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EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS Eur. Trans. Telecomms. 2008; 19:1–13 Published online 23 April 2007 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/ett.1216

Communication Theory Joint channel estimation and equalisation of fast time-varying frequency-selective channels Snjezana Gligorevic∗ German Aerospace Center (DLR), Oberpfaffenhofen, Münchener Str. 20, 82234 Weßling, Germany

SUMMARY This paper addresses joint channel and data estimation for transmission over frequency-selective and timevarying channels based on a Doppler-variant channel impulse response (CIR). A discrete spreading function (DSF), which corresponds to the Doppler Fourier series of the time-varying channel taps, is introduced. The spectral leakage associated with the Fourier series representation is reduced by oversampling in Dopplerdirection. This approach, however, requires joint channel estimation and equalisation (JCE). Therefore, a recursive least squares (RLS) algorithm for the DSF estimation is combined with the data detection in a reduced state diagram. Furthermore, an enhanced metric, previously introduced in De Broeck et al. (Proceedings of Second International Workshop on Multi-Carrier Spread-Spectrum, 1999; published by Kluwer Academic Publishers: Dordrecht, 2000) for time-invariant channels, is applied here for path selection. The performance of this metric is investigated here for time-varying and frequency-selective (TV-FS) channels. The performance of the proposed approach is compared with the JCE algorithm which uses RLS estimation of a CIR by simulations. Copyright © 2007 John Wiley & Sons, Ltd.

1. INTRODUCTION In most wireless communication scenarios, the transmitted signals are distorted due to time-varying and frequencyselective (TV-FS) channels. Data recovery is usually performed under the assumption that the channel is constant for a certain time interval. Still, in some applications, such as underwater acoustic communication, high speed train and automotive, as well as communication between high speed jets especially at low altitude, this assumption does not hold true. In these cases, the approach to channel estimation is crucial for the receiver performance. For example a separate channel estimation based on a known pre- or midamble followed by a separate equalisation performs poorly due to the channel variations. Better performance is achieved if the equalisation is conducted with an adaptive channel tracking algorithm. Different receiver structures for TV-FS channels are discussed in References [1, 2]. An adaptive decision feedback receiver using a maximum likelihood

sequence estimator (MLSE) is presented in Reference [3]. The recursive least squares (RLS) algorithm is applied to channel estimation in a MLSE in References [4, 5]. A Kalman estimator is combined with data detection in Reference [6]. A compromise between the performance and the complexity can be achieved in combination with a sliding window [1, 2, 7]. A common approach to channel estimation is to consider each coefficient of a time-varying CIR (TV-CIR) as a stochastic process. If the time-variation can be modelled as an auto-regressive moving-average (ARMA) process, then the channel can be optimally tracked by a Kalman filter [8, 9]. The least mean squares (LMS) algorithm can also be adapted to perform competitively with the RLS in time-varying channels. In Reference [10], the step-size parameter of the LMS is defined as a discrete-time function and optimised in each iteration step. Some stochastic approaches use only the autocorrelation function (ACF) of the oversampled channel outputs to estimate the TV-CIR. In

* Correspondence to: Snjezana Gligorevic, German Aerospace Center (DLR), Oberpfaffenhofen, Münchener Str. 20, 82234 Weßling, Germany. E-mail: [email protected]

Copyright © 2007 John Wiley & Sons, Ltd.

Received 23 June 2005 Revised 3 December 2005 Accepted 19 December 2006

2

S. GLIGOREVIC

this case, some demands considering correlations properties of the transmitted data sequence have to be met [11, 12]. Another approach is to model the time-variations of the CIR by a set of basis functions. A deterministic channel model can be obtained by representing each channel tap with few coefficients of a chosen basis expansion. In Reference [9], a linearly time-varying model has been used, in Reference [7], time-variations over a short interval are approximated by a small set of polynomial basis functions and in Reference [13], the use of a set of prolate spheroidal sequences has been proposed. In Reference [14], pilot symbol assisted modulation (PSAM) is applied together with the polynomial interpolation of the time-varying coefficients. This constellation allows a TV-CIR to be estimated by the product of a constant interpolation matrix and the fading information at pilot symbol positions. Our objective in this paper is to develop a channel estimator based on the sampled Doppler-variant impulse response, also called spreading function [15]. We assume that the Doppler frequencies of each channel path are unknown. Considering block oriented transmission, we apply sampling in Doppler-direction to obtain discrete Doppler frequencies for the estimation. In this case, the sampled spreading function corresponds to the Doppler Fourier series of the TV-CIR during the considered observation interval. A similar approach is the modelling with a polynomial function which applies windowing to limit the observation interval. Furthermore, the approximation with a polynomial series is related to one point, generally in the middle of the window, and the mean squared identifications error (MSIE) increases with the distance to this point [7]. The performance is also strongly affected by the size of the window and by the order of the polynomial model. However, we consider one data block to be the observation interval. Since the coefficients of a Fourier series are always chosen to independently minimise the squared error between the function and its Fourier series representation, the Doppler Fourier series used in our approach approximate the CIR globally on the whole observation interval, that is data block. An approach similar to ours can be found in Reference [16]. The work in Reference [16] assumes channel path delays to change linearly in order to model a mobile channel as an (almost) periodical function. The estimation is based on previously identified frequencies of the exponential basis sequences: The Doppler frequencies of the channel taps are estimated first by exploiting higher-order cyclic statistics, and thereafter used to model a TV-CIR with a small number of complex exponential functions. If the modelling assumptions are Copyright © 2007 John Wiley & Sons, Ltd.

