A further note on a general likelihood formula for random signals in Gaussian noise

A general likelihood ratio formula of the author's for the detection of signals in independent white Gaussian noise is extended to allow a "one-sided" dependence in which only the future white noise is required to be independent of past signal and noise. The assumption of Gaussian additive noise is also somewhat relaxed. The proof is based on some recent martingale theorems and on the concept of the innovations process.     

Nonsingular detection and likelihood ratio for random signals in white Gaussian noise

This paper is concerned with the mathematical aspect of a detection problem (a random signal in white Gaussian noise). Specifically, we obtain a sufficient condition for nonsingular detection and derive a likelihood-ratio expression in terms of least-mean-square estimates. The problem itself is old, and the likelihood-ratio expression is also well known. The contribution of this paper is a relatively elementary and self-contained derivation of the likelihood-ratio expression as well as the nonsingularity condition.     

A note on conditional moments of random signals in Gaussian noise (Corresp.)

This correspondence generalizes some recent results on relations between minimum-variance estimates of random signals in Gaussian noise and likelihood ratios.     

Expansion of likelihood ratio for signals in Gaussian noise

It is shown that the coefficients of the Taylor expansion of the likelihood ratio for random signals in Gaussian noise are proportional to the mean-square estimates of the successive powers of the signal evaluated at zero. A similar relationship holds for the expansion of the logarithm of the likelihood ratio.     

Array detection of random signals embedded in an additive mixtureof spherically invariant and Gaussian noise

This letter deals with the array detection of a random signal subject to a disturbance modeled as an additive mixture of Gaussian and spherically invariant noise. A detector based on a large sample size approximation of the locally optimum decision statistic is proposed. The implementation of the detection structure requires the knowledge of the correlation matrix of the disturbance at each array element, but is otherwise independent of the noise distribution of the spherically invariant noise component     

The equivalence- singularity problem for Gaussian signals in Gaussian noise (Corresp.)

The structural properties and sample path behavior of Gaussian signals which can and which cannot be perfectly detected in a given Gaussian noise are described.     

Improved energy detector for random signals in gaussian noise

New and improved energy detector for random signals in Gaussian noise is proposed by replacing the squaring operation of the signal amplitude in the conventional energy detector with an arbitrary positive power operation. Numerical results show that the best power operation depends on the probability of false alarm, the probability of detection, the average signal-to-noise ratio or the sample size. By choosing the optimum power operation according to different system settings, new energy detectors with better detection performances can be derived. These results give useful guidance on how to improve the performances of current wireless systems using the energy detector. It also confirms that the conventional energy detector based on the generalized likelihood ratio test using the generalized likelihood function is not optimum in terms of the detection performance.     

Hard-limiting of two signals in random noise

Two sinusoidal signals and Gaussian noise lying in a narrow band are passed through an ideal band-pass limiter that confines the output spectrum to the vicinity of the input frequencies. The output spectrum, consisting of both discrete and continuous components, is studied in terms of its corresponding autocorrelation function. The discrete output components are identified with the output signals and intermodulation products due to interference between the two input signals. The continuous part of the spectrum is associated with the output noise. The effects of limiting are expressed by ratios among the average powers of the output spectral components. Performance curves are given that show signal suppression, the ratio of output to input SNR's, and the relative strength of the intermodulation terms.     

超宽带信号在高斯噪声中传播性能的研究

超宽带无线通信技术使用微弱的、持续时间极短的脉冲进行短距离通信。超宽带信号具有极宽的射频带宽和较低的中心频率，这种特性可使短距无线通信利用较低的发射功率进行高速传输。根据信号的频谱及其在高斯噪声中的传播性能，研究超宽带信号的减少衰减余量特性，给出不同进制波形的PPM调制超宽带信号比特误码率仿真分析。     

Sparse fast Fourier transform for exactly sparse signals and signals with additive Gaussian noise

In recent years, the Fourier domain representation of sparse signals has been very attractive. Sparse fast Fourier transform (or sparse FFT) is a new technique which computes the Fourier transform in a compressed way, using only a subset of the input data. Sparse FFT computes the desired transform in sublinear time, which means in an amount of time that is smaller than the size of data. In big data problems and medical imaging to reduce the time that patient spends in MRI machine, FFT algorithm is not ‘fast’ enough anymore; therefore, the concept of sparse FFT is very important. Similar to compressed sensing, sparse FFT algorithm computes just the important components in the frequency domain in sublinear time. In this work, sparse FFT algorithm is studied and implemented on MATLAB and its performance is compared with Ann Arbor FFT. A filter is used to hash the frequencies in the n dimensional frequency-sparse signal into B bins, where \(B=n/16\). The filter is used for analyzing an important fraction of the whole signal, and therefore, instead of computing n point FFT, B point FFT is computed, and this results in a faster Fourier transform. The probability of success of the implemented algorithm is investigated for noiseless and noisy signals. It is deduced that as the sparsity increases, the probability of perfect transform also increases. If the performances of the algorithm in both cases are compared, it is clearly seen that the performances degrade when there is noise. Therefore, it can be concluded that the algorithm should be improved especially for noisy considerations. The solvability boundary for a constant probability of error is deducted and added to give insight for future studies.     

