On the output autocorrelation function of a single-valued nonlinearity with signal plus noise as input (Corresp.)

A method is given for obtaining series expansions for the coefficients in the double series for the autocorrelation function of the output of a nonlinearity with a signal plus noise as input. Two series are obtained, one in powers and the other in inverse powers of the signal-to-noise ratio. The terms in the series are products of two functions. In these series one function depends on the moments of the signal (or noise) and the other on characteristics of the nonlinearity and the noise (or signal). Differential recursion relationships that simplify evaluation of the nonlinearity-dependent functions are given.     

The Gaussian Nonlinearity's Output Power Spectral Density Due to a Sinusoid Plus Band-Limited Noise

A sinusoid accompanied by stationary, additive, nonzero-mean, band-limited, white Gaussian noise is passed through a Gaussian nonlinear characteristic, and the output power spectral density is evaluated by using known results due to Rice and Atherton. Several characteristics of the Gaussian nonlinearity are revealed in the process, and input tuning is shown to contribute to output noise reduction. The results are applicable in the analysis of tracking systems employing sinusoidal dithers.     

Detection of Modulated Gaussian Signals in Noise

A method using Hermite polynomials is utilized for analyzing the outputs of νth-law devices for inputs of modulated Gaussian signals and zero-mean Gaussian noise. New results are developed for the output correlation functions, and techniques for obtaining closed-form expressions are pointed out. Output signal-to-noise ratios for various input carrier-to-noise-power ratios are calculated, and plots are shown for a single sinusoidal modulating signal. Results are derived to show the existence of a theoretical maximum for the output signal-to-noise ratio. The method can be directly applied for the case of general Fourier expandable signals modulating a Gaussian carrier.     

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.     

A Modified Barrett-Lampard Expansion and Its Application to Bandpass Nonlinearities with Both AM-AM and AM-PM Conversion

A two-dimensional Barrett-Lampard expansion is developed to handle two-dimensional nonlinear systems with random Gaussian inputs. The theory is applied to the problem of evaluating the mean signal and the inphase and quadrature noise correlation functions at the output of a TWT nonlinearity exhibiting arbitrary AM-AM and AM-PM characteristics, when the input consists of a narrow-band signal and Gaussian noise. As a specialized application, the classical problem of determining the coherent and incoherent noise correlation functions at the output of a power-law bandpass nonlinearity is also considered. The results are of interest in assessing the performance of coherent satellite communication channels. The theory developed is accompanied by numerical examples of practical interest.     

Distribution of the filtered output of a quadratic rectifier computed by numerical contour integration

The cumulative distribution of the filtered output of a quadratic rectifier whose input is either narrow-band Gaussian noise or Gaussian noise with a low-pass spectral density is to be computed by numerical quadrature of a Laplace inversion integral along a contour in the complex plane chosen to economize the number of steps. The integrand contains the moment-generating function (mgf) of the output. It is expressed in terms of the Fredholm determinant and the resolvent kernel associated with an integral equation involving the autocovariance function of the input and the impulse response of the output filter. A special case is the power of a mean-zero Gaussian process averaged over a finite interval, and when this process has a rational spectral density, the mgf can be expressed as the ratio of certain finite determinants. By this method distributions are calculated for low-pass noise withRLCand second- and fourth-order Chebyshev spectral densities. For rational input spectral densities but arbitrary positive output filtering and an arbitrary additive input signal, the mgf can be calculated by integrating differential equations of the Kalman-Bucy type.     

Improvement of signal-to-noise ratio in an ideal bandpass limiter with a hysteresis

A sinusoidal signal and Gaussian noise lying in a narrow bandpass through an ideal limiter with a hysteresis, followed by a filter that confines the output to the vicinity of the input frequencies. It is indicated by experiments that a significant improvement in the output SNR at low input SNR can be achieved by this nonlinear device.     

