Description

Joint Data Detection and Channel Estimation for Fading Unknown Time-Varying Doppler Environments

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Transcript

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010 2277
Joint Data Detection and Channel Estimation forFading Unknown Time-VaryingDoppler Environments
Usa Vilaipornsawai and Harry Leib
Abstract
—This work considers a joint channel estimationand data detection technique for Multiple Space-Time TrellisCodes (MSTTCs) operating over unknown time-varying channelswith large Doppler spread. We propose an algorithm, calledDoppler Adaptive Smoothed Data Detection and Kalman Es-timation (DA-SDD-KE), that jointly detects data and estimatesthe channel as well as the time-varying Doppler. In this scheme,an Adaptive Kalman Predictor (AKP) consisting of a KP anda covariance-based Doppler estimator is incorporated into aPer-Survivor Processing (PSP)-based algorithm that utilizes thepast, present and future received symbols for smoothed datadetection. For comparison purposes, we also develop a DopplerAdaptive version of the Delayed Mixture Kalman Filtering(DMKF) technique, referred to as DA-DMKF, where the adaptiveestimations of the channel and the Doppler shift are basedon sequences of importance samples. Moreover, we propose amodel for generating a Rayleigh fading process with time-varyingDoppler using the sum of sinusoids method. The performance of the DA-SDD-KE and DA-DMKF algorithms over channels withconstant, linear and quadratic Doppler functions is evaluatedusing computer simulations, revealing that the DA-SDD-KEalgorithm performs well for all considered Doppler functions,and provides a considerably gain over the DA-DMKF algorithm.
Index Terms
—Per-survivor processing, Kalman estimation,decoding, channel estimation, Doppler estimation, space-timecoding.
I. I
NTRODUCTION
S
PACE-TIME Trellis Codes (STTCs) [1], while crucial tomodern wireless communications, are designed under theassumption of perfect Channel State Information (CSI) at thereceiver. In practice, this assumption is unrealistic and CSIhas to be estimated. One common method to achieve this taskis to use periodically transmitted pilot symbols for channelestimation [2]. This approach, however, leads to loss in powerand bandwidth ef
Þ
ciencies, especially for fast time-varying(high Doppler) fading channels [3].
Paper approved by T. M. Duman, the Editor for Coding Theory andApplications of the IEEE Communications Society. Manuscript receivedAugust 27, 2009; revised January 3, 2010.U. Vilaipornsawai was with the Department of Electrical & ComputerEngineering, McGill University, Montreal, Quebec, Canada, H3A 2A7. Sheis now a research fellow at the Institute for Systems and Robotics, Uni-versity of Algarve, Campus de Gambelas, Faro, Portugal, 8005-139 (e-mail:usa.vilaipornsawai@mail.mcgill.ca).H. Leib is with the Department of Electrical & Computer Engineer-ing, McGill University, Montreal, Quebec, Canada, H3A 2A7 (e-mail:harry.leib@mcgill.ca).Digital Object Identi
Þ
er 10.1109/TCOMM.2010.08.090511
Joint channel estimation and data detection schemes, re-quiring only a small number of pilot symbols for an initialchannel estimate, offer another alternative. Iterative channelestimation and data detection techniques for STTCs based onthe Expectation Maximization (EM) algorithm [4], presentedin [5], [6], belong to this category. In [5], [6] knowledge of normalized Doppler
(with
being the maximumDoppler frequencyand
being the transmitted symbol period)is required to determine the correlation function of the fadingprocess used in channel estimation. The resulting channelestimate is used in data detection, showing that knowledge of
is important for the operation of joint channel estimationand data detection schemes over time-varying fading channels.Per-Survivor Processing (PSP) techniques [7] using multiplehypothetical data sequences (survivors), that also belong to the joint channel estimation and data detection family of schemes,are well suited for high Doppler channels [8]. These PSPalgorithms embed adaptive channel estimation into tree-searchor trellis-search (Viterbi) algorithms [7], [9] where the survivoris used for channel estimation. The development of PSP-based joint channel estimation and data detection algorithms forSTTCs is documented in [8], [10], [11]. Another joint channelestimation and data detection approach is based on particle
