performance comparison of mlse and iterative equalization in fec ...

PERFORMANCE COMPARISON OF MLSE AND ITERATIVE EQUALIZATION IN FEC SYSTEMS FOR PMD CHANNELS T. Schorr+, W. Sauer-Greff+, R. Urbansky+ +

University of Kaiserslautern, Communications Engineering, 67653 Kaiserslautern, Germany, email: [email protected]

Abstract: It is widely known that turbo equalization with low-rate forward error correcting (FEC) codes significantly reduces the bit error ratio (BER) in low signal to noise ratio (SNR) channels, and recently in this workshop it was reported that also for non-linear channels and high-rate codes remarkable turbo decoding gains can be achieved [1]. However, for a required target BER down to 10-16, an additional outer block code, and for high-speed implementation limited analog-to-digital converter (ADC) resolution and internal quantization have to be considered. These aspects will be analyzed and optimized with respect to a typical long-haul fiberoptical transmission system in presence of polarization mode dispersion (PMD).

1

Introduction

Many communication systems are corrupted by nonlinear intersymbol interference (ISI), like wireless communication with non-coherent receivers or high-rate optical data transmission. In the latter, non-linearity and ISI are due to propagation effects along the fiber and due to the use of photo diodes as signal detectors with square law characteristic, see Fig.1. In general, non-linear ISI can be described by a finite state channel model approximating the received signal after the electrical detection [2]. The use of state tables with conditional probability entries enable the application of iterative equalization and decoding, so-called turbo equalization, to non-linear ISI [1] The optimum MLSE receiver minimizing the sequence error probability can be derived from this channel model presented in more details in Section 2 and results in a sampled analog front-end filter delivering a set of sufficient statistics followed by a sequence detector [3]-[9]. Whereas for linear ISI the optimum prefilter consists of a single matched filter sampled at the symbol rate, a bank of matched filters is required for non-linear ISI [10]. For the sake of high-speed implementation, it has to be replaced by a suboptimal noise limiting lowpass filter at the prize of moderate degradations, which can be reduced by oversampling [3], [7], [9]. FEC using Reed-Solomon (RS) codes with 7% overhead rate, which are common in long-haul optical transmission, allow to operate the receiver at a BER up to

10-5, from where the RS decoder will correct to a target BER of 10-16. On the other hand, the turbo coding strategy, which allows for iterative decoding close to the Shannon limit and commonly utilizes low-rate parallel concatenated recursive convolutional codes, also has been proposed for these systems [11], [12]. In this paper we investigate RS codes with a typical high code rate RRS, where in addition to this outer code and its interleaver, turbo equalization is considered. In order to increase the code rate with respect to the classical iterative equalization and decoding approach, we first apply punctured convolutional codes with a code rate of 0.933 [1], [13]. The possibility of changing the code rate simply by modifying the puncturing scheme could be seen as an advantage of punctured convolutional codes. However, as it is expected from theory, suitable block codes of the same code rate require less SNR for the same BER. Therefore we also consider a turbo scheme including a block code. Whereas for punctured convolutional codes equalizer and decoder make use of the same maximum a posteriori probability (APP) symbol-by-symbol detecting algorithm, see Section 3, an APP decoder for block codes requires a different approach [1], [14]. The rest of this paper is organized as follows. We Modulator

d

FEC Coder

a

Pulse Former

s(t)

FEC Decoder

^ a

Transmission fiber - PMD - CD - Kerr nonlinearity - Gain - Noise figure

Receiver

^ d

OA

- Modulation format - Optical power - Chirp - Extinction ratio

Electrical filter

OA

Optical filter

Equalizer & Detector

Electronic signal processing

Bel

- Received Power

Bopt

- OSNR

Fig. 1. Block diagram of optical lightwave system using electronic equalization and FEC coding. outline the channel model, FEC coding, the basic concept of turbo equalization and its extrinsic-informationtransfer (EXIT)-chart analysis. After presenting the considered systems we compare the performance of different turbo equalization schemes with input and internal quantization. The paper concludes with a summary and an outlook on alternative coding schemes.

2

Transmission Channel and Coding

Assuming that additive white Gaussian noise before square law detection, due to amplified spontaneous emission in fiber amplifiers, dominates other noise sources, the detector output r(t) consists of the distorted noiseless signal u(t) and the non-centralized Chi-square distributed noise n(t) r (t ) = u (t ) + n(t ) = 2 2   (1) ui + 2 Re  ∑ ( ni ⋅ ui ) + ∑ ni ,  i ∈{X ,Y }  i ∈{ X ,Y } i ∈{ X ,Y } where X and Y denote the two components of the Jones vector polarization representation, respectively [3], [4], [13], [14], [15]. A widely used model of an ideal system, comprising an optical bandpass filter of rectangular shape with bandwidth B and an electrical lowpass integrate-and-dump filter, results in the probability density function (PDF) of the detector input samples rk=r(kT)