satisfied and frequencies are estimated correctly, the model will be very accurate for fading channels. In practical applications, however, the frequencies will also change with the vehicle’s velocity and some frequencies (reflectors) can suddenly disappear/appear. This may have a significant impact on the algorithm. Furthermore, the basis expansion coefficients can also drift over time and affect the estimation. Thus, the decision feedback equaliser (DFE) is expected to have difficulties in tracking these changes. Possible improvements which may be achieved by combined estimation and data detection are discussed in Reference [16] only briefly. Furthermore, the same technique is used in Reference [17] for blind equalisation and in Reference [18] for estimation of the channel correlation matrix by exploiting distributed training symbols. However, the work in Reference [16] is more relevant for this paper, as it considers an adaptive channel estimation combined with the DFE algorithm in order to equalise the signal. We will use this reference later in Section 7 to emphasise differences between this approach for channel modelling and ours. The same channel modelling as proposed here can be found in Reference [19], applied to separate channel estimation and data detection. However, in Reference [19], the focus is set on the design of the optimal training input and a linear MMSE channel estimator is applied under the assumption that variances of channel coefficients are known. Thus, information about the multipath intensity profile and Doppler power spectrum is exploited in simulations. Important to mention are also the publications [20, 21] as they address problems which occur with the Fourier basis expansion model due to the truncation of the Discrete Fourier Transform (DFT) at the Doppler bandwidth. We overcome this problem by oversampling, that is virtually extending the observation interval for the Fourier series expansion. Omitting Doppler frequency estimation and assuming only a short training sequence at the beginning of one data block, our approach requires JCE to perform well in fast time-varying channels. We combine the RLS algorithm with equalisation in a reduced state diagram. Furthermore, we investigate the applicability of the enhanced metric which considers also the spread energy of the equalised symbol and thus, increases the reliability of data decision in time-invariant channels [22]. Therefore, we simulate its performance in time-varying channels with the RLS algorithm estimating the discrete spreading function (DSF), as well as with the common RLS estimation of the CIR. The paper is organised as follows: We introduce the TV-CIR and the DSF in Section 2 and apply maximum Eur. Trans. Telecomms. 2008; 19:1–13 DOI: 10.1002/ett

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JOINT CHANNEL ESTIMATION AND EQUALISATION

likelihood (ML) channel estimation with a known data sequence in Section 3 in order to observe the error of the Fourier series decomposition. The estimation is improved in Section 4 by oversampling the spreading function in Doppler-direction. A JCE algorithm with an adaptive estimation of the DSF is subject of Section 5. In Section 6, a corresponding extended metric for path selection is introduced. In simulations provided in Section 7, the performance of the JCE algorithm with the DSF estimation is compared with the performance obtained by the JCE based on the CIR estimation. This is done for different channel conditions regarding time-variance and frequencyselectivity. The performance of the DFE using the proposed channel modelling is also taken into consideration. In Section 8, the complexity of the algorithm is evaluated. Section 9 concludes the paper.

2. THE TIME-VARIANT CHANNEL Consider the block diagram of a communication system in Figure 1. The composite channel, consisting of the transmitter filter with the impulse response g(t), the timevarying (physical) channel with the impulse response hc (τ; t) and the receiver filter with the impulse response gˆ (t), is represented by the time-varying impulse response h(τ; t). The relation between the data signal di , with i = 1, . . . , N and N the number of observed data samples, and the noisy baseband channel output yi , sampled at the symbol rate 1/T , can be written as yi =

M ch −1

hm (iT ) · di−m + wi

(1)

m=0

Here, the mth path of the TV-CIR hm (iT ) = h(mT ; iT ) corresponds to the copy of the transmitted signal, which arrives with the delay τm = mT relative to the first received

copy. Commonly, the CIR is limited to the maximum delay τmax , and the discrete TV-CIR is truncated to an order Mch with Mch − 1 =

τ

max

T

(2)

,

where denotes the ceiling function. Thus, the response of the composite channel with delays τm = mT > (Mch − 1)T are going to be neglected in the following. Furthermore, wi represents samples of the additive white Gaussian noise (AWGN) after the receiver filter, with variance σ 2 = N0 /2T , where N0 /2 is the one-sided noise power spectral density. As the channel is unknown, the receiver filter is assumed to be matched to the transmitter filter. The transmitter filter is assumed to be chosen such that uncorrelated samples of the filtered noise are obtained. For a mobile channel the time-variation of hm (t) is induced by the movement of the mobile station. This motion introduces a Doppler shift that describes the time-variation. The Doppler frequency ν is limited by the maximal velocity of the mobile user. For the block oriented signal processing, the Doppler rate Kch = νmax NT

