An Exact Error Probability Analysis of OFDM Systems with Frequency Offset

In this paper, we derive exact closed form bit error rate (BER) or symbol error rate (SER) expressions for orthogonal frequency division multiplexing (OFDM) systems with carrier frequency offset (CFO). We consider the performance of an OFDM system subject to CFO error in additive white Gaussian noise (AWGN), frequency flat and frequency selective Rayleigh fading channels. The BER/ SER performances of BPSK and QPSK modulation schemes are analyzed for AWGN and frequency-flat Rayleigh fading channels while BPSK is considered for frequency-selective Rayleigh fading channels. Our results can easily be reduced to the respective analytical error rate expressions for the OFDM systems without CFO error. Furthermore, the simulation results are provided to verify the accuracy of the new error rate expressions.     

Partial-band jamming rejection of FFH/BFSK with product combiningreceiver over a Rayleigh-fading channel

An exact closed-form expression of the characteristic function is derived for a fast frequency-hopping (FFH) binary orthogonal frequency-shift-keying (FSK) spread-spectrum (SS) communication system. The FFH system employs a product combining receiver over a Rayleigh-fading channel with partial-band jamming and additive white Gaussian noise (AWGN). The derived characteristic function is then used to obtain a compact bit error rate (BER) expression for different diversity levels. Our study shows that there exists an optimum diversity level under certain channel conditions     

Error rate analysis of MDPSK/CPSK with diversity reception undervery slow Rayleigh fading and cochannel interference

The distribution of the phase noise due to additive white Gaussian noise (AWGN) and cochannel interference (CCI) is analyzed for differential phase detection (DPD) and coherent phase detection (CPD) in a very slow nonfrequency selective Rayleigh fading environment. The effects of modulation timing offset between the desired signal and the CCI and of the overall channel filter response are considered. Simple closed-form expressions are derived for ideal selection diversity reception. The derived phase noise distributions are used for evaluating the bit error rate (BER) performance of 2-16DPSK/CPSK assuming square-root raised cosine Nyquist transmit/receive filters. It is found that the BER performance of CPSK is less sensitive to CCI modulation timing offset than DPSK, and that increasing the filter rolloff factor can improve the BER performance due to CCI. Finally, the accuracy of the BER approximation that uses the symbol error rate is discussed     

相干DS-SS/MPSK系统中频差影响的简化评估

该文针对单径慢衰落信道模型下的相干DS-SS/MPSK系统,分析了频差对系统性能的影响,得到了误比特率的封闭表达式。首先推导了匹配滤波解扩符号的信噪比特性和相干相位估计误差的分布;然后给出了衰落信道中误比特率的精确积分式,并利用MPSK条件误符号率的麦克劳林级数形式,推导了AWGN,Nakagami-m衰落和瑞利衰落信道中BPSK,QPSK误比特率的简化逼近式。最后的数值结果表明：逼近式的精度非常高,基本没有损失,可应用于实际系统设计和性能评估。     

Performance Evaluation of Generalized Selection Combiners Over Slow Fading with Estimation Errors

In this paper we derive closed-form expressions for the single-user adaptive capacity of generalized selection combining (GSC) system, taking into account the effect of imperfect channel estimation at the receiver. The channel considered is a slowly varying spatially independent flat Rayleigh fading channel. The complex channel estimate and the actual channel are modelled as jointly Gaussian random variables with a correlation that depends on the estimation quality. Three adaptive transmission schemes are analyzed: (1) optimal power and rate adaptation; and (2) constant power with optimal rate adaptation, and (3) channel inversion with fixed rate. In addition to deriving an exact expression for the capacity of the aforementioned adaptive schemes, we analyze the impact of channel estimation error on the capacity statistics and the symbol error rate for GSC systems. The capacity statistics derived in this paper are the moment generating function, complementary cumulative distribution function and probability density function for arbitrary number of receive antennas. Moreover, exact closed-form expressions for M-PAM/PSK/QAM employing GSC are derived. As expected, the channel estimation error has a significant impact on the system performance.     

Closed-form expressions for the symbol error probability of 3-D OFDM

In this letter, closed-form expressions for the symbol error probability of 3-D orthogonal frequency division multiplexing in additive white Gaussian noise channel are derived. When a 3-D 4-ary constellation is used as signal mapper, we calculate the upper and the lower bound for the error probability, and also provide its approximation. Since decision boundaries of 8- ary constellation form an extended regular hexahedron, an exact symbol error probability can be computed. It is verified that the theoretical error probabilities are very close to or almost the same as simulation results.     