Stochastic gradient adaptation under general error criteria

Examines a family of adaptive filter algorithms of the form W_k+1=W_k+μf(d_k-W_k^t X_k)X_k in which f(·) is a memoryless odd-symmetric nonlinearity acting upon the error. Such algorithms are a generalization of the least-mean-square (LMS) adaptive filtering algorithm for even-symmetric error criteria. For this algorithm family, the authors derive general expressions for the mean and mean-square convergence of the filter coefficients For both arbitrary stochastic input data and Gaussian input data. They then provide methods for optimizing the nonlinearity to minimize the algorithm misadjustment for a given convergence rate. Using the calculus of variations, it is shown that the optimum nonlinearity to minimize misadjustment near convergence under slow adaptation conditions is independent of the statistics of the input data and can be expressed as -p'(x)/p(x), where p(x) is the probability density function of the uncorrelated plant noise. For faster adaptation under the white Gaussian input and noise assumptions, the nonlinearity is shown to be x/{1+μλx²/σ_k ²}, where λ is the input signal power and σ_k² is the conditional error power. Thus, the optimum stochastic gradient error criterion for Gaussian noise is not mean-square. It is shown that the equations governing the convergence of the nonlinear algorithm are exactly those which describe the behavior of the optimum scalar data nonlinear adaptive algorithm for white Gaussian input. Simulations verify the results for a host of noise interferences and indicate the improvement using non-mean-square error criteria     

The effect of a zero memory nonlinearity on the bandlimitedness of contaminated Gaussian inputs (Corresp.)

The bandlimitedness of the output of a time invariant zero memory nonlinearity whose input is the sum of a Gaussian process and another independent random process is discussed. It is shown that, if the nonlinearity is not a polynomlal, or if the Gaussian input component is not bandlimted, then the output is not bandlimited.     

Calculation of Intermodulation from a Single Carrier Amplitude Characteristic

The intermodulation analysis presented in this paper is based on the important observation that any output component of a memoryless nonlinear device can be expressed as an integral of the product of two functions, one being the single carrier amplitude characteristic of the nonlinear device and the other being a function of the statistical parameters of the input signal. The analysis can also be extended to nonlinear devices with AM-PM conversion by expressing the amplitude characteristic as a complex function. The signal-dependent function is given analytically for two simple but important types of signals, for two sinusoidal carriers of equal level, and for Gaussian noise, and it is demonstrated that good agreement has been obtained when the method has been applied to microwave devices like high powerX-band klystrons and traveling wave tubes. It is also shown that the method is convenient for analysis of intermodulation in cascaded nonlinear elements for which the individual single carrier amplitude characteristics are known.     

The Action of Dither in a Polarity Coincidence Correlator

An analysis has been made of how dither added to the input of a polarity coincidence correlator eliminates capture of a weak signal by an unwanted signal. For certain interfering environments the output signal-to-noise ratio of various correlators with three specific input dithers is analyzed. The correlators considered were a polarity coincidence correlator with: 1) no dither, 2) dither of random, amplitude uniformly distributed between the chosen plus-and-minus peak values, 3) sinusoidal dither, and 4) Gaussian dither. Each correlator was analyzed for the cases of Gaussian-noise, rectangular-wave, and sinusoidal interference and compared with the classical correlator. The choice of dither amplitude based upon the measured value of the noise power was studied. A uniformly distributed dither is shown to be slightly superior to sinusoidal dither, and both superior to Gaussian noise.     

Identification of double-valued nonlinearities from their harmonic response

A method is presented for determining an equation for a double-valued nonlinearity from a measurement of its response to a sinusoidal input. It is shown that the nonlinearity can be represented as a series in Cheby?shev polynomials of the first and second kind with argument dependent on the input magnitude. The coefficients of the Cheby?shev-polynomial terms in the series are the measured in-phase and quadrature output-harmonic magnitudes.     

Blind identification of LTI-ZMNL-LTI nonlinear channel models

A simple method is proposed for blind identification of discrete-time nonlinear models consisting of two linear time invariant (LTI) subsystems separated by a polynomial-type zero memory nonlinearity (ZMNL) of order N (the LTI-ZMNL-LTI model). The linear subsystems are allowed to be of nonminimum phase (NMP), though the first LTI can be completely identified only if it is of minimum phase. With a circularly symmetric Gaussian input, the linear subsystems can be identified using simple cepstral operations on a single 2-D slice of the N+1 th-order polyspectrum of the output signal. The linear subsystem of an LTI-ZMNL model can be identified using only a 1-D moment or polyspectral slice if it is of minimum phase. The ZMNL coefficients are not identified and need not be known. The order N of the nonlinearity can, in principle, be estimated from the output signal. The methods are analytically simple, computationally efficient, and possess noise suppression characteristics. Computer simulations are presented to support the theory     

The clipping loss in correlation detectors for arbitrary input signal-to-noise ratios

The detection of stationary Gaussian signals in a background of stationary Gaussian noise by the analog multiplier correlator, the polarity coincidence correlator (PCC), and the sample polarity coincidence correlator (SPCC) is considered. It is assumed that signal and noise have identical normalized autocovariance functions, and they are not cross correlated with each other. The main contributions of this paper are the exact expressions for the output signal-to-noise ratio (SNR) for the correlators mentioned for all values of the input SNR. It is shown that there exists a critical value of the input SNR, such that, whenever this value is exceeded, the PCC output SNR exceeds that of the analog correlator. A sufficient condition for this gain in output SNR is obtained in terms of the input SNR. This result is illustrated for stationary Gauss-Markov processes.     

