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Bit error rate analysis of hybrid selection/maximal-ratio diversity combining with channel estimation error
Abstract
Hybrid selection/maximal-ratio combining (H-S/MRC) diversity is a very important diversity combining technique in wideband systems. It can be employed to reduce the complexity of RAKE receivers, which are needed to achieve acceptable performance on frequency selective fading channels. A method for calculating the bit error rate (BER) of H-S/MRC diversity systems with channel estimation error is introduced. It is applied to derive BER expressions for binary shift keying (BPSK), quadrature shift keying (QPSK) and M-ary quadrature amplitude modulation (M-QAM). Results for selection combining (SC) and MRC are also obtained as two special cases. The effects of the estimation errors on the system performances are studied in detail. The sensitivities of different modulation schemes to channel estimation errors are examined.
... Molisch et al. [63] investigated the impact of additive channel errors on transmit antenna selection for a diversity system but this work did not take into account the system sensitivity to the specific channel estimation (training) scheme employed. In [13], the effect of channel errors was shown for receive antenna selection. An objective in this chapter is to investigate the effect of channel estimation (using training symbols) on transmit antenna selection. ...
... 13. Spectral efficiency for an OSTBC system versus feedback BER. ...
... Studies of the bit error rate (BER) performance with channel estimation errors and diversity combining in fading channels are published in [4]. Characterizations of the indoor propagation channel have been treated before in many other publications. ...
... Some of the references document the behavior of fixed wireless interfaces with the channel variation occurring due to moving scatterers [15]. Others are based on simulations [4,[16][17][18][19][20][21] or calculations [22][23][24] and not on measurements. Channel conditions for TETRA mobile radio use in forests were documented in [25]; however, this environment has different propagation properties. ...
In the 2.45GHz band, indoor wireless off-body data communication by a moving person can be problematic due to time-variant signal fading and the consequent variation in channel parameters. Off-body communication specifically suffers from the combined effects of fading, shadowing, and path loss due to time-variant multipath propagation in combination with shadowing by the human body. Measurements are performed to analyze the autocorrelation, coherence time, and power spectral density for a person equipped with a wearable receive system moving at different speeds for different configurations and antenna positions. Diversity reception with multiple textile antennas integrated in the clothing provides a means of improving the reliability of the link. For the dynamic channel estimation, a scheme using hard decision feedback after MRC with adaptive low-pass filtering is demonstrated to be successful in providing robust data detection for long data bursts, in the presence of dramatic channel variation.
... Equation (22) has several important implications: 1) SSD with non-ideal channel estimation preserves the diversity of the overall system (N d ) despite the fact that only a subset of antenna elements are selected and combined (L d out of N d ) in the presence of estimation error, 2) the quantity C i completely captures the degradation due to the channel estimation, and 3) the quantity I i captures the geometry of the signaling scheme. Similar to (22), the asymptotic behavior of ISIC for large Γ can be obtained from (20) 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.
The conventional single-antenna receiver suffers in wireless fading channels from limitations that preclude deployment of envisioned wireless applications. By increasing complexity, improvements are possible using multi-branch receivers. In particular, smart antenna arrays em- ploy maximal-ratio combining (MRC) or statistical beamforming (BF) to exploit diversity and array gain. However, varying azimuth spread creates unfavourable spatial correlation conditions that diminish these gains, while BF and MRC complexity remains constant. On the other hand, adaptive eigen-combining can yield near-optimum performance for more efficient resource us- age. This motivates our study of maximal-ratio eigen-combining (MREC).
We unravel the relationship between MREC, BF, and MRC performance, and evaluate their complexity. Outage and average error probability expressions are derived for MREC assuming perfectly and imperfectly known channel gains. These results are specialized to MRC and BF, as well as to well-accepted pilot-symbol-based channel estimation techniques. In the process, new performance analyses are provided.
Numerical results for typical urban scenarios with variable correlation demonstrate MREC’s advantages. Existing criteria for optimum eigen-mode selection in MREC are reviewed, and a new adaptation approach that accounts for channel condition, algorithm complexity, resource availability, and intended performance level, is proposed and evaluated.
These single- and multi-branch receivers are then evaluated on a field-programmable gate array (FPGA) in terms of symbol-detection performance and resource and power consumption.
