SPIE Proceedings [SPIE Asia Pacific Optical Communications - Wuhan, China (Thursday 1 November 2007)] Optical Transmission, Switching, and Subsystems V - Volterra based nonlinear equalizer

Volterra based nonlinear equalizer with reduced complexity

Daniel Fritzsche∗a, Leonhard Lischkaa, Dirk Breuera, Christian G. Schäfferb

aT-Systems Enterprise Services GmbH, Goslarer Ufer 35, Berlin, 10589 GERMANY bDresden University of Technology, Communications Laboratory, Dresden, 01062 GERMANY

ABSTRACT

A model for a reduced complexity nonlinear electrical equalizer based on the Volterra theory is presented which can be utilized to mitigate dispersion and other distortions in optical communication systems. In recent years several electronic equalizers like the feed forward equalizer (FFE), the decision feedback equalizer (DFE) and the maximum likelihood sequence estimator (MLSE) were intensively investigated in optical communication systems. Also, nonlinear FFE/DFE structures based on the Volterra theory were proposed. These nonlinear equalizers can mitigate the effects of dispersion much better than the classical FFE/DFE but are more complicated to build. In a practical system the Volterra nonlinearity is therefore limited to second and third order. However, the number of filter coefficients for such a nonlinear FFE of appropriate order (e.g. 5 taps) is still very high for the nonlinear parts. This results in a very high effort in the equalizer control and optimization algorithm and makes a practical implementation questionable. In this Paper we present a reduced model for Volterra based nonlinear equalizers were the order of the nonlinear parts can be set separately from the linear order. This results in less complex filter structures as the number of coefficients is reduced drastically.

Keywords: equalization, Volterra, nonlinear filtering, FFE, DFE

1. INTRODUCTION The use of electronic equalization at the receiver is a very flexible method to mitigate static and time varying fiber distortions. The feed forward equalizer (FFE), the decision feedback equalizer (DFE) and the maximum likelihood sequence estimator (MLSE) are suitable low cost solutions to mitigate the dispersion effects in single mode fiber (SMF) links. Even though a simple FFE can almost double the uncompensated link length in NRZ systems, its capabilities can be limited for other modulation formats e.g. in duo binary systems. To improve the performance of filter based equalizers (FFE/DFE), nonlinear equalizers based on the Volterra theory were proposed1. Using the Volterra theory to describe nonlinear systems, a FIR filter, e.g. a FFE, with linear, quadratic, cubic and even higher order nonlinear terms can be modeled. However, the number of filter coefficients for such a nonlinear FFE of appropriate order (e.g. 5 taps) is very high for the nonlinear parts. In a practical system the Volterra nonlinearity is therefore limited to second and third order, but even then there are 6 linear, 21 quadratic and 56 cubic filter coefficients needed to model a 5 tap Volterra FFE. This high number of coefficients results in a very high effort in the equalizer control and optimization algorithm and makes a practical implementation questionable. The high complexity of the Volterra nonlinear equalizers is due to the fact that all nonlinear parts usually have the same number of tap delays. However, the Volterra theory is flexible to model filters with different number of tap delays in the nonlinear parts. This paper shows how to reduce the number of filter coefficients by individually adjusting the number of delay taps of the Volterra nonlinearities to obtain a less complex nonlinear equalizer. This reduced model allows building FFE/DFE with a high linear but low nonlinear number of tap delays. As a result the number of coefficients is drastically reduced. Using this flexible model the performance of several combinations of linear and nonlinear tap delays is investigated in numerical simulations. It is shown for NRZ and duo binary modulation that even using only the quadratic nonlinearity is sufficient to overcome the limits of a classical FFE/DFE. By separately varying the number of taps of the linear, quadratic and cubic parts of the equalizer structure it is shown in detail which parts influences the overall equalizer performance most. The simulation results show, that the total number of coefficients as well as adders and multipliers in the nonlinear filter can be reduced using a determined combination of linear and nonlinear tap delays while maintaining the same equalizer performance as the classical nonlinear Volterra equalizer. In this case the number of taps in the quadratic and cubic parts is always lower than the linear part.

