Narrowband Gaussian Signals Research Articles

ROC curve measurements using euro-rack modules (and some DIY)

Acoustic signal processing projects encourage DIY electronics as well as hardware designed to expose principles found either in nature or from man-made sources. Searching within the Eurorack format of the modular synthesis community, functional equivalents to more common DIY units are identified. Some signal generators and measurements are still underrepresented commercially and require DIY Eurorack modules to be built. One goal is to use existing Eurorack components where DIY modules are designed as necessary in this format. A Receiver Operating Characteristic curve (ROC) is calculated for three cases using analog energy detection and probability density function (pdf) circuitry. The hypothesis for noise (N) is a fixed Gaussian noise distribution. The three hypotheses for the signal plus noise (S+N) have signals characterized as; a constant sinusoid with random phase, a broadband unknown Gaussian signal, and an unknown narrow band Gaussian signal utilizing a narrow band carrier frequency. Our classroom demonstration involves two loudspeakers, one for noise, the other for signal. The detected microphone’s voltage is bandpass filtered, amplified, squared, integrated and then fed into the pdf circuitry where pulses are counted for 4 seconds in each voltage window. Increasing the signal amplitude will increase the detection index.

Asymptotically Optimal Blind Calibration of Uniform Linear Sensor Arrays for Narrowband Gaussian Signals

An asymptotically optimal blind calibration scheme of uniform linear arrays for narrowband Gaussian signals is proposed. Rather than taking the direct Maximum Likelihood (ML) approach for joint estimation of all the unknown model parameters, which leads to a multi-dimensional optimization problem with no closed-form solution, we revisit Paulraj and Kailath's (P-K's) classical approach in exploiting the special (Toeplitz) structure of the observations' covariance. However, we offer a substantial improvement over P-K's ordinary Least Squares (LS) estimates by using asymptotic approximations in order to obtain simple, non-iterative, (quasi-)linear Optimally-Weighted LS (OWLS) estimates of the sensors gains and phases offsets with asymptotically optimal weighting, based only on the empirical covariance matrix of the measurements. Moreover, we prove that our resulting estimates are also asymptotically optimal w.r.t. the raw data, and can therefore be deemed equivalent to the ML Estimates (MLE), which are otherwise obtained by joint ML estimation of all the unknown model parameters. After deriving computationally convenient expressions of the respective Cram\'er-Rao lower bounds, we also show that our estimates offer improved performance when applied to non-Gaussian signals (and/or noise) as quasi-MLE in a similar setting. The optimal performance of our estimates is demonstrated in simulation experiments, with a considerable improvement (reaching an order of magnitude and more) in the resulting mean squared errors w.r.t. P-K's ordinary LS estimates. We also demonstrate the improved accuracy in a multiple-sources directions-of-arrivals estimation task.

Detection performance analysis of product processing of colinear arrays

Non-uniform linear arrays (NULAs) often achieve better resolution than standard uniform linear arrays (ULAs) with equal numbers of sensors. However, conventional beamforming (CBF) of an NULA received signal leads to the same detection statistic PDF as a ULA with the equal number of sensors, undercutting NULA's improved resolution. Nested and coprime arrays partition an NULA into two subarrays and multiply the subarrays' CBF outputs. This research compares this product processor's detection performance against a CBF detector with an equal number of sensors for a narrowband Gaussian signal in spatially white additive Gaussian noise. The product processor's detection PDF is a scaled product of the detection statistic with modified Bessel functions. Receiver operation characteristics (ROC) curves illustrate that the product processor's performance is inferior to the CBF detector with an equal number of sensors. The detection performance of a product processor matches the CBF detector only for high SNRs and large numbers of sensors. However, in the presence of interferers, the product processor for coprime arrays can outperform both the CBF detector and product processing nested arrays with an equal number of sensors. [Work supported by ONR.]

