Estimating the frequency of a noisy sinusoid by linear regression (Corresp.)

The estimation of the parameters of a sinusoid from observations of signal samples corrupted by additive noise is investigated. At high signal-to-noise ratios the additive noise is viewed as an equivalent phase noise, suggesting frequency and phase estimation by linear regression on the signal phase. The variances of the regression estimates are shown to achieve the Cramer-Rao bounds. A formula for the variance of the regression frequency estimator is derived in terms of the noise power spectrum. A simple formula for the variance with1/f^{2}phase noise is presented.     

Parameter estimation of 2-D random amplitude polynomial-phasesignals

Phase information has fundamental importance in many two-dimensional (2-D) signal processing problems. In this paper, we consider 2-D signals with random amplitude and a continuous deterministic phase. The signal is represented by a random amplitude polynomial phase model. A computationally efficient estimation algorithm for the signal parameters is presented. The algorithm is based on the properties of the mean phase differencing operator, which is introduced and analyzed. Assuming that the signal is observed in additive white Gaussian noise and that the amplitude field is Gaussian as well, we derive the Cramer-Rao lower bound (CRB) on the error variance in jointly estimating the model parameters. The performance of the algorithm in the presence of additive white Gaussian noise is illustrated by numerical examples and compared with the CRB     

Computing Timing Jitter From Phase Noise Spectra for Oscillators and Phase-Locked Loops With White and$1/f$Noise

Phase noise and timing jitter in oscillators and phase-locked loops (PLLs) are of major concern in wireless and optical communications. In this paper, a unified analysis of the relationships between time-domain jitter and various spectral characterizations of phase noise is first presented. Several notions of phase noise spectra are considered, in particular, the power-spectral density (PSD) of the excess phase noise, the PSD of the signal generated by a noisy oscillator/PLL, and the so-called single-sideband (SSB) phase noise spectrum. We investigate the origins of these phase noise spectra and discuss their mathematical soundness. A simple equation relating the variance of timing jitter to the phase noise spectrum is derived and its mathematical validity is analyzed. Then, practical results on computing jitter from spectral phase noise characteristics for oscillators and PLLs with both white (thermal, shot) and$bf 1/f$noise are presented. We are able to obtain analytical timing jitter results for free-running oscillators and first-order PLLs. A numerical procedure is used for higher order PLLs. The phase noise spectrum needed for computing jitter may be obtained from analytical phase noise models, oscillator or PLL noise analysis in a circuit simulator, or from actual measurements.     

An analytical formulation of phase noise of signals with Gaussian-distributed jitter

The output of many oscillatory systems can be approximated by a stochastic square-wave signal with noise-free amplitude and Gaussian-distributed jitter. We present an analytical treatment of the phase noise of this signal with white and Lorentzian jitter spectra. With a white jitter spectrum, the phase noise is nearly Lorentzian around each harmonic. With a Lorentzian jitter spectrum, it is a sum of several Lorentzian spectra, a summation that has a 1/f/sup 4/ shape at far-out frequencies. With a combination of the two, it has 1/f/sup 4/ and 1/f/sup 2/ shapes at close-in and far-out frequencies, respectively. In all cases, the phase noise at the center frequency and the total signal power are both finite. These findings will improve our understanding of phase noise and will facilitate the calculation of phase noise using time- domain jitter analysis.     

An efficient method for nonlinear distortion calculation of the AM and PM noise spectra of FMCW radar transmitters

The demand for autonomous cruise control and collision warning/avoidance systems has increased in recent years. Many systems based on frequency-modulated continuous-wave (FMCW) radar have emerged and are still in development. Due to the high complexity of such systems, the accurate evaluation of the noise spectra in the transmitter chain driven by complex modulated signals is today a severe drawback due to the limitation of simulation tools. In this paper, a method is proposed to compute easily with any commercially available nonlinear simulator, the amplitude and phase modulated signal distortion introduced by the nonlinearities of the transmitter on an FMCW signal. First, the amplitude modulation (AM) and phase modulation (PM) noise spectra of the driving FMCW signal is derived from the knowledge of the continuous wave (CW) AM and PM noise spectra of the voltage-controlled oscillator (VCO), and the modulating saw-tooth signal applied. Using the narrow band envelope concept and a first-order expansion of the nonlinear transfer function of the transmitter, the transfer of the AM and PM noise spectra of the driving FMCW signal through the nonlinear transmitter chain and the resulting output distortion are then computed. This novel approach allows to compute with reduced computation time and very good accuracy the AM/AM, AM/PM, PM/PM, and PM/AM conversion terms in any nonlinear system driven by CW or FMCW signals. This new method has been applied to the characterization of a whole car radar transmitter operating at 77 GHz driven by an FMCW signal issuing from a VCO. A successful comparison between measured and simulated PM-to-AM conversion coefficients of this transmitter is shown, validating the proposed method.     

