Some Results on Nongaussian Signal Detection

We show that the concept of correlation detection of deterministic signals in additive Gaussian noise can be extended in a natural manner to the detection of signals that are transmitted through a "Gaussian" random channel besides being corrupted by additive Gaussian noise. Such situations are typical in communication over scatter-multipath channels (with or without a specular component). In the deterministic case, the receiver essentially crosscorrelates the received signal with the signal before the additive noise was introduced. When a random channel is present, however, this latter signal, i.e., the output of the random channel, is not known at the receiver, However, knowing the statistics of the channel and the noise, the receiver can make an estimate of it from the received signal on the hypothesis that a particular signal was transmitted. The optimum receiver then crosscorrelates this estimate with the received signal.

The use of information theory to design waveforms for the measurement of extended radar targets exhibiting resonance phenomena is investigated. The target impulse response is introduced to model target scattering behavior. Two radar waveform design problems with constraints on waveform energy and duration are then solved. In the first, a deterministic target impulse response is used to design waveform/receiver-filter pairs for the optimal detection of extended targets in additive noise. In the second, a random target impulse response is used to design waveforms that maximize the mutual information between a target ensemble and the received signal in additive Gaussian noise. The two solutions are contrasted to show the difference between the characteristics of waveforms for extended target detection and information extraction. The optimal target detection solution places as much energy as possible in the largest target scattering mode under the imposed constraints on waveform duration and energy. The optimal information extraction solution distributes the energy among the target scattering modes in order to maximize the mutual information between the target ensemble and the received radar waveform. >

The Wigner distribution (WD) is applied to the detection and localization in time, of narrow-band transient signals of unknown waveform in an additive background comprised of quasiharmonic and random components. A traditional method of processing such signals is the spectrogram or short-time power spectrum. Based on an important relationship between the spectrogram and the WD, the application of the WD to the problem is investigated. The smoothed-pseudo-WD is shown to provide superior time and frequency resolutions of the received signal. Based on monitoring the received power in localized regions of the WD spectrum, a detector that achieves good time localization of the transient waveform and is relatively insensitive to changes in signal duration and frequency is obtained. The signal-to-noise ratio of the proposed detection statistic is derived for a transient signal in additive Gaussian noise and its dependence on various signal and detector parameters discussed.

: The report proposes a design of an adaptive receiver for the detection and estimation of nearly periodic signals in additive Gaussian noise. A nearly periodic signal is defined to be a sample function of a Gaussian random process which can be divided into equal length intervals, called periods, in such a manner that the correlation between periods decreases exponentially with their separation. The receiver computes a low signal-to-noise ratio conditional likelihood ratio from which the observer must make decisions. The likelihood ratio is conditional because the receiver estimates any unknown parameters necessary for the computation of the true likelihood ratio. Thus the receiver can only compute a likelihood ratio conditioned upon these estimates being the true values of the unknown parameters. The receiver consists of pre- emphasis filters followed by a comb filter, energy detector, and weighted summation. A theoretical evaluation of the receiver, in terms of ROC curves, is made for the special case of nearly periodic signals with statistically independent equal-strength harmonics in white noise of known power.

In this paper, we put forward a computationally efficient algorithm to estimate the frequency and complex amplitude of a real sinusoidal signal in additive Gaussian noise. The novel method extends an iterative frequency estimator for single complex exponentials that is based on interpolation on Fourier coefficients to the real case by incorporating an iterative leakage subtraction strategy. Simulation results are presented to verify that the proposed algorithm can obtain more accurate estimation than both time and frequency domain parameter estimators in the literature, and the estimation variance of the method sits on the Cramer-Rao lower bound with only a few iterations required.

The linear prewhitening matched filter (PWMF) is the optimal decision function for discriminating exactly-specified signals in additive Gaussian noise. When the signals are less well specified the optimal decision function contains higher than linear terms in the data. Several examples of detection and discrimination tasks are presented for which only the linear term is required and other examples for which higher order terms are necessary. Even in the nonlinear case decision functions can often be approximated by linear operations on the data followed by logical operations. This combination of linear weights plus logic operations is typical of neural network models, which are thought to be elementary models of human processing mechanisms. A number of experiments suggest that there is no decrease in performance of humans for some complex tasks that ideally require such nonlinear operations. However, there are other such tasks where human performance is degraded and this appears to be due more to the complexity of the task and the nature of the correlations in the image than the order of the task. This suggests applications in diagnostic imaging where it might be advantageous for a machine viewer to substitute or work in conjunction with the human observer.