Þ
ltering
1
[12]. In [12], a Delayed Mixture Kalman Filtering(DMKF) technique for single-antenna systems is presented,where importance samples take into account future receivedsymbols. In [13] this approach is applied to STTCs. A KalmanPredictor (KP) is employed for channel estimation in [8],[11], [13], due to its ef
Þ
cient recursive implementation andits tracking abilities [14]. Using a KP requires a state-spacemodel that in turns requires
. The knowledge of
isalso required to determine the step size in the Least MeanSquare (LMS)-based algorithm of [10].In practice
(or
) has to be estimated. For Doppler esti-mation, one can use the Level Crossing Rate (LCR) approachand the covariance approach [15]. Moreover, an asymptoticMaximum Likelihood (ML) estimator based on the Whittleapproximation is proposed in [16] for
estimation. With ashort sequence of channel coef
Þ
cients, the covariance methodperforms better than the asymptotic ML approach [16] andthe LCR approach [15]. With a KP, there are several ways toattack the Doppler estimation problem. One way is to includethe unknown parameters into a state space model. However, inthis case the model becomes nonlinear and the extended KP
1
Also known as sequential importance sampling0090-6778/10$25.00 c
⃝
2010 IEEE
2278 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010
must be used. Initial work in [17] shows that PSP-based jointchannel estimation and data detection, using an Adaptive KP(AKP) for STTC systems, performs well in fading channelswith unknown constant Doppler.In [11], we proposed an algorithm called Smoothed DataDetection and Kalman Estimation (SDD-KE), belonging tothe PSP family, where detection employs the Maximum APosteriori Probability (MAP) criterion with a
Þ
xed delay
,and the channel is estimated by a KP assuming that the
Þ
xed
is known. In this paper, we derive the DopplerAdaptive SDD-KE (DA-SDD-KE) algorithm that does notrequire knowledge of
. This novel algorithm jointly detectsdata, and estimates the channel as well as the
time-varying
Doppler by incorporatingan AKP into the SDD-KE algorithm.We also propose a novel method for generating a Rayleighfading process with time-varying Doppler because of absenceof adequate techniques in the literature.This paper is organized as follows. In Section 2, we discussa general model for Multiple STTC (MSTTC) systems andpresent a technique for generating Rayleigh fading chan-nels with time-varying Doppler. In Section 3 we considercovariance-based
estimation, the AKP and the DA-SDD-KE algorithm. Section 4 presents computer simulation resultsof an MSTTC scheme over fading channels, with constant,linear, and quadratic time-varying Doppler functions, usingthe DA-SDD-KE algorithm. Section 5 presents the DA-DMKFalgorithm, a Doppler adaptive extension of the DMKF of [13] that we developed for comparison with the DA-SDD-KEscheme. Finally, Section 6 presents the conclusions.II. S
YSTEM AND
C
HANNEL
M
ODELS
A. System Model
Consider a wireless communication system with
trans-mit and
receive antennas, employing an MSTTC (e.g thesmart-greedy STTCs of [1] and the STTCs of [13], [18])with
code vectors per trellis branch. The codeword lengthof the MSTTC is
, where
is theinformation frame length, and
is the memory order of thecode. Let
,...,
be the transmitted/received vectorindex, and
,...,
be the trellis stage index. Thecode matrix associated with the
ℎ
trellis stage is
,...,
where
,
∈
,...,
isthe
ℎ
transmitted vector of dimension
. The
ℎ
receivedvector
,...,
is given by
√
,
,...,
(1)where
is transmit symbol energy per antenna,
,
is the
channel matrix affecting
, whose elementsare independent identically distributed zero-mean circular-symmetric complex Gaussian random variables of unit vari-ance. The additive noise
is a zero-mean circular-symmetriccomplex Gaussian vector with covariance matrix
where
denotes the