∗ email: [email protected]; phone +49 30 3497 4364; fax +49 30 3497 4956; www.t-systems.com

Optical Transmission, Switching, and Subsystems V,edited by Dominique Chiaroni, Wanyi Gu, Ken-ichi Kitayama, Chang-Soo Park,

Proc. of SPIE Vol. 6783, 67831R, (2007) · 0277-786X/07/$18 · doi: 10.1117/12.745320

Proc. of SPIE Vol. 6783 67831R-1

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/07/2013 Terms of Use: http://spiedl.org/terms

2. VOLTERRA THEORY The Volterra theory2 is a pure mathematical model that allows describing nonlinear systems. The output y[n] of a discrete nonlinear Volterra system can be described as

[ ] [ ] [ ] [ ]∑∑ ∑= = =

−⋅⋅−⋅=p

i

N

vi

N

vii vnxvnxvvhny

i

i1 01

01

1

1

,, KKL (1)

where p is the order of the Volterra system, Ni the number of delay elements, hi[…] is the discrete Volterra kernel of order i and x[n] is the input signal. A Volterra system can be viewed as a combination of linear, quadratic and higher order nonlinear system parts. An example of a third order Volterra system is shown in Fig. 1.

The corresponding formula is given in (2) using the symmetry property of Volterra systems (v2 = v1…Np). This symmetry property allows to eliminate redundant Volterra kernels and to reduce the complexity of the whole system.

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ] [ ]∑ ∑ ∑

∑ ∑

∑

= = =

= =

=

−−−+

−−+

−=

3

1

3

12

3

23

2

1

2

12

1

1

03213213

021212

0111

,,

,

N

v

N

vv

N

vv

N

v

N

vv

N

v

vnxvnxvnxvvvh

vnxvnxvvh

vnxvhny

(2)

The structure shown above can be extended towards higher order nonlinear parts, but due to the high complexity practical systems are limited to third order. Note that the number of delay elements N of different system parts can be set individually.

3. VOLTERRA BASED NONLINEAR EQUALIZER WITH REDUCED COMPLEXITY The Volterra based equalizer is an extension of the classical FFE/DFE where the equalization is done by filtering the time continuous input signal. In the FFE the input signal is delayed by N delay taps with uniform delay times T. The output signal y(t) is the sum of the individual delayed signals x(t) weighted with different coefficient e0…N. An FFE with three taps with T = Tbit/2 is shown on the left side of Fig 2. In a DFE with M tap delays the signal is filtered with coefficients b1…M in a feedback path shown as on the right side of Fig. 2. Because of the decision circuit the DFE is referred to as a nonlinear equalizer. However the actual filtering in the FFE as well as in the feedback path of the DFE is linear. To obtain the best equalization result the FFE is combined with a DFE. A 6-tap FFE and 1-tap DFE is sufficient for optical systems in the most cases. Both equalizers can be extended towards nonlinear filtering using the Volterra theory. In 1 Volterra based nonlinear equalizers were investigated for different modulation formats in optical communication systems. There the nonlinear equalizer is described with the number of tap delays and the order of the

h3

h1

h2[ ]nyquadratic

system part

[ ]nx

linear system part

cubic system parth3h3

h1h1

h2h2[ ]nyquadratic

system part

[ ]nx

linear system part

cubic system part

Fig. 1. Third order Volterra system



nonlinearity. The number of taps is always equal for the linear and all nonlinear parts. However, the Volterra theory allows designing more flexible structures.

Within our proposed model a nonlinear FFE is declared as an FFE[N1,N2,N3] which means it has a linear order N1, quadratic order N2 and cubic order N3. An FFE[5,0,0] would be equivalent to a classical FFE of order 5. The advantage of this declaration is that the filter order of each nonlinear part (N1, N2 and N3) can be set separately. An example of a FFE[2,1,0] is shown in Fig. 3. There are two tap delays in the linear part and one in the quadratic part, the cubic part, however, is not present. The quadratic tap section was drawn separately to show the different order, but is normally included in the linear stage, no additional taps are required. However, additional multipliers and adders are required in the quadratic part. Using the symmetry property the weighting coefficient e10 is redundant as it has the same effect as e01 and is therefore left out.

Mathematically the system is described in (3), where ev1 and ev1,v2 are the coefficients of the linear and quadratic part, respectively. One can imagine extending just the linear part while keeping the number of tap delays of the quadratic part low.

