An alternative expression for the symbol-error probability of MPSK in the presence of I/Q unbalance

This letter provides a new expression for the probability of the phase angle between two vectors perturbed by Gaussian noise with in-phase/quadrature-phase (I/Q) unbalance, in order to simplify the analysis of the error probabilities of M-ary phase-shift keying (MPSK). The error probabilities, symbol-error rates, or bit-error rates for MPSK with I/Q balance or unbalance, can be presented straightforwardly from the derived expression. Because the newly derived result is provided in terms of the conventional first-order Gaussian Q-function and the joint Gaussian Q-function with a correlation coefficient dependent on M, it readily allows rapid evaluation for various cases of practical interest.     

SEP performance of coherent MPSK over fading channels in the presence of phase/quadrature error and I-Q gain mismatch

This letter presents a joint approach to the symbol-error probability (SEP) of coherent M-ary phase-shift keying in a situation where the phase error, quadrature error, and in-phase-quadrature (I-Q) gain mismatch problems take place all concurrently over an additive white Gaussian noise and arbitrary fading channel. A set of equations that characterizes the conditional SEP on an instantaneous fading signal-to-noise ratio is derived in the form of the Craig representation.     

On the performance of DQPSK communication systems impaired bytiming error, mixer imbalance, and frequency nonselective slow Rayleighfading

In this paper, the error performance of the differential detection scheme is assessed for differentially encoded quaternary-phase-shift-keying (DQPSK) and π/4-shifted-DQPSK signals, It is assumed that the receiver suffers from mixer imbalance and timing error impairments. Expressions for the system bit-error rate (BER) are obtained when the channel is free of fading and when the channel suffers from frequency nonselective (flat) slow Rayleigh fading in the face of additive white Gaussian noise. To arrive at the desired expressions, the probability density function (pdf) of the phase angle between a pair of Gaussian random vectors is used to obtain the probability of bit error conditioned on the channel fading and timing error. The resulting expressions are then averaged over the nonnegligible timing error and the channel fading to arrive at the desired expressions for the system BER. Finally, with the aid of numerical analysis and in the presence of the aforementioned impairments, a number of conclusions are drawn and the performance of differential receivers for DQPSK and π/4-shifted-DQPSK modulation schemes are compared. It is shown here that the performance of a DQPSK receiver in the absence of fading and timing error is substantially impaired by mixer imbalance for BERs less than 10^-4. In the presence of fading, the performance is noticeably degraded due to mixer imbalance when a nonzero timing error is present     

New exponential bounds and approximations for the computation of error probability in fading channels

We present new exponential bounds for the Gaussian Q function (one- and two-dimensional) and its inverse, and for M-ary phase-shift-keying (MPSK), M-ary differential phase-shift-keying (MDPSK) error probabilities over additive white Gaussian noise channels. More precisely, the new bounds are in the form of the sum of exponential functions that, in the limit, approach the exact value. Then, a quite accurate and simple approximate expression given by the sum of two exponential functions is reported. The results are applied to the general problem of evaluating the average error probability in fading channels. Some examples of applications are also presented for the computation of the pairwise error probability of space-time codes and the average error probability of MPSK and MDPSK in fading channels.     

Trellis Coded Modulation for 4800-9600 bits/s Transmission Over a Fading Mobile Satellite Channel

The combination of trellis coding and MPSK signaling with the addition of asymmetry to the signal set is discussed with regard to its suitabllity as a modulation/coding scheme for the fading mobile satellite channel. For MPSK, introducing nonuniformity (asymmetry) into the spacing between signal points in the constellation buys a further improvement in performance over that achievable with trellis coded symmetric MPSK, all this without increasing the average or peak power, or changing the bandwidth constraints imposed on the system. Whereas previous contributions have considered the performance of trellis coded modulation transmitted over an additive white Gaussian noise (AWGN) channel, the emphasis in this paper is on the performance of trellis coded MPSK in the fading environment. The results will be obtained by using a combination of analysis and simulation. It will be assumed that the effect of the fading on the phase of the received signal is fully compensated for either by tracking it with some form of phase-locked loop or with pilot tone calibration techniques. Thus, our results will only reflect the degradation due to the effect of the fading on the amplitude of the received signal. Also, we shall consider only the case where interleaving/deinterleaving is employed to further combat the fading. This allows for considerable simplification of the analysis and is of great practical interest. Finally, the impact of the availability of channel state information on average bit error probability performance is assessed.     

Composite Phase PDF in Gamma Shadowed Nakagami Fading Channel

This paper presents a new Fourier series form probability density function (PDF) for the phase of the received signal over gamma shadowed Nakagami-m fading channel with additive Gaussian noise. This PDF is further used to evaluate the average symbol-error probability of M-ary CPSK. An alternative new closed-form expression for bit error-rate of binary CPSK is also derived and numerically compared with the corresponding result based on the Fourier series form phase PDF. We show that the two expressions are indeed numerically equivalent.     