Optimization of Trellis Codes with Multilevel Amplitude Modulation with Respect to an Error Probability Criterion

An easily computed upper bound on the error probability of a data communications system with combined trellis coding and multilevel/phase modulation is derived, assuming an additive white Gaussian noise channel and maximum-likelihood decoding. This bound is used to search for codes obtained by set-partitioning that minimize the bound for a fixed number of trellis states. Only amplitude modulated signals typically used in voiceband modem applications are considered. The signal levels that minimize the error probability bound subject to an average power constraint are presented for some specific codes.     

毫米波热辐射阵列的空间谱估计阵列误差模型研究

将空间谱估计算法应用于毫米波热辐射阵列接收系统对目标进行超分辨率探测,需要解决目标热辐射信号弱及阵列误差使性能下降这两个主要问题.为此,建立了低信噪比阵列误差模型,该模型综合考虑了通道幅相误差对信源和通道噪声的影响,推导出由它们所造成的阵列接收数据自相关矩阵的特征值扰动范围和信号特征向量空间误差距离.利用该模型提出了一种通道幅相误差校正算法,使得高分辨率的空间谱估计算法能够很好地应用于毫米波热辐射阵列接收系统,并通过实验验证了新模型及校正算法的正确性和有效性.     

The discrete polynomial-phase transform

The discrete polynomial-phase transform (DPT) is a new tool for analyzing constant-amplitude polynomial-phase signals. The main properties of the DPT are its ability to identify the degree of the phase polynomial and to estimate its coefficients. The transform is robust to deviations from the ideal signal model, such as slowly-varying amplitude, additive noise and nonpolynomial phase. The authors define the DPT, derive its basic properties, and use it to develop computationally efficient estimation and detection algorithms. A statistical accuracy analysis of the estimated parameters is also presented     

Synthetic instruments extract masked signal parameters

We have reviewed the signal conditioning required to obtain high-quality spectral measurements with an FFT-based spectrum analyzer. The conditioning includes data windowing and extended length. Folded windows are required at the input to the transform, and significant ensemble averaging of power spectrum and cross power spectrum at the output of the FFT is needed. We also commented on the effect of cross coupling between signal and noise terms on the variance of the spectral output terms when the signal contains both spectral lines and additive noise. Finally, we presented techniques based on cross spectra. In particular, the normalized cross spectra known as the coherence function was shown as a technique to separate spectral components traceable to the input signal from those not traceable to the input signal.     

Determination of amplitude levels of the piecewise constant signal by using polynomial approximation

A new approach to determining the amplitude levels of piecewise constant signal has been proposed that is based on using its multiplicative model and solving the problem of polynomial approximation. In case of the absence of noise, the statement of polynomial approximation problem is based on the requirement of exact match of the current signal value with the amplitude value of one of its levels. In case of the presence of ordinary additive noise, the problem statement is based on the least squares criterion, while the solution of problem is presented in the analytical form. For the case of the presence of pulse-type noise, the problem statement is based on the minimum duration criterion, while the problem solution is achieved numerically by an appropriate functional minimization in unknown amplitudes of levels. The case of binary piecewise constant signal is considered in detail. The results of numerical simulation are presented for the cases, where the binary signal is distorted by ordinary additive noise with Gaussian distribution law and the pulse-type noise with the Cauchy distribution law.     