(3)

describes the time-variations of the channel within one data block of duration NT , with N the number of data samples and νmax the maximum Doppler frequency. If the timevariance of the channel taps within this interval cannot be neglected, then N · Mch samples of the TV-CIR are to be estimated from N received samples available. Thus, additional information will be required. Due to the limitations of the physical channel hc (τ; t) in delay and Doppler frequency the spreading function S(τ; ν) seems to be appropriate for the description of the TV-FS channel. It is linked to the CIR by the Fourier transform over the absolute time t [23] ∞ S(τ; ν) =

hc (τ; t)e−j2πνt dt

and

(4)

−∞

∞ Figure 1. Continuous-time model of a baseband time-varying communication system. Copyright © 2007 John Wiley & Sons, Ltd.

hc (τ; t) =

S(τ; ν)e+j2πνt dν

(5)

−∞

Eur. Trans. Telecomms. 2008; 19:1–13 DOI: 10.1002/ett

4

S. GLIGOREVIC

Considering discrete delays, we use the following relation: +∞ S(mT ; ν) = Sm (ν) = hm (t)e−j2πνt dt,

(6)

given in Equation (8), the received signal can be written as yi =

M−1 m=0

di−m ·

K

ξk (i) · Sm,k + wi

(9)

k=−K

−∞ 2πk

with m = 0, . . . , Mch − 1. Note that after filtering and windowing the received signal, the assumption of the limitation in delay and Doppler frequency becomes void for the composite channel. Regarding the time limitation, the values obtained by sampling in the Doppler frequency domain are sufficient to describe hm (t) completely. Taking into account the physical limitation in Doppler frequency, we consider only discrete Doppler frequencies νk = k/NT , with k = −Kch , . . . , Kch . Without loss of generality, we assume the CIR to be observed on the interval (0, NT ). The sampling values Sm,k = S(mT ; k/NT ) of the spreading function represent the coefficients of the Fourier series of hm (t) given by

Sm,k

1 = NT

NT

2πk

hm (t)e−j NT t dt

(7)

with ξk (i) = ej N i . By considering the maximum possible channel delay and the highest possible velocity for the considered scenario, M and K can be chosen in order to ensure that M Mch and K Kch . Then, the TV-CIR can be reconstructed from M · (2K + 1) estimated timeinvariant coefficients. Defining the vector S, with spreading coefficients Sm,k , S=(S ˆ 0,(−K) , . . . , Sm,k , . . . , S(M−1),K )T

(10)

the received signal can be given by y = D · S + w,

(11)

with y = (y0 , y1 , . . . , yN−1 )T and w = (w0 , w1 , . . . , wN−1 )T . The rows of the data matrix D are given by d(i) = di · ξ(i), . . . , di−m · ξ(i), . . . , di−M+1 · ξ(i) (12)

0

As the Doppler spectrum is unlimited due to the windowing of the received time function, the channel representation by the bounded DSF is not entirely correct. The function hˆ m (t) reconstructed from the Fourier coefficients, that is

hˆ m (t) =

K ch

2πk

Sm,k ej NT t

(8)

k=−Kch

is necessarily periodical in t on the interval of the length NT, which is not the case with hm (t). The resulting difference between hm (t) and hˆ m (t) corresponds to the error due to the truncation in Doppler-direction.

where ξ(i) = (ξ−K (i), . . . , ξK (i)). Thus, the timevariations, determined through k and i in the exponent function of Equation (9), are now contained in the data matrix D . Assuming the data matrix D to be known, the ML [24] criterion can be applied. More specifically, for uncorrelated Gaussian noise samples the ML approach leads to the least squared error (LSE) criterion. In order to get unique solution [25] S˜ = (DH D)−1 DH y,

(13)

the matrix DH D should be regular, resulting in the constraint that the length of the data sequence fulfils the inequality N > M · (2K + 1)

(14)

3. CHANNEL ESTIMATION In the following, we denote by M and K the number of channel paths and the maximum Doppler rate to be estimated, respectively, and use the expression DSF for a set of spreading coefficients Sm,k , with m = 0, . . . , M − 1 and k = −K, . . . , K. According to the channel approximation Copyright © 2007 John Wiley & Sons, Ltd.

For underspread channels [26] this is always the case. Consequently, with their positions determined by the bandwidth and the data length, the spreading coefficients are estimated in a way to minimise the squared error between the reconstructed TV-CIR and the true one for the duration of the considered data block. Since the function Eur. Trans. Telecomms. 2008; 19:1–13 DOI: 10.1002/ett

5


4. OVERSAMPLING FOR REDUCTION OF FOURIER SERIES DECOMPOSITION ERROR

(a) 1

m

Re{h (t)}

0.5 0 −0.5 −1 0

1

h0(t) estimated h40 (t) 20 0 h (t) 1 estimated h1(t)

60

80

100

120

140

160

An oversampling by a factor of g > 1 in Doppler frequency direction corresponds to the Fourier series decomposition on the interval of the length g · N. Now, the number of ¯ + 1, with K ¯ = K · g discrete frequencies increases to 2K and the expression for the received values becomes yi =

Im{hm(t)}

0.5

M−1

di−m

−0.5

20

40

60

80

100

120 128

140

160

t/T

(b)

(15)

2πki

with ξk (i) = e gN . Consequently, the data matrix D ¯ + 1) matrix with rows given by expands to a M × (2K d(i) = (di · ξ (i), . . . , di−m · ξ (i), . . . , di−M+1 · ξ (i)) and the condition in Equation (14) becomes j