2bT

1e

2bT

0e

00e 11e01e

( )2

+

( )2

)(tx

)(tx

)(ty

2bT

2e2bT

2bT

1e1e

2bT

2bT

0e0e

00e00e 11e11e01e01e

( )2( )2

++

( )2( )2

)(tx

)(tx

)(ty

2bT

2bT

2e2e

Fig. 3. Nonlinear equalizer model: FFE[2,1,0] with linear order 2 and quadratic order 1, the quadratic stage was drawn separately to show the different order, but is normally included in the linear stage, no additional taps are required.

2bT

1e0e

+

2bT

2e2bT

3e

)(tx

)(ty

2bT

2bT

1e1e0e0e

++

2bT

2bT

2e2e2bT

2bT

3e3e

)(tx

)(ty

--+

2b

+

2bT

1b2bT

)(tyDFE

)(tx

)(txDFE

--+

2b2b

++

2bT

2bT

1b1b2bT

2bT

)(tyDFE

)(tx

)(txDFE

Fig. 2. FFE and DFE



( ) ( )

( ) ( )∑ ∑

∑

= =

=

−−+

−=

1

0

1

21,

2

01

1 12

21

1

1

v vvvv

vv

TvtxTvtxe

Tvtxety

(3)

A DFE[2,1,0] is described in a similar way using (4).

( ) ( )

( ) ( )∑ ∑

∑

= +=

=

−−+

−=

1

1

1

121,

2

11

1 12

21

1

1

v vvvv

vv

TvtxTvtxb

Tvtxbty

DFEDFE

DFEDFE

(4)

Reducing the number of taps in the higher order system parts is critical as the number of weighting coefficients increases dramatically as shown in Fig. 4. Using an equal number of taps in the linear and nonlinear parts even a Volterra based FFE with only 5 taps has more than 80 coefficients. This increases the complexity of the physical design and the control algorithms as plenty of weighting coefficients as well as adders and multipliers are needed for the nonlinear parts.

For example a there are 6 linear, 21 quadratic and 56 cubic Volterra filter coefficients needed to model a 5 tap FFE, while the reduced model allows to build a FFE[5,4,3] with linear order 5, quadratic order 4 and cubic order 3 which results in 6, 15 and 20 coefficients only. Using the reduced model the influence of the number of tap delays of each nonlinear part on the equalization performance can be investigated separately. In the next section it is investigated how the individual nonlinear parts influence the whole equalizer. The aim here is to find a reduced complexity equalizer while maintaining the same performance.

4. SIMULATION RESULTS The performance of the reduced equalizer is evaluated in numerical simulations. To investigate the equalizer performance in general, quasi-analytical simulations are carried out. A 10 GBit/s NRZ or duobinary modulated chirp-free signal (1024 PRBS sequence) is transmitted over uncompensated SSMF (D = 16.5 ps/nm/km, S = 0.056 ps/nm2/km, α = 0.23 dB/km, n2 = 2.6·10-20 m2/W, Aeff = 80 µm2). The receiver consists of a photodiode, a low pass filter and the nonlinear Volterra based equalizer. The BER is estimated from the deterministic worst-case eye-opening assuming Gaussian noise distribution (Q-factor). Then the optical signal to noise ratio required (OSNRreq) for a BER = 10-9 is calculated.

1 2 3 4 5 6 70

20

40

60

80

100

120

linear part quadratic part cubic part

Num

ber o

f coe

ffici

ents

Number of Tap delays 1 2 3 4 5 6 7

0

10

20

30

40Nu

mbe

r of c

oeffi

cien

ts

Number of Tap delays Fig. 4. Number of coefficients for the linear, quadratic and cubic filter parts of the Volterra FFE (left) and DFE (right)



To compare the performance of the nonlinear Volterra Equalizer with a linear FFE/DFE the OSNRreq for FFE and FFE/DFE combinations with different number of delay taps are shown in Fig. 5. For the FFE 6 or 7 of tap delays are sufficient and more do not bring much additional benefit. The performance can only be increased significantly using an additional DFE. The results of a second and third order Volterra FFE with equal number of tap delays are shown on the left and right side of Fig. 6, respectively. Using the nonlinear elements the equalizer performance can be increased to the cost of an increased number of filter elements. The performance increase, however, is very small with respect to the increase in complexity. Only the use of an additional DFE can increase the performance significantly. But also in this case a linear DFE seems to be sufficient in combination with a Volterra FFE.