Closed-Form Expressions for the Exact Symbol Error Probability of 32-Cross-QAM in AWGN and in Slow Nakagami Fading

Novel closed-form exact expressions for the average symbol error probability of 32-cross-QAM in an additive white Gaussian noise channel and in a slow, flat Nakagami fading channel with L-branch maximal ratio combining diversity are derived     

Performance of RS coded M-ary modulation with and without symbol overlapping

In this paper, we present analytical bit error probability results for M-ary modulation concatenated with Reed Solomon (RS) codes. The analysis of bit error probability is nontrivial as the number of bits per symbol for the RS codes may not be an integer multiple of the number of bits per symbol for a modulation symbol. We propose a Markov chain technique which allows analytical evaluation of the bit error probability for such cases. The performance of RS coding with coherent biorthogonal, coherent/non-coherent orthogonal modulation over an additive white Gaussian noise (AWGN) channel is evaluated. Simulation of the bit error probability of RS code concatenated with a Nordstrom Robinson (NR) code as an inner code is performed and compared with the case of biorthogonal modulation. From the results, we notice that a stronger inner code gives better bit error probability. In addition, the throughput of the coded system with biorthogonal modulation over an AWGN channel is discussed. For a Rayleigh flat fading and block fading channel, we analyze the bit error probability of RS codes concatenated with biorthogonal modulation. From the result, we notice that a stronger outer code gives a better bit error probability for the case of Rayleigh flat fading channel.     

Error probabilities of two-dimensional M-ary signaling in fading

A new exact method for computing the average symbol error probability of two-dimensional M-ary signaling in slow fading is presented. The method is generally applicable to polygonal decision regions. The exact average symbol error rate of coherent 16-star-QAM is obtained and compared to that of 16-rectangular-QAM. New results for optimum ring ratios of 16-star-QAM in additive white Gaussian noise (AWGN) and in slow fading are also given     

Performance analysis of a decision-feedback coherent code tracking loop for pilot-symbol-aided DS/SS systems

In this paper, the performance of a decision-feedback coherent code tracking loop for pilot-symbol-aided (PSA) direct-sequence spread-spectrum systems, considering the effect of both symbol and channel estimation errors, is analyzed. An analytical closed-form expression on the steady-state jitter variance in additive white Gaussian noise (AWGN) environments using the linear analysis model is first derived. Then, the upper and lower bounds of the jitter variance in fading environments are obtained based on the analysis of the loop bandwidth variation according to the input signal level. Finally, a second-order coherent tracking loop is designed, and its performance in both AWGN and fading environments is evaluated, along with verification by computer simulations. It is demonstrated that analytical results are very close to the simulation results in AWGN environments and that the jitter variance in fading environments is located between analytical upper and lower bounds with its position depending on the fading rate.     

A closed-form expression for the exact BER of generalized PAM and QAM constellations

We derive an explicit closed-form expression for the exact bit-error rate (BER) computation of generalized hierarchical M-ary pulse amplitude modulations over additive white Gaussian noise (AWGN) and fading channels. This expression can also be used to obtain the exact BER of generalized hierarchical quadrature amplitude modulations (square and rectangular). For the AWGN case, these expressions are in the form of a weighted sum of complementary error functions, and are solely dependent on the constellation size M, the carrier-to-noise ratio, and a constellation parameter which controls the relative message importance.     

Performance of GMSK in a land mobile radio channel

Discriminator detection of Gaussian minimum shift keying (GMSK) in a cellular mobile-communication channel is analyzed. The channel is modeled as a frequency-selective fast Rayleigh fading channel corrupted by additive white Gaussian noise (AWGN) and co-channel interference (CCI). A closed-form expression for the probability of error is derived. Numerical computation is used to obtain the GMSK bit error rate (BER) performance for various combinations of channel parameters. These results show that GMSK gives slightly better performance compared to that for π/4-quadrature phase shift keying (QPSK) previously reported in the literature     

Error probability of binary phase shift keying in Nakagami-m fadingchannel with phase noise

The error performance of coherent detection of binary phase shift keying (BPSK) signals with noisy phase reference is analysed for a flat Nakagami-m fading channel and in the presence of additive white Gaussian noise (AWGN). By assuming Gaussian and Tikhonov probability density functions (PDFs) for the phase error, closed-form expressions for the average bit error rate (BER) are derived     

High accuracy frequency offset correction with adjustable acquisition range in OFDM systems

A new carrier frequency offset estimation scheme in orthogonal frequency-division multiplexing (OFDM) systems is proposed in this paper. Both the carrier frequency offset acquisition and tracking are based on a fixed-length training-symbol-block, which consists of multiple small identical training symbols. When each training symbol is shortened, the number of training symbols in the training-symbol-block should be increased accordingly to keep the total training-symbol-block length fixed. The proposed scheme extends Moose's estimator, where the estimation error is only dependent on total training symbol energy and cannot be reduced any more, once the total training symbol energy is determined. The proposed scheme can shorten each training symbol in a training-symbol block and select an appropriate estimator simultaneously, which can lead to further reduction of estimation error and increase of acquisition range, even with the total training-symbol-block energy being fixed. Performance analyzes for the proposed scheme in both the additive white Gaussian noise channel (AWGN) and the multipath channel are also presented in this paper. All estimators in the proposed scheme are conditionally unbiased, and simulation results demonstrate that they can work well both in the multipath channel and in the AWGN channel.     