−1 0

ξk (i) · Sm,k + wi

¯ k=−K

m=0

0

¯ K

1

¯ + 1) N > M · (2K

Re{hm(t)}

0.5 0

−0.5 −1 0

h (t) 0 estimated h (t) 0 20 40 h1(t) estimated h (t)

60

80

100

120

140

160

1

1

Im{h (t)} m

0.5 0

Applying oversampling with g = 1.3 to the example in Figure 2(a) yields the improved estimation in Figure 2(b). Since no optimisation is done in the extended part of the observation interval, N i < g · N, large deviations from hm (t), m = 0, 1, may appear here. However, the differences between the reconstructed and the measured functions at the borders of the original interval are reduced. Figure 3 shows the mean squared identification error (MSIE) [2, 7] given by

−0.5 −1 0

20

40

60

80

100

t/T

120 128 140

MSIEh = E h˜ m (iT ) − hm (iT )2

160


(17)

0.12

Figure 2. ML estimation over one block. (a) Without oversampling, that is g = 1.0; (b) With oversampling, g = 1.3.

g=1.0 g=1.1 g=2.0 g=1.8 g=1.3 g=1.6 g=1.4 g=1.2

0.1

h

0.08

MSIE

given by the Fourier series is always periodical on the observation interval, a cyclic repetition of the observation interval generally causes errors in the reconstructed hˆ m (t) at the beginning and the end of the interval. This can be observed in the example in Figure 2(a). Here, the measured two channel taps h0 (t) and h1 (t) have the maximum Doppler rates Kch0 = 0.9 and Kch1 = 1.7. We have set M = 2 and K = 3. It can be observed that the estimated functions try to maintain the periodicity on the given time interval and therefore differ from the measured tap functions. In the following section, we introduce oversampling in Doppler-direction to overcome this problem.

(16)

0.06

0.04

0.02

0 0

5

10

15

20

E / N [dB] b

0

Figure 3. MSIE over SNR for specified channel parameters and different values of oversampling factor g. Eur. Trans. Telecomms. 2008; 19:1–13 DOI: 10.1002/ett

6

S. GLIGOREVIC

simulated versus Eb /N0 for different values of the oversampling factor g. Here, E[·] denotes the expectation value and h˜ m (iT ) is the CIR reconstructed from the ¯ + 1) coefficients of the DSF. The MSIE estimated M · (2K is averaged over 1000 channels with τmax = 2T and Kch slowly varying between 1.1 and 1.3. The length of one data block is N = 1200. Obviously, the number of unknown parameters increases with g, but already a small oversampling factor reduces the error of the Fourier series representation. Thus, in further simulations in Section 7 we fix the oversampling factor to g = 1.2. Generally, the estimation of the DSF provides no information about the time trajectory of the channel taps outside the considered observation interval. Therefore, this approach is not suitable for data equalisation with channel estimation based on an initial training sequence only. The ML estimation, though, can be applied directly, if the training sequence is sent parallel to the data. Thus, this method is directly suitable for the Universal Mobile Telecommunications System (UMTS). A possible application for the UMTS Terrestrial Radio Access (UTRA) Frequency Division Duplex (FDD) uplink was presented in Reference [27].

[28], could have been used as well. Also possible is the T-algorithm [29] which retains a variable number of paths in each iteration step depending on the defined threshold parameter. After initial estimation S˜ of the spreading coefficients vector, the training sequence is extended with the hypotheses of a data symbol and the channel estimation is updated by using the RLS algorithm. This procedure is continued until a certain number of paths P is reached. With each following data hypothesis, the number of paths is increased (in case of BPSK doubled), but only the best P paths chosen according to some metric will be retained. A separate estimation of the spreading coefficients is calculated for each path, assuming that the hypothetical data sequence is correct, and then used for the metric calculation of the corresponding path. The data sequence corresponding to the path with the smallest metric at the end of the tree will represent the equalised data sequence. Note that the metric for JCE depends on the channel estimation and is therefore not additive. Thus, the Viterbi equalisation does not apply here. Assuming uncorrelated noise samples and no a priori information about the channel available, the metric for the path selection is defined by the actual squared error of the estimated received sequence

5. JOINT CHANNEL ESTIMATION AND EQUALISATION In this section we propose a joint channel estimation and equalisation (JCE) algorithm which continuously adapts the channel estimate in a reduced state equaliser. An estimation of the spreading coefficients initialised with a training sequence is valid only for the duration of this training sequence. If no additional information is available, using this channel estimate to equalise the following data symbols would lead to detection errors due to the time-varying channel. The optimal approach for JCE is a tree structure with all possible hypotheses of a transmitted data sequence. As the number of paths in the tree increases exponentially with the length of the data sequence, this optimal approach has a prohibitive complexity. Therefore, it is often substituted with a suboptimal solution where only a small number of hypotheses given by paths is followed through the tree. There are different approaches for the reduction of a full-state diagram. In the simulations, we apply the single survivor per state approach, known as per survivor processing (PSP) [5]. However, a multiple survivor per state approach as in Reference [22], or as the M-algorithm Copyright © 2007 John Wiley & Sons, Ltd.