MREC flexibility is shown to yield near-optimum performance for half of the hardware and power requirements of MRC, or, equivalently, a doubling of the number of users which can be handled with the same hardware. Smarter antennas, i.e., array receivers aware of the channel- statistics, resource availability, and required performance, can thus be deployed.
Finally, for code-division multiple access (CDMA) systems, we specify an eigen-combining approach. A recently-developed signal despreading method, which eliminates the intended signal, is exploited for interference-plus-noise correlation matrix calculation. After some trans- formations, combining can once again be relegated to a few eigen-modes, for lower complexity and improved performance.
In the 2.45 GHz band, indoor wireless off-body data communication by a moving person can be problematic due to time-variant signal fading and the consequent variation in channel parameters. Off-body communication specifically suffers from the combined effects of fading, shadowing, and path loss due to time-variant multipath propagation in combination with shadowing by the human body. Measurements are performed to analyze the autocorrelation, coherence time, and power spectral density for a person equipped with a wearable receive system moving at different speeds for different configurations and antenna positions. Diversity reception with multiple textile antennas integrated in the clothing provides a means of improving the reliability of the link. For the dynamic channel estimation, a scheme using hard decision feedback after MRC with adaptive low-pass filtering is demonstrated to be successful in providing robust data detection for long data bursts, in the presence of dramatic channel variation.
In the paper, we investigate the selection algorithm of hybrid selection and maximum-ratio combining systems. Most existing algorithms require channel coefficients of all receive antennas. It is difficult to satisfy for reduced number of RF chains. Therefore, a new selection algorithm is proposed so that the number of required channel coefficients equals that of RF chains. The presented algorithm uses the remainder antenna to substitute the worse one whose channel coefficient is estimated. Simulation results show that the performance of proposed algorithm is greatly close to that of optimal selection, especially for larger number of RF chains. The proposed algorithm is asymptotic-optimal in performance; moreover, it doesn't need excessive channel estimation.
Diversity techniques will play a key role in next generation wireless communication systems, thus system design will benefit from a clear understanding of how these techniques affect system performance. To this aim we propose simple bounds, optimized within a given class, on the symbol error probability (SEP) in the presence of non-ideal channel estimation for arbitrary two-dimensional signaling constellations. Unlike known bounds, the optimized simple bounds are tight for all signal-to-noise ratios (SNRs). In addition, these bounds are easily invertible, which enables us to obtain bounds on the symbol error outage (SEO) and SNR penalty. As an example application for digital mobile radio, we consider the SEO in log-normal shadowing for both maximal ratio combining with unequal branch power profile and subset microdiversity. The reported lower and upper bounds are extremely tight, that is, within a fraction of a dB from each other.
The performance of a multicarrier direct sequence code division multiple access (MC-DS-CDMA) system in fading environment has been extensively analysed in the literatures. The analysis that is available in the literatures did not take into account the impact of jamming and channel estimation errors on the performance of the system. In this study, closed-form expressions of the average bit error probability (BEP) of the MC-DS-CDMA system with channel estimation errors in the presence of partial band, broadband and multitone jamming are derived in a Rayleigh fading channel. The analysis shows that the magnitude of the normalised correlation between the actual and the estimated complex fading channel gain, A , greatly affects the performance of the MC-DS-CDMA system. For large signal-to-noise ratio, the average BEP with A =1 and without multiple access interference (MAI) tends to zero. When A ¿1, the BEP of the system exhibits an error floor even without MAI. This error floor increases as the number of users increases. It is also shown that increasing the number of the system subcarriers enhances the system performance against jamming.
In this paper, we analyze the error probability performance of an uncoded multicarrier frequency-hopping coherent binary phase-shift keying (MC-FH/BPSK) signal transmitted over a Rayleigh fading channel with multitone interference (MTI) and channel estimation errors. The channel estimation errors are modeled as Gaussian random variables. The performance in presence of worst case MTI is evaluated as a function of signal to noise ratio (SNR), signal to jamming ratio, and channel estimation errors. The effect of channel estimation errors on the error probability performance is investigated in presence and absence of MTI. It is shown, from the analytical results that when channel estimation errors are not neglected, the asymptotic error probability performance for high SNR exhibits an error floor even in the absence of MTI. Worst case MTI causes large increase in this error floor. It is also shown that increasing the subband order L enhances the system performance against multitone interference.
