Combined Block-Turbo Coding and RBF-Based ...

ICICS-PCM 2003 15-18 December 2003 Singapore

3A7.10 Combined Block-Turbo Coding and RBF-based Equalization with Decision Feedback M. Noorbakhsh, K. Mohamed-Pour K.N.Toosi Univ. of Technology, Tehran, Iran P.O.Box: 16315-1355, [email protected]

decided symbols. Such a decision feedback RBFbased equalizer is introduced in [6]. The system of concatenated RBF-based equalizer and BTC is introduced in [7], while joint BTC and RBF-based equalizer is introduced in [8]. This paper introduces a different design for combination of RBF-based equalizer and BTC. The channel situation used in [8] is slowly fading channel so that it is possible to use M-QAM modulations. Here, we consider a faster fading channel and use 4,8-PSK modulations, and show the gain of the combined RBF-based equalizer and BTC versus a non-combined system and also the system of DFE equalizer and BTC. In part II, the RBF-based equalization is reviewed. In part III, the system of joint RBF-based equalizer and BTC is introduced and simulation results are presented in part IV.

Abstract The combination of the Radial Basis Function (RBF) based-equalizer with decision feedback and block turbo decoder is studied. In block turbo coding (BTC) which is based on iterative decoding of a product code, the RBF-based equalizer enters the iteration loop. There is some improvement in performance of this new scheme compared with the case that they are used individually, and also the system of the decision feedback equalizer and block turbo decoder. Key words: Turbo code, Channel Equalization, Radial Basis Function Networks I. Introduction It has been shown that if the equalizer incorporates in the iteration loop of the (turbo) decoder and exchanges information with the constituent decoders, so that the decoders help the equalizer and vise versa, the performance will be improved. The system of combined equalization and error correcting decoding is named “Turbo Equalization” and studied in papers [1-3]. In these papers single convolutional coding or convolutional turbo coding is considered and the optimum equalizer (ML equalizer, based on the trellis detection) is used which is much more complex than the decision feedback equalizer (DFE). If a high throughput modulation such as 16-QAM, 64-QAM or 256-QAM is used, there are a huge number of states in the trellis equalizer so that it becomes impractical. Therefore combination of the decision feedback equalizer and turbo coding is introduced for BPSK [4], and M-QAM modulations [5]. If we use the symbolby-symbol MAP/ML equalizer, the equalization performance is improved with respect to DFE. (Trellis equalization is based on the sequence MAP/ML decision.) It is shown that such an optimum symbol-by-symbol equalizer has the structure of the Radial Basis Function (RBF) Network, which is a class of Neural Network systems [6]. Although the problem of numerous states, when using a high throughput modulation is seen in this equalizer, but the complexity can be dramatically reduced if we use the previously 0-7803-8185-8/03/$17.00 © 2003 IEEE

II. RBF-based Equalization Consider the discrete baseband channel model with coefficients {hk} so that the equalizer input after demodulation and match filtering is:

vk =

L2

∑h I

i = − L1

i k −i

+ nk

(1)

where Ik is the transmitted symbols of an M-ary modulation with symbols s1,…,sM, and nk is the channel complex white Gaussian noise. The parameters L1 and L2 are in such a way selected so that the maximum of hk occurs at k=0. For the symbol

decision

ˆI , k

the

vector

v k = [v k v k +1 … v k + m-1 ] and the decision vector Iˆ k = [ˆI k −1 ... ˆI k -n ] as feedback are used. The block diagram of the RBF-based equalizer is shown in Fig. 1. In the RBF-based equalizer, the radial basis functions fi(vk) for each symbol of the M-ary modulation is computed as following. Ns

f i ( v k ) = ∑ p ij (k ) exp(− v k − c ij (k ) / w ) 2

j=1

i=1,..,M (2) Where cij(k) are the RBF centers and w is the RBF width.