Sample size required in error-rate measurement on fading channels

This paper presents analytical and numerical results on the sample size required to achieve a specified root mean square (RMS) error in estimating the error rate for flat fading channels having complex Gaussian statistics. The analysis shows that for the large sample sizes normally used in estimating error rates, k, the required sample size normalized to the required sample size for independent symbol fading, can be expressed in the form k=1+dβ where d is the symbol rate normalized to the Doppler spread of the channel. For a given modem, β is a function of the error probability and the order of diversity. It is shown that if the Doppler spread measure used is proportional to the RMS Doppler spread, P will be relatively insensitive to the shape of the Doppler power spectrum. Numerical results are presented for Lth order diversity reception of binary phase shift keying (PSK), differential PSK, and frequency shift keying (FSK) signals and for five different Doppler power spectra. Ideal maximal ratio combining is assumed for the PSK modem, and square law combining is assumed for the DPSK and FSK modems     

Symbol-error probability and bit-error probability for optimum combining with MPSK modulation

New expressions are derived for the exact symbol error probability and bit-error probability for optimum combining with multiple phase-shift keying. The expressions are for any numbers of equal-power cochannel interferers and receive branches. It is assumed that the aggregate interference and noise is Gaussian and that both the desired signal and interference are subject to flat Rayleigh fading. The new expressions have low computational complexity, as they contain only a single integral form with finite limits and finite integrand.     

Multiple-symbol differential detection for MPSK space-time block codes: decision metric and performance analysis

Approximately 3 dB signal-to-noise ratio (SNR) loss is always paid with conventional differential space-time block codes (STBCs), compared with coherent STBCs. In this paper, a multiple-symbol differential detection (MSDD) technique is proposed for M-ary phase-shift keying (PSK) STBCs. The new scheme can greatly narrow the 3-dB performance gap by extending the observation interval for differential decoding. The technique uses maximum-likelihood sequence detection instead of traditional symbol-by-symbol detection, and is carried out on the slow, flat Rayleigh fading channel. A generalized decision metric is derived for an observation interval of arbitrary length. It is shown that for a moderate number of symbols, MSDD provides approximately 1.5 dB performance improvement over conventional differential detection. In addition, a closed-form pairwise error probability and approximate bit-error probability (BEP) are derived for multiple-symbol differential binary PSK STBC. Results show that the theoretical BEP matches simulation results well. The BEP is shown to converge asymptotically with the number of symbols in the observation interval to that of the differential scheme with coherent detection.     

Impact of diversity reception on fading channels with codedmodulation. I. Coherent detection

We address the problem of designing and analyzing the performance of a coded modulation scheme for the fading channel when space diversity is used. Under fairly general conditions, a channel affected by fading can be turned into an additive white Gaussian noise (AWGN) channel by increasing the number of diversity branches. Consequently, it can be expected (and is indeed verified by our analyses and simulations) that a coded modulation scheme designed to be optimal for the AWGN channel also will perform asymptotically well on a fading channel with diversity. This paper presents bounds on the bit-error probability of a system with coded modulation and diversity for space- and time-correlated Rician flat fading. Specifically, we derive a new method which allows evaluation of the pairwise error probability extremely easily, as well as accurately and computationally fast. The accuracy achieved improves considerably on the widely used, but rather loose Chernoff bound. Starting from this analysis, we study the asymptotic behavior of the fading channel with diversity as the number of diversity branches increases, and we address the effects of diversity on coded modulation performance and design criteria, including the effect on interleaver depth (which affects the total delay of the system)     

New analytical probability of error expressions for classes of orthogonal signals in Rayleigh fading

New exact expressions for the symbol-error rate and bit-error rate of coherent 3-ary and 4-ary orthogonal and transorthogonal signaling in slowly fading Rayleigh channels are derived. New exact error probability expressions for coherent 6-ary and 8-ary biorthogonal signaling in slow Rayleigh fading are also presented. The use of these exact expressions as accurate approximations for the error rates of M-ary orthogonal, biorthogonal, and transorthogonal signaling with arbitrary M is illustrated.     

Performance analysis of space-time MMSE multiuser detection for coded DS-CDMA systems in multipath fading channels

This paper investigates the use of convolutional coding in space-time minimum mean-square-error (MMSE) multiuser-based receivers over asynchronous multipath Rayleigh fading channels. We focus on the performance gain attained through error control coding when used with binary-phase-shift-keyed modulation (BPSK) and multiuser access based on direct sequence-code-division multiple access (DS-CDMA). In our analysis, we derive an approximation for the uncoded probability of bit-error in multipath fading channels. This bit-error rate (BER) approximation is shown to be very accurate when compared to the exact performance. For a convolutionally coded system, we obtain a closed form expression for the bit-error rate upper bound. This error bound is noted to be tight as the number of quantization levels increased beyond eight. Using our theoretical results, we obtain an estimate for the achieved user-capacity that accrues due to error control coding. It is found that using convolutional coding with 3-bit soft-decision decoding, a user-capacity gain as much as 300% can easily be achieved when complete fading state information plus ideal channel interleaving are assumed.     

A useful integral for wireless communication theory and its application to rectangular signaling constellation error rates

A useful integral, representing the average over Rayleigh fading of the product of two Gaussian Q-functions, is solved in closed-form. A closed-form solution for the symbol-error probability of general rectangular quadrature amplitude modulation in Rayleigh fading is derived.     

