Performance analysis of equal-gain diversity receivers over generalized Gamma fading channels

Generalized Gamma (GG) distribution is a generic model that covers many well-known fading distributions as special cases. This paper deals with the performance analysis of L-branch equal gain combining (EGC) receivers operating over GG fading channels. For these receivers and by using convergent infinite series approach, the probability of error (P_e) can be formulated in the form of an infinite series. The coefficients of P_e series can be derived by calculating complicated integrations over the fading envelope distribution. In this paper, it is shown that the required integrations for the case of GG distribution have a complex closed-form in terms of Meijer's G function, and then, a new approximation method is developed for computation of them. The proposed method only needs mean and variance of the fading envelope; hence it has low complexity and eliminates the need for calculation of complex functions. The presented numerical examples show that the developed method can approximate the required parameters and also the individual coefficients accurately and this accuracy increases with the increase of L. The proposed method is applied to analyze the probability of error performance of the L-branch EGC receiver with both coherent phase shift keying (CPSK) and frequency shift keying (CFSK) modulation schemes under different GG channel conditions. Also the effect of gain unbalance between diversity branches on the probability of error is investigated.     

Phase-Locked Loop Operation in the Presence of Impulsive and Guassian Noise

A first-order phase-locked loop with detuning is considered in the presence of white Gaussian noise and random amplitude impulsive noise with Poisson times. The stochastic equation for the phase error density is of infinite order, but when the stationary mod-2π phase density is represented by a Fourier series, a linear second-order difference equation is the Fourier coefficients results. The difference equation is solved numerically, and the phase error density is generated from the Fourier series. This method uses no approximations and is valid for any impulsive amplitude probability density.     

The Use of the Fourier-Bessel Series in Calculating Error Probabilities for Digital Communication Systems

Using the theory of spherical symmetric random vectors one can find an expression for the error probability of a wide variety of digital communications systems. These expressions, however, are in the form of Bessel integrals which are usually difficult to solve. In this paper we show how the Fourier-Bessel series can be used to solve the integrals numerically. The calculation error is found to depend on two series parameters which can be manipulated to make the error arbitrarily small. Two examples are used to show the utility of the technique. In the first the probability of error for a CPSK communications system operating in Gaussian noise and cochannel interference is found. In the second the error performance for a multilevel ASK communications system operating in the same corrupting environment is determined. The Fourier-Bessel series technique is a valuable practical tool for solving these and other signal detection problems.     

Further results on the Beaulieu series

A frequent problem in digital communications is the computation of the probability density function (PDF) and cumulative distribution function (CDF), given the characteristic function (CHF) of a random variable (RV). This problem arises in signal detection, equalizer performance, equal-gain diversity combining, intersymbol interference, and elsewhere. Often, it is impossible to analytically invert the CHF to get the PDF and CDF in closed form. Beaulieu (1990, 1991) has derived an infinite series for the CDF of a sum of RVs that has been widely used. We rederive his series using the Gil-Pelaez (1951) inversion formula and the Poisson sum formula. This derivation has several advantages including both the bridging of the well-known sampling theorem with Beaulieu's series and yielding a simple expression for calculating the truncation error term. It is also shown that the PDF and CDF can be computed directly using a discrete Fourier transform.     

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 analytical method for linear combining with application to FFHNCFSK receivers

A method for obtaining the probability density function (PDF) and the cumulative density function (CDF) of sum-independent random variables is presented. The method is capable of determining the PDF and CDF of this sum for an input consisting of any combination of a signal tone, white Gaussian noise, and multiple interfering tones. It is based upon circularly symmetric function theory, Fourier-Bessel series, and Fourier series. To illustrate this method, applications are presented for a fast frequency-hopped noncoherent frequency-shift-keyed communications system. From the PDF and CDF of the received signal, performance values such as the error probability for demodulation, the probability of detection and false alarm for coarse-time synchronization and the mean and variance of timing-error estimates for fine-time synchronization are obtained     

Probability density of the phase of a random RF pulse in the presence of Gaussian noise

In this paper, we show that the probability density function (pdf) of the phase of a random nonstationary radio frequency (RF) impulse signal perturbed by Gaussian noise does not depend on the multiple time derivatives of the signal amplitude and phase at a given time instance and is the Bennett's pdf. Employing this revealing, we derive an alternative form of the conditional pdf of the phase representing it with the von Mises/Tikhonov pdf conditional on the envelope, signal-to-noise ratio (SNR), and signal phase. We expend this new form to the Fourier series and investigate for the locked and unlocked reference phases. The error probability for the phase to exceed a threshold is also analyzed.     