Noise-Error Determination of Combinational Circuits by Walsh Functions

The stochastic behavior of digital combinational circuits is analyzed by the use of Walsh functions. An n-input Boolean function is represented as a Walsh series and the error caused by noise is measured in terms of a distance which is the fraction of the time that the system output due to noise-corrupted signal differs from that due to signal alone. It is shown that the error can be expressed as the sum of two parts: one part depends only on noise statistics, and the other on both signal and noise. Some interesting properties of both parts are discussed and typical examples are given.     

Quantization noise in single-loop sigma-delta modulation withsinusoidal inputs

An exact nonlinear difference equation is derived and solved for a simple sigma-delta modulator consisting of a discrete-time integrator and a binary quantizer inside a single feedback loop and an arbitrary input signal. It is shown that the system can be represented as an affine operation (discrete-time integration of a biased input) followed by a memoryless nonlinearity. An extension of the transform method for the analysis of nonlinear systems is applied to obtain formulas for first- and second-order time-average moments of the binary quantization noise, including the sample mean, energy, and autocorrelation. The results are applied to the special case of a sinusoidal input signal to evaluate these time averages and the power spectrum. In the limit of large oversampling ratios, the marginal moments behave as if the quantization noise had a uniform distribution. The spectrum is neither white nor continuous, however, even in the limit of large oversampling ratios     

Quasilinear analysis of a nonlinear a.c. channel

The letter shows how the response to a sinusoidal and/or Gaussian input for an a.c. channel containing a modulator, nonlinearity and demodulator in cascade can be obtained. Results for the modified nonlinearity and the equivalent gain to the signal are tabulated for various situations.     

Interference suppression by biased nonlinearities

It is shown that, when the input signal-to-interference ratio (SIR) is small, a biased nonlinearity that has a dead zone below some threshold value can provide a large enhancement in output SIR and signal-to-interference-plus-intermodulation ratio (SIIMR) if the threshold is set equal or close to the amplitude of the strong interference. In the absence of noise, the output SIR and SIIMIR are virtually independent of the input SIR, however low the latter may be. It Is shown that, for a dead-zone limiter, the output SIIMR in this case is -4.93 dB regardless of the input SIR. Under strong interference, any noise present in the input reduces the SIR and SIIMR improvement, but a biased nonlinearity can still provide an output SIR and SIIMR superior to the input SIR, i.e. an output SIIMR 6.35 dB below the input signal-to-noise ratio (SNR) instead of 6.02 dB below the input SIR as is the case with hard limiting     

Semianalytic BER evaluation by simulation for noisy nonlinearbandpass channels

The bit error rate (BER) of channels including a memoryless bandpass nonlinearity is evaluated by simulation. This would typically be an onboard travelling wave tube (TWT) amplifier in a satellite repeater, when the noise at the input of such a nonlinear element is non-negligible. The usual evaluation technique, error counting, requires a large amount of computer time if small error probabilities are to be estimated. It is shown that faster semianalytic procedures can be used, provided that a proper model for the nonlinear element is adopted. The output process is decomposed into a signal component plus an additional term representing an equivalent noise component, and an equivalent nonlinear transformation, relating the input useful signal to the output signal component, is derived. In addition, several modes for the probability density function (PDF) of the uplink noise component at the output of the transmission chain are discussed and compared. The procedure has been tested on a transparent satellite link using 4-CPSK modulation format. The results compare well with those of the error-counting technique if a composite rectangular PDF with exponential tails as adopted     

Series expansion for two input describing functions

The letter shows that, by using a Taylor-series expansion for a single-valued nonlinear characteristic, expressions for its two input describing functions can be obtained as infinite series. The method is applicable to sinusoidal or Gaussian inputs in the presence of any other independent signal with a known probability-density function.