Turbo-synchronization : an EM algorithm-based ...

5 downloads 0 Views 156KB Size Report
ABSTRACT. This paper is devoted to turbo-synchronization, i.e. the use of soft information delivered by a turbo receiver to estimate system parameters like the ...
EUROPEAN COOPERATION IN THE FIELD OF SCIENTIFIC AND TECHNICAL RESEARCH

COST 273 TD(03) 017 Barcelona, Espana January 15-17, 2003

———————————————— EURO-COST

———————————————— SOURCE: Communications and Remote Sensing Laboratory Universite catholique de Louvain Belgium

Turbo-synchronization : an EM algorithm-based approach

C. Herzet, V. Ramon, M. Moeneclaey, L. Vandendorpe Communications and Remote Sensing Laboratory Universite catholique de Louvain 2 place du Levant 1348 Louvain-la-Neuve Belgium Phone: +32 10 47 80 66 Fax: +32 10 47 20 89 Email: {herzet,ramon,vdd}@tele.ucl.ac.be

ABSTRACT This paper is devoted to turbo-synchronization, i.e. the use of soft information delivered by a turbo receiver to estimate system parameters like the carrier phase and the timing delay. It is shown how the iterative ExpectationMaximization (EM) algorithm may provide a maximum likelihood estimation of those parameters. A structure in which EM iterations are combined with those of a turbo receiver is then proposed. More particularly, phase-aided timing recovery and joint timing and phase recovery will be considered and compared. In each case performance of the proposed synchronizer will be illustrated through simulation results : the estimator mean and mean squared error as well as the bit error rate reached by the synchronized system will be reported. 1. INTRODUCTION Since the discovery of turbo codes by Berrou [1], the so-called turbo principle has been extended to a number of operations to be performed by a receiver notably joint decoding and demodulation, joint decoding and equalization. In addition to detection and decoding, a receiver has also to estimate a number of synchronization parameters in order to work properly. A consequence of the low-SNR operating point in turbo receivers is that classical estimators fail to provide good estimates of the synchronization parameters. However, such iterative receivers are able to deliver soft information on bits or symbols. It would therefore be relevant to try using this soft information in order to help the synchronizer. The idea of using soft information to estimate parameters has already been applied in a number of contributions. Reference [2], for instance, proposes to combine soft decision-directed carrier phase estimation with turbo decoding. In [3] the carrier phase synchronizer is embedded in a maximum a posteriori decoder and exploits the extrinsic information generated at each turbo iteration. Paper [4] presents a unifying framework for ML synchronization in turbo systems by means of the expectation-maximization (EM) algorithm [5]. In this paper, we will focus both on phase-aided timing recovery and on joint timing and phase recovery in turbo receivers. In those two cases, the proposed synchronizer will be derived from the EM algorithm. The sequel of this paper will then be organized as follows. In section 2, the considered transceiver will be presented. EM algorithm principles will be introduced in section 3 and will be applied to synchronization in section 4. The implementation of the proposed EM-based synchronizers in a turbo receiver will then be discussed in section 5. Finally, in section 6, the performance of the synchronizers will be illustrated through simulation results.

bits

Encoder

Interleaver

Mapper

Channel

Waveform

δ(t − τ )

u(t)

ak s(t)

r(t)

v(t)

ej θ

Fig. 1. Transmitter. 2. SYSTEM MODEL In this paper we will focus on a bit-interleaved coded modulation (BICM) scheme. The transmitter (Fig. 1) is then made up of a binary convolutional encoder and a constellation mapper separated by a bit interleaver. In the baseband formalism, the signal at the transmitter output may then be written as X s(t) = ak u(t − kT ), (1) k

where ak ’s are complex symbols belonging to constellation alphabet A, T is the symbol period and u(t) is a unit energy square-root raised-cosine pulse with roll-off α. Assume that s(t) is sent over an AWGN channel introducing a time delay τ and a carrier phase offset θ, the received signal is X r(t) = ak u(t − kT − τ ) ejθ + v(t), (2) k