An approximate analytical framework for performance analysis of equal gain combining technique over independent Nakagami,Rician and Weibull fading channels

In this paper, an approximate analytical method for performance analysis of equal gain combiner (EGC) receiver over independent Nakagami and Rician fading channels is presented. We use a convergent infinite series approach which makes it possible to describe the probability of error of EGC receiver in the form of an infinite series. In this paper, we develop a new approximation method for computation of the required coefficients in this series which lets us to derive simple analytical closed-form expressions with good accuracy compared with the exact results existing in the literatures. Our proposed approximation method only needs the mean and the variance of the fading envelope, which are known for various fading distributions, and hence, bypasses the required integration over the fading envelope distribution while computing the required coefficients. This feature lets us to extend our approximation method for performance analysis of EGC receiver over independent Weibull fading channels where the required integration has not any closed-form or tabulated solutions. To give an application of our developed method, we analyze the probability of error of an EGC receiver for binary, coherent PSK (CPSK) modulation over independent Nakagami, Rician and weibull fading channels and study the effect of the fading conditions on the system performance.     

Performance evaluation of generalized selection diversity systems over Nakagami‐m fading channels

The generalized selection combining (GSC) scheme that adaptively combines a subset of M strongest paths out of L available diversity paths finds applications in several wideband receivers and broadband wireless communications. In this paper, exact closed‐form expressions for the moment generating function (MGF), the probability density function (PDF) and the cumulative density function (CDF) of the GSC(M, L) output signal‐to‐noise ratio (SNR) in independent and identically distributed (i.i.d) Nakagami‐m fading channels are derived while the fading index is a positive integer. These expressions hold for any M and L and provide a comprehensive framework for performance analysis including the derivation of closed‐form formulas for the average symbol error probability (ASEP) of a broad class of binary and M‐ary modulations, mean combined SNR and the outage probability of GSC(M, L) receiver structures. When the Nakagami‐m fading index is not an integer, the MGF of GSC(M, L) output SNR is derived as an (M − 1)‐fold infinite series. With this MGF, analytical expressions for both the outage probability and error rates can be readily obtained. An easily programmable recursive solution of the MGF of GSC(M, L) output SNR is also outlined for both the positive integer and noninteger fading severity index cases. Copyright © 2002 John Wiley & Sons, Ltd.     

基于FPGA的2CPSK调制解调系统的设计

二进制绝对相位键控（2CPSK）,是数字通信系统中使用的一种信号调制方式。2CPSK的基本原理是利用数字基带信号控制载波相位的变化来达到传送数字信息的目的。这里详细介绍了基于FPGA的2CPSK信号的原理、设计和调试,并且通过QUARTUSII软件,在基于FPGA的实验板上设计了一种新的2CPSK信号的调制解调系统。并调试出结果,结果表明此设计能有效简化传统系统的设计,缩短整个系统设计周期,具有良好的实用性和较强的可操作性。     

Digital transmission through a land mobile satellite channel

An analytical derivation of the probability of bit error noncoherent frequency-shift keying (FSK) and coherent phase-shift keying (PSK) signals transmitted through a land-mobile satellite channel is described. The channel characteristics used in the analysis are based on a recently developed model which includes the combined effects of fading and shadowing. Analytical expressions for the probability of bit error of FSK and coherent phase-shift keying (CPSK) signals are obtained. The results show that large amounts of signal-to-noise ratio (SNR) are required to compensate for the combined effect of fading and shadowing. An analytical expression for the irreducible probability of bit error of a CPSK signal due to phase variations caused by fading and shadowing is derived. The results described should be useful in the design of land mobile satellite communication systems     

Effects of Jitter and Cycle Slipping of Phase Reference Upon Unique Word Missed Detection in QPSK Systems

The effects of phase jitter and cycle slipping of the recovered carrier upon unique word (UW) missed detection in QPSK systems are investigated. Schrempp et al. have already presented the relationship between the UW missed detection probability (UWMP) and the bit error probability, which is simple and useful as long as the phase reference is noiseless. However, when the authors recently made some theoretical and experimental assessments of UW missed detection, they found that the above disturbances of the recovered carrier significantly affected the UW missed detection performance. Specifically, the actual UWMP was often worse than that predicted hy Schrempp et al. In fact, a bottoming effect takes place in the UWMP versus UW length characteristic. In this paper, after obtaining the conditional symbol and bit error probabilities for the recovered carrier with some phase error, we discuss the nature of phase fluctuation and cycle slipping for the recovered carrier. Consequently, we derive general expressions for the UWMP in both CPSK and DCPSK transmissions. The UWMP's are numerically calculated, and are then compared to our experimental data.     