˜ 2 (i) = y(i) − Di · S

(18)

Here, y(i) = (y0 , y1 , . . . , yi )T , S˜ is the estimated spreading ¯ + 1) matrix of vector and Di the upper (i + 1) × M · (2K 2 the data matrix D. The term y(i) is the same for all paths in the state diagram and does not influence the path decision. By omitting this term the metric is given by

2 (i) = −2Re y(i)H · Di · S˜ + Di · S˜ = −2Re

i =0

yH

(19)

i d() · S˜ 2 (20) ˜ · d() · S + =0

In each step i, only the rows 0, . . . , i of the data matrix D and the values y0 , . . . , yi of the received vector contribute to the metric. The estimation of the spreading function in each path is based on the assumption of uncorrelated noise samples and on the data hypothesis belonging to the considered path to be correct. Thus, we can insert the ML estimation of ˜ = the spreading vector given by Equation (13), that is S(i) H H −1 (Di Di ) Di y(i) into Equation (19). Herewith, the metric Eur. Trans. Telecomms. 2008; 19:1–13 DOI: 10.1002/ett

7


in Equation (19) becomes

˜ − S˜ H (i)DH Di S(i) ˜ (i) = −Re 2y(i)H · Di · S(i) i

−1 H D ) D y(i) (21) = −Re y(i)H Di (DH i i i As shown in the Appendix, this metric can be calculated recursively by ∗

(i) = (i − 1) + Re{e (i|i)e(i|i − 1)}

(22)

where e(i|i − 1) is the a priori estimation error given by ˜ − 1) e(i|i − 1) = yi − d(i)S(i

(23)

and e(i|i) the a posteriori estimation error given by ˜ e(i|i) = yi − d(i)S(i)

(24)

5.1. RLS algorithm for estimation of the spreading coefficients The channel estimation is started under the assumption that no channel statistic is available. Furthermore, as we have a time-invariant filtering problem, the RLS algorithm matches the Kalman algorithm when an appropriate initialisation is provided [25]. The system is given by the equation yi = d(i)S(i) + wi

(25)

with the spreading vector S(i) = (S0,−K¯ (i), . . . , Sm,k (i), . . . , SM−1,K¯ (i))T . Since the spreading coefficients are constant over one transmission block, the equation S(i) = S(i − 1),

∀ i = 0, . . . , N − 1

d(i)P(i − 1)d H (i) + 1

P(i − 1)d H (i)


6. EXTENDED METRIC In the ith step of the proposed JCE algorithm, the path selection according to the metric in Equation (21) is based on the hypothetical data matrix Di where the actual data symbol d˜ i is weighted with the first channel tap ¯ M−1 2 2 h0 ≈ K ¯ ξk (i) · S0,k . If h0 < m=1 hm , more k=−K energy of d˜ i is contained in the following received values yi+1 , yi+2 , . . . , yi+M−1 . Generally, due to the reduced number of paths, the interference of the current symbol di on the subsequent symbols di+1 , . . . , di+M−1 is not automatically taken into account in a suboptimal algorithm. Consequently, the simple metric in Equation (21) does not exploit the entire information. Especially in highly frequency-selective channels with strong signal reflections, a significant improvement can be achieved if the metric is extended to comprise also the subsequent M − 1 received values. Here, we combine the extended metric with the channel estimation based on the DSF for applications in time-varying channels. To obtain the extended metric for the ith time step, it is assumed that the estimation of the S˜ m,k (i), as well as the already detected data symbols di+m−j with m − j < 0, are correct, to then calculate the squared distance between yi+m and y˜ i+m . The estimation of the future received value yi+m , with 0 < m M − 1, is given by y˜ i+m =

M−1

di+m−j

¯ K

ξk (i + m) · S˜ j,k (i)

(30)

¯ k=−K

where data symbols d˜ i+m−j with m − j > 0 are unknown. We model these unknown data symbols, d˜ i+ ∈ {−1, 1}, as statistically independent random variables with E[d˜ i+ ] = 0

(27)

where the Kalman gain is given by 1

(29)

(26)

˜ − 1)) ˜ = S(i ˜ − 1) + K(i)(yi − d(i)S(i S(i)

K(i) =

P(i) = P(i − 1) − K(i)d(i)P(i − 1)

j=0

is valid. The RLS equations for the estimation of the spreading coefficients are summarised as follows: The spreading vector estimation update is calculated by

˜ − 1) + K(i)e(i|i − 1) = S(i

and the a priori estimation error by Equation (23). The covariance matrix is updated by

(28)

and

E[d˜ i+l · d˜ i+ ] = ES δ( − )

(31)

where ES denotes the symbol energy. Under this assumption, the interference of a long CIR can be considered as a superposition of many statistically independent additive noise terms. Consequently, the noise added to each data symbol can be approximated by Gaussian noise. Eur. Trans. Telecomms. 2008; 19:1–13 DOI: 10.1002/ett

8

S. GLIGOREVIC

The interference introduced by the unknown symbols is considered as additional noise wi+m for yi+m with m−1

wi+m =

d˜ i+m−j

¯ K

ξk (i + m) · S˜ j,k

(32)

¯ k=−K

j=0

With Equation (31) the variance σ 2 (l = i + m) is given by 2 K¯ m−1 N 0 2 ˜ (33) + ES ξk (l) · Sj,k σ (l) = 2 ¯ j=0 k=−K for i + 1 < l i + M − 1 and σ 2 (l) = N0 /2 for 0 < l i as before. The squared error is then weighted by the respective variance, that is the new metric becomes E (i) = (i) · 2ES /N0 + µ(i)

(34)

where (i) is the simple metric calculated by Equation (21) and the additional metric term µ(i) is given by µ(i) =

M−1 m=1

E yi+m − y˜ i+m 2 σ 2 (i + m)

(35)

By distinguishing between known and unknown data symbols in the second term and exploiting Equation (31), the simplified expression given in Equation (36) is obtained. Note that the recursion rule in Equation (22) remains valid. The additional term µ(i) affects only the actual decision and hence, it has to be discarded after the path selection in the ith step is done.

