Complex Additive White Gaussian Noise Research Articles

Joint source-channel coding via model predictive control

We propose a joint source-channel coding (JSCC) scheme based on model-predictive control (MPC) for encoding an analog discrete-time source over a memoryless communication channel. The decoder is assumed to be fixed and given, and the MPC encoder is set to minimize the frequency-weighted mean-squared error (FWMSE) while satisfying a channel input constraint. We derive analytic expressions for the associated finite-horizon optimization MPC problem for an asymptotic (in time) average or sum channel input constraint. For the case of complex additive white Gaussian noise channel and an M-quadrature-amplitude modulation (M-QAM) decoder, these expressions can be written explicitly, which allows for an efficient numerical solution. For the case M=4 and considering a frequency weighting criterion that models the sensitivity of the human hearing, we show by simulations that our proposed scheme yields an FWMSE significantly smaller than that yielded by the corresponding non-JSCC scheme (a 2-bit uniform quantizer in tandem with a 4-QAM channel encoder and decoder).

Modulated Sparse Superposition Codes for the Complex AWGN Channel

This paper studies a generalization of sparse superposition codes (SPARCs) for communication over the complex additive white Gaussian noise (AWGN) channel. In a SPARC, the codebook is defined in terms of a design matrix, and each codeword is a generated by multiplying the design matrix with a sparse message vector. In the standard SPARC construction, information is encoded in the locations of the non-zero entries of the message vector. In this paper we generalize the construction and consider modulated SPARCs, where information in encoded in both the locations and the values of the non-zero entries of the message vector. We focus on the case where the non-zero entries take values from a phase-shift keying (PSK) constellation. We propose a computationally efficient approximate message passing (AMP) decoder, and obtain analytical bounds on the state evolution parameters which predict the error performance of the decoder. Using these bounds we show that PSK-modulated SPARCs are asymptotically capacity achieving for the complex AWGN channel, with either spatial coupling or power allocation. We also provide numerical simulation results to demonstrate the error performance at finite code lengths. These results show that introducing modulation to the SPARC design can significantly reduce decoding complexity without sacrificing error performance.

Maximum-Likelihood, Magnitude-Based, Amplitude and Noise Variance Estimation

Maximum likelihood (ML) amplitude and noise variance estimation without having to jointly estimate the frequency and the phase and based only on information from the noisy received signal magnitude, is studied for a single sinusoid in complex additive white Gaussian noise. This estimation problem is equivalent to the classic problem of parameter estimation for the Rician distribution. While solving the likelihood equation is impossible in general, we propose a new approach based on a large argument approximation. For the case with known noise variance, a closed-form ML amplitude estimator is obtained, which outperforms the conventional root-mean-square estimator. For the case with unknown noise variance, the closed-form joint amplitude and noise variance estimators obtained do not require prior knowledge of one another.

On Capacity-Achieving Distributions for Complex AWGN Channels Under Nonlinear Power Constraints and Their Applications to SWIPT

The capacity of a complex and discrete-time memoryless additive white Gaussian noise (AWGN) channel under three constraints, namely, input average power, input amplitude and output delivered power is studied. The output delivered power constraint is modelled as the average of linear combination of even moments of the channel input being larger than a threshold. It is shown that the capacity of an AWGN channel under transmit average power and receiver delivered power constraints is the same as the capacity of an AWGN channel under an average power constraint. However, depending on the two constraints, the capacity can be either achieved by a Gaussian distribution or arbitrarily approached by using time-sharing between a Gaussian distribution and On-Off Keying. As an application, a simultaneous wireless information and power transfer (SWIPT) problem is studied, where an experimentally-validated nonlinear model of the harvester is used. It is shown that the delivered power depends on higher order moments of the channel input. Two inner bounds, one based on complex Gaussian inputs and the other based on further restricting the delivered power are obtained for the Rate-Power (RP) region. For Gaussian inputs, the optimal inputs are zero mean and a tradeoff between transmitted information and delivered power is recognized by considering asymmetric power allocations between inphase and quadrature subchannels. Through numerical algorithms, it is observed that input distributions (obtained by restricting the delivered power) attain larger RP region compared to Gaussian input counterparts. The benefits of the newly developed and optimized input distributions are also confirmed and validated through realistic circuit simulations. The results reveal the crucial role played by the energy harvester (EH) nonlinearity on SWIPT and provide new engineering guidelines on how to exploit this nonlinearity in the design of SWIPT modulation, signal and architecture.