BER expressions for differentially detected π/4 DQPSK modulation

Closed form bit-error rate (BER) expressions for differentially detected π/4-shifted differentially encoded quadrature phase-shift keying (QPSK) modulation (π/4 DQPSK) are derived for both additive white Gaussian noise (AWGN) and Rayleigh-fading channels. The derivations are carried out in an exact and most general manner in that in-phase (I) and quadrature (Q) channel bit-error probabilities P_cl and P_cQ are separately obtained in terms of the same-quadrature and cross-quadrature noise Correlation functions, including a measure of noise nonstationarity. We then specialize the general expressions for uncorrelated noise and equal noise powers in successive symbol periods to obtain a useful bit-error probability expression for the AWGN channel in the form P_e≈Q(√(1.1716·E_b/N₀)) where Q(·) is the Gaussian distribution Q-function and E_b /N₀ is the bit energy-to-noise density ratio. Exact BER expressions for the Rayleigh-fading channel that involve the noise parameters are also given and are extended to the case of L-fold diversity combining     

On the general BER expression of one- and two-dimensional amplitude modulations

Quadrature amplitude modulation (QAM) is an attractive technique to achieve high rate transmission without increasing the bandwidth. A great deal of attention has been devoted to the study of bit error rate (BER) performance of QAM, and approximate expressions for the bit error probability of QAM have been developed in many places in the literature. However, the exact and general BER expression of QAM with an arbitrary constellation size has not been derived yet. We provide an exact and general closed-form expression of the BER for one-dimensional and two-dimensional amplitude modulations, i.e., PAM and QAM, under an additive white Gaussian noise (AWGN) channel when Gray code bit mapping is employed. The provided BER expressions offer a convenient way to evaluate the performance of PAM and QAM systems for various cases of practical interest. Moreover, simple approximations can be found from our expressions, which are the same as the well-known approximations, if only the dominant terms are considered.     

Phase noise sensitivity analysis of lattice constellation

Based on the assumption of large number of constellation points and high signal-to-noise ratio (SNR), phase noise sensitivity of lattice constellation is analyzed. The upper bound of symbol error rate (SER) in additive white Gaussian noise (AWGN) channel is derived from pairwise error probability. For small phase noise, phase noise channel is transformed to AWGN channel. With the aid of Wiener model, the obtained upper bound can be extended to phase noise channel. The proposed upper bound can be used as performance criterion to analyze the sensitivity of phase noise in multi-dimensional lattice constellation. Simulation results show that with the same normalized spectral efficiency, higher dimensional lattice constellations are more sensitive than lower ones in phase noise channel. It is also shown that with the same dimension of constellation, larger normalized spectral efficiency means more performance loss in phase noise channel.     

Power-efficient M-ary PSSK Communications with Uncertain Phase Errors

M-ary phase-silence-shift-keying (PSSK), one of power-efficient modulation schemes, has enhanced symbol error rate (SER) performance by 6 dB compared with that of the conventional phase-shift-keying (PSK). Phase tracking problem of the PSSK with phase-locked-loop (PLL) is discussed in this paper. Capacity of the PSSK for the additive white Gaussian noise (AWGN) channel and Rayleigh fading channel is analyzed. Capacity loss of the PSSK due to the PLL is small, which tells us that the PSSK demodulator can be practically implemented.     

MIMO multichannel beamforming: SER and outage using new eigenvalue distributions of complex noncentral Wishart matrices

This paper analyzes MIMO systems with multichannel beamforming in Ricean fading. Our results apply to a wide class of multichannel systems which transmit on the eigenmodes of the MIMO channel. We first present new closed-form expressions for the marginal ordered eigenvalue distributions of complex noncentral Wishart matrices. These are used to characterize the statistics of the signal to noise ratio (SNR) on each eigenmode. Based on this, we present exact symbol error rate (SER) expressions. We also derive closed-form expressions for the diversity order, array gain, and outage probability. We show that the global SER performance is dominated by the subchannel corresponding to the minimum channel singular value. We also show that, at low outage levels, the outage probability varies inversely with the Ricean A*-factor for cases where transmission is only on the most dominant subchannel (i.e. a singlechannel beamforming system). Numerical results are presented to validate the theoretical analysis.     

An Improved Approximation for the Gaussian Q-Function

We present a novel, simple and tight approximation for the Gaussian Q-function and its integer powers. Compared to other known closed-form approximations, an accuracy improvement is achieved over the whole range of positive arguments. The results can be efficiently applied in the evaluation of the symbol error probability (SEP) of digital modulations in the presence of additive white Gaussian noise (AWGN) and the average SEP (ASEP) over fading channels. As an example we evaluate in closed-form the ASEP of differentially encoded QPSK in Nakagami-m fading.