The Distribution of Error Probability for Rayleigh Fading and Gaussian Noise

The distribution function of the probability of error in the presence of Rayleigh fading and Gaussian noise is determined for the basic binary modulation schemes of coherent frequencyshift keying (CFSK), noncoherent frequency-shift keying (NCFSK), differential phase-shift keying (DPSK), and coherent phase-shift keying (CPSK). General expressions for the distribution function of error probability are also derived when linear maximal-ratio diversity combining is employed. Results are given for various values of average error probability and various orders of diversity.     

Performance analysis of equalized OFDM systems in Rayleigh fading

Channel estimation is usually needed to compensate for the amplitude and phase distortions associated with a received orthogonal frequency-division multiplexing (OFDM) waveform. This paper presents a systematic approach for analyzing the bit-error probability (BEP) of equalized OFDM signals in Rayleigh fading. Closed-form expressions for BEP performance of various signal constellations [phase-shift keying (PSK), differential phase-shift keying (DPSK), quaternary phase-shift keying (QPSK)] are provided for receivers that use a linear pilot-assisted channel estimate. We also derive the optimal linear channel estimates that yield the minimum BEP and show that some previous known results are special cases of our general formulae. The results obtained here can be applied to evaluate the performance of equalized single-carrier narrowband systems as well.     

Analysis of a two-branch maximal ratio and selection diversitysystem with unequal SNRs and correlated inputs for a Rayleigh fadingchannel

An analytical expression for the probability density function of the signal-to-noise ratio (SNR) at the output of a two-branch maximal ratio and selection diversity system is given. The two branches are assumed to be Rayleigh fading, correlated, as well as of unequal average SNRs. Measurements of the cumulative distribution functions after selection and maximal ratio combining were made in Rayleigh fading channels and compared with the analytical results. Also presented are the exact analytical average probabilities of symbol error for coherent binary phase-shift keying and coherent quaternary phase-shift keying before and after two-branch maximal ratio combining for a slow and flat fading correlated Rayleigh channel     

APSK receiver for the cancellation of SPM-induced phase shift

An amplitude and phase-shift keying receiver is proposed. The receiver reduces the penalties caused by the self-phase modulation-induced phase shift by employing three 1-bit delayed interferometers and a switch for each differential phase-shift keying tributary. The performance of the proposed receiver is verified numerically for single-channel and wavelength-division-multiplexing transmissions.     

Exact BER analysis of OSTBCs in spatially correlated MIMO channels

The bit-error rate (BER) performance of orthogonal space-time block codes (STBCs) in correlated multiple-input multiple-output (MIMO) channels is studied. We first derive closed-form exact BER equations for pulse amplitude modulation, quadrature amplitude modulation, and phase-shift keying constellations in a correlated Rayleigh MIMO channel. The BER expressions can be easily evaluated without any numerical methods. Then we consider correlated MIMO channels where the line of sight components exist and they suffer from shadowing, namely, correlated shadowed Rician MIMO channels. For three practical channel scenarios, the BER is analyzed, and closed-form BER equations are presented.     

Error Rate of Quadrature Subbranch Hybrid Selection/ Maximal-Ratio Combining in Rayleigh Fading

The symbol-error rate (SER) of a quadrature subbranch hybrid selection/maximal-ratio combining (QBHS/MRC) scheme for 1-D modulations in Rayleigh fading is examined. At the receiver, N diversity branches are split into 2N in-phase and quadrature subbranches. Traditional hybrid selection/maximal-ratio combining (H-S/MRC) is then applied over the 2N subbranches. M-ary pulse amplitude modulation, including coherent binary phase-shift keying, with QBHS/MRC is studied. The SER performances of QBHS/MRC and H-S/MRC are compared. Results show that QBHS/MRC systems outperform traditional H-S/MRC systems     

Linear diversity analyses for M-PSK in Rician fading channels

Symbol and bit error rates of M-ary differentially encoded/differentially decoded phase-shift keying (MDPSK) and coherent M-ary phase-shift keying (M-PSK) over slow, flat, Rician fading channels are derived when linear diversity combining is applied to combat degradation due to fading. These closed-form solutions are general enough to cover several cases of nondiversity, additive white Gaussian noise (the nonfading mode), Rayleigh fading, mixtures of Rayleigh and Rician fading (the mixed mode), and Rician fading. The results presented here can also be applied to predict the error-rate performance when recent transmit diversity techniques are employed. The solutions for the nonuniform fading profile are included as well. Error probabilities are graphically displayed for both modulation schemes.     