The upper bound of multi-source DOA information in sensor array and its application in performance evaluation

Direction of arrival (DOA) estimation has been discussed extensively in the array signal processing field. In this paper, the authors focus on the multi-source DOA information which is defined as the mutual information between the DOA and the received signal contaminated by complex additive white Gaussian noise. A theoretical expression of DOA information with multiple sources is derived for the uniform linear array. At high SNRs and under the sparse-source assumption obtained is the upper bound of DOA information contained in K sparse sources which can be regarded as the sum of all single-source information minus the uncertainty of sources’ order logK!. Moreover, because of the uncertainty of multi-sources’ order, the posteriori probability distribution of DOA no longer obeys single peak Gaussian distribution so that the mean square error is unsuitable in evaluating the performance of multi-dimensional parameter estimation. Consequently, entropy error (EE) is used as a new performance evaluation metric, whose relationship with DOA information is given.

Closed expression of source signal's DOA information in sensor array

In this study, a novel closed expression of direction of arrival (DOA) information (DI) in a single-source scenario is proposed, which is more convenient than the theoretical expression to evaluate the performance of DOA estimation. Firstly, the authors give a uniform linear array system model, where the DI is expressed as the mutual information between the source and the received signal with complex additive white Gaussian noise. Then the posterior probability density function of DOA is deduced. Moreover, by dividing the integral interval into a signal part and a noise part, a novel closed approximate expression of DI is derived. It contains the posteriori entropy in low and high signal-to-noise ratio (SNR) cases and the normalised probability. The normalised probability is obtained through a normalisation procedure on the basis of the correlation functions. As special cases, the closed expression of DI in low and high SNR cases is given and some interesting relationships between physical quantities are revealed. Finally, numerical results are presented to verify the effectiveness of the proposed approximate closed expression.

Closed-Form Asymptotic Approximation of Target’s Range Information in Radar Detection Systems

The echoes of the radar systems can provide useful information about the target, including range, velocity, shape, and angular direction. Extensive studies have utilized the information of the target to improve system performance, whereas the problem of a favorable closed-form asymptotic approximation of the target's range information (RI) is seldom investigated. In this paper, we address the problem of obtaining a closed-form asymptotic approximation of the target's RI in all SNR regions for radar detection systems. The RI is formulated as the mutual information (MI) between the random range and the received signals with complex additive white Gaussian noise (CAWGN). The basis of our scheme is to employ the a posteriori probability density function (PDF) of the range to extract RI from the echoes. We show the a posteriori PDF is a function of the autocorrelation function (ACF) of signal and the cross-correlation function (CCF) of signal-noise. By dividing the integration interval of the a posteriori PDF into the signal-noise interval and noise interval, a closed-form asymptotic approximation of RI is derived based on the probability of range distinguishability and the normalized a posteriori entropies of low and high signal-to-noise ratio (SNR). The results also reveal an interesting relationship between the RI and the uncertainty of the target. As special cases, the closed-form approximations of RI in low and high SNR are obtained. Moreover, for the problem of slightly looser in medium SNR approximation, a better approximation is derived based on a linear model approach. Numerical results are presented to validate the proposed theory and verify the effectiveness of the proposed approximation.

Frequency domain model of power amplifier for OFDM signals

In this paper, we present a theoretical model operating in the frequency domain of the effects of a non-linear power amplifier on the bit error performance of orthogonal frequency-division multiplexing (OFDM) transmission systems. Based on Saleh model (Saleh, 1981), we model the in-band distortion caused by the power amplifier by an equivalent complex gain and additive white Gaussian noise. We characterize analytically the parameters of the proposed model as function of the back-off factor and the underlying parameters of the power amplifier. We demonstrate the accuracy of the proposed model by mean of numerical simulations of the bit error performance over an additive white Gaussian noise channel.

Quasi-Concavity for Gaussian Multicast Relay Channels.

Standard upper and lower bounds on the capacity of relay channels are cut-set (CS), decode-forward (DF), and quantize-forward (QF) rates. For real additive white Gaussian noise (AWGN) multicast relay channels with one source node and one relay node, these bounds are shown to be quasi-concave in the receiver signal-to-noise ratios and the squared source-relay correlation coefficient. Furthermore, the CS rates are shown to be quasi-concave in the relay position for a fixed correlation coefficient, and the DF rates are shown to be quasi-concave in the relay position. The latter property characterizes the optimal relay position when using DF. The results extend to complex AWGN channels with random phase variations.

An Approximate Cramer–Rao Lower Bound for Multiple LFMCW Signals

In this paper we focus on deriving an approximate Cramer–Rao lower bound (CRLB) for the parameters of a multicomponent linear frequency modulated continuous wave (LFMCW) signal corrupted by complex additive white Gaussian noise. The approximation is necessary due to the discontinuities inherent in the mathematical model of the instantaneous phase of each LFMCW signal model. By comparing our approximate bound to a simulation of the maximum likelihood estimator (MLE) of the LFMCW parameters, we confirm our analysis. In general, the CRLB is a useful tool for feasibility studies or in evaluating the degree of suboptimality that non-MLE methods exhibit. For passive detection and estimation of LFMCW signals, the Generalized Likelihood Ratio Test and the associated MLE are difficult to implement in practice, primarily due to their large computational requirements. So, lower bounds on performance, such as those provided by the CRLB, are necessary to evaluate suboptimal methods that are more suited for practical implementations.