Some criteria for frequency domain design of signals imbedded in Gaussian noise

The design of time-limited binary signals imbedded in colored Gaussian noise resulting from passing white noise through a channel of known transfer function is investigated. It is shown that, for a class of channels whose magnitude characteristic is expressible as the ratio of polynomials with the numerator divisible by and of lower or the same degree as the denominator, the output noise kernel (correlation function) and its inverse will have Fourier transforms which are reciprocals of each other. The above assumption results in an expression for probability of error which is independent of the received waveform and depends solely on the transmitter signal-to-noise ratio. The additional freedom of design afforded by the above enables one to impose further restrictions in the form of energy concentration in the frequency domain on the received signal without deterioration of system performance. Some such bandwidth constraints and their associated optimum signal designs are considered in this paper.     

Detection of weak random signals in IID non-Gaussian noise

This paper considers the detection of weak random signals in circularly symmetric, independent, identically distributed noise. Locally optimum detectors and ad hoc nonlinearities are considered, with asymptotic expressions provided for evaluation of detection performance. The analytical expressions are used to evaluate the robustness of detectors to mismatch in the noise models. Finite-sample Monte Carlo simulation results indicate the reliability of these asymptotic measures in cases of practical interest. The results show that, as has been found for detection of weak known signals in non-Gaussian noise, reasonably configured ad hoc nonlinearities are nearly optimum and robust to modest errors in the noise statistics     

Filtering random noise from deterministic signals via datacompression

We present a novel technique for the design of filters for random noise, leading to a class of filters called Occam filters. The essence of the technique is that when a lossy data compression algorithm is applied to a noisy signal with the allowed loss set equal to the noise strength, the loss and the noise tend to cancel rather than add. We give two illustrative applications of the technique to univariate signals. We also prove asymptotic convergence bounds on the effectiveness of Occam filters     

Gaussian Cramer-Rao bound for direction estimation of noncircular signals in unknown noise fields

This paper focuses on the stochastic Cramer-Rao bound (CRB) on direction of arrival (DOA) estimation accuracy for noncircular Gaussian sources in the general case of an arbitrary unknown Gaussian noise field parameterized by a vector of unknowns. Explicit closed-form expressions of the stochastic CRB for DOA parameters alone are obtained directly from the Slepian-Bangs formula for general noncircular complex Gaussian distributions. As a special case, the CRB under the nonuniform white noise assumption is derived. Our expressions can be viewed as extensions of the well-known results by Stoica and Nehorai, Ottersten et al., Weiss and Friedlander, Pesavento and Gershman, and Gershman et al. Some properties of these CRBs are proved and finally, these bounds are numerically compared with the conventional CRBs under the circular complex Gaussian distribution for different unknown noise field models.     

Robust Wiener filters for random signals in correlated noise (Corresp.)

Minimax robust Wiener filtering is considered for the case in which the signal and noise spectral-density matrix is not completely specified. Results are obtained for spectral-density matrix classes which are defined by upper and lower bounds on the components of the matrix. These results form an extension of earlier results on robust Wiener filtering for the case of uncorrelated signals and noise.     

On the performance of ultra-wide-band signals in Gaussian noise anddense multipath

An ultra-wide-band (UWB) signal is characterized by a radiated spectrum with a very wide bandwidth around a relatively low center frequency. In this paper, we study the reduced fading margin property of UWB signals. To evaluate the fading margin, we compare the performance of UWB signals in an environment with only additive white Gaussian noise (AWGN) versus the performance of UWB signals in a dense multipath environment with AWGN. The assumption here is that the presence of multipath causes a small increase in the signal-to-noise ratio required to achieve reasonable levels of bit error rate. A numerical example confirms this assumption, more specifically, the example shows that to achieve a bit error rate equal to 10^-5, we require about 13.5 dB in the AWGN case and about 15 dB in the multipath case, resulting in a fading margin of just 1.5 dB. This small fading margin can be understood by the ability of the UWB signal to resolve the dense multipath     

Spectral density of random signals corrupted by multiplicative noise

A simple method of evaluating the spectral density of stationary random signals corrupted by multiplicative noise is presented. It is assumed that the multiplicative noise is also stationary, and that it is statistically independent of the signal.     

Cramer-Rao bound for Gaussian random processes and applications toradar processing of atmospheric signals

Calculations of the exact Cramer-Rao bound (CRB) for unbiased estimates of the mean frequency, signal power, and spectral width of Doppler radar/lidar signals (a Gaussian random process) are presented. Approximate CRBs are derived using the discrete Fourier transform (DFT). These approximate results are equal to the exact CRB when the DFT coefficients are mutually uncorrelated. Previous high SNR limits for CRBs are shown to be inaccurate because the discrete summations cannot be approximated with integration. The performance of an approximate maximum likelihood estimator for mean frequency approaches the exact CRB for moderate SNR and moderate spectral width