identity matrix.With multiple antenna systems in quasi-static or indepen-dent fading environments,the channel coef
Þ
cients are assumedto be fully correlated or independent in time, respectively, andthey are assumed to be spatially independent. Such models areconvenient for design and analysis. However, fading channelswith Doppler spread represent more realistic wireless linkssince they model time-correlated fading with coef
Þ
cients thatchange continuously. In this work, we consider Doppler fadingchannels of communication systems where the transmitter,receiver and/or surrounding objects move with time-varyingspeed, resulting in a time-varying Doppler frequency
[19].The decoder employs the DA-SDD-KE algorithm. This al-gorithm, while utilizing future received symbols for smootheddata detection as in the SDD-KE algorithm [11], does notassume perfect knowledge of the Doppler as in [11]. Inthe DA-SDD-KE algorithm, we incorporate an AKP into theSDD-KE scheme where the survivor sequences are used by theAKP for channel tracking and
estimation. Hence, unlikethe SDD-KE, the new algorithm not only estimates jointly thechannel matrices and transmitted data sequence, but also
.The following notations will be used throughout this pa-per. Let
,
,
∗
, denote the transpose, Hermitian, andconjugate operations on a vector or matrix. For an
-statetrellis diagram, a branch at stage
connects a state
∈
to a state
∈
where
⊆
,...,
with
, and
⊆
is the set of states in
for which there are branches connecting to thesestates terminating at
∈
. Denote by
and
ℂ
the set of all possible
and
, respectively. Furthermore,
ℂ
⊂
ℂ
denotes a set of code matrices associatedwith branches at stage
whose terminating state is
, and
,
,
,...,
,
denotes a code matrix at stage
associated with a branchconnecting states
and
, where
,
,
,...,
, is the code vector at discretetime
. Let
be a code matrix sequence,
be a codevector sequence,
ℂ
and
be the set of possible
and
, respectively. Furthermore,
,...,
is the received matrix corresponding to trellis stage
, and
,
denote the sequence of received matrices and vectors,respectively. Let
ℂ
,
⊂
ℂ
be the set of sequences
for
≥
such that
,
.Furthermore,
,
is a code matrix at trellis stage
associated with a survivor at
,
,
is the sequenceof code matrices corresponding to the survivor at
, and
,
is the detected sequence.
B. A Technique for Generating Time-Varying Doppler FadingChannels
In [20], Jakes proposed a model (that became very popular)for generating a time-correlated Rayleigh fading process basedon a Sum of Sinusoids (SoS) technique. In [21], differentgeneration methods for fading channels with constant Dopplershifts, using the SoS technique, are compared in terms of complexity and performance. One of the results of [21]recommends the model from [22] for Rayleigh fading channelsimulation because of its superior performance (while slightlymore complex) over other models. Note that all techniques inthe literature for Rayleigh fading generation assume a constant
. However, channels with time-variant Doppler that modelsituations where the transmitter, receiver and/or surroundingobjects move with time-varying speed are also of interest
VILAIPORNSAWAI and LEIB: JOINT DATA DETECTION AND CHANNEL ESTIMATION FOR FADING UNKNOWN TIME-VARYING DOPPLER
...
2279
[23]–[25]. For example, [26], [27] consider non-stationarityissues associated with channels for vehicle-to-vehicle com-munications where each vehicle can move with time-varyingspeeds and/or directions, and hence can experience time-variant Doppler effects. From the literature it is not clear howsuch time-varying Doppler channels can be generated. In thispaper we show that substituting a time-varying
in placeof a constant
in a technique for generating a fading processwith constant Doppler is not proper. The key point that veryoften is missed is that the Doppler shift should be treated asan instantaneous frequency. In [28], the authors point out toinaccurate methods of generating time-varying carrier offsetsin the literature where
is applied in place of a constant