0 20 40 60 80 100 120 140 160 180 200

16

18

20

22

24

without FFE FFE[2] FFE[3] FFE[4] FFE[5] FFE[6] FFE[7] FFE[8] FFE[9] FFE[10]

OSNR

requ

ired

for B

ER=1

.0E-

9 [d

B]

Fiber length [km]0 20 40 60 80 100 120 140 160 180 200

16

18

20

22

24 nur Faser FFE[6] DFE[1] FFE[6] DFE[2] FFE[6] DFE[3]

OSNR

requ

ired

for B

ER=1

.0E-

9 [d

B]

Fiber length [km]Fig. 5. Performance of linear FFE/DFE for NRZ modulation

0 20 40 60 80 100 120 140 160 180 200

16

18

20

22

24 without FFE FFE[2]-NL[2] FFE[3]-NL[2] FFE[4]-NL[2] FFE[5]-NL[2] FFE[6]-NL[2] FFE[7]-NL[2]

OSNR

requ

ired

for B

ER=1

.0E-

9 [d

B]

Fiber length [km]0 20 40 60 80 100 120 140 160 180 200

16

18

20

22

24 without FFE FFE[2]-NL[3] FFE[3]-NL[3] FFE[4]-NL[3] FFE[5]-NL[3] FFE[6]-NL[3] FFE[7]-NL[3]

OSNR

requ

ired

for B

ER=1

.0E-

9 [d

B]

Fiber length [km]Fig. 6. Performance of quadratic and cubic Volterra FFE and NRZ modulation



With duobinary modulation the dispersion tolerance is generally increased but the linear FFE does not increase this tolerance much further. However, for duobinary modulation the nonlinear FFE performs much better than the linear FFE. This is due to the nonlinear signal parts in duobinary signals after direct detection, which can be equalizaed better using nonlinear equalizers1. With respect to a 2dB back-to-back (BTB) OSNR penalty the increase in the dispersion limit over uncompensated transmission for the linear, quadratic and cubic FFE is 15%, 35%, 45% for duobinary modulation while for NRZ it is 92%, 107%, 115%. This means that the nonlinear FFE is better suited for duobinary modulation as a third order Volterra FFE has three times higher increase in the dispersion limit as the linear FFE. For NRZ modulation however, the nonlinear FFE is not worth the increased complexity.

First the influence of the second order Volterra nonlinearity on the equalization of duobinary modulated signals is investigated in more detail. A typical structure of a 6th order FFE combined with a first order DFE is extended by quadratic parts in the FFE. Using duobinary modulation the uncompensated link length can be up to 200 km until an OSNRreq of 18 dB is reached (Fig. 9). Using the linear equalizer brings only a small additional distance (18 dB OSNRreq at 230 km) as expected. However the uncompensated link length can be increased using quadratic parts. With a FFE[6,3,0] DFE[1,0,0] the link length can be increased up to 270 km. Using the FFE[6,6,0]DFE[1,0,0] the link length can be increased further at the cost of much more filter coefficients. It is interesting to note that the FFE[6,5,0] DFE[1,0,0] with 28 coefficients performs almost as good as the FFE[6,6,0] DFE[1,0,0] with 35 coefficients. That is 20% less coefficients while maintaining the same system performance.

0 50 100 150 200 250 300 35014

16

18

20

22

24

without FFE FFE[6] FFE[6]-NL[2] FFE[6]-NL[3]

OSNR

requ

ired

for B

ER=1

.0E-

9 [d

B]

Fiber length [km] Fig. 8. Performance with duobinary modulation

0 20 40 60 80 100 120 140 160 180 200 220 240

16

18

20

22

24

nur Faser FFE[6]-NL[2], DFE[1] FFE[6]-NL[2], DFE[2] FFE[6]-NL[2], DFE[3]

OSNR

requ

ired

for B

ER=1

.0E-

9 [d

B]

Fiber length [km]0 20 40 60 80 100 120 140 160 180 200 220 240

16

18

20

22

24 without FFE FFE[6]-NL[2], DFE[1]-NL[1] FFE[6]-NL[2], DFE[2]-NL[2]

