The problem of signal design for bandwidth-efficient multiple access (BEMA) over additive white Gaussian noise (AWGN) channels is addressed under quality of service (QoS) requirements specified by asymptotic effective energies (AEEs). The AEE characterizes the bit error rate (BER) in the low-noise regime, but in contrast to BER, it is tractable and amenable to analysis and signal design. We adopt the BEMA strategy of bandwidth conservation where users are detected successively using minimum mean-squared error decision feedback (MMSE-DF) detection and where signals are designed in a greedy fashion for one user at a time, in the reverse order in which the users are detected. The signal design method proposed here is based on an exact characterization of how a signal update for one user affects the issue of preserving bandwidth with the addition of signals for subsequent users. A geometric insight in the construction of good signal sets and significant improvements in bandwidth over full-rank or orthogonal signaling are obtained. The main result of this paper can hence be seen as providing a tight upper bound on the minimum signature sequence dimension or rank (and hence bandwidth) needed to satisfy individual, possibly distinct user QoS constraints specified in terms of the AEE measure.

This correspondence highlights the performance limitation of linear minimum mean-squared error (mmse) detection in underdetermined vector Gaussian channels (as in overloaded code-division multiple-access (CDMA) systems) where the number of symbols (users) exceeds the signal space dimension (spread factor). It is shown that for such a simple receiver it is not possible to construct signal sets (or spreading codes) to even satisfy the basic requirement that every user's symbol error probability decays exponentially as noise power vanishes. This result holds for arbitrary received energies, modulation schemes, and any strictly underdetermined system with a finite signal space dimension and a finite number of users

This correspondence characterizes the performance limitation of the well-acclaimed generalized Welch-bound equality (WBE) signals under linear minimum mean-squared error (MMSE) detection. It is analytically proven, and experimentally verified, that when users in an overloaded but finite-size code-division multiple-access (CDMA) system are allocated such signals, the error rate of at least one user "floors" (i.e., it cannot be driven to zero even in the absence of additive noise), independently of the symbol energies. If, in addition, all users are received with equal energy, then the error rate of every user "floors".

Nonstationary complex random signals are in general improper (not circularly symmetric), which means that their complementary covariance is nonzero. Since the Karhunen-Loeve (K-L) expansion in its known form is only valid for proper processes, we derive the improper version of this expansion. It produces two sets of eigenvalues and improper observable coordinates. We then use the K-L expansion to solve the problems of detection and estimation of improper complex random signals in additive white Gaussian noise. We derive a general result comparing the performance of conventional processing, which ignores complementary covariances, with processing that takes these into account. In particular, for the detection and estimation problems considered, we find that the performance gain, as measured by deflection and mean-squared error (MSE), respectively, can be as large as a factor of 2. In a communications example, we show how this finding generalizes the result that coherent processing enjoys a 3-dB gain over noncoherent processing.

Some quadratic time-frequency representations (TFRs) may be called time-varying spectrum estimators. They are derived from first principles, and they turn out to be time-varying multiwindow spectrum estimators. In special cases they are time-varying spectrograms that may be written as Fourier transforms of lag-windowed, time-varying correlation sequences or as spectrally smoothed time-varying periodograms. These are not ad-hoc variations on stationary ideas to accommodate time variation. Rather, they are the only variations one can obtain for time-varying spectrum analysis.

Knowledge of the power spectrum of a stationary random sequence can be used for quantizing the signal efficiently and with minimum mean-squared error. A multichannel filter is used to transform the random sequence into an intermediate set of variables that are quantized using independent scalar quantizers, and then inverse-filtered, producing a quantized version of the original sequence. Equal word-length and optimal word-length quantization at high bit rates is considered. An analytical solution for the filter that minimizes the mean-squared quantization error is obtained in terms of its singular value decomposition. The performance is characterized by a set of invariants termed second-order modes, which are derived from the eigenvalue decomposition of the matrix-valued power spectrum. A more general rank-reduced model is used for decreasing distortion by introducing bias. The results are specialized to the case when the vector-valued time series is obtained from a scalar random sequence, which gives rise to a filter bank model for quantization. The asymptotic performance of such a subband coder is derived and shown to coincide with the asymptotic bound for transform coding. Quantization employing a single scalar pre- and postfilter, traditional transform coding using a square linear transformation, and subband coding in filter banks, arise as special cases of the structure analyzed here.