√ where j = −1 and v(t) is the complex envelope of an additive white Gaussian noise with passband power spectral density N0 /2. At the receiver, after anti-aliasing filtering, r(t) is sampled at a rate of 1/Ts (with Ts < T /(1 + α) ) leading to samples X rl , r(lTs ) = ak u(lTs − kT − τ ) ejθ + vl , (3) k

where vl is a white Gaussian noise with variance 2N0 /Ts . The resulting samples rl are passed through a discretetime matched filter i.e. X y(kT + τ ) = rl u∗ (lTs − kT − τ ). (4) l

Then, sufficient statistics are obtained by correcting the carrier phase offset which affects the matched filter outputs, z(kT + τ ) = y(kT + τ ) e−jθ . (5) Finally, we assume that statistics z(kT + τ ) are processed in a turbo demodulator. Such a device introduced in [6] performs iterative joint demodulation and decoding through the exchange of extrinsic information between a soft-input soft-output (SISO) demodulator and a SISO decoder.

We see from (5) that the computation of sufficient statistics z(kT + τ ) requires both the knowledge of the timing delay τ and the carrier phase offset θ. The problem addressed in the sequel will therefore be the estimation of those synchronization parameters. Two cases will be considered. First, we will assume that the carrier phase offset is known and that only parameter τ has to be estimated. Secondly, the case of joint timing and phase recovery will be considered.

transmitted data symbols (a0 , a1 , ..., aK−1 ) ∈ AK , the parameter vector b to be estimated only contains the symbol timing τ in the phase-aided case and contains both parameters τ and θ in the case of the joint synchronization. Observation vector r contains the values of all the samples rl . Let us consider the joint synchronization case. Using the expression of the received samples rl and neglect˜ the log-likelihood function ing terms independent of b, ˜ ln p(r|a, b) present in (10) can then be written as

3. ML ESTIMATION AND EM ALGORITHM Let r denote a random vector and let b indicate a deterministic vector of parameters to be estimated from the observation of the received vector r. Assume that r also depends on a random nuisance vector a independent of b and with a priori probability density function p(a). The maximum likelihood estimate of b is then the value ˆ which maximizes the probability of observing vector b r i.e. ˆ = arg max{ln p(r|b)}, ˜ b (6) ˜ b

where ˜ = p(r|b)

Z

˜ p(a) da, p(r|a, b)

(7)

a

˜ is a trial value of b. The EM algorithm is a method and b which allows to resolve iteratively the maximization problem defined in (6). Formally, if we set z , [rT , aT ]T , ˆ (n) defined the EM algorithm states that the sequence b by Z (n−1) ˆ (n−1) ) ln p(z|b) ˜ dz,(8) ˜ ˆ p(z|r, b Q(b, b ) = z

ˆ (n) b

˜ b ˆ (n−1) )}, arg max{Q(b,

=

˜ b

(9)

converges under fairly general conditions towards the ML estimate (6). In the particular case of parameter vector b independent of vector a, it can be shown [4] that the Q-function defined in (8) reduces to Z (n−1) ˜ ˆ ˆ (n−1) ) ln p(r|a, b) ˜ da. Q(b, b ) = p(a|r, b a

(10) The solution of (6) can then be found iteratively by only ˆ (n) ) and the logusing posterior probabilities p(a|r, b ˜ likelihood function ln p(r|a, b). 4. SYNCHRONIZATION AND EM ALGORITHM In this section we will apply the EM framework described in the previous section to the particular case of synchronization for a BICM transmission. In this context, the vector a contains the values of the K unknown

˜ = Re ln p(r|a, b)

nK−1 X k=0

o

˜

a∗k y(kT + τ˜) e−j θ

(11)

where y(kT +˜ τ ) corresponds to the matched filter output evaluated at kT + τ˜. Let us define for each transmitted symbol ak Z 4 ˆ (n−1) ) = ˆ (n−1) ) da ηk (r, b ak p(a|r, b a ∈AK X ˆ (n−1) ). (12) = a p(ak = a|r, b a ∈A