Uncoded and coded performance of MFSK and DPSK in Nakagami fadingchannels

The author presents uncoded and coded performance results for noncoherent M-ary frequency-shift keying (MFSK) and differentially coherent binary phase-shift keying (DPSK) in a slow nonselective Nakagami-m (1960) fading channel. He gives simple expressions for the asymptotic slopes of probability of bit error for large signal-to-noise ratio and shows that the effective order of diversity compared to an uncoded Rayleigh channel is the product of two parameters, one for the channel and one for the code. He also compares the uncoded Nakagami-m results to those of the Rician channel in order to show performance differences between these two generalized fading channel models     

Error rates of DPSK systems with MIMO EGC diversity reception over Rayleigh fading channels

This paper analyzes the average bit error probability (BEP) of the differential binary and quaternary phase-shift keying (DBPSK and DQPSK respectively) with multiple-input multiple-output (MIMO) systems employing postdetection equal gain combining (MIMO EGC) diversity reception over Rayleigh fading channels. Finite closed-form expressions for the average BEP of DBPSK and DQPSK are presented. Two approaches are introduced to analyze the error rate of DQPSK. The proposed structure for the differential phase-shift keying (DPSK) with MIMO EGC provides a reduced-complexity and low-cost receiver for MIMO systems compared to the coherent phase-shift keying system (PSK) with MIMO employing maximal ratio combining (MIMO MRC) diversity reception. Finally, a useful procedure for computing the associated Legendre functions of the second kind with half-odd-integer order and arbitrarily degree is presented.     

A unified performance analysis of digital communication with dualselective combining diversity over correlated Rayleigh and Nakagami-mfading channels

Using a simple finite integral representation for the bivariate Rayleigh (1889) cumulative distribution function previously discovered by the authors, we present expressions for the outage probability and average error probability performances of a dual selective diversity system with correlated slow Rayleigh fading either in closed form (in particular for binary differential phase-shift keying) or in terms of a single integral with finite limits and an integrand composed of elementary (exponential and trigonometric) functions. Because of their simple form, these expressions readily allow numerical evaluation for cases of practical interest. The results are also extended to the case of slow Nakagami-m fading using an alternate representation of the generalized Marcum (1950) Q-function     

MFSK-based modulations for hybrid DS/FH-CDMA systems operating inslowly Rayleigh fading channels

This paper presents an analytical evaluation of a direct-sequence/frequency-hopped code-division multiple-access (DS/FH-CDMA) system in a Rayleigh fading environment. The modulations under consideration are noncoherent M-ary-frequency-shift keying (MFSK) and an MFSK-based joint-frequency phase modulation utilizing differential binary phase-shift keying (PSK). Multiple-access interference has been taken into account, and the spectral efficiency for uncoded as well as convolutionally coded systems has been calculated     

Exact probability of error of BPSK communication links subjected to asynchronous interference in Rayleigh fading environment

We derive a new expression for the exact bit-error probability for the detection of a coherent binary phase-shift keying signal experiencing a number of asynchronous interferers having unequal power levels in Rayleigh fading channels. The new expression is readily computed in terms of the coefficients of a Hermite polynomial.     

Radial basis function-assisted turbo equalization

This paper presents a turbo equalization (TEQ) scheme, which employs a radial basis function (RBF)-based equalizer instead of the conventional trellis-based equalizer of Douillard et al. (1995). Structural, computational complexity, and performance comparisons of the RBF-based and trellis-based TEQs are provided. The decision feedback-assisted RBF TEQ is capable of attaining a similar performance to the logarithmic maximum a posteriori scheme in the context of both binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulation, while achieving a factor 2.5 and 3 lower computational complexity, respectively. However, there is a 2.5-dB performance loss in the context of 16 quadrature amplitude modulation (QAM), which suffers more dramatically from the phenomenon of erroneous decision-feedback effects. A novel element of our design, in order to further reduce the computational complexity of the RBF TEQ, is that symbol equalizations are invoked at current iterations only if the decoded symbol has a high error probability. This techniques provides 37% and 54% computational complexity reduction compared to the full-complexity RBF TEQ for the BPSK RBF TEQ and 16QAM RBF TEQ, respectively, with little performance degradation, when communicating over dispersive Rayleigh fading channels.     