A general representation is presented for quadratic estimators of the far-field frequency-wavenumber spectrum. The quadratic estimators are constructed from linear transformations of the array data. These linear transformations contain the windows normally associated with multiple window spectral estimators and act as multidimensional baseband beamformers. It is the selection of these windows that is most important in the design of quadratic spectral estimators. A discussion is presented of the two dimensional periodogram and the two dimensional Thomson (1982) estimator as prototypical examples of quadratic estimators.>

A simple linear procedure is given to compute the cross-variance sequence associated with the outputs of two rational digital filters that are driven by the same white noise sequence. Such a computation appears in the study of digital filters, Wiener filters, noise-variance estimation, low order approximations, and in the study of multichannel systems. Fast algorithms based on Euclid's algorithm are introduced to solve the linear systems of equations involved. The special case of the autocovariance computation is reviewed. Alternate polynomial representations are shown to share the same properties as the matrix equations.>

The inverse eigenvalue problem for real symmetric Toeplitz matrices is defined. A Newton-Raphson-type algorithm is developed for the solution of the problem. The algorithm converges unsafeguarded in all the computed cases and shows the typical behavior of Newton-type algorithms: in general quadratic convergence, linear convergence near double roots. Examples of dimension 10 and 20 are presented. Known sufficient conditions for inverse eigenvalue problems of real symmetric matrices are discussed.>

A real symmetric polynomial Q(z) can be factored into the product A(z)A(z/sup -1/) if Q(z) is nonnegative on the unit circle. The authors pose a constrained minimization problem that results in the correct factorization in this case and gives an approximation to Q(z) if Q(z) does not satisfy the nonnegativity condition.>

The development of VLSI has changed the important parameters in signal processing algorithms and structures. Technology developments are dictating new and different criteria for "efficient" realizations. In VLSI the number of computations performed per output sample is not enough to measure the "goodness" of a realization/implementation. This work is an attempt to quantify the important parameters in VLSI implementations for DSP problems and apply the theory to the task of digital filtering.

An orthogonal filter structure is presented, whose synthesis is based on a generalized Levinson algorithm. This structure has a ladder-like flow graph topology. It is modular and has a high degree of computational parallelism. Implementation guidelines in the framework of parallel data processing are considered. A closed form for the round off noise error variance is given, from which it appears that the structure will be particularly useful for the implementation of narrow bandwidth filters.

Good digital filter realizations for hardware implementation in narrowband filtering applications should have low roundoff noise, low coefficient sensitivity, and freedom from overflow oscillations. The hardware implementations presented here incorporate the above properties into a highly modular structure which can perform computations in pipeline fashion. That is, after an initial delay an output sample is obtained for each input sample. The realization and implementations discussed here contain more multipliers than direct form realizations. However, by using shorter word lengths because of their increased performance and distributed arithmetic implementations (instead of multiplier structures) these implementations can have less total hardware complexity.

Spectral pole-zero models are used as a method of reducing the number of parameters needed to represent certain classes of spectra, e.g., spectral envelopes having deep valleys. We present here an iterative algorithm for Autoregressive-Moving Average (ARMA) models that is based on a repeated use of Levinson's algorithm to obtain both poles and zeros. The procedure allows one to increase the number of poles and zeros iteretively, similar to linear prediction methods. Empirical results indicate that a slow growth of the orders of the denominator and numerator polynomials usually produces better ARMA models.

Normal realizations of IIR digital filters possess many desirable properties including freedom from overflow oscillations, low coefficient sensitivity, and invariance of form and roundoff noise under frequency transformations of the filter. Normal realizations are closely related to minimal roundoff noise structures. Given a state variable model {A,b,c,d} of a filter, we define a normal structure as a filter whose A matrix satisfies AAT=ATA. This paper develops design equations and roundoff noise equations for normal and minimal roundoff noise filters in terms of the poles and zeros of the filter transfer function. The development has geometric interpretations which yield additional insights into low roundoff noise structures.

Most of the literature dealing with overflow oscillations in fixed-point digital filters has considered the direct form realization exclusively. In direct forms the question of whether oscillations are possible depends on the location of the poles of the filter. We show that it is possible to eliminate overflow oscillations, regardless of pole locations, by considering more general realizations. A sufficient condition is given for a state variable realization (using two's complement arithmetic) of any order to be free of overflow oscillation. A simple characterization of this condition is given for second order filters. Among second order realizations which meet the condition are so-called normal realizations and structures which minimize roundoff noise.

We have outlined a synthesis procedure which minimizes the output noise in cascade and parallel connections of second-order sections caused by roundoff of internal storage registers. Theorem 1 characterizes the optimal fi Iter and dictates the synthesis procedure. The output noise is made up of two terms. The dominant noise term for block optimal fi lters is independent of bandwidth unl ike the usual cascade and paralle I connections of second-order di rect forms. For narrow band filters improvements of 30 to 40 db are not uncommon. Numerical results yield excellent agreement between theory and simulation.

A formulation of finite-memory information processing problems is presented. The total state space of the system, including the "memory" of the source and processor, is assumed to be finite. A cost functional is specified over the trajectories of the system and a variational approach is used to minimize cost. There results a two-point boundary value problem and an associated improvement algorithm. Special attention is given to two types of cost functionals: finite-time problems, and time-average problems over an infinite time interval. Several examples are included.

The problem of sequentially deciding upon one ofmmutually exclusive and exhaustive hypotheses has been studied in detail for the case ofm = 2. This correspondence presents the structure for the Bayesm-ary sequential detector. Computational results depend on a property called subspace separability, which allows one to reduce the dimensionality of the essential averaging process. Numerical results are presented for the case of Gaussian observation statistics andm = 3.