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     

BPSK Bit Error Outage over Nakagami-m Fading Channels in Lognormal Shadowing Environments

In this letter, we address the problem of finding a tractable expression for the bit-error outage (BEO) defined as the probability to observe a given average bit error rate (BER) over a fading channel in a shadowing environment. Our contribution is two-fold: (1) a simple yet tight approximation of the bit error probability (BEP) for binary phase shift keying (BPSK) over a frequency-flat Nakagami-m fading channel is derived, which (2) facilitates the derivation of a tight lower bound of the BEO in presence of lognormal shadowing in closed form. Theoretical results are corroborated by means of simulation results, confirming the tightness of the bounds.     

Average error probability of digital cellular radio systems using MRC diversity in the presence of multiple interferers

The performance of digital cellular radio systems employing maximal ratio combining diversity is analyzed in a flat-fading channel with cochannel interference and additive white Gaussian noise. It is assumed that the desired signal may experience Rice fading (due to the presence of a line-of-sight component), while the interferers are Rayleigh-faded and may have similar or dissimilar average powers. Exact expressions are derived for the average symbol-error probability of M-ary phase-shift keying modulation in the presence of multiple independent Rayleigh-faded interferers.     

Double differential MPSK on the fast Rician fading channel

An analysis of adjacent and nonadjacent double differential M -ary PSK (M-ary D²PSK) on the fast Rician fading channel is presented. The probability density function (PDF) of the first-order phase error (FOPE) and the probability density functions of the second-order phase error (SOPE) for both adjacent and nonadjacent D²PSK on a fading channel have been derived. Asymptotic approximations for these PDFs are proposed in the case of channels with weak fading (including the additive white Gaussian noise (AWGN) channel) and high signal-to-noise ratio. For nonadjacent D²PSK with independent FOPEs a simplified PDF of SOPE has been obtained. The derived PDFs allow calculation of error performance of D²PSK, as well as of DPSK, on the Rician channel including the AWGN channel and the Rayleigh channel and verification of some results reported in the literature earlier. It is shown that nonadjacent D²PSK can be optimized by choosing a proper separation between the first-order phase differences     

Probability of Block Error for Very Slow Rayleigh Fading in Gaussian Noise

Expressions for probability of block error,P_{f}(M,N), the probability of more thanMerrors in a block ofNdigits, are derived for binary signaling over a channel with very slow nonselective Reyleigh fading and additive Gaussian noise. The analysis is applicable to noncoherent FSK, coherent FSK, and coherent PSK signaling. An asymptotic expression for the block error probability, suitable for high signal-to-noise ratio, is also derived. Computations ofP_{f}(0,N)are presented for a broad range of error probabilities and block sizes, and the characteristics of block error probabilities for the fading and nonfading cases are compared.     

Multiple trellis coded frequency and phase modulation

Multiple trellis coded modulation of constant envelope frequency and phase modulated signal sets (MTCM/FPM) is investigated for performance on the additive white Gaussian noise (AWGN) channel and on the one-sided normal, Rayleigh and Rician fading channels. The Nakagami- m fading model is used as an alternative to the Rician fading model to calculate the error probability upper bound for trellis-coded schemes on the fading channel. The likeliness and the disparity between the upper bounds to the error probability for the two fading models are discussed. The design criteria for the one-sided normal fading channel, modeled by the Nakagami-m distribution, are observed to be the same as those for the Rayleigh-fading channel. For the MTCM/FPM schemes, it is demonstrated that the set partitioning designed to maximize symbol diversity (optimum for fading channels) is optimum for performance on the AWGN channel as well. The MTCM/FPM schemes demonstrate improved performance over MTCM/MPSK schemes and TCM/FPM schemes on the AWGN channel and the fading channel     

“Turbo DPSK”: iterative differential PSK demodulationand channel decoding

Multiple symbol differential detection is known to fill the gap between conventional differential detection of MPSK (M-DPSK) and coherent detection of M-PSK with differential encoding (M-DEPSK). Emphasis has been so far on soft-input/hard-output detectors applied in uncoded systems. In this paper, we investigate a receiver structure suitable for coded DPSK signals on static and time-varying channels. The kernel is an a posteriori probability (APP) DPSK demodulator. This demodulator accepts a priori information and produces reliability outputs. Due to the availability of reliability outputs, an outer soft-decision channel decoder can be applied. Due to the acceptance of a priori information, if the outer channel decoder also outputs reliability information, iterative (“turbo”) processing can be done. The proposed “APP DPSK demodulator” uses linear prediction and per-survivor processing to estimate the channel response. The overall transmission scheme represents a type of serial “turbo code,” with a differential encoder concatenated with a convolutional code, separated by interleaving. The investigated system has the potential to improve the performance of coherent PSK without differential encoding and perfect channel estimation for fading cases! Only a small number of iterations are required. The receiver under investigation can be applied to several existing standards without changing the transmission format. Results are presented for uncoded and convolutionally coded 4-DPSK modulation transmitted over the Gaussian channel and the Rayleigh flat-fading channel, respectively