Improved bounds for coherent M-ary PSK symbol error probability

The M-ary phase-shift keying (MPSK) symbol error probability bounds used in the past are simple to calculate, but not particularly accurate. More recently, very tight bounds have been proposed, but they are not true bounds since they are based on an arbitrary truncation of an infinite series expansion. Two new upper and lower bounds that are straightforward and very tight, even as the signal energy goes to zero, are derived by the authors     

Bit error probability for coherent M-ary PSK systems

A simple technique for evaluating the bit-error probability of coherent M-ary phase-shift keying (PSK) with any bit-mapping is proposed. Closed-form expressions are given in terms of error functions for half- and quadrant-plane probabilities in the decision space, avoiding the numerical evaluation of complicated integrals that occurs in the direct method. Bit error probability expressions for M-ary PSK with Gray, natural binary, and folded binary bit-mappings are derived     

A new formula for MDPSK symbol error probability

This article presents an extremely compact new formula for the symbol error probability in M-ary differential phase shift keying (MDPSK) that is extraordinarily similar to its counterpart for M-ary phase shift keying (MPSK). This result follows from a new form for a generic error probability associated with the phase angle between two vectors perturbed by Gaussian noise. The new form also leads to a new expression for the Marcum Q-function. As special cases, the generic error probability is noted to also contain MPSK, CPFSK, and digital FM. Finally, it is pointed out that the generic error probability simplifies the error probability expressions in other situations such as nonorthogonal signaling, maximum-likelihood differential detection of DPSK with block-by-block detection, and Gray coding of MDPSK     

On symbol error probability bounds for ISI-like channels

Some issues with Forney's upper and lower bounds (1972) for the symbol error probability in systems with memory (e.g., intersymbol interference channels) have been pointed out in the literature. We expound on these issues. For the upper bound, we show that, although the most commonly cited proofs are not logically consistent, the bound is true for more general conditions. The reasoning leading to the lower bound is shown to be flawed and, in general,to lead to invalid lower bounds. We suggest a lower bound based on Mazo's bound (1975) as an alternative     

M-ary symbol error outage over Nakagami-m fading channels in shadowing environments

This letter addresses the problem of finding a tractable expression for the symbol error outage (SEO) in flat Nakagami-m fading and shadowing channels. We deal with M-ary phase shift keying (M-PSK) and quadrature amplitude modulation (M-QAM) which extends our previous results on BPSK signaling. We propose a new tight approximation of the symbol error probability (SEP) holding for M-PSK and M-QAM signals which is accurate over all signal to noise ratios (SNRs) of interest. We derive a new generic expression for the inverse SEP which facilitates the derivation of a tight approximation of the SEO in a lognormal shadowing environment.     

The error probability of M-ary PSK block coded modulation schemes

A tight upper bound on the decoding error probability is derived for block-coded modulation structures where an M-ary phase shift keying (M-PSK) signal constellation is employed. This bound, called a tangential sphere bound, is tight for very low (as well as for high) signal-to-noise ratios (SNRs). Berlekamp's tangential union bound, previously derived for binary codes, can be derived for an M-PSK block coded modulation structure as well. However, it is proven that our tangential sphere bound is tighter than Berlekamp's (1980) tangential bound. For particular schemes, it is shown that for low SNRs our bound is considerably tighter than the tangential bound. As one of the examples, a multistage decoder is considered     

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.     

On the probability of symbol error in Viterbi decoders

An upper bound is derived on the probability that at least one of a sequence of B consecutive bits at the output of a Viterbi (1979) decoder is in error. Such a bound is useful for the analysis of concatenated coding schemes employing an outer block code over GF(2^B) (typically a Reed-Solomon (RS) code), an inner convolutional code, and a symbol (GF(2^B)) interleaver separating the two codes. The bound demonstrates that in such coding schemes a symbol interleaver is preferable to a bit interleaver. It also suggests a new criterion for good inner convolutional codes     

Exact symbol error probability of m-psk for multihop transmission with regenerative relays

We determine the exact symbol error probability of M-ary phase shift keying (M-PSK) for multihop communication systems with regenerative relays, where the source terminal transmits data to the destination terminal via a set of intermediate relay stations, which perform hard decisions on the received symbols before forwarding them to their respective successor node. Both, time-invariant additive white Gaussian noise channels as well as frequency-flat fading channels are considered and we derive generic expressions, which might be easily evaluated numerically or even be given in closed-form for various cases.     

On the error probability of binary and M-ary signals in Nakagami-m fading channels

In this letter, we present new closed-form formulas for the exact average symbol-error rate (SER) of binary and M-ary signals over Nakagami-m fading channels with arbitrary fading index m. Using the well-known moment generating function-based analysis approach, we express the average SER in terms of the higher transcendental functions such as the Gauss hypergeometric function, Appell hypergeometric function, or Lauricella function. The results are generally applicable to arbitrary real-valued m. Furthermore, with the aid of reduction formulas of hypergeometric functions, we show previously published results for Rayleigh fading (m=1) as special cases of our expressions.     