Probability of error analysis of predetection generalized selection combining receivers with correlated unbalanced Nakagami branches

In this paper we determine the probability of error of a predetection generalized selection combining (GSC) receiver with correlated and unbalanced diversity branches in a Nakagami‐m multipath fading channel. We start by finding the joint probability density function (PDF) of the decision variables. This involves the derivation of the joint PDF of the L largest random variables (L maxima) of an input population of N > L correlated nonidentically distributed random variables, based on the statistics of the input population. The results obtained are then used in the derivation of the error probabilities of noncoherent FSK (NCFSK) and DPSK receivers. Copyright © 2006 John Wiley & Sons, Ltd.     

Performance of Multiphase CPSK Systems with Intersymbol Interference

An upper bound on the probability of error for multiphase coherent phase-shift-keying (CPSK) systems with intersymbol interference and additive Gaussian noise is presented. The bound is derived through the Chernoff bounding technique and is given by a simple closed-form expression. To demonstrate its applicability the bound is computed for quaternary and octonary CPSK systems with various "typical" pulse responses. Graphical comparisons with other measures of the probability of error show the relative efficacy of the bound.     

Analysis of equal gain diversity on Nakagami fading channels

An infinite series for the complementary probability distribution function (CDF) of the signal-to-noise ratio (SNR) at the output of L -branch equal-gain (EG) diversity combiners in Nakagami (1960) fading channels is derived. The bit error rate for a matched filter receiver is analyzed for the L-branch EG combiner and different fading parameters. Both coherent phase shift keying (CPSK) and differential coherent phase shift keying (DCPSK) are considered. The effects of gain unbalance between branches on the probability distribution of the SNR and on the bit error rates are investigated. Bit error rate results are also obtained for coherent and noncoherent reception of frequency shift keying (FSK). The effects of gain unbalances on FSK modulations are also investigated. Bit error rates for EG combining on Rayleigh fading channels are obtained for L>2. These results are presented as a special case of the more generalized Nakagami fading model     

Unified analysis of EGC diversity over Weibull fading channels

In this paper, the performance analysis based on PDF approach of an L ‐branch equal gain combiner (EGC) over independent and not necessarily identical Weibull fading channels is presented. Several closed‐form approximate expressions are derived in terms of only one Fox H‐function as PDF, cumulative distribution function, and moments of the EGC output Signal‐to‐noise ratio (SNR), outage probability, amount of fading, channel capacity, and the average symbol error rate for various digital modulation schemes. All results are illustrated and verified by simulations using computer algebra systems.     

Error probability of coherent PSK and FSK systems with multiple cochannel interferences

The symbol error probability of binary and quaternary CPSK and binary CPSK systems with multiple cochannel interferences and Gaussian noise is investigated. Numerical results obtained by using the Monte Carlo method are presented as an example.<>     

An orthonormal Laguerre expansion yielding Rice's envelope densityfunction for two sine waves in noise

Through an orthonormal Laguerre expansion, expressions are derived for a lesser known Rician probability distribution-the probability density function (PDF) of the envelope of two fixed-amplitude randomly phased sine waves in narrowband Gaussian noise-and for the integral of the density, the cumulative distribution function (CDF). The principal formula derived has been checked analytically, numerically, and (approximately) graphically. Analytically, the moment-generating function for the PDF of the square of the envelope has been found to be a three-term product of elementary functions times an I₀ Bessel function (and thus to be in closed form); in confirmation, the same result has been secured via another, more direct route     

Performance of coherent phase-shift-keyed systems with intersymbol interference

In order to optimize the design of a coherent phase-shift-keyed (CPSK) system, it is necessary to estimate the amount of degradation produced by intersymbol interference. In this paper, an upper bound to the probability of error ofm-ary CPSK systems is given when an ideal CPSK signal is passed through a linear time-invariant noisy filter. This bound can be used to estimate the maximum amount of deterioration produced by intersymbol interference, and hence to choose the filter and channel parameters appropriately. It is assumed that allmsymbols have equal a priori probabilities and that the noise is Gaussian.     

Error probability evaluation of optical systems disturbed by phasenoise and additive noise

A direct and efficient method for evaluation of the error probability of optical heterodyne receivers in the presence of phase noise is presented. A closed form expression for the statistics of the decision variable, including photodetector shot noise and thermal noise from electronic circuitry, is derived. The analysis assumes simple integrating filters in the receiver and is based on a power series expansion of the filtered phase noise. The error probability is calculated using a saddle point approximation which is numerically simple and gives accurate results. The optimal prefilter bandwidth for best phase noise rejection is easily determined