On updating signal subspaces

The authors develop an algorithm for adaptively estimating the noise subspace of a data matrix, as is required in signal processing applications employing the `signal subspace' approach. The noise subspace is estimated using a rank-revealing QR factorization instead of the more expensive singular value or eigenvalue decompositions. Using incremental condition estimation to monitor the smallest singular values of triangular matrices, the authors can update the rank-revealing triangular factorization inexpensively when new rows are added and old rows are deleted. Experiments demonstrate that the new approach usually requires O(n²) work to update an n×n matrix, and that it accurately tracks the noise subspace     

Digital differentiation of noisy data measured through a dynamicsystem

The design of discrete time differentiating filters, in the presence of colored noise and nonneglectable transducer dynamics, is investigated. The signal and noise are described by autoregressive moving average (ARMA) models, possibly with poles on |z|=1. The mean square error (MSE) optimal filter, based on a discrete time approximation of the derivative operator, is given by a spectral factorization and a linear polynomial equation     

A new extrapolation algorithm for band-limited signals using theregularization method

The ill-posedness of the extrapolation problem in the presence of noise is considered. A stable algorithm is constructed by solving a Fredholm equation based on a regularization method. The algorithm appears relatively robust, since the noise η_δ(t ) is taken as a function in L²[-T,T](T>0) such that the error energy ∫|η_δ(t)|² dt⩽δ², where integration is from - T to T, and the constructed extrapolation uniformly converges to the desired signal over (-∞, +∞) as δ→0. An estimate for the error energy of the constructed extrapolation over (-∞, +∞) and for the absolute error between the constructed extrapolation and the desired signal over (-∞, +∞) are presented     

Maximizing the output energy of a linear channel with a time- andamplitude-limited input

The problem of maximizing the output energy of a linear time-invariant channel, given that the input signal is time and amplitude limited, is considered. It is shown that a necessary condition for an input μ to be optimal, assuming a unity amplitude constraint is that it satisfy the fixed-point equation=sgn [F(μ)], where the functional F is the convolution of μ with the autocorrelation function of the channel impulse response. It is also shown that all solutions to this equation for which |μ|=1 almost everywhere correspond to local maxima of the output energy. Iteratively recomputing μ from the fixed-point equation leads to an algorithm for finding local optima. Numerical results are given for the cases where the transfer function is ideal low-pass and has two poles. These results support the conjecture that in the ideal low-pass case the optimal input signal is a single square pulse. A generalization of the preceding fixed-point condition is also derived for the problem of maximally separating N outputs of a discrete-time, linear, time-invariant channel     

Error probability of M-ary FSK with differential phasedetection in satellite mobile channel

Formulas are derived for the error probability of M-ary frequency shift keying (FSK) with differential phase detection in a satellite mobile channel. The received signal in this channel is composed of a specular signal, a diffuse signal, and white Gaussian noise; hence, the composite signal is fading and has a Rician envelope. The error probability is shown to depend on the following system parameters: (1) the signal-to-noise ratio; (2) the ratio of powers in the specular and diffuse signal components; (3) the normalized frequency deviation; (4) the normalized Doppler frequency; (5) the maximum normalized Doppler frequency; (6) the correlation function of the diffuse component, which depends on the normalized Doppler frequency and the type of the antenna; (7) the number of symbols; and (8) the normalized time delay between the specular and diffuse component (t _d/T) where 1/T is the symbol rate. Except for T_d/T, all normalized parameters are the ratios of the parameter value and symbol rate. The Doppler frequency depends on the velocity of the vehicle and the carrier frequency. The error probability is computed as a function of the various parameters. The bit error probability is plotted as a function of signal-to-noise ratio per bit and other system parameters     

On the design of maximally sparse beamforming arrays

The problem of designing nonuniformly spaced arrays is formulated as one of constrained optimization in which the cost function is chosen to select the array with the minimum number of elements. The response of the array is controlled by a set of inequality point response constraints. It is shown that a suitable cost function for this problem is the l_p quasi-norm for 0<p<1 and that there exists a number p₁ such that for all 0<p<p₁ the resulting array is maximally sparse. Furthermore it is shown that a solution to the problem lies at an extreme point of the simplex formed by the point constraints. A simplex search algorithm is described which will converge to a local minimum of the cost function on this simplex. The algorithm is applied to the design of sparse linear and planar arrays     

Solution and applications of the Lyapunov equation for controlsystems

Recent advances in control systems analysis and design have implied new uses for the Lyapunov equation of the form AX+XA^T+Q=0. Implementation requirements for the incorporation of the use of Lyapunov equations in practical design, however, point out the need for a set of specialized numerical procedures. This special set of numerical procedures must efficiently solve large, sparse Lyapunov equations, solve sets of Lyapunov equations that differ only in the coefficient matrix Q, and provide good low rank estimates of the Lyapunov equation solution X in the case where low rank approximations are applicable. Discussions of the motivations for the solution of these problems and of candidate solution approaches are provided     