Decision feedback differential detection of differentially encoded16APSK signals

Multiple-symbol differential detection (DD) with reference signal estimation based on the feedback of past detected symbols is presented for differentially encoded 16-level amplitude/phase shift keying (16DAPSK). A suboptimal decision is derived. The approximate analysis of the bit-error rate (BER) performance taking into account the decision error propagation effect is presented in an additive white Gaussian noise (AWGN) channel and the BER performance is compared with those of 16DPSK and 16 quadrature amplitude modulation (16QAM)     

Symbol error probability analysis of a multiuser detector for M-PSKsignals based on successive cancellation

A narrow-band multiuser receiver based on successive signal detection and subtraction, is considered. The symbol error probability (SEP) for M-PSK modulated signals is evaluated and analytical approximations for the SEP of the individual signals are presented and compared with results obtained from simulations. For geometrically related signal amplitudes, a constant minimum distance can be guaranteed independent of the number of signals. The required amplitude ratio is shown to be related to M and the number of co-channel signals. Optimizing the transmitted power for the different signals while ensuring the same SEP is then addressed and closed-form expressions of the signal amplitude ratios are derived. The effect of inaccurately estimated signal parameters due to noise is also analyzed. SEP results are presented for synchronous signals in an additive white Gaussian noise environment     

Analysis of DFT-based frequency excision algorithms fordirect-sequence spread-spectrum communications

The capacity of direct-sequence spread-spectrum modulation to reject narrow-band interference can be significantly improved by eliminating narrow-band energy at the receiver in a process called frequency excision. This paper considers several algorithms that operate on the real-time discrete Fourier transform (DFT) of the received signal to perform frequency excision. The case in which only the signal and additive white Gaussian noise (AWGN) are present at the receiver is considered as a means of comparing the relative performance of different algorithms that operate without knowledge of the power spectral density of the interference. An approach for analysis, using the postcorrelation signal-to-noise ratio (SNR) as the figure of merit, is presented that is valid for a broad class of spreading modulations. First, the algorithm that sets a fixed fraction of the frequency domain record to zero is examined using rank-order statistics as an analytical tool. This result is then generalized to confirm previous estimates of SNR degradation for the algorithm that sets all values that exceed a threshold to zero. These results are again generalized to apply to the algorithm that sets a fixed fraction of the band to a fixed amplitude while retaining phase information in an algorithm called fraction clip. The relative performances of several clip algorithm options are derived as special cases. Finally, a performance measure of the algorithms in the presence of multiple narrow-band interference is provided and illustrated with an example     

Adaptive Spectrum Sensing Algorithm in Cognitive Ultra-wideband Systems

Energy detection is a simple spectrum sensing technique that compares the energy in the received signal with a threshold to determine whether a primary user signal is present or not. Setting the threshold is very important to the performance of the spectrum sensing. This paper proposes an adaptive spectrum sensing algorithm where an optimal decision threshold of energy detection is derived based on minimizing the weighted sum of probabilities of detection and false alarm. Since the optimal decision threshold is dependent on the noise power and signal power, a simple, practical frequency domain approach is devised to estimate both. The algorithm can be used for the detection of various kinds of signals without any prior knowledge of the signal, channel or noise power, and is able to adapt to noise fluctuation. Simulations for detecting narrow-band and wideband signals (phase shift keying signal, frequency shift keying signal, orthogonal frequency division multiplexing signal) and ultra-wideband (UWB) signals (direct sequence spread spectrum signals) in an IEEE 802.15.3a UWB band are presented. The results show that the proposed algorithm has excellent robustness to noise uncertainty and outperforms the existing spectrum sensing algorithms in the literature.     

Noncausal 2-D spectrum estimation for direction finding

A two-dimensional noncausal autoregressive (NCAR) plus additive noise model-based spectrum estimation method is presented for planar array data typical of signals encountered in array processing applications. Since the likelihood function for NCAR plus noise data is nonlinear in the model parameters and is further complicated by the unknown variance of the additive noise, computationally intensive gradient search algorithms are required for computing the estimates. If a doubly periodic lattice is assumed, the complexity of the approximate maximum likelihood (ML) equation is significantly reduced without destroying the theoretical asymptotic properties of the estimates and degrading the observed accuracy of the estimated spectra. Initial conditions for starting the approximate ML computation are suggested. Experimental results that can be used to evaluate the signal-plus-noise approach and compare its performance to those of signal-only methods are presented for Gaussian and simulated planar array data. Statistics of estimated spectrum parameters are given, and estimated spectra for signals with close spatial frequencies are shown. The approximate ML parameter estimate's asymptotic properties, such as consistency and normality, are established, and lower bounds for the estimate's errors are derived, assuming that the data are Gaussian     