Average error probability for quadriphase modulated DS-SSMA communications through Nakagami fading channel

An asynchronous direct sequence spread spectrum multiple access system using quadrature phase shift keying modulation through Nakagami'sm-distributed fading channel is considered for nondiversity reception. Approximation to the average error probability is evaluated in two steps. Using the Gauss quadrature rule, the moments of the multiple interferences are used to evaluate the conditional probability conditioned on a fixed fading amplitude of the desired signal. The probability density function of the desired signal is Nakagami distributed. The average error probability which is the expected value of the conditional probability is evaluated by the trapezoidal integration method. This method provides a good approximation to the average error probability for a small to a fairly large number of users. Numerical results are presented for a set of Gold code of code length 127.     

Average digit error probability in decoding a linear block code

An upper bound on the average digit error probability of a linear block code is given which is dependent only upon the minimum distance of the code. The tightness of this bound is also demonstrated, and an example is given where knowledge of the average digit error probability is important.     

Simulated symbol error probability for a matched filterdifferential detector with variable delay for continuous phasemodulation

A matched-filter differential detector for continuous phase modulation is proposed and simulated error probability for this detector is presented. The detector consists of a conventional differential detector operating with a delay, a bank of matched filters, and a Viterbi processor to perform sequence decision. The author shows that quaternary schemes outperform binary schemes, when the schemes are compared in a power-bandwidth tradeoff. Schemes that lose about 3.4 dB compared to coherently detected minimum shift keying (MSK) at 10^-3 on a Gaussian channel, but with a bandwidth of about 0.6 times the MSK bandwidth, are presented     

Exact average symbol error probability of optimum combining with arbitrary interference power

A new expression is derived for the exact average symbol error probability for optimum combining with M-ary phase-shift keying. The expression adds to the significant body of work in this field by handling interferers with arbitrary power levels. The expression involves a single integral with finite limits and finite integrand. A closed-form expression is also derived for the average symbol error probability for binary phase-shift keying.     

Exact symbol error probability of a Cooperative network in a Rayleigh-fading environment

In a distributed spatial diversity wireless system, not all antennas are located at one station as in classical transmit diversity systems, but are dispersed at different, possibly mobile, stations in the network. Transmit diversity is created when the selected stations assist a sender by relaying its information signal to the destination. In this letter, we present an exact average symbol error rate analysis for the distributed spatial diversity wireless system with K amplifying relays in a Rayleigh-fading environment. The average symbol error rate formula allows us to clearly illustrate the advantage that the distributed diversity system has in overcoming the severe penalty in signal-to-noise ratio caused by Rayleigh fading. Using simple bounds on the probability of error, we show that the cooperative network presented in this letter achieves full diversity order.     

Virtual branch analysis of symbol error probability for hybridselection/maximal-ratio combining in Rayleigh fading

We derive analytical expressions for the symbol error probability (SEP) for a hybrid selection/maximal-ratio combining (H-S/MRC) diversity system in multipath-fading wireless environments. With H-S/MRC, L out of N diversity branches are selected and combined using maximal-ratio combining (MRC). We consider coherent detection of M-ary phase-shift keying (MPSK) and quadrature amplitude modulation (MQAM) using H-S/MRC for the case of independent Rayleigh fading with equal signal-to-noise ratio averaged over the fading. The proposed problem is made analytically tractable by transforming the ordered physical diversity branches, which are correlated, into independent and identically distributed (i.i.d.) “virtual branches,” which results in a simple derivation of the SEP for arbitrary L and N. We further obtain a canonical structure for the SEP of H-S/MRC as a weighted sum of the elementary SEPs, which are the SEPs using MRC with i.i.d. diversity branches in Rayleigh fading, or equivalently the SEPs of the nondiversity (single-branch) system in Nakagami fading, whose closed-form expressions are well-known. We present numerical examples illustrating that H-S/MRC, even with L≪N, can achieve a performance close to that of N-branch MRC     

A Laguerre polynomial-based bound on the symbol error probability for adaptive antennas with optimum combining

We derive a simple closed-form upper bound on the symbol error probability for coherent detection of M-ary phase-shift keying using antenna arrays with optimum combining, in the presence of multiple uncorrelated equal-power cochannel interferers and thermal noise in a Rayleigh fading environment. The new bound, based on Laguerre polynomials, is valid for an arbitrary number of antenna elements as well as arbitrary number of interferers, and it is proven to be asymptotically tight. Comparisons with Monte Carlo simulation are also provided, showing that our bound is useful in many cases of interest.     

On the symbol error probability of general order rectangular qam in nakagami-m fading

Recently, Beaulieu, following an ingenious concept, presented a closed-form expression for a useful integral, which was used for the evaluation of the symbol error probability (SEP) of general order rectangular quadrature amplitude modulation (QAM) in slow Rayleigh fading. In this letter, these results are extended to Nakagami-m fading channels, deriving a novel closed-form formula for the average over Nakagami-m fading of the product of two Gaussian Q-functions, which can be efficiently used to study the impact of fading severity on the error performance of general rectangular QAM constellations