Statistical properties of timing jitter due to data echo in digitalmodem receivers

Consideration is given to the influence of the noise and data sequences present in the received data signal on a nondecision-aided timing recovery scheme in digital modem receivers, It is known that white noise is not particularly disturbing for timing recovery, whereas data signals such as local echo (or residual echo) introduce a bias, called jitter, into the sampling recovered phase. It is shown that when the disturbing data signal has power P less than the power S of the useful signal (whose timing must be recovered), the jitter is sinusoidal with amplitude proportional to the ratio P/S. In the opposite situation, the bias increases indefinitely with time     

Phase-difference distribution for a narrow-band non-Gaussian noiseand related statistics

A probability density function _Pm(R₁,R₂,Δ) is presented for a narrowband noise process in which R₁ and R₂ are two envelope samples and Δ is the phase difference. For m=1 the process is Gaussian, but for m=2,3, etc., it is non-Gaussian. New second-order statistical properties are identified for it as well as the density function for the resulting envelope when a signal is added to the noise. These results are given, though the major concern is with the density of the phase difference Δ and the density of &thetas;, the response of an FM detector fed with the noise     

Roundoff error analysis of the discrete Wigner distribution usingfixed-point arithmetic

The issue of roundoff noise effects in the implementation of the discrete Wigner distribution using fixed-point arithmetic is addressed. The sign-magnitude number representation is assumed throughout the analysis. The measure of roundoff noise effects in an algorithm is the output noise-to-signal ratio. Using a statistical model, an analytical expression of the noise-to-signal ratio is derived as a function of the wordlength b and the transform length N. The noise-to-signal ratio is obtained by evaluating the signal and noise powers at different points in the algorithm, then reflecting to the output both signal and noise powers. Based on the derived noise-to-signal ratio is is noted that if the transform length is doubled, then) one additional bit is required in the wordlength to maintain a constant noise-to-signal ratio. It is demonstrated through the software simulations that the predicted noise-to-signal ratio is a good closed-form estimate of the `true' roundoff error. It is also found from the simulation that the wordlength b and the transform length N=2^v must satisfy the condition b- v⩾4     

N-delta and differential average signal processors:detailing of their signal and noise response

A detailed and complete derivation of the impulse response (and transfer function) for the most commonly used charge-coupled-device (CCD) output signal processing techniques is presented. A basic mathematical model is discussed for 2-δ double sampling and is then generalized to n-sample systems. The method is adapted to show how the impulse response for the differential averaging technique is derived. The relationship between n-δ and differential average is illustrated mathematically. Simple signal gain derivations using the previously derived impulse responses are given. White, 1/f , and 1/f² noise power are discussed in light of the derivation. Graphs illustrating the noise dependence of both n-δ and differential average processor types are included     

Theory of linewidth for multielectrode laser diodes with spatiallydistributed noise sources

A general theory of linewidth for single-frequency semiconductor lasers is presented. The effects of spatially distributed noise sources together with spatially varying carrier and photon densities and injection current are analyzed in a rigorous manner by solution of the scalar wave equation. A new rate equation for the electric field is derived, in which the longitudinal effects are represented in the form of the weight functions C_N(z) and C_S(z). These functions express the sensitivity of the (output) field to local changes in carrier and photon density at the position z. For Fabry-Perot laser's the z dependence of the C factors is shown to be negligible, in agreement with the fact that spatial hole burning is not considered to be important for Fabry-Perot lasers. For distributed-feedback (DFB) lasers, however, the z dependence is shown to be very significant     

Maximum likelihood estimation of the parameters of discretefractionally differenced Gaussian noise process

A maximum-likelihood estimation procedure is constructed for estimating the parameters of discrete fractionally differenced Gaussian noise from an observation set of finite size N. The procedure does not involve the computation of any matrix inverse or determinant. It requires N²/2+O(N) operations. The expected value of the loglikelihood function for estimating the parameter d of fractionally differenced Gaussian noise (which corresponds to a parameter of the equivalent continuous-time fractional Brownian motion related to its fractal dimension) is shown to have a unique maximum that occurs at the true value of d. A Cramer-Rao bound on the variance of any unbiased estimate of d obtained from a finite-sized observation set is derived. It is shown experimentally that the maximum-likelihood estimate of d is unbiased and efficient when finite-size data sets are used in the estimation procedure. The proposed procedure is extended to deal with noisy observations of discrete fractionally differenced Gaussian noise     