On the Gaussian Pulse Propagation Through Multilayer Graphene Plasmonic Waveguides—Impact of Electrostatic Screening and Frequency Dispersion on Group Velocity and Pulse Distortion

In this paper, we analytically describe the characteristics of propagating and diffusive plasma waves in multilayer graphene heterostructures targeted toward implementing on-chip interconnects. When the effect of collision-dominated resistance of the graphene sheet is negligible, the plasma waves propagate through the media without pulse reshaping. However, in the presence of electron scatterings due to phonons and charged impurities in the sample, the plasma waves acquire a diffusive character and undergo significant frequency-dependent pulse reshaping. We analytically model the propagation speed and pulse-width distortion of a narrow-band Gaussian signal centered at THz frequencies by accounting for the finite electrostatic screening between the multiple layers of graphene. The consequences of signal distortion and attentuation on limiting the signal bit-rate and the impact on energy dissipation are quantified. A partition length of the plasmonic waveguide that distinguishes between $LC$ - and $RC$ -dominated regimes of signal transport is derived. For on-chip interconnect application, the upper bound on the propagation length of plasma waves is determined by both the energy dissipation and the bit-rate requirement of the interconnect infrastructure. Upon comparison with on-chip electrical interconnects, we identify the energy budget that can be allocated toward plasmonic transceivers, such that the overall plasmonic communication is more energy efficient compared to electrical communication.

Use of the kurtosis statistic in the frequency domain as an aid in detecting random signals

Power spectral density estimation is often employed as a method for signal detection. For signals which occur randomly, a frequency domain kurtosis estimate supplements the power spectral density estimate and, in some cases, can be employed to detect their presence. This has been verified from experiments with real data of randomly occurring signals. In order to better understand the detection of randomly occurring signals, sinusoidal and narrow-band Gaussian signals are considered, which when modeled to represent a fading or multipath environment, are received as non-Ganssian in terms of a frequency domain kurtosis estimate. Several fading and muitipath propagation probability density distributions of practical interest are considered, including Rayleigh and log-normal. The model is generalized to handle transient and frequency modulated signals by taking into account the probability of the signal being in a specific frequency range over the total data interval. It is shown that this model produces kurtosis values consistent with real data measurements. The ability of the power spectral density estimate and the frequency domain kurtosis estimate to detect randomly occurring signals, generated from the model, is compared using the deflection criterion. It is shown, for the cases considered, that over a large range of conditions, the power spectral density estimate is a better statistic based on the deflection criterion. However, there is a small range of conditions over which it appears that the frequency domain kurtosis estimate has an advantage. The real data that initiated this analytical investigation are also presented.

Calculating Detection Probabilities for Systems Employing Noncoherent Integration

Cumulative probability distributions that occur in radar and sonar detection problems are calculated directly from the characteristic function by using a Fourier series. The error in the result is controlled by two parameters which can be adjusted to suit the application. The technique is applied to the problem of determining the detection performance of consecutive discrete Fourier transforms (DFTs) for a narrowband Gaussian signal with a rectangular spectrum. Since the characteristic function is used directly in its product form this technique does not suffer from the numerical problems associated with the partial fraction approach. The technique can handle many different problems in a single computational structure making it a valuable tool in system performance studies.

Delay estimation using narrow-band processes

Array processing of narrow-band Gaussian signals is studied with emphasis on delay estimation. The Barankin bound is used to examine the effect of ambiguity on mean-square measurement error. When the bound is plotted as a function of signal-to-noise ratio one observes a distinct threshold. Above the critical signal-to-noise ratio the lower bound on mean-square error is given by the Cramér-Rao inequality, which is approached by the Barankin inequality under these conditions. Below the threshold the Barankin bound can exceed the Cramér-Rao bound by large factors. The relative magnitude of the bounds in that region depends critically on the ratio of signal center frequency to signal bandwidth.

