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Effect of imperfect channel estimation on the performance of selection combining in Rayleigh fading channels
Abstract
The paper presents a general analysis on the performance of selection combining (SC) systems in Rayleigh fading channels with imperfect channel estimation (ICE). The channel estimate and the actual fading are modeled as complex joint Gaussian random variables. Simple single integral formulas with finite integration limits are derived for the symbol error probability (SER) of an arbitrary two-dimensional (2D) modulation format, and a closed-form expression is obtained for the SER of M<sub>s</sub>-PAM. These error probability expressions are then applied to three types of channel estimation errors possibly encountered in practical systems to study their impact on the performance of selection diversity. Other applications of this general analysis include semi-analytical simulation of practical communication systems.
... Then, the SNR penalty, β BPSK A,GSNC , can be computed, using (21) and (24) as ...
In modern wireless systems employing diversity techniques, combining all the available diversity branches may not be feasible due to complexity and resource constraints. To alleviate these issues, subset diversity (SSD) systems have been proposed. Here, we develop a framework for evaluating the symbol error probability for antenna SSD, where the signals from a subset of antenna elements are selected and combined in the presence of channel estimation error. We consider independent identically distributed Rayleigh fading channels and use an estimator structure based on the maximum likelihood (ML) estimate which arises naturally as the sample mean of N<sub>p</sub> pilot symbols. The analysis is valid for arbitrary two-dimensional signaling constellations. The expressions give insight into the performance losses of non-ideal SSD when compared to ideal SSD. Due to estimation error, these losses occur in branch combining as well as in branch selection. However, our analytical results show that the practical ML channel estimator still preserves the diversity order of an ideal SSD system with N<sub>d</sub> branches. Finally, we investigate the asymptotic signal-to-noise ratio penalty due to estimation error.
Diversity techniques play a key role in modern wireless systems, whose design benefits from a clear understanding of how these techniques affect system performance. To this aim we propose a simple class of bounds, whose parameters are optimized, on the symbol error probability (SEP) for detection of arbitrary two-dimensional signaling constellations with diversity in the presence of non-ideal channel estimation. Unlike known bounds, the optimized simple bounds are tight for all signal-to-noise ratios (SNRs) of interest. In addition, these bounds are easily invertible, which enables us to obtain bounds on the symbol error outage (SEO) and SNR penalty. As example applications for digital mobile radio, we consider the SEO in log-normal shadowing and the SNR penalty for both maximal ratio diversity, in the case of unequal branch power profile, and subset diversity, in the case of equal branch power profile, with non-ideal channel estimation. The reported lower and upper bounds are extremely tight, that is, within a fraction of a dB from each other. United States. Office of Naval Research (Presidential Early Career Award for Scientists and Engineers [PECASE] N00014-09-1-0435) National Science Foundation (Grants ECCS- 0636519 and ECCS-0901034) Institute of Advanced Study Natural Science & Technology Fellowship Startup’08 project funded by the University of Ferrara FP7 European project OPTIMIX (Grant Agreement 214625)
The paper examines the impact of Gaussian distributed weighting
errors (in the channel gain estimates used for coherent combination) on
both the output statistics of a hybrid selection/maximal-ratio (SC/MRC)
receiver and the degradation of the average symbol-error rate (ASER)
performance as compared with the ideal case. New expressions are derived
for the probability density function, cumulative distribution function
and moment generating function (MGF) of the coherent hybrid SC/MRC
combiner output signal-to-noise ratio (SNR). The MGF is then used to
derive exact, closed-form, ASER expressions for binary and M-ary
modulations in conjunction a nonideal hybrid SC/MRC receiver in a
Rayleigh fading environment. Results for both selection combining (SC)
and maximal-ratio combining (MRC) are obtained as limiting cases.
Additionally, the effect of the weighting errors on both the outage rate
of error probability and the average combined SNR is investigated. These
analytical results provide insights into the tradeoff between diversity
gain and combination losses, in concert with increasing orders of
diversity branches in an energy-sharing communication system
The performance of various coherent 8-ary and 16-ary modulations
in additive white Gaussian noise (AWGN) and slowly fading channels are
analyzed. New expressions for the exact symbol error rates (SER) in
fading with diversity combining are derived for any two-dimensional
signaling format having polygonal decision boundaries. Maximal ratio
combining, equal gain combining, and selection combining are considered.
The SER formulas obtained make it possible for the first time to
optimize parameters of various constellations precisely and to
determine,which constellation has the lowest probability of error. For
example, a star constellation such as that specified in the CCITT V.29
standard can be improved by adjusting the amplitude ratios of the points
in the constellation to save about 0.63 dB power in AWGN without
sacrificing the phase error tolerance, while maintaining the same error
rate. The sensitivity of each constellation to phase error is presented
and comparisons are made. Six 8-ary signal sets and 11 16-ary signal
sets are examined using the new symbol error probability formulas to
determine best signal sets for fading channels
In a fading channel, maximal ratio diversity combilling improves the average signal-to-noise ratio over thatof a single branch in proportion to the number of diversity branches combined. However, its main advantage is the reduction of the probability of deep fades. The effect of Gaussian errors in the combiner weighting factors on the probability distribution of the output signal-to-noise ratio is computed. The limits on allowable error for a specified probability of fades below any given level are indicated. The results are applied to a mobile radio example in which the weighting factor is determined from a pilot transmitted along with the signal. To keep the pilot from overlapping the signal, they are separated either in frequency or in time. In this case the Gaussian error is due to decorrelation of the pilot from the signal either because their frequency separation or their time separation is too large.
