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Joint channel estimation and symbol detection in Rayleigh flat-fading channels with impulsive noise

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IEEE COMMUNICATIONS LETTERS, VOL. 1, NO. 1, JANUARY 1997 19
Joint Channel Estimation and Symbol Detection inRayleigh Flat-Fading Channels with Impulsive Noise
Xiaodong Wang and H. Vincent Poor,
Fellow, IEEE
Abstract—
A new technique for joint channel estimation andsymbol detection in the Rayleigh ﬂat-fading channels with impul-sive noise is developed. This technique involves an approximationto the likelihood statistics for such channels, which in turn isbased on the Masreliez approximation of nonlinear ﬁltering. It isseen that the proposed detector outperforms the detector basedon the Kalman ﬁlter.
Index Terms—
Channel estimation, impulsive noise, Masrelieznonlinear ﬁltering, Rayleigh ﬂat-fading.
I. I
NTRODUCTION
M
ANY mobile communication channels can be modeledas Rayleigh ﬂat-fading channels. Coherent detection of the digital signal transmitted over such channels has betterpower efﬁciency, and it eliminates the error ﬂoor exhibitedby the conventional DPSK receivers [6]. The MAP sequencedetection receiver requires a different Kalman ﬁlter for eachpossible symbol sequence [3]. The complexity of the coherentreceiver can be reduced if symbol-by-symbol detection isused [3] or the channel memory is ﬁnite [6]. Several recentworks have addressed the symbol-aided coherent detection infading channels [4], [5], where channel estimation is assistedby periodically inserted pilot symbols. Some low complexitysequence detectors that jointly estimate the channel and datasymbols in the per-survivor fashion have also been proposed[2], [10].So far most of the work on coherent detection in fadingchannels assumes that the additive channel noise is whiteand Gaussian. However, the ambient noise in many practicalmobile communication channels is impulsive, resulting fromvarious natural and manmade impulsive sources. The purposeof this letter is to develop a technique for coherent detectionin Rayleigh ﬂat-fading channels with additive impulsive noise.II. S
YSTEM
M
ODEL
Consider a sequence of uncoded -ary PSK symbolstransmitted through a Rayleigh ﬂat-fading channel with addi-tive impulsive noise. The symbol transmitted during the thsignaling interval is , for , where. Assuming symbol-rate sampling, the received signal sample in the th signaling
Manuscript received October 10, 1996. This work was supported by theU. S. Army Research Ofﬁce under Grant DAAH04-93-G-0219. The associateeditor coordinating the reveiew of this letter and approving it for publicationwas Dr. Y. Bar-Ness.The authors are with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544 USA.Publisher Item Identiﬁer S 1089-7798(97)01343-4.
interval is [10](1)where is the Rayleigh fading gain process—that is, eachis a complex Gaussian random variable with independentand identically distributed (i.i.d.) in-phase and quadrature (I/Q)components; is an i.i.d. complex random process, and theI/Q components of each are i.i.d. with a general pdf. Usingreal vector notation hereafter, (1) can be written as(2)wherewith the superscripts and representing the I/Q componentsof the complex signals. The fading coefﬁcients thatapproximate the statistical properties of the Rayleigh fadingmodel can be generated by feeding white complex Gaussiannoise through a linear ﬁlter, expressed in a state-space form as(3)(4)where is a vector of state variables:, , and arematrix state-space parameters; and . Note thatwith appropriate choice of parameters, the model given by (3)and (4) can represent any process with a rational spectrum,or any ARMA process. Moreover, most covariance stationaryprocesses have spectra that can be approximated arbitrarilyclosely by rational spectra with appropriate choice of orders.Substituting (4) into (2) we obtain(5)III. O
PTIMUM
D
ETECTOR
S
TRUCTURE
Deﬁne , . Themaximum-likelihood sequence detector (MLSD) chooses thesymbol sequence that maximize the likelihood function,i.e.,(6)
1089–7798/97$10.00
1997 IEEE
20 IEEE COMMUNICATIONS LETTERS, VOL. 1, NO. 1, JANUARY 1997
The density function can be written as(7)Now using the state measurement model of (5) we have(8)where , and is the state predic-tion density. It can be shown from the model of (3) and (4)that the satisﬁes a joint recursion with thestate ﬁltering density , given by [8](9)(10)with initial condition . We seefrom above that the optimum detection statistic for themaximization problem (6), can be computed by determiningfrom the recursion (8)–(10).