. However, in [28], a single-carrier frequency-offset modelis considered, while in our work we consider a technique forgenerating a Rayleigh fading process with a prescribed timevarying Doppler shift based on the SoS method.Extending the method of [22] to time-varying Doppler fre-quency (instantaneousfrequency)
, results in a continuoustime fading process
,
given by (2)-(6), with
,
,
(7)where
,
,
,
and
,
are uniformly distributed on
,
and independent for all
,...,
,
,...,
and
,...,
. Note that (3) reduces to (4) since the phaseat
associated with
∫
∞
can be absorbed inthe random phase
,
. Similarly, we obtain (6) from (5).For a constant Doppler frequency, i.e.
, we have
∫
and the model (2)-(6) is reduced to the onepresented in [22].
Theorem 1:
If
→ ∞
, then the process
,
tends tobe Gaussian.Proof of this theorem can be found in Appendix A.
Theorem 2:
The fading process
,
speci
Þ
ed by (2)-(6) has the properties shown in (8)-(12), where
,
∫
and
is the zero-order Bessel function of the
Þ
rst kind.Proof of this theorem can be found in Appendix B.From Theorem 2, if the Doppler shift is time-invariant, i.e.
and
∫
is used in the generationof
,
(2), then we have
,
, and (11) reducesto
,
,
∗
which is the well knownresult for constant Doppler. For a linear time-varying Doppler
, we have
,
, and (11) reducesto
,
,
∗
. Note that if this
is applied in place of
in the model for generatingthe fading process with constant Doppler [22], or equivalently
is used in place of
∫
in (2), then it can beshown (using the approach in the Proof of Theorem 2) thatthe autocorrelation function of the process is
, rather than
. This shows that a directsubstitution of
for
in the model for constant Doppleris not a proper way to generate time-varyingDoppler channels.Next, we consider the generation of discrete-time fadingcoef
Þ
cients using (2)-(6). First, consider
(13)where
, and
is the normalizedDoppler shift function. For a polynomial Doppler function of degree
, we have
∑
where
takes realvalue. Then,
in this case is given by
(14)where
. From (13) and (14), we have
∫
∑
. To generate the discrete-timefading process
,
,
∣
with
given by (14),we use
ℜ
,
√
,
,
(15)
ℑ
,
√
,
,
(16)Non-polynomial forms for
can be approximated bypolynomials to synthesize a large variety of normalizedDoppler functions
.The validity of the discrete-time model obtained by sam-pling the continuous-time model is investigated in [29] forconstant Doppler, and shown to depend on BER
ß
oor
,the roll-off factor
of the transmitted square-root raisedcosine pulse shape, and the constant
. A fading processwith autocorrelationfunction
∗
′
′
andJakes U-shape power spectrum is considered. With a given
,it is shown in [29] that as
increases, a larger
is required.In this paper we use the same approach as in [29] toinvestigate the validity of the discrete-time model for time-varying Doppler channels with QPSK and coherent detection.Appendix C shows that the BER
ß
oor associated with the
ℎ
information symbols
as a function of the instantaneousDoppler
and
is
≈
.
(17)Note that (17) is approximately the same for linear andquadratic Doppler functions
∑
consid-ered in Section IV and the Doppler function with
considered in [30, Case 8]. Moreover, it is also approximatelythe same as [29, Eqn. (63)] for constant Doppler. Hence,for time-varying Doppler channels and a given
, we canuse (17) or equivalently the result of [29] for a constant
to determine the roll-off factor
for aspeci
Þ
c
. For time-varying Doppler channels,pulse shaping and the receiver front-end should be designedfor the maximum instantaneous normalized Doppler (the worst
2280 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010
,
ℜ
,
ℑ
,
(2)
ℜ
,
√
∞
,
,
(3)
√
,
,
,
≤
∞
(4)
ℑ
,
√
∞
,
,
(5)
√
,
,
,
≤
∞
(6)
,
(8)
ℜ
,
ℜ
,
ℑ
,
ℑ
,
,
(9)
ℜ
,
ℑ
,
ℑ
,
ℜ
,
(10)
,
,
∗
,
(11)
,
,
(12)case) since it requires the largest
for a given
. The discretetime fading model in our work assumes such proper pulseshaping and receiver front-end.III. T
HE
DA-SDD-KE A
LGORITHM
This section presents the DA-SDD-KE algorithm that in-cludes an AKP for channel and Doppler estimations in theSDD-KE scheme of [11]. We consider the use of covariance-based
estimation in the AKP, and its integration in SDD-KE algorithm.