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)
Optimum diversity receivers that operate in the presence of estimation errors are proposed. Previous work has mainly examined the performance of maximal ratio combining (MRC) with estimation errors. It is shown that MRC is not optimal when estimation errors occur. Moreover, it is shown that the optimum diversity receiver in the presence of estimation errors can be derived by using knowledge of the channel estimate statistics. Numerical results show that the optimum diversity receiver can perform as much as 2.0 dB in signal-to-noise ratio better than the conventional MRC receiver in some cases.
Single- and Multi-carrier Quadrature Amplitude Modulation Principles and Applications for Personal Communications, WLANs and Broadcasting L. Hanzo Department of Electronics and Computer Science, University of Southampton, UK W. Webb Motorola, Arlington Heights, USA formerly at Multiple Access Communications Ltd, Southampton, UK T. Keller Ubinetics, Cambridge Technology Centre, Melbourn, UK formerly at Department of Electronics and Computer Science, University of Southampton, UK Motivated by the rapid evolution of wireless communication systems, this expanded second edition provides an overview of most major single- and multi-carrier Quadrature Amplitude Modulation (QAM) techniques commencing with simple QAM schemes for the uninitiated through to complex, rapidly-evolving areas, such as arrangements for wide-band mobile channels. Targeted at the more advanced reader, the multi-carrier modulation based second half of the book presents a research-orientated outlook using a variety of novel QAM-based arrangements.
* Features six new chapters dealing with the complexities of multi-carrier modulation which has found applications ranging from Wireless Local Area Networks (WLAN) to Digital Video Broadcasting (DVB)
* Provides a rudimentary introduction for readers requiring a background in the field of modulation and radio wave propagation
* Discusses classic QAM transmission issues relevant to Gaussian channels
* Examines QAM-based transmissions over mobile radio channels
* Incorporates QAM-related orthogonal techniques, considers the spectral efficiency of QAM in cellular frequency re-use structures and presents a QAM-based speech communications system design study
* Introduces Orthogonal Frequency Division Multiplexing (OFDM) over both Gaussian and wideband fading channels
By providing an all-encompassing self-contained treatment of single- and multi- carrier QAM based communications, a wide range of readers including senior undergraduate and postgraduate students, practising engineers and researchers alike will all find the coverage of this book attractive.
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 bit-error rate (BER) performance of multilevel quadrature amplitude modulation with pilot-symbol-assisted modulation channel estimation in static and Rayleigh fading channels is derived, both for single branch reception and maximal ratio combining diversity receiver systems. The effects of noise and estimator decorrelation on the received BER are examined. The high sensitivity of diversity systems to channel estimation error is investigated and quantified. The influence of the pilot-symbol interpolation filter windowing is also considered.
Several methods of diversity combining for a Rayleigh-faded
channel are evaluated and compared. The methods considered are, for
coherent reception, maximal ratio combining (MRC), selection combining
(SC), and a generalization of SC, whereby the two (three) signals with
the two (three) largest amplitudes are coherently combined. We will call
this method second (third) order SC, and denote it SC2 (SC3). Similar
techniques are also investigated for noncoherent reception, with equal
gain combining (EGC) replacing MRC, and noncoherent versions of SC2 and
SC3. Numerical results indicate that SC2 and SC3 significantly enhances
the bit-error rate (BER) performance relative to that achievable with
SC, and under certain conditions approaches the performance achieved by
MRC or EGG. The performance enhancement of SC2 and SC3 is especially
noticable for noncoherent reception, where EGC is seen to provide the
best performance only for low BER values. In fact, when the BER is 10
<sup>-3</sup> or greater, SC2 and SC3 performed comparably to EGG, and
in some cases performed better than EGC
This book is intended to provide a comprehensive coverage of digital communication systems for senior-level undergraduates, first-year graduate students, and practicing engineers. Even though the emphasis of the book is on digital communications, necessary analog fundamentals are included, since analog waveforms are used for the radio transmission of digital signals. Contents: Signals and spectra, Formatting and baseband transmission, Bandpass modulation and demodulation, Communications link analysis, Channel coding, Modulation and coding tradeoffs, Synchronization, Multiplexing and multiple access, Spread-spectrum techniques, Source coding, Encryption and decryption, A review of Fourier techniques, Fundamentals of statistical decision theory.