Performance degradation in a quadrature receiver for cw signals corrupted by multipath

In active sonar applications, a quadrature receiver can be employed to detect the presence of an underwater reflector. This well‐known receiver performs optimally when the input waveform is a coherent signal (i.e., constant but unknown amplitude and phase) embedded in white, Gaussian, zero‐mean noise. Unfortunately, when an attempt is made to detect a distributed highlight reflector in a multipath environment by using a sinusoidal transmit signal, the various multipath returns may overlap, producing a nonresolvable multipath situation and causing the amplitude and phase of the signal to become random functions of time. Under these circumstances, the detection performance of the quadrature receiver is degraded compared to its performance with a coherent signal. It is the intent of this paper to theoretically determine the degradation in the receiver’s output signal‐to‐noise ratio (SNR) when the input waveform consists of a narrowband Gaussian signal embedded in white Gaussian noise. The output SNR’s dependence on the correlation characteristics of the input signal is clearly shown.

A maximum likelihood ratio alternative to the multiple coherence statistic

The multiple coherence function has been suggested for application in multiple receiver detection of common narrowband acoustic signals. As an alternative the maximum likelihood ratio detection statistic is developed for multiple receiver detection of narrowband Gaussian signals in additive white Gaussian noise. For the assumptions of rectangular signal spectrums and sufficiently general nonlinear or random time varying propagation channel models, the maximum likelihood ratio statistic is shown to be equivalent |Σ|/Πtσij where Σ = [σij] is the cross-power spectral density matrix estimate within the signal spectral band. A Monte-Carlo computer simulation procedure is discussed. Receiver Operating Characteristic (ROC) curves resulting from the simulation are presented which suggest that in general the maximum likelihood ratio statistic is superior in detection capability to the multiple coherence statistic.

Power Spectrum of Hard-Limited Gaussian Processes

The power spectral density at the output of an ideal hard limiter (one-bit quantizer) is examined when the input is driven by a narrowband gaussian signal plus an additive gaussian noise that consists of a broadband background component plus narrowband interference. Assuming that the input signal-to-noise power ratio is small by virtue of the large bandwidth of the observed broadband noise, calculations are made of the average output signal power, the average output noise power in the signal band, and the average power of the strongest intermodulation product. The results support the intuitive conclusion that spectrum analyzer performance is degraded by the presence of the limiter and that this degradation is more pronounced when a strong narrowband interfering signal is present. They also indicate that the degradation can be minimized by making the bandwidth observed by the limiter sufficiently wide that the broadband noise power dominates both the signal and interference powers. In particular, for a typical example, the signal-to-noise power ratio measured in the signal band is degraded by less than about 1.3 dB by the presence of the limiter and the ratio of output signal power to power of the strongest intermodulation product is greater than about 14.5 dB as long as the broadband noise power exceeds the interfering-signal power.

The Effect of Multiple CW and FM Signals Passed Through a Hard Limiter or TWT

This study reviews the analysis of the output autocorrelation function and spectral density for two CW signals which are passed through a hard limiter, describes the experimental results for two CW signals passed through a TWT, and analyzes the output autocorrelation function and spectral density for two FM signals, four CW signals, and four FM signals passed through a hard limiter. The principal analytical result shows that the output spectrum of multiple FM signals has the properties of multiple CW signals, as far as the centers of the cross-modulation spectrum are concerned, and the properties of narrowband Gaussian signals as far as the spectrum spreading is concerned, although the mechanism by which the spectral spreading occurs is different. The principal engineering result is the channel spacing for two, three, and four FM signals which will yieldnegligible third- and fifth-order crosstalk, when the input signals are passed through a hard limiter or TWT. The analysis of the output autocorrelation function and spectral density for an angle modulated signal when passed through a limiter is a contribution of the author.

Phase Principle for Detecting Narrow-Band Gaussian Signals

This paper describes a phase principle for detecting a weak, narrow-band Gaussian signal in the presence of receiver noise. The phase principle leads to a phase detector which performs 2.5 db worse than the time-honored square-law detector when gain fluctuations are negligible. However, when gain fluctuations are significant the phase detector can perform better than the square-law detector. The phase principle can be implemented by using radio interferometer type receivers or mono-pulse radar-type receivers.