˜ by (11) Using this definition and replacing ln p(r|a, b) in (10), we get ˜ b ˆ (n−1) ) = Re Q(b,

nK−1 X

o

˜

ˆ (n−1) ) y(kT + τ˜) e−j θ ηk∗ (r, b

k=0

(13) At each iteration the EM algorithm requires to find the estimates τˆ(n) and θˆ(n) which maximize (13). Due to ˜ b ˆ (n−1) ), the solution of the particular structure of Q(b, this maximization problem leads to a decoupled solution K−1 X ∗ ˆ (n−1) ηk (r, b ) y(kT + τ˜) (14) τˆ(n) = arg max τ˜

θˆ(n) = arg

nK−1 X k=0

k=0

ˆ (n−1) ) y(kT + τˆ(n) ) ηk∗ (r, b

o

(15)

Notice the similarity between data-aided (DA) maximumlikelihood expressions [7] and (14) and (15) : the latter are formally obtained from the former by simply replacing the actual symbols ak by their respective a posteriˆ (n−1) ). These mean values ori expected values ηk (r, b are not constellation points but rather weighted averages over all constellation points according to the posterior ˆ (n−1) ). The proposed EM-based probabilities p(ak |r, b synchronization method may therefore be seen as a “soft decision” directed method. In the same way we may derive a phase-aided timing estimator. At each iteration the new timing estimate is

2.a Estimator Mean

then given by τˆ

= arg max Re τ˜

nK−1 X

ˆ (n−1) ) y(kT ηk∗ (r, b

+ τ˜) e

−jθ

k=0

o

(16)

Mean of the estimated timing (−)

(n)

0.5 0.4 0.3 0.2 0.1

ˆ (n−1) ) ' p(ak |r, b

M Y

m=1

ˆ (n−1) ), p(cm k |r, b

(17)

where M is the number of bits contained in symbol ak , th m ˆ (n−1) ) is the bit cm k is the m bit of ak and p(ck |r, b a posteriori probabilities delivered by the SISO decoder at turbo iteration n. Using approximation (17), we may perform a new EM-step at each turbo iteration and therefore merge the synchronization iterations (EM algorithm) into those of detection (turbo demodulation).

0

0.1

0.2 0.3 Normalized timing offset (−)

0.4

0.5

0.4

0.5

0.4

0.5

2.b Timing estimator mean quadratic error

0

10

CR bound −1

10

−2

10

−3

10

−4

10

−5

10

0

0.1

0.2 0.3 Normalized timing offset (−) 2.c BER

0

10

−1

10 BER (−)

Thanks to the particular structure of the log-likelihood functions (11), we see from (12) that only the marginal ˆ (n−1) ) are required to posterior probabilities p(ak |r, b (n−1) ˜ b ˆ compute Q(b, ). The latter probabilities are however not directly available since the considered turbo demodulation scheme since it computes a posteriori probabilities on bits rather than on symbols. However, if we assume that, thanks to the presence of the interleaver, the bits transmitted in one symbol are independent, symbol posterior ˆ (n−1) ) may be approximated as probabilities p(ak |r, b

Mean quadratic error of the estimated timing (−)

0

5. EM ALGORITHM IMPLEMENTATION IN TURBO RECEIVERS

iteration 1 iteration 3 iteration 6 iteration 12

−2

10

−3

10

Perfect synchronization performance

−4

10

0

0.1

0.2 0.3 Normalized timing offset (−)

Fig. 2. Estimator mean, estimator variance and BER versus normalized timing offset for Eb /N0 = 4dB. new parameter estimates are computed using the posterior soft information delivered by the turbo demodulator. Statistics z(kT + τ ) are then recomputed according to the new estimates. 6.1. Phase-aided timing estimator