In this paper, we study the effect of imperfect channel estimation (ICE) on the performance of M-ary phase shift keying (M-PSK) with maximum ratio combining (MRC) in generalized Rician fading channels. Using decision variable (DV)-based and signal-to-noise ratio (SNR)-based moment generating function (MGF) approaches, error and outage probability formulas, and SNR statistics for the M-PSK MRC receiver are derived, taking into account the effects of ICE and all relevant system and channel parameters. We analytically quantify the average SNR (ASNR) loss of M-PSK caused by ICE, and provided a unified MGF expression for ICE and different M's. In addition, we point out a major approximation for a popular approach used in the literature to evaluate the adverse effect of Gaussian weighting errors.
The analytical framework reported in X. Dong et al. (May 2003) for calculating the symbol error rate (SER) of two-dimensional (2-D) signaling in Rayleigh fading with channel estimation errors is further developed to address the more general case of frequency-flat Ricean fading. We show that in the presence of channel estimation errors, the SER of arbitrary 2-D signaling with polygonal decision regions in Ricean fading can be expressed as a two-fold proper integral with finite integration limits, which is suitable for numerical evaluation. Moreover, this new analysis is general in the sense that it is applicable to any channel estimation scheme where the estimated and the actual channel gains are jointly complex-Gaussian. The effect of static channel estimation errors and dynamic channel estimation errors introduced by pilot symbol assisted modulation (PSAM) and minimum mean square error (MMSE) channel estimations are studied using the newly derived SER formula. The effect of Doppler frequency shift in the line-of-sight (LOS) component of the channel on the error performance is investigated in our analysis. The analytical and numerical results presented in this work provide a useful tool on choosing suitable signaling formats and optimizing parameters in the communication system design.
A general analytical framework for evaluating the performance of practical coherent two-dimensional (2-D) signaling in frequency flat Rayleigh fading with channel estimation is proposed in this paper. A new and simple analytical expression for the symbol error rate (SER) of an arbitrary 2-D constellation in Rayleigh fading in the presence of channel estimation errors is presented. This framework is applicable to many current channel estimation methods such as pilot symbol aided modulation and minimum mean square error estimation where the fading estimate is a complex Gaussian variable correlated with the channel fading. The sensitivity of various 2-D signaling formats to static and dynamic channel amplitude and phase estimation errors in Rayleigh fading can be easily studied using the derived formula. The new exact SER expression makes it possible to optimize constellation parameters and various parameters associated with channel estimation schemes. It also provides insights into choosing an appropriate signaling format for a fading environment with practical channel estimation methods used at the receiver.
A new channel estimator, termed as the sinc interpolator, based on
the pilot symbol assisted modulation channel sounding is proposed for
Rayleigh fading channels. This method uses sinc function to estimate
channel state information. A Wiener filtering estimator known to have
optimum performance needs a priori information of autocorrelation
function of the channel gain, Doppler frequency and signal-to-noise
ratio to optimize filter coefficients. But the proposed method does not
need a priori information while its performance is similar to that of
the Wiener filtering estimator
The author presents a simple integral expression for calculating
the exact probability of a symbol error for an arbitrary array of signal
points. The integrand contains only elementary functions and the range
of integration is finite. The approach is introduced by applying it to
M -ary phase shift keying (MPSK). The special case of
M =2 gives novel and possibly useful expressions for calculating
the Gaussian tail probability function and the related complementary
error function. The approach is outlined for polygonal decision regions,
and results are given for 16-point signal constellations. A method of
obtaining, not exact, but even simpler and highly accurate expressions
for symbol error probability when the latter is less than a few
hundredths is presented
This paper derives probability density functions (PDFs) to
describe the behavior of data demodulated with imperfect channel
estimation. The PDFs apply to cases of flat Rayleigh-fading,
multiple-diversity reception Rayleigh-fading, and flat Ricean-fading.
These PDFs can be used to determine analytically both the symbol error
rate and bit error rate for multi-amplitude constellations with in-phase
and quadrature components, thus avoiding the need for extensive computer
simulation
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
A new method for calculating symbol error probabilities of two-dimensional signalings in Rayleigh fading with channel estimation errors, " accepted for publication in IEEE Trans
- X Dong
- N C Beaulieu
X. Dong and N. C. Beaulieu, " A new method for calculating symbol error probabilities of two-dimensional signalings in Rayleigh fading with channel estimation errors, " accepted for publication in IEEE Trans. Commun., 2004. (Accessible at http://www.ee.ualberta.ca/˜xdong/channel_err.pdf).