. This beamformer can provide wide antenna tunable bandwidth and intrapulse beam forming     

Experimental versus theoretical rates for photoassociation and forformation of ultracold molecules

We report on the spectra of the long-range attractive molecular states of the Cs₂ dimer below the dissociation limits 6s+6p _1/2 and 6s+6p_3/2 which are accessible by molecular photoassociation of cold Cs atoms. For the states O_u⁺ and O_g^-(6s+6p_3/2), we have performed trap-loss measurements, which are in good agreement with the theoretical calculations, performed in a perturbative approach. For the O_g^- and 1_u(6s+6p_3/2) states, we observe, after spontaneous decay of the electronically excited molecules, the formation of translationally cold molecules. A rate of formation of cold molecules of the order of ~10⁶ molecules per second is obtained in the case of the state O_g^$     

Optical multiplication using a bisected intersecting waveguide

We present an analysis of a bisected intersecting waveguide in periodically poled lithium niobate that can produce and isolate the second-order optical product E(ω₁+ω₂)~E₁(ω₁ )E₂(ω₂) free from the near degenerate second harmonics. Model calculations of such a device fabricated in quasi-phase matched (QPM) LiNbO₃ predict a conversion efficiency of ≈1.5%/W, about 15% that of a straight waveguide of the same length, and a crosstalk of ≲-30 dB     

Multivariate regression estimation of continuous-time processesfrom sampled data: local polynomial fitting approach

Let (Y,X)={Y(t),X(t),-∞j) be a renewal point processes on (0,∞), with a finite mean rate, independent of (Y,X). We consider the estimation of regression function r(x₀, x₁,...,x_m-1; τ₁,...,τ_m) of ψ(Y(τ_m)) given (X(0)=x₀, X(τ₁)=x₁,...,X(τ_m-1)=_x-1 ) for arbitrary lags 0<τ₁<...< τ_m on the basis of the discrete-time observations {Y(t_j),X(t_j),t_j)_j=1ⁿ . We estimate the regression function and all its partial derivatives up to a total order p⩾1 using high-order local polynomial fitting. We establish the weak consistency of such estimates along with rates of convergence. We also establish the joint asymptotic normality of the estimates for the regression function and all its partial derivatives up to a total order p⩾1 and provide explicit expressions for the bias and covariance matrix (of the asymptotically normal distribution)     

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     

Nondegenerate optical Kerr effect in semiconductors

We calculate the nondegenerate bound electronic nonlinear refractive index n₂(ω₁;ω₂) (i.e., an index change at frequency ω₁ due to the presence of a beam at frequency ω₂) in semiconductors. We calculate this nonlinearity and its dispersion using a Kramers-Kronig transformation on the calculated nondegenerate nonlinear absorption spectrum due to two-photon absorption, electronic Raman and optical Stark effects. The calculated n₂ values and their dispersion are compared to new experimental values for ZnSe and ZnS obtained using a 2-color Z-scan     

Stimulated Raman effect and four-wave mixing in a hollow waveguide

A theoretical and experimental investigation of stimulated Raman scattering and four-wave mixing in a gas-filled hollow waveguide was performed. The four-wave mixing processes, specifically the generation of the third Stokes component at ω₃ when ω₃ =2ω₂-ω₁, show a strong resonance-like behavior as a function of wavelength. These resonances are caused by the phase matching required for efficient four-wave mixing, because the hollow waveguide supports only certain propagation directions leading to the development of waveguide modes. The theoretical results are compared with experimental data for TEM₀₀ and TEM₁₀ fundamental dye-laser beams generating third Stokes wavelengths between 2.6 and 7.5 μm     

Strongly consistent online forecasting of centered Gaussianprocesses

An estimator Eˆ(d_n,n) of the conditional expectation E[X_n+1|X_n,...,X(n-d_n+1)] in a centered, stationary, and ergodic Gaussian process {X_i}_i with absolutely summable Wold coefficients is constructed on the basis of having observed X₁,...,X_n . For a suitable choice of the length dn→∞ (n→∞) of the past covered by the conditional expectation, it is established that |Eˆ(d_n,n)-E[X_n+1|X_n ,...,X(n-d_n+1)]|→0 with probability 1. In addition, sufficient conditions for |E[X_n+1|X_n,X _n-1,...]-E[X_n+1|X_n,...,X(n-d_n +1)]| →0 to hold with probability 1 are given, that is, conditions under which Eˆ(d_n,n) can be used as a strongly consistent forecaster for |E[X_n+1|X_n,X _n-1,...]     

Some new constructions for simplex codes

Three constructions for n-dimensional regular simplex codes α_i, 0⩽i⩽n, are proposed, two of which have the property that α_i for 1⩽i⩽n is a cyclic shift of α₁. The first method is shown to work for all the positive integers n=1,2,... using only three real values. It turns out that these values are rational whenever n+1 is a square of some integer. Whenever a (v,k,λ) cyclic (or Abelian) difference set exists, this method is generalized so that a similar method is shown to work with ν=n (the number of dimensions)     

Relationship between and measurement of differential scatteringcoefficient (σ0) and bidirectional reflectancedistribution function (BRDF)

The microwave and optical scattering formulations are reviewed, a relationship between them at the same wavelength is presented, and the measurement conditions for obtaining range-independent and equipment-independent values are discussed. It is shown that the physical mechanism of scattering of electromagnetic beams from a surface can be quantified by either a differential scattering coefficient (σ⁰) or a bidirectional reflectance distribution function (BRDF). For a diffuse surface, σ⁰=4π BRDF cos &thetas;_i cos &thetas;_r, where &thetas;_i and &thetas;_r are the incident and reflected angles, respectively, from the surface normal. The geometrical constraints inferred in the measurement of coherent electromagnetic wave scattering from relatively small objects are easily satisfied to obtain range- and equipment-independent values, but care in experimental design is required when dealing with large surfaces. More care is also required when dealing with surfaces entailing specular features in addition to diffuse features