6. SIMULATION RESULTS In this section the performance of the proposed synchronization methods will be studied through simulation results. At the transmitter, we consider a rate- 12 non-systematic convolutional encoder with polynomial generators (g1 , g2 ) = (5, 7)8 and use 16-QAM modulation. A mapping proposed by ten Brink in [6] and referred to as medium unconditioned bit-wise mutual information mapping is used. The pulse waveform is a square-root raised cosine with roll-off 0.2. The interleaver is totally random and a different permutation is used at each frame. At the receiver, the matched filter outputs are computed from the samples thanks to an interpolator and a discrete-time matched filter. The considered interpolator [7] is designed in order to minimize, on the bandwidth of the useful signal s(t), the quadratic error between the ideal interpolator frequency response and the interpolator frequency response. The number of taps of the interpolator is set to 21. The parameters to be estimated are initialized to 0 at the first iteration. At each turbo iteration, the

Fig. 2 represents the phase-aided timing synchronizer performance (mean and mean squared error) as well as the bit error rate (BER) for Eb /N0 = 4dB and for a normalized timing offset τ /T ranging from 0 to 0.5. The dashed curves represent the data-aided (DA) CramerRao bound in Fig. 2.b and the BER reached by a perfectly synchronized system in Fig. 2.c. The simulations have been run for frames of 500 16-QAM symbols and 12 turbo iterations have been performed. We solve (16) by applying a Newton-Raphson method to the derivative of the function to be maximized. We notice that the timing estimation is unbiased and reaches the Cramer-Rao bound for values of τ /T up to about 0.4. In almost all this range, the estimation error is therefore small enough for the system to reach the BER of a perfectly synchronized system. For values of τ /T greater than 0.4, the estimator is biased and its variance moves away from the Cramer-Rao bound. This may be explained by observing that successive estimates τˆ(n) sometimes converge towards T −τ instead of

3.a Estimator Mean

4.a Timing estimator mean Mean of the estimated timing (−)

0.2

0.1

0

Mean quadratic error of the estimated timing (−)

0.5

iteration 1 iteration 3 iteration 6 iteration 12

0

0.5

1

1.5

2

2.5 Eb/N0 (dB)

3

3.5

4

4.5

3.b Timing estimator mean quadratic error

0

10

CR bound −1

10

−2

10

−3

10

−4

10

−5

10

0

0.5

1

1.5

2

2.5 Eb/N0 (dB)

3

3.5

4

4.5

5

0.2 0.1

0

0.05

0.1

0.15

0.2 0.25 0.3 Normalized timing offset (−)

0.35

0.4

0.45

0.5

0.35

0.4

0.45

0.5

0.35

0.4

0.45

0.5

4.b Timing estimator mean quadratic error

0

10

CR bound −1

10

−2

10

−3

10

−4

10

0

0.05

0.1

0.15

0.2 0.25 0.3 Normalized timing offset (−) 4.c BER

0

10

10

−1

−1

10 BER (−)

10 BER (−)

0.3

0

3.c BER

0

iteration 1 iteration 3 iteration 6 iteration 12 iteration 18

0.4

5 Mean quadratic error of the estimated timing (−)

Mean of the estimated timing (−)

0.3

−2

10

−2

10

−3

−3

10

10

Perfect synchronization performance

Perfect synchronization performance

−4

10

−4

10

0

0.5

1

1.5

2

2.5 Eb/N0 (dB)

3

3.5

4

4.5

5

Fig. 3. Estimator mean, estimator variance and BER versus Eb /N0 for normalized timing offset τ /T = 0.25. to the actual timing τ . The estimate mean remains however positive, which expresses that the system converges more often to the actual timing offset than to T − τ . In particular, the mean curve does not cancel out for τ /T = 0.5T unlike the non data-aided (NDA) estimator mean curve. This may be explained by the presence of the encoder in the transmission scheme. Indeed, it limits the number of possible transmitted sequences and therefore enables the proposed synchronizer to benefit from an a priori information about the transmitted sequence. Our timing estimator can then be regarded as a “codeaided” synchronizer i.e. is an intermediate case between a non data-aided and a data-aided estimator. Fig. 3.a, 3.b and 3.c represent the mean, the variance and the BER obtained for a normalized timing offset τ /T = 0.25 and for Eb /N0 -ratios ranging from 0dB to 5dB. The parameters are the same as for the previous simulations. We see that the synchronizer and the turbo demodulator are actually complementary. Indeed, on the one hand, the synchronizer requires a low BER in order to deliver good-quality estimates (i.e unbiased and with small variance). On the other hand, the turbodemodulator needs a good timing estimate in order to decrease the BER. This is the reason why the estimate variance starts converging to the Cramer-Rao bound for an Eb /N0 -ratio located in the so-called “water-fall” region. In the same way, it explains why the turbo demod-