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.
The method used in Aghamohammadi and Meyr (1990) for finding the
error probability of linearly modulated signals on Rayleigh
frequency-flat fading channels has been applied to the more general case
of Ricean fading. A signal received on a fading channel is subject to a
multiplicative distortion (MD) and to the usual additive noise.
Following a compensation of the MD, the signal provided to the detector
may be thought to include only a single additive distortion term
(“final noise”), which comprises the effects of the original
additive noise, the MD, and the error in MD compensation. An exact
expression for the probability density function of the final noise is
derived. This allows calculation of error probability for arbitrary
types of linear modulations. Results for many cases of interest are
presented. Furthermore, as special cases of Ricean fading, error
probability for Rayleigh fading and non-fading channels are obtained
which either match the results or complete the approximate derivations
formerly known from the literature
Thesis (M. Sc.)--Queen's University at Kingston, 1995. Includes bibliographical references.
A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science, Dept. of Electrical and Computer Engineering, University of Alberta. Thesis (M. Sc.)--University of Alberta, 2003. Includes bibliographical references.
Tabulation of definite and indefinite integrals of products of error function with elementary and transcendental functions
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
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
Using the notion of the “spacing” between ordered
exponential random variables, a performance analysis of the generalized
selection combining (GSC) diversity scheme over Rayleigh fading channels
is presented and compared with that of the conventional maximal-ratio
combining and selection combining schemes. Starting with the moment
generating function (MGF) of the GSC output signal-to-noise ratio (SNR),
we derive closed-form expressions for the average combined SNR, outage
probability, and average error probability of a wide variety of
modulation schemes operating over independently, identically distributed
(i.i.d.) diversity paths. 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 non-i.i.d.
diversity paths
We determine the bit-error rate (BER) of multilevel quadrature
amplitude modulation (M-QAM) in flat Rayleigh fading with imperfect
channel estimates, Despite its high spectral efficiency, M-QAM is not
commonly used over fading channels because of the channel amplitude and
phase variation. Since the decision regions of the demodulator depend on
the channel fading, estimation error of the channel variation can
severely degrade the demodulator performance. Among the various fading
estimation techniques, pilot symbol assisted modulation (PSAM) proves to
be an effective choice. We first characterize the distribution of the
amplitude and phase estimates using PSAM. We then use this distribution
to obtain the BER of M-QAM as a function of the PSAM and channel
parameters. By using a change of variables, our exact BER expression has
a particularly simple form that involves just a few finite-range
integrals. This approach can be used to compute the BER for any value of
M. We compute the BER for 16-QAM and 64-QAM numerically and verify our
analytical results by computer simulation. We show that for these
modulations, amplitude estimation error leads to a 1-dB degradation in
average signal-to-noise ratio and combined amplitude-phase estimation
error leads to 2.5-dB degradation for the parameters we consider
We use a novel virtual branch technique to succinctly derive the mean and variance of the combiner output signal-to-noise ratio for hybrid selection/maximal-ratio combining in a multipath-fading environment
This paper introduces two general relationships concerning the
output signal-to-noise ratio distribution and the bit error performance
of the nonideal maximal ratio combiner. The imperfections are modeled as
Gaussian distributed weighting errors in the channel gain estimates used
for coherent combination. The general form for the bit error rate is
applicable to any modulation format
The author presents pilot-symbol-assisted modulation (PSAM) on a
solid analytical basis, a feature missing from previous work.
Closed-form expressions are presented for the bit error rate (BER) in
binary-phase-shift-keying (BPSK) and in quadrature-phase-shift-keying
(QPSK), for a tight upper bound on the symbol error rate in 16
quadrature-amplitude-modulation (16-QAM), and for the optimized receiver
coefficients. The error rates obtained are lower than for differential
detection for any combination of signal-to-noise ratio (SNR) and Doppler
spread, and the performance is within 1 dB of a perfect reference system
under slow-fading conditions and within 3 dB when the Doppler spread is
5% of the symbol rate