Block-error rate model for DPSK in Rayleigh- and sub-Rayleigh-fading channels

Wireless transmission channels experience substantial signal degradation due to irregularities in the propagation path that impact synchronization between two communicating ends. Consequently, differential and noncoherent techniques that require no such synchronization have gained wide popularity. However, the transmitted data are still highly prone to transmission errors and appropriate error-control mechanisms have to be employed. Knowledge of the error statistics becomes crucial when designing these systems. A block-error rate (BLER) characteristic defines the error distribution within data blocks and in this paper, we propose a finite-state Markov chain-based method to evaluate BLERs in slow frequency-nonselective Rayleigh-fading channels for differential phase-shift keying. The methodology is then extended to the cases of faster Rayleigh and sub-Rayleigh fading. The model is verified by means of numerical simulation. The proposed model possesses many advantages of the well-developed Markov theory and may be easily extended to other modulation schemes such as noncoherent frequency shift keying.     

Low complexity decision-aided decoder for turbo DPSK

A new decision-aided soft a posteriori probability (APP) algorithm for iterative differential phase-shift keying (DPSK) signal demodulation/decoding in a Rayleigh flat-fading channel is presented. Compared with conventional APP algorithms for iterative DPSK, the new algorithm results in considerably lower decoding cost yet can achieve nearly the same performance     

Decision-Directed Automatic Gain Control for MAPSK Systems

An automatic gain control (AGC) loop is presented for use withM-ary amplitude- and phase-shift keying (MAPSK) systems. The gain control amplifier is regulated by an error signal formed by the difference between the estimated amplitude level and the received amplitude level. The AGC performance is thus independent of the short-term average received signal energy. AGC loop analysis and simulation is presented forM-ary amplitude-shift keying (MASK) and quadrature amplitude-shift keying (QASK). The AGC is shown to have a negligible degradation on the symbol probability of error for most practical cases. A generalized AGC for an arbitrary MAPSK system is also presented.     

Multiple-symbol differential detection with interference suppression

A multiple-symbol differential detector is formulated for M-ary differential phase-shift keying modulation where the channel state information is unknown to the receiver. The maximum-likelihood decision statistic is derived for the detector, and its performance is demonstrated by analysis and simulation. Under the Gaussian assumption for the aggregate interference plus noise, an exact expression for the symbol pairwise error probability is developed for M-ary differential phase-shift keying modulation over a diversity, slow-fading Rayleigh channel in the presence of an interference source. A simpler expression of the pairwise error probability is developed for the asymptotic case of large signal-to-noise ratio and small signal-to-interference ratio. It is shown that with an increasing observation interval, the performance of the differential detector over an unknown channel approaches that of optimum combining with known channel.     

Performance of multicarrier CDMA with DPSK modulation anddifferential detection in fading multipath channels

Multicarrier (MC) direct sequence (DS) code division multiple access (CDMA) with differential phase-shift keying (DPSK) modulation and differential detection is proposed. Transmitted data bits are differentially encoded after serial-to-parallel conversion to a number of parallel streams. On each branch, encoded bits are direct sequence spread spectrum (SS) modulated and transmitted using different carriers. The system is analyzed with a differential detector in static Rayleigh fading multipath channel, in fast Rayleigh fading multipath channel and for variable overlapping between carrier spectra in static fading channel. Closed-form expressions are derived for the error probability and evaluated for many cases. The performance is compared to that of a system using phase-shift keying (PSK) with conventional matched filter (CMF) coherent receiver. For static fading channel, the error probability performance of the differential detector is close to that of CMF receiver. For fast fading, the performance degrades slightly with increasing fading rate. Finally; successive carriers of the system are allowed to overlap with various overlapping percentages. The condition of a single path can be achieved by increasing both the number of carriers and the separation between successive carriers. Also, for each number of carriers, there exists an optimum overlapping percentage at which the system performance is optimized. The performance of the proposed DPSK with differential detection system is close to that of PSK with CMF receiver, but the former is simpler to implement     

A closed-form expression for the symbol-error rate of M-ary DPSK in fast Rayleigh fading

The error performance of M-ary differential phase-shift keying on Rayleigh fading channels is widely analyzed in the framework originated by Pawula, and the results usually take the form of an integral. In this letter, we formulate the same problem in a different framework, and solve it by using a linear prediction-based technique resulting in a simple solution in closed form.     

Performance analysis of optimum combining for dual-antenna diversity with multiple interferers in a Rayleigh fading channel

We study the performance analysis of optimum combiner for M-ary phase-shift keying (MPSK) with multiple interferers in a flat Rayleigh fading channel. The first-order approximation is considered for the dual-antenna diversity reception. We derive the analytical solutions for the ordered mean eigenvalues of interference-plus-noise covariance matrix and a simple closed-form expression for the average symbol error rate of MPSK in the case of multiple cochannel interferers.