A. Doppler Shift Estimation
Let
,
,
,
∗
denote the autocorrelation func-tion for lag
of the channel coef
Þ
cients associated withreceive antenna
and transmit antenna
, and
,
denotethe estimate of
,
based on
,
,
,
...
,
. Sincea covariance-based
estimator operates well over shortsequences of channel coef
Þ
cients [16], only a small numberof pilot symbols are required to provide an initial estimatefor
. Hence, we employ this method as in [24]. The
estimate is given by
∈ℱ
∑
∑
∑
(18)
where
ℱ
is a set of
candidates, and for
,
,...,
with
,
∑
ℜ
,
,
∗
,
(19)where
,
,
,
,
,
,
are channel estimates avail-able from the AKP in the DA-SDD-KE algorithm. Comparedto [24], (19) introduces an exponential forgetting factor
≤
to allow for tracking in time-varying
estimation.With
(19) is a conventional sample autocovariancefunction [31, pp. 321-323].A forgettingfactor has been widelyused in Recursive Least Square (RLS) problems [32, Chapter9], allowing past data to be weighted less and hence enhancingtracking capabilities. From (19),
,
can be updated from
,
using the equation at the top of the next page,where later we use
because a phase ambiguity resistantMSTTC with
is employed, and in the DA-SDD-KEalgorithm the
is updated at trellis states rather than withintrellis branches.
B. Adaptive Kalman Predictor (AKP)
In this section, we present the KP structure embedded in theAKP. We use a state space Auto-Regressive model of order 3,AR(3), matched to the Jakes U-shape power spectrum. Hencethe AR(3) model must depend on
and it is given by [8],
(20)where
,
,
, and
⎛⎝
⎞⎠
,
(21)with
provided by the Doppler estimator, and AR(3)coef
Þ
cients
obtained as in [11].
VILAIPORNSAWAI and LEIB: JOINT DATA DETECTION AND CHANNEL ESTIMATION FOR FADING UNKNOWN TIME-VARYING DOPPLER
...
2281
,
∑
,
ℜ
,
,
∗
∣
∣
′
Σ
∣
∗
∣
(22)
Σ
∣
Σ
∣
Σ
∣
(23)
⎡⎢⎢⎣
Σ
∣
∗
′
√
Σ
∣
⎤⎥⎥⎦
⎛⎜⎝⎡⎢⎣
√
√
Σ
∣
√
Σ
∣
⎤⎥⎦⎞⎟⎠
(24)
∣
,
(25)
Σ
∣
⎛⎝
,
,
,
,
,
,
,
,
,
⎞⎠
,
(26)
,
⎧⎨⎩∑
,
∑
,
∗
,
.
(27)We use the square root Kalman
Þ
lter [14, pp. 150] to avoidnumerical errors, as shown in (22)-(24), where
denotesa triangularization by means of the
decomposition. Theinitial conditions for the AKP are obtained using the initialvalue
as shown in (25) and (26), where
,
is the autocorrelation function of the AR(
) process, as shownin (27) [33, pp.838]. For a given
,
,
,
,we solve for
,
,
,
,
using (27) with
,
= 1, and process noise power
∑
,
.Subsequently, we use
,
,
and
to denote
,
and
estimated from the sequence of channel estimates asso-ciated with the pilot sequence of length
, respectively. Also,we use
,
,
and
to denote
,
and
estimatedfrom the sequence of channel estimates associated with codedsequence
concatenated with the pilot sequence. Fig. 1illustrates the structure of the AKP considered in this work,where dash lines are used to underline the quantities associatedwith the training mode, while solid lines are used for thedata-aided mode. During the training mode, the algorithmstarts by calculating the AR(3) coef
Þ
cients
basedon
. Using the AR(3) model, the KP estimates thechannel matrices
,
based on the pilot sequence of length
.Then,
,
are used to calculate the autocorrelation function
,
,
for lag
,
,...,
which are fed tothe
estimator of Section III-A to provide
. During
,
,
,
,
,
,
,
AR(3)Coef
Þ
cientsCalculationKPOperation
Estimation
111222Training ModeData-aided Mode
Fig. 1. Structure of an adaptive Kalman predictor for Doppler shiftestimation.
the data-aided mode, we set
,
,
,
,
,
,
,
, and using
, the AR(3) model isupdated for the
Þ
rst time based on the coef
Þ
cients
.

Search

Similar documents

Tags

Related Search

Ofdm Channel Estimation and CfoComputer Aided Detection and Diagnosis for MeAutomated Data Delivery and Processing for DiError Detection And CorrectionData Mining and Neural NetworksData Structures and AlgorithmsData Warehousing and Data MiningFault Detection and Diagnosis in Systems/struComputer-Aided Detection and Diagnosis (CAD) Fraud Detection And Prevention

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks

SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...Sign Now!

We are very appreciated for your Prompt Action!

x