Exploiting high-order phase-shift keying modulation and direct-detection in silicon photonic systems.

A computational approach to evaluate the bit-error ratio (BER) in silicon photonic systems employing high-order phase-shift keying (PSK) modulation formats is presented. Specifically, the investigated systems contain a silicon based optical interconnect, namely a strip silicon photonic waveguide or a silicon photonic crystal waveguide, and direct-detection receivers suitable to detect PSK and amplitude-shaped PSK signals. The superposition of a PSK signal and complex additive white Gaussian noise passes through the optical interconnect and subsequently through two detection-branch receivers. To model the signal propagation in the silicon optical interconnects we used a modified nonlinear Schrödinger equation, which incorporates all relevant linear and nonlinear optical effects and the mutual interaction between free-carriers and the optical field. Finally, the BER is calculated by applying a frequency-domain approach based on the Karhunen-Loève series expansion method. Our computational studies of the BER reveal that the optical power, type of PSK modulation, waveguide length, and group-velocity are key factors characterizing the system BER, their influence on BER being more significant in a photonic system with larger nonlinearity. In particular, our analysis shows that the system performance is affected to a much larger extent when the signal propagates in the slow-light regime, despite the fact that this regime allows for a significantly reduced length of optical interconnects.

Parameters estimation of noise amplitude modulation signal

This study proposes a maximum likelihood (ML) estimator for parameter estimation for a complex noise amplitude modulation (NAM) signal. The NAM signal is a general expansion of the exponential signal in which the noise is not circularly symmetric. In the proposed estimator, a non-parametric modified interpolation on Fourier coefficients method is first applied to produce a coarse estimation. Then, a gradient descent method is applied to refine the parameters. The modulation noise to complex additive white Gaussian noise ratio, which is an implicit unknown parameter, is calculated and refreshed at each iteration. For signal-to-noise ratios above the threshold, the variances of the coarse estimation are derived and shown to be independent of the modulation index. The probabilities of a divergence in different gradient descent methods are derived; the pure Newton's method is adopted based on the consideration of both computational complexity and practical applications. Monte Carlo simulations verify the variances and the threshold effect of the coarse estimation; the variances of the ML estimator are demonstrated to asymptotically approach the Cramer–Rao bounds.

Data-classification-based SNR estimation for linearly modulated signals

We present a new numerical approach to SNR estimation of linearly modulated signals after passing through a complex additive white Gaussian noise (AWGN) channel. This classified data (CD) based SNR estimator is particularly suitable for both constant and non-constant modulus constellations, including BPSK, M-PSK and M-QAM. In essence, the received data will be first classified into a number of classes, and then a look-up table (LUT) is searched to find an entry that closest matches with the classified data; this matched result in LUT corresponds to the SNR value of the received data. The performance of the proposed estimator in terms of accuracy and complexity is evaluated by numerical simulations and compared with a few well-known estimators such as moment and maximum likelihood based estimators.

Moment-Based Joint Estimation of Ricean K-Factor and SNR Over Linearly-Modulated Wireless SIMO Channels

In this paper, we develop a new joint estimator for the Ricean K-factor and the signal-to-noise ratio (SNR) when multiple antenna elements receive linearly-modulated signals with a complex additive white Gaussian noise spatially uncorrelated. The fourth-order cross-moments and the second-order moments of the received signal at a single-input multiple-output (SIMO) system are considered. The SNR is deduced by estimating the powers of the useful signals and the noise. While the K-factor is computed by estimating the kurtosises of the transmitted data and the Ricean channel. The new joint estimator is non-data-aided and does not require the a priori knowledge of the modulation type or order. The performances of this algorithm are investigated in terms of normalized root mean square error over different modulated transmissions and compared with the data-aided auto-correlation function (DA-ACF) based joint estimator extended to a SIMO configuration. Simulation results show that our approach outperforms the DA-ACF estimator.

Frequency Estimation of Single-Tone Sinusoids Under Additive and Phase Noise

We investigate the performance of main frequency estimation methods for a single-component complex sinusoid under complex additive white Gaussian noise (AWGN) as well as phase noise (PN). Two methods are under test: Maximum Likelihood (ML) method using Fast Fourier Transform (FFT), and the autocorrelation method (Corr). Simulation results showed that FFT-method has superior performance as compared to the Corr-method in the presence of additive white Gaussian noise (affecting the amplitude) and phase noise, with almost 20dB difference.