1

decision, if pij(k)=Pr(Ik=sij) , cij(k)=vk(sij), and w = 2σ 2n , where sij is all Ns possible values of Ik with the constraint of Ik=si, and cij(k) is the value of vk from (1) by setting Ik=sij and ignoring the noise, and σ 2n is the channel noise variance. If we don’t use the previously decided symbols Iˆ k as m + L + L −1 feedback, N s = M 1 2 . Using Iˆ k , Ns is m + L −1 exponentially reduced to N s = M 1 . Computational complexity of the RBF symbol-bysymbol equalizer with feedback is much less than the trellis sequence equalizer. Furthermore despite the trellis equalizer, the RBF equalizer simply gives a soft output value, which is necessary for soft decision decoding or turbo decoding. As fi(vk) is proportional to Pr(Ik=si|vk), the RBF soft output can be computed as:

Fig. 1: RBF-based equalizer

The symbol decision ˆI k is made as

ˆI = s , k i0

i 0 = arg max f i ( v k )

(3)

i

According to (1) Ik influences [ v k − L1 ... v k + L 2 ] .

M

Thus the maximum of m is selected as m=L2+1. We may compromise between complexity and performance in such a way that if we select a smaller value for m, the performance degrades, but the computational complexity is reduced exponentially. The transmitted symbols that influence the equalizer decision ˆI k are:

~ Ik =

∑ s f (v i =1 M

i i

∑ f (v i =1

i

k

k

) (4)

)

Logarithmic RBF equalizer To avoid the multiplication and exponential operations in (2) for complexity reduction, we can use the logarithmic version of the RBF equalizer [7,8]. If we define the Jacobean function J ( x 1 ,..., x K ) = ln(e x1 + ... + e x K ) (5) the logarithmic RBF functions are computed as (6), and maximization of ln f i ( v k ) is equivalent to

I k = [I k − L 2 ... I k -1 I k ... I k + m + L1 −1 ] . Thus n=L2 is selected. Selection of m and n more than the mentioned values has no use. The structure of the RBF equalizer is identical to the optimum symbol-by-symbol maximum a posteriori (MAP) or Bayesian equalizer, if the parameters pij(k), cij(k), and w are properly selected [6]. It can easy be shown that maximization of fi(vk) is equivalent to maximization of Pr(Ik=si| vk) , which is the MAP

maximization of g i ( v k ) defined as (7).

2 2 ln f i ( v k ) = J ln p i1 (k ) − v k − c i1 (k ) / w ,..., ln p iN s (k ) − v k − c iNs (k ) / w  , i=1,..,M  

(6)

g i ( v k ) = ln f i ( v k ) − ln p11 (k ) p iN (k ) 2   p (k ) 2 = J ln i1 − v k − c i1 (k ) / w ,..., ln s − v k − c iNs (k ) / w  p11 (k )   p11 (k )

(7)

using the function g ( x ) = ln(1 + e − x ) , which can be implemented with a simple look up table. The other advantage of the logarithmic approach is that the turbo decoder gives the values of ln(pij/p11) as the feedback to the equalizer. Thus relation (7) is more suitable than relation (2) for

Where p11 (k ) = Pr( I k = s 1 , I k +1 = s 1 ,..., I k + m + L −1 = s 1 ) and s1 is the first modulation symbol corresponding to all zero bits. As -|x1 − x 2 | J(x 1 , x 2 ) = max (x 1 , x 2 ) + ln(1 + e ) , the Jacobean function can recursively be computed 1

2

(presented in part V) the hard and soft symbol estimations nearly have a same performance. Thus we issue the hard symbol estimation. Each log2M ‘L’ obtained in (8) is compared to zero to estimate the corresponding bit. Then using the constellation bit map, these log2M bits are mapped to a symbol as a hard estimate. L = α(L ext1 + L ext 2 ) + L ch (8) (Lch is the channel LLR, and Lext1 and Lext2 are the extrinsic LLRs of the constituent decoders.) Also the priori probability mass functions pij(k) of the equalizer, corresponding to the non-decided symbols [ I k ... I k + m + L1 −1 ] are determined with the

the case of combination of equalization and turbo decoding. III. Combined BTC and RBF-based Equalization Consider the block turbo coding system introduced in [1,2,5]. For each complex-valued MQAM/PSK symbol, log2(M) channel LLRs must be computed and fed to the turbo decoding process shown in Fig.2. In [2,5] the cases of M-QAM modulation are considered, where the channel LLRs corresponding to symbol bits are functions of either inphase or quadrature components. For the case of M-PSK, the channel LLRs are functions of the symbol phase and like the cases of M-QAM, piece-wise approximation for them can be used. Lpri(di)