=

∑

f ( rk | u k ) =

1 2σ

r u  k

2

k

D−2 4

⋅e

 rk + uk   −  2σ 2 

 ⋅ I D / 2 −1  

rk ⋅ u k

σ

2

  , (2)

where uk is the noiseless part of the samples rk, In(ξ) denotes the nth order modified Bessel function of first kind and the degree of freedom D calculates to D=2(BT+1). Furthermore, a basic first order PMD model also used in [13] assumes a differential group delay (DGD) ∆τ, 0≤∆τ100, ln(f(r|u)) was approximated as proposed in [13]. Fig. 5 shows the OSNR penalty to achieve a BER of 10-5 without outer RS code and the OSNR penalty to achieve a BER of 10-16 with an outer RS code from Galois field GF(28), with an overall code rate of 0.9 and without input quan-

tization. The penalties are normalized to the PMD free case with code rate 0.9 and BER=10-16.

12

T urb o 1 6 b its T urb o 4 bits T urb o 3 bits ML S E 1 6 b its ML S E 4 bits ML S E 3 bits

10 12

8 OSNR penalty

10

OSNR penalty

8 6

6 4 2

4

0 2

-2 MLSE T urb o MLSE T urb o

0 -2 -4

0

0 .5

1

w/o R S w /o R S /w R S /w R S

-4

0

0 .5

1

1.5

2

∆τ / T

1 .5

2

∆τ / T

Fig. 5. OSNR penalty without quantization for turbo equalization vs. MLSE without outer RS code at BER=10-5 and with outer RS code at BER=10-16.

Fig. 6. Turbo equalization with V=4 bit ADC and internal quantization with outer RS code at BER=10-16 10

8

6 OSNR penalty

Turbo equalization with a maximum of 10 iterations provides a gain from 3 to 4dB compared to MLSE equalization. Iteration gains are already achieved for less than 10 iterations, however, 8 iterations are required to obtain a difference of less than 0.5dB compared to 10 iterations. Simulations using the SOVE algorithm ([13], [17]) indicated only a very poor performance with a gap of up to 3dB to the Max-Log algorithm. Since the SOVE still requires a high effort for such a low gain, it is not considered further. A more realistic scenario includes the usage of an ADC combined with a histogram based MLSE channel model [3], [6]. We assume an ADC resolution of V=4 bits and internal quantization in the turbo decoder of Q=16, 4 and 3 bits for the LLR values and histogram entries. The results including an outer RS code are plotted in Fig. 6. An internal quantization of 4 bits and 3 bits leads to a degradation of approximately 0.75 and 1.5dB, respectively. There is also a slight degradation using the Viterbi equalizer with quantized path metrics. The loss caused by V=4 bit input quantization and Q=16 bit internal quantization is about 0.15dB compared to the case without any quantization, which complies with the results in [14]. Note that this performance gap between the curve of turbo equalization with RS code in Fig. 5 and the curve “Turbo 16 bits” in Fig. 6 is nearly invisible. Another interesting question is how to split up the rates RTE and RRS for an overall rate R=0.9 to optimize the performance. The two scenarios RTE=0.93, RRS=0.97 and RTE=0.96, RRS=0.94 are depicted in Fig. 7, together with the performance of turbo equalization with a memory 2 code, i.e. the codes of Fig. 3b). The results demonstrate the benefits of a more powerful memory 3 code in combination with a high-rate outer code.

4

2 R = 0 .9 3 TE R = 0 .9 6 TE R = 0 .9 3 TE R = 0 .9 3

0

-2

-4

TE

0

0 .5

1

m em 3 m em 3 m em 2 m e m 3 c han. e s t.1 0 4 b its

1 .5

2

∆τ / T

Fig. 7. OSNR penalty turbo equalization with different RTE and memory Because of time variance of the PMD channel it is important to know the influence of a finite number of training symbols for channel estimation. The “boxed” curve in Fig. 7 depicts the performance of turbo equalization with 104 training symbols using the histogram based estimation of conditional probabilities proposed in [3], which shows a degradation from 0.4 up to 0.92dB. A realistic assumption is a channel being constant for 0.1ms at 10.7Gbit/s and thus providing 106 bits for channel estimation. Simulations of turbo equalization with this constraint have shown a loss of at most 0.05dB compared to perfect channel knowledge.

8

Conclusions

The concept of iterative equalization and decoding has been outlined in detail for use in non-linear ISI systems and compared with the two separate steps of equalization and decoding. The superior performance of turbo equalization has been confirmed for realistic internal word lengths and ADC resolutions in high-speed systems. Assuming a margin of 2dB for the MLSE/FEC combination in fiber-optical systems corrupted by PMD, the tolerable DGD can be increased from 0.5T to 0.8T by use of turbo equalization. However, its high-speed im-

plementation is challenging and the MLSE approach is less demanding. Alternatively, large block size LDPC codes achieve a performance close to the Shannon limit and they also allow for SISO decoding. They have been included in telecommunications standards and high-speed hardware implementations have been reported [23], [24]. Also, the design of LDPC codes with turbo equalization has been considered [25]. Therefore, ongoing work concentrates on iterative equalization and decoding with specifically designed LDPC codes.

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