This paper derives the performance of the joint radar and communication system (JRCS). Firstly, for the radar application of the JRCS, we derive the mutual information (MI) between target and received signal in additive Gaussian noise. For communication, we derive the communication data rate that the JRCS can achieve. Then, under partial coexistence between radar and communication bands, the MI and communication data rate for the JRCS are derived. In terms of the overlap between the received radar and communication signals, the successive interference cancellation (SIC) scheme is adopted. Finally, the provided numerical results demonstrate the performances for the JRCS.

One approach to the analysis of the high signal-to-noise ratio (SNR) capacity of noncoherent wireless communication systems is to ignore the noise component of the received signal in the computation of its differential entropy. In this paper, we consider the error incurred by this approximation when the transmitter and the receiver have one antenna each and when the noise has a Gaussian distribution. We consider the complex and real cases, and we show that when the probability density function (pdf) of the signal component of the received signal is piecewise differentiable, the approximation error decays as 1/SNR, which tightens the available result that the error decays as o(1). In addition, we consider the special instance in which the signal component of the received signal corresponds to a signal transmitted over a channel with a Gaussian fading coefficient. For that case, we provide explicit expressions for the first nonconstant term of the Taylor expansion of the differential entropy, and we invoke Schwartz's inequality to obtain an efficiently computable bound on it. Our results are supported by numerical examples.

A method of detecting arbitrary random signals in the presence of additive Gaussian noise is discussed. The method is based on eigendecomposition of the noise probability subspace. This decomposition leads to unique detector structure, invariant with respect to the signal distribution. The resulting detection scheme uses a variable number of terms per decision, but any decision made is exactly the same as the one produced by the optimal detector. The scheme is found to be computationally efficient, and well suited for generalization to the case of unknown (or only partially known) statistics. The efficiency of the algorithm is demonstrated in a typical example, with the aid of computer simulations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A theoretical result for processing signals that have unknown distributions and priors in white Gaussian noise

Optimal detection of a known FSK-modulated binary signal in additive white Gaussian noise using a matched filter receiver requires knowledge of second-order noise statistics. The dependence on noise statistics causes the probability of detection to be sensitive to errors in the noise variance value. This makes the optimal scheme of limited use in situations where the noise statistics are unknown and cannot be estimated reliably. We propose an alternative approach, which has a more easily calculated test statistic than the optimal method and yields a constant false detection rate, regardless of noise statistics. While this approach has sub-optimal detection probability, it is significantly advantageous in applications where a primary interest is to limit false detections strictly. In addition, it allows one to determine easily the required detection length (in symbols) to achieve a desired performance level.

The statistical theory of signal detection and estimation has been applied to problems in large array seismology. Using this theory the structure of the optimum detector for a known signal and long observation time in additive Gaussian noise is derived. The array processing filter employed by the optimum detector is known as the maximum-likelihood filter. This filter also has the property that it provides a minimum-variance unbiased estimate for the input signal when it is not known, which is the same as the maximum-likelihood estimate of the signal if the noise is a multidimensional Gaussian process. A series of experiments was performed using data from the large aperture seismic array to determine the effectiveness of the maximum-likelihood method relative to simpler methods such as beam-forming. These results provide significant conclusions regarding the design and processing of data from large seismic arrays. The conventional and high-resolution estimation of the frequency-wavenumber spectrum of the background microseismic noise is also presented. The diffuse structure of this spectrum is shown to aid in explaining the relative performance of array processing methods.

Optimum processors for detection of known and random signals in noise using an array of sensors are derived. It is shown how the Schwartz inequality may be used to solve the problem of maximizing the detection index in both cases. This approach provides both the optimum filter functions and the performance of the optimum processor simply and directly. Narrow-band signal assumptions are avoided.

The optimum detection of binary antipodal signals in additive white Gaussian noise (AWGN) channels with Gaussian channel estimation error has been studied in prior work. In this paper, we present the optimum detector based on the maximum- likelihood criterion for binary orthogonal signals in the presence of Gaussian distributed channel estimation error and AWGN. It is shown that the optimum detector is a linear combination of the optimum coherent and optimum noncoherent detectors. We derive the exact closed-form expression of the average bit error probability of the proposed optimum detector in Rayleigh fading channels with AWGN. It is found that if the variance of channel estimation error for a given average SNR is greater than a threshold, then orthogonal signalling outperforms antipodal modulation, and the analytical expression of this threshold is derived.

Estimation of parameters of two-dimensional random amplitude chirp signal in additive noise