LLR Mapper Demodulated Input Signal

Lext(di)

Lpost(di)

Lpost(dn) to Decision

Lch(dn)

Lch(di)

We can enter the RBF-based equalizer in the decoding iteration loop as shown in Fig.3. The extrinsic LLR from turbo decoding improves the channel LLR, and by using the addition of these LLRs, the symbol estimation for the previously

ln

LLR mapping

turbo

)

m + L1 −1

Mod _ k

 =0

t =1

(9)

∏ ∏ P(Bit

t

(I k +  ))

i=0 i =1

(11)

(12)

We substitute the extrinsic LLR values of (10) for the equation (12).

DeInterleaver

Lext1 Lpri2 SISO SISO DEC. 1 DEC. 2

Lch

Previously decided symbols Symbol Estimation

k +

P(Bit t (I k + ) = i) 0 = P(Bit t (I k + ) = 0) L(Bit t (I k + ))

Lpri1

Priori probabilities

 =0

To compute g i ( v k ) in (7), we have p ij (k ) m+ L1 −1 Mod _ k P(Bit t (I k + )) = ∑ ∑ ln ln p11 (k ) P(Bit t (I k +  ) = 0) t =1  =0

Iˆ k = [ˆI k −1 ... ˆI k -n ] is decided symbols, performed; while in the separated equalization and decoding system, hard decisions are used as the decision feedback regardless of coding. The extrinsic information of the turbo decoder is sum of the extrinsic LLRs produced by the constituent decoders and attenuated by a factor α like the system described in [5]. The factor α increases to one as the iterations go on. As mentioned in [5], symbol estimation can be hard or soft. For the simulated channel and transceiver condition

RBF Equalizer

constituent

where Mod_k=log2(M) is the number of bits carried by a symbol, and Bitt(I) is the tth bit carried by the symbol I. An LLR type variable corresponding to the bit b can be interpreted as L(b) = ln[P(b = 1) / P(b = 0)] . For each bit, the turbo decoder extrinsic LLR is L(b) = α(L ext1 + L ext 2 ) (10)

Fig. 2: Block Turbo Decoding

Demodulated input signal

∏ P( I

=

Lext(dn)

of

m + L1 −1

p ij (k ) =

Lpri(dn)

SISO COLUMN DEC.

SISO ROW DEC.

extrinsic information decoders. We can write

+

Lext2

x

Interleaver

+

α

Fig. 3: Combined the RBF-based equalizer and block turbo decoder

3

Lpost2 Decision

conventional DFE at Pb=10-3. Combining the equalization and decoding gives about 0.8 dB gain for the RBF-based equalizer, and 0.5 dB for the DFE.

IV. Simulation Results For simulation, we have considered a multipath Rayleigh fading channel in urban condition in the GSM band as specified in [9] with the delay power mentioned in table I. Two systems BTC(64,51)24PSK with the symbol rate of 0.5 Msymb/s, and the mobile velocity v=20 m/s, and BTC(63,50)28PSK with the symbol rate of 0.27 Msymb/s, and v=10 m/s are considered. The constituent code (63,50) is obtained by shortening of the extended BCH code (64,51). As in the fading condition, the decision directed adaptive channel tracking (based on the LMS algorithm) may not be so reliable, we have considered a meadamble training sequence per each burst for the channel estimation. Based on the estimated channel coefficients, the equalizer parameters are set. The DFE equalizer with the linear feedforward and feedback filters is adjusted as mentioned in [10]. The RBF-based equalizer parameters are cij(k)=vk(sij), which can be obtained from (1) after channel estimation, and w = 2σ 2n , which can be estimated from the smoothed sequence of

10

10

B it E rror R ate

10

10

10

0

1: 2: 3: 4:

-1

-2

-3

-4

1 10

10

3

4

-5

2

-6

8

10

12

14 E b/N 0 (dB )

16

18

20

Fig. 4: BTC(64,51)2-4PSK performance for the fading channel with the normalized doppler frequency Fdop/Fsymbol rate=1.33e-4. 10

