CPSK signaling over hard limited channels in additive Gaussian noise and intersymbol interference

A method is presented for analyzing the performance of M-ary coherent phase-shift-keyed (CPSK) signals which have been transmitted through a hard limiting channel in the presence of linear intersymbol interference preceding the limiter and additive Gaussian noise both preceding and following it. The probability of error is obtained in the form of an infinite series as a function of the expected values of trigonometric functions of the interference. The characteristic function of the interference is expanded in a power series and used to obtain these expected values. The resulting error probability is obtained as the product of two absolutely convergent series. Numerical examples are presented to illustrate the technique.     

An analytic expression for the magnitudes of the signal and intermodulation outputs of an ideal hard limiter,assuming n input signals plus gaussian noise

An expression is derived that gives the output fundamental and intermodulation product magnitudes for an ideal hard limiter, for the case of n input signals plus Gaussian noise. The expression is derived by building upon previous work by Shimbo⁴ to account for the Gaussian noise input. The expression is implemented in a computational tool. Results are generated and are compared with data generated by Jain⁵ for three-signal suppression, and by Blachman¹¹ for the functional relationship between the output and input SNR for a bandpass limiter. In all cases, the data generated agree extremely well with published results.     

On the capacity of a twisted-wire pair: peak-power constraint

The capacity of a twisted-pair channel (TPC) with a peak-power constraint imposed on the transmitted signal is considered. Bounds on the capacity are evaluated for two models of the near-end crosstalk (NEXT). The NEXT is modeled either by a peak-limited signal governed by the same probability law characterizing the transmitted signal, or by a Gaussian process possessing the same power spectral density as that of the transmitted signal. White Gaussian noise is also present in addition to the NEXT interference. The important case of NEXT-dominated interference for which the bounds do not depend on the specific pulse-shape used, and therefore apply also for the optimized pulse amplitude modulation (PAM) scheme, is particularly examined. Optimized channel symbol rates and bounds on the number of information bits per channel symbol are evaluated using a certain PAM format and a suboptimal receiver. A comparison between optimally coded and uncoded PAM schemes is performed for the NEXT-dominated TPC. The results presented extend those reported in a previous work by I. Kalet and S. Shamai (see ibid., vol.38, p.379-83, Mar. 1990) on the capacity of a TPC in which a Gaussian model for both the transmitted and the NEXT signals is adopted and only an average power constraint on both signals is imposed     

Error bounds for the white Gaussian and other very noisy memoryless channels with generalized decision regions

For a class of generalized decision strategies, which afford the possibility of erasure or variable-size list decoding, asymptotically tight upper and lower error bounds are obtained for orthogonal signals in additive white Gaussian noise channels. Under the hypothesis that a unique signal set is asymptotically optimal for the entire class of strategies, these bounds are shown to hold for the optimal set in both the white Gaussian channel and the class of input-discrete very noisy memoryless channels.     

Cramer-Rao bounds in the parametric estimation of fadingradiotransmission channels

The context of this paper is parameter estimation for linearly modulated digital data signals observed on a frequency-flat time-selective fading channel affected by additive white Gaussian noise. The aim is the derivation of Cramer-Rao lower bounds for the joint estimation of all those channel parameters that impact signal detection, namely, carrier phase, carrier frequency offset (Doppler shift), frequency rate of change (Doppler rate), signal amplitude, fading power, and Gaussian noise power. Time-selective frequency-flat fading is modeled as a low-pass autoregressive multiplicative distortion process. In particular, the important case of “slow” fading, with the multiplicative process remaining constant over the whole data burst, is specifically discussed. Asymptotic expressions of the bounds, valid for a large observed sample or for high signal-to-noise ratio (SNR), are also derived in closed form. A few charts with numerical results are finally reported to highlight the dependence of the bounds on channel status (SNR, fading bandwidth, etc.)     

The Effect of Non-Gaussian Noise on the Performance of Binary CPSK System

Upper and lower bounds on the probability of error of binary CPSK system subject to additive non-Gaussian noise and intersymbol interference are derived and the bounds are evaluated for square-wave signaling when the signal is pasSed through filters of Butterworth and Chebyshev transmission characteristics. It is assumed that the distribution of the non-Gaussian noise is characterized by skewness and kurtosis. These parameters express the effect of the limiter on the channel noise. For square-wave signaling it is shown that the skewness has no effect on the performance of the system, and it is also shown that the kurtosis always degrades the probability of error of the system and it increases linearly with the kurtosis.     

