Peter F. Swaszek

The good, bad and ugly: distributed detection of a known signal in dependent Gaussian noise

Most results about quantized detection rely strongly on an assumption of independence among random variables. With this assumption removed, little is known. Thus, in this paper, Bayes-optimal binary quantization for the detection of a shift in mean in a pair of dependent Gaussian random variables is studied. This is arguably the simplest meaningful problem one could consider. If results and rules are to be found, they ought to make themselves plain in this problem. For certain problem parametrizations (meaning the signals and correlation coefficient), optimal quantization is achievable via a single threshold applied to each observation-the same as under independence. In other cases, one observation is best ignored or is quantized with two thresholds; neither behavior is seen under independence. Further, and again in distinction from the case of independence, it is seen that in certain situations, an XOR fusion rule is optimal, and in these cases, the implied decision rule is bizarre. The analysis is extended to the multivariate Gaussian problem.

On the performance of serial networks in distributed detection

Distributed detection on the serial (or tandem) topology is considered with the probability of error performance criterion. Previously published efforts, while presenting probability of detection versus false alarm results, limited the number of array elements to two or three. For the detection of known, equally likely signals in additive, symmetric noise, the author presents simple recursive expressions for the threshold values and the performance of the system. Examples for known signals in Gaussian and Laplacian noise show the degradation in performance due to the array structure. >

Asymptotic performance of unrestricted polar quantizers (Corresp.)

Unrestricted polar quantizers (UPQs) were previously introduced to overcome the weak mean-squared error performance of polar coordinate quantizers for the independent bivariate Gaussian source when the number of levels was small ( bits /dimension). An asymptotic ntean-squared error analysis of the UPQs with a general circularly symmetric source is presented. The performance of the UPQs is shown to be asymptotically within 0.17 dB of that of the optimum bivariate quantizer. The results are extended to dimensions greater than two. Examples for the independent Gaussian source are included.

Locally optimal detection in multivariate non-Gaussian noise

The detection of a vanishingly small, known signal in multi-variate noise is considered. Efficacy is used as a criterion of detector performance, and the locally optimal detector (LOD) for multivariate noise is derived. It is shown that this is a generalization of the well-known LOD for independent, identically distributed (i.i.d.) noise. Several characterizations of multivariate noise are used as examples; these include specific examples and some general methods of density generation. In particular, the class of multivariate densities generated by a zero-memory nonlinear transformation of a correlated Gaussian source is discussed in some detail. The detector structure is derived and practical aspects of obtaining detector subsystems are considered. Through the use of Monte Carlo simulations, the performance of this system if compared to that of the matched filter and of the i.i.d. LOD. Finally, the class of multivariate densities generated by a linear transformation of an i.i.d, noise source is described, and its LOD is shown to be a form frequently suggested to deal with multivariate, non-Gaussian noise: a linear filter followed by a memoryless nonlinearity and a correlator.

A vector quantizer for the Laplace source

The low complexity, nearly optimal vector quantizer (VQ) is a generalization of T. R. Fischers (1986) pyramid VQ and is similar in structure to the unrestricted polar quantizers previously presented for the independent Gaussian source. An analysis of performance is presented with results for both the product code pyramid VQ and the unrestricted version. This analysis, although asymptotic in nature, helps to demonstrate the performance advantages of the VQ. Implementation issues of the VQ are discussed. Nonasymptotic results are considered. In particular, the author presents an approximate design algorithm for finite bit rate and demonstrates the usefulness of this VQ through several example designs with Monte Carlo simulations of performance. For the restricted form (the pyramid VQ), the author provides further implementational information and low dimension analytical results. >

Multidimensional spherical coordinates quantization

Several investigators have considered polar coordinates quantization of a circularly symmetric source; in particular, the independent bivariate Gaussian source. Their schemes quantize the polar coordinates independently in an attempt to reduce the mean-square error below that of an analogous rectangular coordinates quantizer yet retain an implementation simpler than that of the optimal bivariate quantizer. The design of a spherical coordinates quantizer in k dimensions with k>2(k=2 matches published results) is considered. Examples are presented along with comparisons to the rectangular (one-dimensional) and optimal schemes.

Uniform Spherical Coordinate Quantization of Spherically Symmetric Sources

For signal quantization with the minimum mean square error criterion, optimum spherical coordinate quantizers have been shown to outperform optimum rectangular coordinate qnantizers for many spherically symmetric sources. In this paper, spherical coordinate quantizers are designed and analyzed under the added constraint that each scalar quantizer is a uniform (equal step-size) quantizer. It is shown that for small block length, uniform spherical coordinate quantizers outperform the uniform rectangular coordinate quantizer as well as the uniform qnantizer consisting of a tesselation of the minimum inertia, space filling polytope.

Deflection-optimal data forwarding over a Gaussian multiaccess channel

We look at the simple scenario where multiple sensors make conditionally independent observations of a binary source and process the measurement data using a function U(x) before forwarding them to a fusion center via a Gaussian multiaccess channel. Subject to a total power constraint, we obtain the optimal U(x) that maximizes the deflection; the latter can also be interpreted as the output signal-to-noise ratio for an equivalent binary detection problem. The shape of the optimal function only depends on the probability density function of the observation noise, which we assume symmetric around zero, while the height is scaled by the allowed transmission power. We emphasize that the optimal function herein is derived for an arbitrary distribution of the observation noise. It reduces to a tanh(middot) function when the observation noise is additive Gaussian, which has been studied in the literature.

A lower bound on the error probability for signals in white Gaussian noise

The performance of specific signal constellations in digital communications problems is often described through use of the union bound, an upper bound which is asymptotically tight with respect to the signal-to-noise ratio. It is not unnatural to ask how tight this result is for small to medium values of signal-to-noise ratio; one way to answer this question is to develop an asymptotically tight lower bound. Such is the goal of the present paper. >

How often is hard-decision decoding enough?

The problem of decoding binary linear block codes has received much attention; the two extremes are optimal, high-complexity soft-decision (or maximum-likelihood) decoding and lower performance, much lower complexity hard-decision (or algebraic) decoding. This article considers a class of decoders which first implements hard-decision decoding; second, tests to see if that is enough, that its result matches the result of soft-decision decoding; and third, continues to search if a match is not found. The advantage of such a testing procedure is that if the hard-decision decoding result is found to be enough (called a success for the test), then the computational effort expended by the decoder is low. The performance, as measured by the probability of a success, of a variety of simple tests of the hard-decision codeword are analyzed.