OSNR

requ

ired

for B

ER=1

.0E-

9 [d

B]

Fiber length [km]Fig. 7. Performance of quadratic Volterra FFE with linear and quadratic DFE and NRZ modulation



On the right side of Fig. 9 the performance of some other combinations are shown. The FFE[5,4,0] DFE[1,0,0] performs as good as the FFE[5,5,0] DFE[1,0,0] and the FFE[7,5,0] DFE[1,0,0] is nearly as good as the FFE[7,7,0] DFE[1,0,0] but again needs many fewer coefficients (about 35% less). A very interesting result is the FFE[6,3,0] DFE[1,0,0] which performs as good as the FFE[5,5,0] DFE[1,0,0] and reaches a OSNRreq of 18 dB at about 270 km although it needs only 17 filter coefficients. That is again 38% less coefficients than the classical nonlinear filter (a FFE[5,5,0] DFE[1,0,0] has 27 coefficients) at the same system performance. Note, that an equalizer with a smaller nonlinear order also needs less multiplication elements (Fig.3) which reduces the overall equalizer complexity enormously.

Based on the results above the maximum compensateable dispersion per filter coefficient was calculated for several combinations and is shown on the left side of Fig. 10. The marked groups represent equalizers which have the same performance, but different a number of coefficients due to a reduction of the nonlinear tap delays. This shows that using the proposed model the complexity of the Volterra based equalizers can be reduced while maintaining the same performance. However, as shown on the right side of Fig. 10 the compensated dispersion per coefficient will steadily decrease using more coefficients.

5 10 15 20 25 30 35 40 45 503600

3800

4000

4200

4400

4600

4800

Disp

ersio

n [p

s/nm

]

Number of coefficients 5 10 15 20 25 30 35 40 45 50

50

100

150

200

250

300

350

400

450

500

Disp

ersio

n [p

s/nm

] per

coef

ficie

nt

Number of coefficients Fig. 10. Compensated dispersion per coefficient for duobinary modulation

0 50 100 150 200 250 300 350

16

18

20

22

24 uncompensated FFE[6,0,0] DFE[1,0,0] FFE[6,1,0] DFE[1,0,0] FFE[6,2,0] DFE[1,0,0] FFE[6,3,0] DFE[1,0,0] FFE[6,4,0] DFE[1,0,0] FFE[6,5,0] DFE[1,0,0] FFE[6,6,0] DFE[1,0,0]

OSNR

requ

ired

[dB]

Fiber Length [km] 0 50 100 150 200 250 300 350

16

18

20

22

24 uncompensated FFE[7,7,0] DFE[1,0,0] FFE[7,5,0] DFE[1,0,0] FFE[5,5,0] DFE[1,0,0] FFE[5,4,0] DFE[1,0,0] FFE[6,3,0] DFE[1,0,0]

OSNR

requ

ired

for B

ER=1

.0E-

9 [d

B]

Fiber Length [km]

Fig. 9. Performance of quadratic Volterra FFE with linear DFE for duobinary modulation



5. CONCLUSION A reduced model for Volterra based nonlinear equalizers is presented which allows setting the order of the nonlinear parts of the filter separately. This results in a reduced filter with fewer coefficients and therefore a reduced complexity. Numerical simulations show, that the equalizer performance can be increased using nonlinear parts. Using the reduced model the performance can be as good as the classical Volterra equalizer but needs less filter coefficients.

REFERENCES

1 C. Xia, W. Rosenkranz: “Performance Enhancement for Duobinary Modulation Through Nonlinear Electrical Equalization” ECOC 2005, Tu4.2.3 2 M. Schetzen: “The Volterra and Wiener Theories of Nonlinear systems” Krieger Publishing Company, 2006 3 C. Xia, W. Rosenkranz: “”Nonlinear Electrical Equalization for Different Modulation Formats With Optical Filtering” Journal of Lightwave Technology, Vol. 25, No. 4, April 2007



Documents

SPIE Proceedings [SPIE Asia Pacific Optical Communications - Wuhan, China (Thursday 1 November 2007)] Optical Transmission, Switching, and Subsystems V - Volterra based nonlinear equalizer