µ(i) =

M−1 m=1

7. SIMULATION RESULTS Simulations presented in this section are intended to illustrate the performance of the proposed JCE using DSF approach in comparison with the common RLS estimation of the CIR, which is provided jointly with the data detection in a reduce state diagram. ¯ + 1) DSF The JCE algorithm estimates C = M(2K coefficients. In a time-invariant channel, with K = 0 the estimation reduces to the CIR estimation. Setting K > 0 would lead to a larger number of unknown parameters and thus, to an inferior performance for low Eb /N0 . We assume that a signal transmitted at high velocities would not have significant delay spread (considering for example aeronautical en-route communication, a line-ofsight is always present but signal reflections are rather negligible) and a highly frequency-selective channel would not entail a very wide Doppler spread (indoor or urban area). Thus, for equalisation test purposes three different channels have been simulated:

CH-10 with dominant frequency-selectivity, CH-100 with dominant time-variance, CH-40 where time-variance and frequency-selectivity are balanced. Table 1 summarises the parameters of the test channels. CH-100 is generated for the vehicle velocity of ‘only’ 100 km/h for two reasons: first to have a similar number of unknown parameters and thus, comparable performance as for CH-10 and CH-40 and second to enable a comparison with the simulation results in Reference [16]. Furthermore,

 2 ¯ M−1 K 1  ˜ di+m−j ξk (i + m)Sj,k (i) · yi+m − σ 2 (i + m) ¯ j=m k=−K

  ∗ ¯ M−1 ¯ m−1 K K   − 2Re yi+m − di+m−j ξk (i + m)S˜ j,k (i) · S˜ j,k (i)ξk (i + m)E[d˜ i+m−j ]    + E 

¯ j=m k=−K

¯ m−1 K



d˜ i+m−j ξk (i + m)S˜ j,k (i) 

¯ j=0 k=−K

¯ j=0 k=−K

¯ m−1 K

∗ 

d˜ i+m−j ξk (i + m)S˜ j,k (i) 

¯ j=0 k=−K

2 2 ¯ M−1 ¯ m−1 K K (i + m)S (i + m)S ˜ j,k (i) + ES ˜ j,k y − d ξ ξ i+m i+m−j k k M−1 ¯ j=m ¯ j=0 k=−K k=−K = 2 ¯ m−1 K m=1 N0 ξk (i + m)S˜ j,k 2 + ES ¯ j=0 k=−K Copyright © 2007 John Wiley & Sons, Ltd.

(36)


9


Table 1. Equaliser test channels.

0

10

CH-100

CH-40

CH-10

Velocity (km/h) Number of taps Mch Approx. Doppler rate Kch νmax · τmax Number of DSF coeff. estimated C

100 2 10 0.0083 50

40 5 4 0.0133 55

10 9 1 0.0075 45

the test channels are generated with a flat multipath intensity profile, which generally presents the worst case scenario for equalisation. Nevertheless, a distinct improvement is expected to be achieved here with the extended metric. For all test channels, the simple and the extended metrics are simulated also for JCE with CIR estimation. Besides, the performance of the proposed JCE method with only one path is compared with the DFE using the DSF. Differentially encoded Binary Phase-Shift Keying (DBPSK) modulation without channel coding has been considered in the simulations. In non-fading channels, the bit error rate (BER) obtained with DBPSK will be doubled in comparison with BPSK. However, if the signal is faded, the JCE algorithm is susceptible to the burst errors caused by the possible estimation of the inverted tap functions. Such errors are avoided with DBPSK. The symbol rate is set to 10 kbit/s, the carrier frequency to 900 MHz. Unless otherwise stated, the JCE simulations are provided with P = 4 paths in the reduced state diagram. For the DSF estimation, the oversampling factor is set to g = 1.2 and for the CIR estimation, the forgetting factor of the RLS algorithm is set to = 0.9. In each block of length N = 1200, a training sequence of length L = 50 is sent for initialisation of the RLS

−1

10

BER

Channel

−2

10

DFE with DSF JCE with DSF, P=1 JCE with CIR, P=4 JCE with DSF, P=4 JCE with CIR +ext.metric JCE with DSF+ext.metric

−3

10

6

7

8

9

10 11 Eb / N0 [dB]