DPSK Signaling Over Hard Limiting Channel in the Presence of Intersymbol Interference

This paper considers the performance evaluation of binary differentially-phase-shift-keyed signals which have been transmitted through a hard limiting channel in the presence of intersymbol interference and additive Gaussian noise. An expression for the probability of error is derived which is in the form of an infinite series. In the analysis, the effects of Gaussian noise in the uplink and downlink are taken into consideration. The intersymbol interference is assumed to be present in the uplink only. In order to illustrate the applicability of the method, numerical results are computed for a channel which employs a Butterworth filter.     

α稳定噪声中基于正态变换的次优接收机

通信信道中的冲激噪声通常可由对称的α稳定分布(SαS)描述。由于SαS的概率密度函数(PDF)无闭式表达,因此基于Neyman-Pearson准则的解析的最优接收机也不存在。采用解析模型——双参数柯西-高斯混合模型(BCGM)近似SαS的PDF,导出了基于正态化变换的零记忆非线性(NZMNL)限幅器,NZMNL限幅器使得α稳定噪声变换为标准的正态分布噪声。分析了基于该限幅器次优接收机的误码率性能,仿真结果表明对于噪声参数α(1.52.0]时,所提出接收机的性能优于经典的柯西接收机和钳位接收机。     

GMSK with frequency-selective Rayleigh fading and cochannelinterference

The author derives a formula for error probability of partial-response continuous-phase modulation with differential phase detector and limiter discriminator detector in a multipath Rayleigh fading channel, taking into account frequency-selective fading, cochannel interference, Doppler frequency shift, and additive Gaussian noise while the receiver rejects a specified amount of adjacent channel interference. A formula for the error floor is also presented. Numerical results are presented for Gaussian minimum shift keying with a premodulation normalized filter bandwidth of 0.25. Under mild channel conditions and low energy-to-noise ratios, the best detector is an optimized two-bit differential detection; otherwise the best detector is the limiter discriminator detector     

On the Use of Polarity Coincidence Correlation Techniques for Differential Phase-Shift-Keyed Demodulation

The use of polarity coincidence correlation techniques can allow efficient use of digital logic elements for binary differential phase-shift-key demodulation. For the case of sinusoidal signals in the presence of additive Gaussian noise it is shown that an ideal hard limiter at the input to the demodulator adds no degradation to the bit error rate performance.     

Baseband pulse shaping to improve M-ary FSK in satellitemobile systems

Baseband pulse shaping is proposed to improve the performance of M-ary FSK with limiter discriminator integrator detection in a satellite mobile channel. Numerical results obtained for binary, quaternary, and octonary signaling with a Gaussian and a third-order Butterworth IF filter indicate that pulse shaped signals perform significantly better than frequency shift keying (FSK) signals. The error rate floor in the presence of a Doppler shift is also analyzed. Numerical results indicate that the error rate floor of pulse-shaped signals is significantly lower than that of FSK signals     

Improved error probability evaluation methods for direct detection optical communication systems

The problem of average error probability evaluation for direct detection binary optical communications in the presence of avalanche gain, intersymbol interference, and colored additive Gaussian noise is considered. Tight new upper and lower bounds, together with a modified Gaussian quadrature rule based on approximate moments, are derived and evaluated. The bounds are found to be much tighter than the Chernoff bound though only slightly more complex to evaluate, and can be used as approximations to the error probability in most cases of practical interest. Taken together the new bounds and the modified Gaussian quadrature rule form a comprehensive set of performance evaluation tools offering a judicious balance between complexity and accuracy.     

Receiver Structure for Saturating Channels

A Maximum Likelihood receiver for binary phase-shift keyed signals is derived for the case when the channel includes a bandpass nonlinearity preceded and followed by sources of additive noise. This optimum receiver is then approximated for the case of a limitertype of channel and is seen to be a nonlinear function of both pre- and post-limiter signal to noise ratios and a normalized limiter softness factor λ^*. The probability of error for this receiver is evaluated for the case of a large time-bandwidth productWTusing a central limit theorem argument.     