0

0.05

0.1

0.15

0.2 0.25 0.3 Normalized timing offset (−)

Fig. 4. Timing estimator mean, timing estimator mean squared error and BER versus normalized timing offset for Eb /N0 = 4dB and θ = 15 degrees. ulator reaches the same BER as the perfectly synchronized system only when the synchronizer has converged to the Cramer-Rao bound. Note however that one may observe that the convergence to such a BER requires more iterations than for a perfectly synchronized system since in the considered system the receiver has to perform both the detection and the synchronization. 6.2. Joint timing and phase estimator Fig. 4 and 5 illustrate the performance of the joint timing and phase synchronizer. Mean and mean squared error of the timing estimate, respectively in Fig. 4.a and Fig. 4.b, as well as BER in Fig. 4.c are shown for Eb /N0 = 4dB, θ = 15 degrees and for a normalized timing offset τ /T ranging from 0 to 0.5. The dashed curves in Fig. 4.b and 4.c have respectively the same meaning as in Fig. 2.b and 2.c. Simulations have still been run for frames of 500 16-QAM symbols but this time 18 turbo iterations have been performed. Again, we solve (14) by applying a Newton-Raphson method to the derivative of the function to be maximized. We notice that after 18 iterations the timing estimate is unbiased and reaches Cramer-Rao bound for values of τ /T up to about 0.35. For those values the reached BER almost matches that of a perfectly synchronized system.

5.a Phase estimator mean

6.a Timing estimator mean

30

20 15 10

0.2 0.15 0.1 0.05 0

0

0.5

1

1.5

2

0 −5 −10

0

0.05

0.1

0.15

0.2 0.25 0.3 Normalized timing offset (−)

0.35

0.4

0.45

0.5

5.b Phase estimator mean quadratic error

0

10

CR bound

−1

10

2.5 EbNo (dB)

3

3.5

4

4.5

5

6.b Timing estimator mean quadratic error

0

10

CR bound −1

10

−2

10

−3

10

−4

10

0

0.5

1

1.5

2

2.5 EbNo (dB)

3

3.5

4

4.5

5

3

3.5

4

4.5

5

6.c BER

0

10 −2

10

−1

10 −3

10

BER (−)

Mean quadratic error of the estimated phase (−)

iteration 1 iteration 3 iteration 6 iteration 12 iteration 18

0.25

5

Mean quadratic error of the estimated timing (−)

Mean of the estimated phase (degrees)

25

Mean of the estimated timing (−)

iteration 1 iteration 3 iteration 6 iteration 12 iteration 18

−2

10

−4

10

−3

10

Perfect synchronization performance −4

0

0.05

0.1

0.15

0.2 0.25 0.3 Normalized timing offset (−)

0.35

0.4

0.45

0.5

10

0

0.5

1

1.5

2

2.5 EbNo (dB)

Fig. 5. Phase estimator mean and phase estimator mean squared error versus normalized timing offset for Eb /N0 = 4dB and θ = 15 degrees.

Fig. 6. Timing estimator mean, timing estimator mean squared error and BER versus Eb /N0 for a normalized timing offset τ /T = 0.25 and for θ = 15 degrees.