A. Example—Gaussian Measurement Noise Case
Suppose that . In this case, the densitiesappearing in (8)–(10) are Gaussian densities and so are charac-terized by their means and covariance matrices. In particular,we have(11)(12)(13)where ,for and can be computed recursively by theKalman ﬁler [1]; ,. It then follows straightforwardly that the MLSD givenby (6) chooses the symbol sequence that minimizes alikelihood metric deﬁned as(14)IV. ACM P
REDICTOR
-B
ASED
D
ETECTION
We now consider the situation in which the measurementnoise sequence in (5) has a general marginal densityfunction . A particular useful approximation for the linearstate and measurement relationship described by (3) and (5)is the
Masreliez approximation
, which has been applied in[7]–[9] to the problem of obtaining ﬁnite-dimensional ap-proximation to the optimum ﬁlter/predictor with non-Gaussianmeasurement noise. In particular, it is assumed that(15)Let . Then the measurement prediction density (8)can be rewritten as(16)Using the Masreliez approximation (15) we have(17)with and . Thus it followsfrom (16) that the measurement prediction density can bewritten as(18)where the function is given by(19)with denoting the Cholesky factor of the matrix . More-over, satisﬁes a nonlinear joint recursionwith given by [7](20)(21)(22)(23)where(24)(25)This structure is known as an
approximate conditional mean
(ACM) ﬁlter. It follows that in this case the MLSD given by (6)chooses the symbol sequence that minimizes a likelihoodmetric deﬁned as(26)The computational complexity of the MLSD is ,which is certainly prohibitive for any practical considerations.Next we consider two suboptimal detectors based on thelikelihood metric in (26).
WANG AND POOR: RAYLEIGH FLAT-FADING CHANNELS 21
Fig. 1. Performance of the ACM-ﬁlter-based detector.
V. S
IMPLER
D
ETECTOR
S
TRUCTURES
The simplest suboptimal receiver is obtained when it isassumed that at time all previously transmitted symbolsare known [3]. Then only the likelihood metricneeds to be evaluated for each of the possiblevalues of , and it is easily seen that the solution is given by(27)where in (27) and are explicitly expressed as a functionof (through the matrix ) conditioned on .Using the likelihood metric (26), a per-survivor sequencedetector (e.g., [2], [10]) can also be developed.
B. Example—QPSK Signaling with Impulsive Noise
Suppose and are independent i.i.d. processeswith the sample density function given by
1
(28)We consider QPSK signaling, i.e., . It iseasily seen from (3) and (5) that in this case the I/Q processesof the received complex signal are independent. The densitycan be written as(29)where , and(30)
1
That is,
0
.
Fig. 1 shows the simulated performance of the ACM-ﬁlter-based detector versus the performance of the Kalman-ﬁlter-based detector, assuming is known at time (). The channel is modeled by a third-order Butterworth ﬁlter, with cutoff frequency equal to themaximum channel Doppler shift normalized to the symbolrate , which is used as a measure of the fading rate. isthe number of diversity channels. It is seen that the proposeddetector outperforms the detector based on the Kalman ﬁlterby about 3 dB in this example.VI. C
ONCLUSION
We have developed a technique for joint channel estimationand symbol detection in Rayleigh ﬂat-fading channels withimpulsive noise. This technique, which is based on the ACMmethod of nonlinear ﬁltering, is seen to outperform conven-tional methods based on Kalman ﬁlters with little attendantincrease in complexity.R
EFERENCES[1] B. D. O. Anderson and J. B. Moore.
Optimal Filtering
. EnglewoodCliffs, NJ: Prentice-Hall, 1979.[2] A. N. D’Andrea, A. Diglio, and U. Mengali, “Symbol-aided channelestimation with nonselective Rayleigh fading channels,”
IEEE Trans.Commun.
, vol. 44, pp. 41–48, Feb. 1995.[3] R. Haeb and H. Meyr, “A systematic approach to carrier recovery anddetection of digitally phase modulated signals on fading channels,”
IEEE Trans. Commun.
, vol. 37, pp. 748–754, July 1989.[4] G. T. Irvine and P. J. McLane, “Symbol-aided plus decision-directedreception for PSK/TCM modulation on shadowed mobile satellite fadingchannels,”
IEEE J. Select. Areas Commun.
, vol. 10, pp. 1289–1299, Oct1992.[5] Y. Liu and S. D. Blostein, “Identiﬁcation of frequency nonselectivefading channels using decision feedback and adaptive linear prediction,”
IEEE Trans. Commun.
, vol. 43, pp. 1484–1492, Feb./Mar./Apr. 1995.[6] J. H. Lodge and M. L. Moher, “Maximum likelihood sequence estima-tion of CPM signals transmitted over Rayleigh ﬂat-fading channels,”
IEEE Trans. Commun.
, vol. 38, pp. 787–794, June 1990.[7] C. J. Masreliez, “Approximate non-Gaussian ﬁltering with linear stateand observation relations,”
IEEE Trans. Automat. Contr.
, vol. AC-20,pp. 107–110, Jan. 1975.[8] H. V. Poor, “Detection of stochastic signals in non-Gaussian noise,”
J. Acoust. Soc. Amer.
, vol. 94, pp. 2838–2850, Nov. 1993.[9] R. Vijayan and H. V. Poor, “Nonlinear techniques for interferencesuppression in spread spectrum systems,”
IEEE Trans. Commun.
, vol.38, pp. 1060–1065, July 1990.[10] G. M. Vitetta and D. P. Taylor, “Maximum likelihood decoding of uncoded and coded PSK signal sequences transmitted over Rayleighﬂat-fading channels,”
IEEE Trans. Commun.
, vol. 44, pp. 2750–2758,Nov. 1995.

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