12

13

14

15

Figure 5. BER performance in CH-10 channel. ¯ ˜ algorithm. We set S(−1) = 0M(2K+1)×1 and P(−1) = ¯ ¯ M(2K+1)×M(2 K+1) . Here, 0 represents a zero-matrix, I an I identity matrix, with the respective dimensions given by their exponents. The initial data matrix D contains symbols of the known or virtual training sequence for i 0 and zeros for i < 0. The metric (0), the Kalman gain K(0) and the a priori estimation error e(0| − 1) are set to zero at the beginning of each block. Figure 4 shows an example of the measured envelope of the CH-100 channel and the estimation obtained with the proposed algorithm at the end of one data block with P = 4 if no noise is available. The BER performances for different test channels are shown in Figures 5–7. As can be seen in Figure 5, which 0

10

10

−1

10

−10 BER

Rayleigh Envelope in dB

0

−20

−2

10

−30

DFE with DSF JCE with DSF, P=1 JCE with CIR, P=4 JCE with DSF, P=4 JCE with CIR+ext.metric JCE with DSF+ext.metric

h0(t) estimated h0(t)

−40

h1(t) estimated h (t) −50 0

−3

1

200

400

600 t/T

10

800

Figure 4. Original and estimated envelope. Copyright © 2007 John Wiley & Sons, Ltd.

1000

1200

6

7

8

9

10 11 E / N [dB] b

12

13

14

15

0

Figure 6. BER performance in CH-40 channel. Eur. Trans. Telecomms. 2008; 19:1–13 DOI: 10.1002/ett

10

S. GLIGOREVIC

0

0

10

10

CH−100, P=2 CH−100, P=4 CH−100, P=8 CH−40, P=2+ext.metric CH−40,P=4+ext.metric CH−40, P=8+ext.metric

−1

10

−1

BER

BER

10

approx.DFE in [16]

−2

10

DFE with DSF JCE with DSF, P=1 JCE with CIR, P=4 JCE with CIR+ext.metric JCE with DSF, P=4 JCE with DSF+ext.metric

−3

10

6

7

8

9

10 11 Eb /N0 [dB]

−2

10

12

13

14

15

Figure 7. BER performance in CH-100 channel.

shows the results for the test channel CH-10, the proposed method does not show any advantages over the estimation of the CIR in slow time-varying channels. Due to the significant inter-symbol-interference (ISI), JCE profits from the extended metric despite the slow time-variations in the CIR. In CH-40 channel given in Figure 6, the DSF estimation outperforms CIR estimation for higher Eb /N0 . The extended metric yields a gain of at least 4 dB even for the CIR estimation. Comparing JCE with DSF performance with the simple metric in CH-40 with the performance in CH-100 in Figure 7 verifies the capability of our approach to exploit the time-diversity. Furthermore, due to the considerable ISI and hence larger multipath diversity, a significantly higher performance gain is achieved with the extended metric for higher signal-to-noise ratios in CH-10 and CH-40 compared to CH-100. In all three channels, the DFE based on DSF estimation performs poorly. JCE with only one path performs comparable, although the decision is based on the updated channel estimation for the two hypotheses of the current symbol. Since the estimation in both algorithms is based on the DSF, ‘virtual’ frequencies have been used instead of the true ones. This differs from the DFE applied in Reference [16], where the Doppler frequencies are estimated before. Overall, this shows that the channel variations cannot be easily adapted if the past channel course is already determined. Considering the CH-100 channel, we have the same Doppler frequency of ν = 83.33 Hz as used in BER simulations presented in Reference [16], but with K = 10 the Doppler rate is doubled. To indicate the differences in the performance between these two approaches, we approximate the required BER curve by halving the Copyright © 2007 John Wiley & Sons, Ltd.

−3

10

0

20

40 60 Length of the training sequence

80

100

Figure 8. Impact of the training sequence length on the JCE performance.

symbol-error-rate (SER) of QPSK modulation simulated over Eb /N0 in Reference [16] and add the resulting BER curve to Figure 7. Despite a higher Doppler rate and a shorter training sequence (L = 200 as in Ref. [16]), the JCE algorithm based on the DSF outperforms the DFE with estimated Doppler frequencies in Reference [16]. It also shows that a reliable performance in fast time-varying channels is achievable with the DSF approach. According to the results for DFE, the DSF seems to be unsuitable for separate estimation and equalisation. However, an appropriate application for slow fading channels is proposed in Reference [19]. Here, the optimal design of training parameters requires L = 2M(2K + 1) pilot symbols for channel identification. Note also that no oversampling is considered here. Contrary to this application, the JCE algorithm does not fail even if only a short training sequence is used. Figure 8 shows the BER performance in CH-100 with the simple metric and in CH-40 with the extended metric for different lengths of the training sequence, in case of P = 2, 4 and 8 and for Eb /N0 = 10 dB. It can be seen that the joint approach barely degrades when reducing the length of the training sequence. Only JCE with extended metric and with more paths, that is data hypotheses considered, profits slightly from the longer training sequence. 8. COMPLEXITY ISSUES While algorithms for tracking of TV-CIR update a vector of length M in each step, in our approach an update of Eur. Trans. Telecomms. 2008; 19:1–13 DOI: 10.1002/ett