10

0

1: 2: 3: 4:

-1

2

Min v k − c ij (k ) . The DFE and

10 B it Error R ate

i, j

RBF equalizer have the same orders: m=2 and n=2. (i.e. for the symbol decision ˆI k , the input

10

10

vector v k = [v k v k +1 ] and the decision vector

10

Iˆ k = [ˆI k −1 ˆI k -2 ] as feedback are used.) Figures 4 and 5 show the RBF/DFE-based equalization and turbo decoding performance for the BTC(64,51)2-4PSK and BTC(63,50)2-8PSK systems respectively. In the case of BTC(64,51)24PSK, RBF based equalizer has about 2 dB gain with respect to the conventional DFE at the bit error rate of Pb=10-3. Combining the equalization and decoding gives about 0.5 dB gain at Pb=10-3 for both DFE and RBF-based equalizers.

10

In dep end ent D FE (2,2), T urb o D ec. Joint D FE (2,2), T urbo D ec. In dep end ent R B F (2,2), Tu rb o D ec. Joint R B F (2,2), Tu rb o D ec.

-2

4

-3

-4

3

-5

1

2

-6

10

12

14

16 18 E b/N 0 (dB )

20

22

24

Fig. 5: BTC(63,50)2-8PSK performance for the fading channel with the normalized doppler frequency Fdop/Fsymbol rate=1.23e-4.

V. Conclusion The RBF-based equalizer has a superior performance with respect to the conventional DFE equalizer. The decision feedback used in the RBFbased equalizer reduces the complexity exponentially. When RBF / DFE equalizer enters the decoding iteration loop, further gain is achieved.

TABLE I: Delay Power for the wireless fading channel used in the simulation. Delay (micro sec) 0 0.2 0.5 1.6 2.3 5.0

Ind ep en dent D F E(2,2), Turb o D ec. Joint D F E (2,2), Turbo D ec. Ind ep en dent R B F(2,2), T urbo D ec. Joint R B F(2,2), T urb o D ec.

Normalized Mean Power (dB) -3 0.5 -2 -6 -8 -10

References [1] R.M.Pyndiah, “Near-optimum decoding of product codes: Block Turbo Codes”, IEEE Trans. Comm. Aug 1998, p.1003-1010. [2] R.Pyndiah, A.Picart, A.Glavieux, “Performance of block turbo coded 16-QAM and 64-QAM modulations “, IEEE GLOBECOM95, Nov 1995, pp.1039-1044.

In the case of BTC(63,50)2-8PSK, RBF based equalizer has about 1 dB gain with respect to the

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[3] D. Raphaeli, Y.Zarai, “Combined turbo equalization and turbo decoding”, IEEE GLOBECOM 1997, pp. 639-643. [4] D.Raphaeli, A. Saguy, “Linear equalization for turbo equalization A new optimization criterion for determinig the equalizer taps”, Proceedings of the 2nd Symposium on Turbo Codes & Related Topics, Brest, France, September 2000, pp.371-374. [5] M. Noorbakhsh, K. Mohamed-pour, “Combined turbo equalization and block turbo coded modulation”, IEE proceedings on communications, accepted for publication. [6] S.Chen, B.Mulgrew, S.McLaughlin,“Adaptive bayesian equalizer with decision feedback”, IEEE Trans. Signal Processing, vol. 41, Sept. 1993, pp. 29182927. [7] M. S. Yee, T. H. Liew, L. Hanso, “Block turbo coded burst-by-burst adaptive radial basis function decision feedback equalizer assisted modems”, Proceeding of IEEE Vehicular Tech. Conf., vol .3, Sept. 1999, pp. 1600-1604. [8] M.S. Yee, T.H. Liew, L. Hanso, “Burst-by-burst adaptive turbo coded radial basis function-assisted decision feedback equalization”, IEEE Trans. Communications, vol. 49, no. 11, Nov. 2001, pp. 19351945. [9] M.C.Jeruchim, P.Balaban, K.S. Shanmugan, “Simulation of Communication Systems, Modeling, Methodology, and Techniques”, Kluwer Academic Publishers, 2000. [10] J.G. Proakis,“Digital Communications”, McGrawHill, 1995, third edition.

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