High-Throughput Cooperative Communication with Interference Cancellation for Two-Path Relay in Multi-Source System

Relay-based cooperative communication has become a research focus in recent years because it can achieve diversity gain in wireless networks. In existing works, network coding and the two-path relay scheme is exploited to deal with the increase in network size and the half-duplex nature of relay, respectively. To further improve bandwidth efficiency, we propose a novel cooperative transmission scheme which combines network coding and the two-path relay scheme together in the multi-source system. Due to the utilization of two-path relay, our proposed scheme achieves full-rate transmission. Adopting complex field network coding (CFNC) at both the sources and the relays ensures that symbols from different sources are allowed to be broadcasted in the same time slot. We also adopt physical-layer network coding (PNC) at the relay nodes to deal with the inter-relay interference caused by the two-path relay. With careful process design, our scheme can achieve the ideal throughput up to 1 symbol per source per time slot (sym/S/TS). Furthermore, the theoretical analysis provides a method to estimate the symbol error probability (SEP) and throughput in complex additive white Gaussian noise (AWGN) and Rayleigh fading channels. The simulation results verify the improvements achieved by the proposed scheme.

A two-stage autocorrelation method for frequency estimation of sinusoidal signal

A two-stage autocorrelation method is proposed for frequency estimation of a complex sinusoidal signal in complex additive white Gaussian noise. In the first stage, a function about signal frequency is constructed via an autocorrelation procedure. By the Least-Square (LS) principle, the signal frequency is estimated from the function. In the second stage, the estimated error of signal frequency in the first stage is defined. Applying the Taylor series, a function about the estimated error of signal frequency is conducted from the autocorrelation function. Applying the LS estimation once more, the estimate of estimated error of frequency in the first stage is obtained. Then the fine estimate of signal frequency is obtained. Simulation results show that the performance of proposed method can approach the Cramer–Rao lower bound in the whole frequency range.

Improved SNR Estimation Technique for BPSK and QPSK Signals

To improve the estimation accuracy of non-data-aided (NDA) signal-to-noise ratio (SNR) estimators at low SNR value, A novel estimation technique for binary phase-shift keying and quadrature phase-shift keying signals in complex additive white Gaussian noise channel is proposed. The mathematical relation between SNR and the ratio of two simple statistical computations is derived, then SNR is determined by looking up a table. Its accuracy surpasses other NDA estimators, approaching closely to the Cramer-Rao lower bound at SNR > 5dB.

Cramér–Rao lower bound for non-data-aided carrier phase estimation from general M-ary phase-shift keying modulation signals over flat Rayleigh fading channel

The authors derive the true Cramér–Rao lower bound (CRLB) for a non-data-aided (NDA) carrier phase estimation from general M-ary phase-shift keying (M-PSK) modulation signal over flat Rayleigh fading channels (FRFC). The numerical Monte Carlo method is used to compute the true CRLB from general M-PSK modulation signals over FRFC with complex additive white Gaussian noise. The efficiency of the proposed method is made obvious through the assessment of the true CRLBs from both symmetric and asymmetric 8-PSK modulation signals for various observation window sizes. The simulation results show that optimal NDA carrier phase recovery is harder to acquire from symmetric modulation signals for signal-to-noise ratio (SNR) range up to 5 dB, but depends on the constellation density, only for SNR range superior to 5 dB. For both constellations the CRLB decreases as long as the observation window size increases. The derived bound provides an efficient standard for evaluating the performance of any unbiased NDA carrier phase estimator from general M-PSK modulation signals over FRFC. The method presented here stands useful for general M-ary quadrature amplitude modulation signals as well.

LQG Control Approach to Gaussian Broadcast Channels With Feedback

A code for communication over the k-receiver complex additive white Gaussian noise broadcast channel (BC) with feedback is presented and analyzed using tools from the theory of linear quadratic Gaussian optimal control. It is shown that the performance of this code depends on the noise correlation at the receivers and it is related to the solution of a discrete algebraic Riccati equation. For the case of independent noises, the sum rate achieved by the proposed code, satisfying average power constraint P, is characterized as 1/2 log(1+Pφ), where the coefficient φ ∈ [1,k] quantifies the power gain due to the presence of feedback. This includes a previous result by Elia and strictly improves upon the codes by Ozarow and Leung and by Kramer. When the noises are correlated, the prelog of the sum capacity of the BC with feedback can be strictly greater than 1. It is established that for all noise covariance matrices of rank r the prelog of the sum capacity is at most k-r+1 and, conversely, there exists a noise covariance matrix of rank r for which the proposed code achieves this upper bound. This generalizes a previous result by Gastpar et al. for the two-receiver BC.