11


¯ + 1) coefficients is required. A similar C = M · (2K observation holds true for the approach of Reference [16], where the channel coefficients vector is q · M long, with q the number of frequencies of the exponential basis sequences. Furthermore, in a practical application C would probably turn out to be larger than M · q. Still, the latter approach implies an estimation of the frequencies. The proposed estimation of the DSF in a reduced state diagram needs to store in the ith step P · (1 + C + i) values. Here, i corresponds to the length of the currently detected data sequence, ‘1’ to the current metric value and P to the number of paths. If decisions are taken according to the extended metric, the number of values to be stored does not change. Still, while the simple recursive metric in Equation (22) requires 2 · (C + 1) additions and (2 · C + 1) multiplications, the extended metric requires additionally (M − 1) · (C + 1) additions and (M − 1) · (C + 2) multiplications. An a priori knowledge about the position of reflectors or delay profile could be used to reduce the number of parameters to be estimated, thus having an impact on the storage and number of computations. Note also, that the estimated coefficients comprise not only the Doppler shift but also any other drift in the frequency, thus making the synchronisation at the receiver redundant. 9. CONCLUSION By exploiting the fact that the DSF represents the Fourier series of the time-variations of the CIR within an observation interval, we developed an algorithm for robust estimation of TV-FS channels. Starting from the representation with the DSF, we analysed a method for joint adaptive estimation of the DSF and data equalisation. An adaptive channel estimation was provided in a reduced state diagram, where the metric for the path selection evaluates the squared distance between the received signal and its hypotheses. By extending the metric in order to consider the interference of the actual data symbol appearing with later received symbols a more reliable data decision is achieved. In comparison with JCE based on the common RLS estimation, the proposed approach showed improved performance in channels with significant time-variance. Furthermore, any available a priori information about the channel allows a complexity reduction by predetermining the DSF and enabling an energy location. Simulation results show that in time-varying channels with significant ISI, the JCE can benefit from the extended metric, even if the common RLS estimation of the CIR is applied. Copyright © 2007 John Wiley & Sons, Ltd.

APPENDIX Below, the recursion formula in Equation (22) is derived. The abbreviations b(i) = DH i y(i)

and

Qi = DH i Di

(37)

allow us to write the following relations: b(i) = b(i − 1) + d H (i) · yi Qi = Qi−1 + d (i)d(i) H

(38) (39)

¯ + 1) vector given by Equation Here, d(i) is a 1 × M · (2K (12). The matrix inversion lemma implies −1 Q−1 i = Qi−1 −

−1 H Q−1 i−1 d (i)d(i)Qi−1 H d(i)Q−1 i−1 d (i) + 1

(40)

Hereby, Q−1 (i) corresponds to the covariance matrix P(i) and the vector K(i) =

H Q−1 i−1 d (i) H d(i)Q−1 i−1 d (i) + 1

(41)

to the gain vector K(i) of the RLS algorithm. Thus, we obtain P(i) = P(i − 1) − K(i)d(i)P(i − 1)

(42)

K(i) = P(i − 1)d H (i)(d(i)P(i − 1)d H (i) + 1)−1

(43)

and

It follows from Equation (43) that P(i − 1)d H (i) = K(i) · (d(i)P(i − 1)d H (i) + 1)

(44)

Considering this while multiplying Equation (42) with d H (i) leads to P(i)d H (i) = K(i) · (d(i)P(i − 1)d H (i) + 1) −K(i)d(i)P(i − 1)d H (i) = K(i)

(45)

The estimated spreading vector then becomes ˜ = P(i)b(i) = P(i)(b(i − 1) + d H (i) · yi ) S(i) Eur. Trans. Telecomms. 2008; 19:1–13 DOI: 10.1002/ett

12

S. GLIGOREVIC

= P(i − 1)b(i − 1) −K(i)d(i)P(i − 1)b(i − 1) + K(i)yi

7.

˜ − 1) + K(i)(yi − d(i)S(i ˜ − 1)) = S(i ˜ − 1) + K(i)e(i|i − 1) = S(i

8.

(46) 9.

The metric in Equation (21) is then given by "

10.

= −Re{(bH (i − 1) + yi∗ d(i))(! S(i − 1)

11.

S(i) (i) = −Re bH (i)!

+ K(i)e(i|i − 1))} = −Re{(b

H

12.

(i − 1)! S(i − 1) + yi∗ d(i)! S(i − 1) H

+ bH (i)P(i)d (i)e(i|i − 1)}

13.

= (i − 1) − Re{yi∗ d(i)! S(i − 1) + S˜ H (i)d H (i)e(i|i − 1)} = (i − 1) − Re{y(i)2 − yi∗ (yi − d(i)! S(i − 1)) + S˜ H (i)d H (i)e(i|i − 1)} = (i − 1) − y(i)2 # $ " + Re yi∗ − S˜ H (i)d H (i) e(i|i − 1)

14.

15.

(47)

16.

Neglecting the term −y(i)2 , which is common for all paths, we obtain

17.

(i) = (i − 1) + Re{e∗ (i|i)e(i|i − 1)}

18. 19.

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AUTHOR’S BIOGRAPHY Snjezana Gligorevic received her Dipl.-Ing. Univ. degree in Electrical Engineering from the Technical University of Darmstadt, Germany, in 1998. In 2004, she received her Dr.-Ing. (Ph.D.) degree from University of Ulm, Germany, for her work in the field of channel estimation and equalisation for fast time-varying channels. Since 2005 she is a member of the research staff at the Institute of Communications and Navigation of DLR, Oberpfaffenhofen.