Spectrum estimation of short-time stationary signals in additivenoise and channel distortion

In this work, spectrum estimation of a short-time stationary signal that is degraded by both channel distortion and additive noise is addressed. A maximum likelihood estimation (MLE) algorithm is developed to jointly identify the degradation system and estimate short-time signal spectra. The source signal is assumed to be generated by a hidden Markov model (HMM) with state-dependent short-time spectral distributions described by mixtures of Gaussian densities. The distortion channel is linear time-invariant, and the noise is Gaussian. The algorithm is derived by using the principle of expectation-maximization (EM), where the unknown parameters of channel and noise are estimated iteratively, and the short-time signal power spectra are obtained from the posterior sufficient statistics of the source signal. Other spectral representation parameters, such as autoregressive model parameters or cepstral parameters, are obtained by minimum mean-squared error (MMSE) estimation from the power spectral estimates. The estimation algorithm was evaluated on simulated signals at the signal-to-noise ratios (SNRs) of 20 dB down to 0 dB, where it produced convergent estimation and significantly reduced spectral distortion     

Cramer-Rao bounds and maximum likelihood estimation for randomamplitude phase-modulated signals

The problem of estimating the phase parameters of a phase-modulated signal in the presence of colored multiplicative noise (random amplitude modulation) and additive white noise (both Gaussian) is addressed. Closed-form expressions for the exact and large-sample Cramer-Rao Bounds (CRBs) are derived. It is shown that the CRB is significantly affected by the color of the modulating process when the signal-to-noise ratio (SNR) or the intrinsic SNR is small. Maximum likelihood type estimators that ignore the noise color and optimize a criterion with respect to only the phase parameters are proposed. These estimators are shown to be equivalent to the nonlinear least squares estimators, which consist of matching the squared observations with a constant amplitude phase-modulated signal when the mean of the multiplicative noise is forced to zero. Closed-form expressions are derived for the efficiency of these estimators and are verified via simulations     

Digitally Phase Modulated (DPM) Signals

Some properties of digitally phase modulated (DPM) signals are presented. Phase modulation with (overlapping) pulses generated by a digital FIR filter belong to this class, which may be considered to be a practical approximation to continuous phase modulated (CPM) signals. The power spectra of DPM signals are derived analytically. The ability of these signals to operate through an additive white Gaussian noise channel is assessed by calculating their minimum Euclidean distance. Their noise and spectral properties are found to be similar to those of CPM signals. Assuming that a Viterbi decoder is used to resolve symbol interference, the out-of-band power tends to decrease as the pulse duration increases, and the noise immunity is enhanced. At the same time the receiver complexity grows exponentially. Hence, noise immunity and spectrum compactness are achieved at the cost of higher received complexity. Modems for DPM signals are believed to be easier to implement than those for CPM signals. This is because filter design is simple and a residual carrier component can be retained to facilitate carrier regeneration. Furthermore, the accumulated carrier phase does not need to be continuously evaluated in order to perform matched filtering. The analytic results derived are supported by measurements and simulations.     

Frequency or phase modulation with a noise carrier

The behavior of an ideal angle-modulation system in which the usual sinusoidal carrier has been replaced by a narrow-band Gaussian random noise or by an amplitude-limited narrow-band Gaussian random noise is studied. The use of a noise carrier results in an additive output noise (in addition to the usual output noise due to the noise in the channel, which is neglected here). It is assumed that the system is to be used to transmit a single-sideband multiplex signal consisting of many adjacent, narrow-band channels, and that the same output signal-to-noise ratio is desired in each of these channels. Subject to these assumptions, it turns out that the phase modulator need have a peak phase index of only about 1 radian. Further, this phase modulator should be driven by a band-pass modulating signal with an appropriately shaped spectrum, rather than by a base-band signal. Based on results of S. O. Rice, curves are presented relating the output signal-to-noise ratio, the required channel bandwidth, the bandwidth of the intelligence transmitted over the system, the required frequency translation of a baseband multiplex signal before modulation, and the peak phase index required of the modulator; all three bandwidths are normalized to the half-bandwidth of the noise carrier.