If we stop at iteration 12, like in the simulations for the phase-aided timing synchronizer, the timing estimate is unbiased only for τ /T less than 0.3 and even moves away from the Cramer-Rao bound for τ /T greater than 0.25. Fig. 5.a and 5.b respectively show the mean and mean squared error of the phase estimate, for the same simulation parameters as in Fig. 4. We notice that after 18 iterations this estimator is unbiased and reaches its Cramer-Rao bound for values of τ /T slightly less than 0.4. It clearly appears from Fig. 4 that the final performance (mean, mean squared error, BER) obtained with the joint timing and phase synchronizer is the same as with its phase-aided counterpart. We also observe that the timing and phase estimates for a given timing offset attain their respective Cramer-Rao bound at approximately the same iteration and for the same value of τ /T . This can be explained by the fact that the evolution of those two estimates depend on each other since θˆ(n) is computed from that of τˆ(n) , as we can see from (15). Fig. 6.a and 6.b respectively illustrate the mean and mean squared error of the timing estimate and Fig. 6.c shows the BER obtained for a normalized timing offset τ /T = 0.25, θ = 15 degrees and for values of Eb /N0 ranging from 0dB to 5dB. The other simulation parameters are again the same. Fig. 7.a and 7.b show the mean and mean squared error of the phase estimate

also for τ /T = 0.25, θ = 15 degrees and values of Eb /N0 between 0dB and 5dB. It appears that this estimator is unbiased for an Eb /N0 -ratio greater than 1.5dB but reaches Cramer-Rao bound only for Eb /N0 greater than 3.5dB. Again, we notice that the joint timing and phase synchronizer needs more iterations than its phaseaided counterpart to reach the BER of a perfectly synchronized system. Besides timing and phase estimates reach their respective Cramer-Rao bounds at the same value of Eb /N0 -ratio. 7. CONCLUSION In this paper we derived synchronizers from the EM algorithm. It is shown that those synchronizers are actually well-suited to the soft iterative structure of the turbo systems: they can be embedded in the receiver without significant increase of the complexity and may take benefit from the available soft information. Simulation results show that the proposed estimators are unbiased and reach the Cramer-Rao bound over a wide range of timing offsets. Over this range, the bit error rate reached by the synchronized system does not suffer from any degradation with respect to a perfectly synchronized system. It also appears that only a few more iterations are required in order that the performance of the joint timing

7.a Phase estimator mean

Global Communications Conference, GLOBECOM’, Sidney, Australia, Nov. 1998, pp. 579–584.

30 iteration 1 iteration 3 iteration 6 iteration 12 iteration 18

Mean of the estimated phase (degrees)

25 20

[7] H. Meyr, M. Moeneclaey, and S. Fetchel, Digital Communication Receivers : Synchronization, Channel Estimation and Signal Processing, Wiley Series in Telecommunications and Signal Processing, USA, 1998.

15 10 5 0 −5 −10

0

0.5

1

1.5

2

2.5 EbNo (dB)

3

3.5

4

4.5

5

4

4.5

5

7.b Phase estimator mean quadratic error

0

10 Mean quadratic error of the estimated phase (−)

CR bound

−1

10

−2

10

−3

10

−4

10

0

0.5

1

1.5

2

2.5 EbNo (dB)

3

3.5

Fig. 7. Phase estimator mean and phase estimator mean squared error and BER versus Eb /N0 for a normalized timing offset τ /T = 0.25 and for θ = 15 degrees. and phase synchronizer match that of the phase-aided timing synchronizer. 8. REFERENCES [1] C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: Turbo codes,” in IEEE Transactions on Communications, Oct. 1996, vol. 44, pp. 1261– 1271. [2] V. Lottici and M. Luise, “Carrier phase recovery for turbocoded linear modulations,” in IEEE International Conference on Communications, ICC’, New-York, USA, Apr. 2002. [3] W. Oh and K. Cheun, “Joint decoding and carrier phase recovery algorithm for turbo codes,” IEEE Communications Letters, vol. 5, no. 9, pp. 375–377, Sept. 2001. [4] N. Noels, C. Herzet, A. Dejonghe, V. Lottici, H. Steendam, M. Moeneclaey, M. Luise, and L. Vandendorpe, “Turbo-synchronization: an em algorithm interpretation,” submitted to IEEE International Conference on Communications, ICC’, 2003. [5] A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum-likelihood from incomplete data via the em algorithm,” J. Roy. Stat. Soc., vol. 39, no. 1, pp. 1–38, Jan. 1977. [6] S. ten Brink, J. Speidel, and J. C. Yan, “Iterative demapping and decoding for multilevel modulation,” in IEEE