Limiter-differential detection of a frequency-hopped CPFSKdiversity system in partial-band jamming

The error probability achieved by a differential detector with a bandpass limiter preceding the receiver is analyzed for a slow-frequency-hopped CPFSK diversity waveform transmitted over a partial-band noise jamming channel, and is compared to the system's performance without the bandpass limiter. The system's thermal noise is not neglected in the analysis. In principle, each bit is repeated on L different hops, and for the FH/CPFSK system analyzed, these repetitions are combined to yield a soft decision. The main result is that a diversity gain for error rate improvement in worst-case partial-band jamming is realized with the detector preceded by a limiter, but not without the limiter. This is shown by considering the error probability for L=2 in comparison with that for L=1     

The optimum constant false alarm probability detector for relatively coherent multichannel signals in Gaussian noise of unknown power

In this paper, the method of "most powerful similar tests" is used to obtain the optimum (largest probability of detection) constant false alarm probability detector for multichannel signals received in the presence of additive Gaussian noise of unknown power. The signals are assumed to contain a common random phase angle, and hence are relatively coherent over the multiple channels. The noise is assumed to be correlated from channel to channel. The performance of the optimum detector is calculated. Finally, for illustrative purposes, the technique is applied to the detection of a signal in the presence of a jammer, and to the detection of a single channel signal in white and colored noise of unknown power.     

Intersymbol interference on a continuous-time Gaussian channel

The effect on the probability of error from intersymbol interference caused by a repeated use of a continuous-time Gaussian channel is examined. The model assumes that signals that are power constrained are put through a linear filter after which Gaussian noise is added to produce the received signal. One can view the linear filter as a frequency constraint on the inputs. Because of the memory in the filter, past inputs affect the filter output for the current input. It is shown that this interference can be treated as an additional input whose reproducing kernel Hilbert space (RKHS) norm is upper bounded by a function that tends to 0 with increasing block length. Describing the channel in RKHS terms, we show that the exponential error bounds obtained by Gallager are unchanged if one takes intersymbol interference into account.     

Error Probabilities for Binary Direct-Sequence Spread-Spectrum Communications with Random Signature Sequences

Binary direct-sequence spread-spectrum multiple-access communications, an additive white Gaussian noise channel, and a coherent correlation receiver are considered. An expression for the output of the receiver is obtained for the case of random signature sequences, and the corresponding characteristic function is determined. The expression is used to study the density function of the multiple-access interference and to determine arbitrarily tight upper and lower bounds on the average probability of error. The bounds, which are obtained without making a Gaussian approximation, are compared to results obtained using a Gaussian approximation. The effects of transmitter power, the length of the signature sequences, and the number of interfering transmitters are illustrated. Each transmitter is assumed to have the same power, although the general approach can accommodate the case of transmitters with unequal powers.     

Two-Dimensional Polynomial Phase Signals: Parameter Estimation and Bounds

This paper considers the problem of parametric modeling and estimation of nonhomogeneous two-dimensional (2-D) signals. In particular, we focus our study on the class of constant modulus polynomial-phase 2-D nonhomogeneous signals. We present two different phase models and develop computationally efficient estimation algorithms for the parameters of these models. Both algorithms are based on phase differencing operators. The basic properties of the operators are analyzed and used to develop the estimation algorithms. The Cramer-Rao lower bound on the accuracy of jointly estimating the model parameters is derived, for both models. To get further insight on the problem we also derive the asymptotic Cramer-Rao bounds. The performance of the algorithms in the presence of additive white Gaussian noise is illustrated by numerical examples, and compared with the corresponding exact and asymptotic Cramer-Rao bounds. The algorithms are shown to be robust in the presence of noise, and their performance close to the CRB, even at moderate signal to noise ratios.     

Random-Coding Lower Bounds for the Error Exponent of Joint Quantization and Watermarking Systems

We establish random-coding lower bounds to the error exponent of discrete and Gaussian joint quantization and private watermarking systems. In the discrete system, both the covertext and the attack channel are memoryless and have finite alphabets. In the Gaussian system, the covertext is memoryless Gaussian and the attack channel has additive memoryless Gaussian noise. In both cases, our bounds on the error exponent are positive in the interior of the achievable quantization and watermarking rate region.     

Error performance analysis of an energy sequence estimation receiver for binary FSK on frequency-selective fading channels

Energy detection of frequency-shift keying (FSK) signals is the optimum noncoherent detection technique yielding a minimum bit-error rate (BER) on the additive white Gaussian noise channel. It is usually used when carrier phase estimation is difficult. When FSK signals are passed through a frequency-selective multipath fading channel, the multipath delay in the channel results in a multitone waveform being received during each signaling interval. An effective practical sequence estimation receiver for this situation is proposed. It makes use of energy detectors followed by a Viterbi (1979) or sequence estimator that uses an energy difference metric. Statistical properties of the detection variable are derived. Analytical upper and lower bounds on the BER are derived and the results compared with computer simulations to show the effectiveness of the technique.