Convergence of filters with applications to the Kalman-Bucy case

For each N, and each fixed time T, a signal X^N and a `noisy' observation Y^N are defined by a pair of stochastic difference equations. Under certain conditions (X^N, Y^N) converges in distribution to (X, Y, where dX(t)= f(t, X(t))dt+dV( t), dY(t)=g(t, X( t))dt+dW(t). Conditions are found under which convergence in distribution of the conditional expectations E{F(X^N)|Y^N} to E{F(X)|Y} follows, for every bounded continuous function F. The case in which the conditional expectations still converge but the limit is not E{ F(X)|Y} is also studied. In the situation where f and g are linear functions of X, an examination of this limit leads to a Kalman-Bucy-type estimate of X ^N which is asymptotically optimal and is an improvement on the usual Kalman-Bucy estimate     

Polyspectrum modeling using linear or quadratic filters

The polyspectrum modeling problem using linear or quadratic filters is investigated. In the linear case, it is shown that, if the output pth-order cumulant function of a filter, driven by a white noise, is of finite extent, then the filter necessarily has a finite-extent impulse response. It is proved that every factorable polyspectrum with a non-Gaussian white noise can also be modeled with a quadratic filter driven by a Gaussian white noise. It is shown that, if the quadratic filter has a finite-extent impulse response, then the output pth-order cumulant function is of finite extent; and if the output pth-order cumulant function of a quadratic filter is of finite extent, then the impulse response may or may not be of finite extent. It is shown that there exist finite and infinite extent p th-order cumulant functions that are not factorable but can be modeled with quadratic filters     

Optimization of signal sets for partial-response channels. II.Asymptotic coding gain

For Pt. I see ibid., vol.37, no.5, p.1327-141 (1991). For a linear, time-invariant, discrete-time channel with a given transfer function H(f), and information rate R bits/ T, where T is the symbol interval, an optimal signal set of length K is defined to be a set of 2^RK inputs of length K that maximizes the minimum l₂ distance between pairs of outputs. The author studies the minimum distance between outputs, or equivalently, the coding gain of optimal signal sets as K→∞. He shows how to estimate the coding gain, relative to single-step detection, of an optimal signal set length K when K is large     

Bounding the capacity of saturation recording: the Lorentz modeland applications

The authors consider the problem of bounding the information capacity of saturation recording. The superposition channel with additive Gaussian noise is used as a model for recording. This model says that for a saturation input signal, x(t) (i.e., one that can assume only one of two levels), the output can be expressed as y(t)=x˜(t)+z(t ) where x˜(t) is a filtered version of the input x(t) and z(t) is additive Gaussian noise. The channel is described by the impulse response of the channel filter, h(t), and by the autocorrelation function of the noise. A specific example of such a channel is the differentiated Lorentz channel. Certain autocorrelation and spectrum expressions for a general Lorentz channel are derived. Upper and lower bounds on the capacity of saturation recording channels are described. The bounds are explicitly computed for the differentiated Lorentz channel model. Finally, it is indicated how the derived bounds can be applied in practice using physical measurements from a recording channel     

1/f frequency noise effects on self-heterodyne linewidthmeasurements

The effects of a 1/f frequency noise on self-heterodyne detection are described, and the results are applied to the problem of laser diode linewidth measurement. The self-heterodyne autocorrelation function and power spectrum are evaluated for both the white and the 1/ f components of the frequency noise. From numerical analysis, the power spectrum resulting from the 1/f frequency noise is shown to be approximately Gaussian, and an empirical expression is given for its linewidth. These results are applied to the problem of self-heterodyne linewidth measurements for coherent optical communications, and the amount of broadening due to 1/f frequency noise is predicted     

Properties of polarization noise caused by external disturbances insingle-mode fiber transmission systems

The properties of polarization noise caused by external disturbances in intensity modulation/direct detection transmission systems that have polarization-dependent optical devices are analyzed. It is experimentally shown that the noise consists of additive noise and multiplicative noise. The multiplicative noise directly disturbs signals, even if the external disturbance is of a low frequency. The signal to polarization noise ratio easily reaches the ratio at which flickers obstruct the picture in an analog TV transmission system. The polarization noise caused by external disturbances is also discussed. It is shown that disturbances to the fiber can make the maximum noise n ² times greater than the noise caused by a single disturbance, where n is the number of disturbances     

Exact maximum likelihood time delay estimation for shortobservation intervals

An exact solution is presented to the problem of maximum likelihood time delay estimation for a Gaussian source signal observed at two different locations in the presence of additive, spatially uncorrelated Gaussian white noise. The solution is valid for arbitrarily small observation intervals; that is, the assumption T≫τ _c, |d| made in the derivation of the conventional asymptotic maximum likelihood (AML) time delay estimator (where τ _c is the correlation time of the various random processes involved and d is the differential time delay) is relaxed. The resulting exact maximum likelihood (EML) instrumentation is shown to consist of a finite-time delay-and-sum beamformer, followed by a quadratic postprocessor based on the eigenvalues and eigenfunctions of a one-dimensional integral equation with nonconstant weight. The solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities. Finally, the performance of the EML and AML estimators is compared by means of computer simulations