Delayed Dual-tap Equalizer for FD CD Compensation

Although the delayed single-tap equalizer already tries to compensate not only the phase but also the group delay of the CD channel at each discrete frequency point fk, which is the center of each frequency subband, it can perform this for the group delay only in a coarsely quantized way, i.e. in steps ofM T /2. This quantization can be reduced by introducing the delayed dual-tap equal-izer which reduces the step-size to M T /4. This is achieved by utilizing two taps with a delay ofM T /2in between, thereby, interpolating the group delay compensation betweenℓuM T /2and (ℓ_u±1)M T /2. This interpolation is specially useful in those frequency sub-bands where the quan-tization error due to rounding (3.45) is almostM T /2.

The design of the delayed dual-tap equalizer is originated from the delayed single-tap concept (Sec. 3.5.2). There we observe the quantization error defined as

qk =ℓk−ℓ^′_k (3.47)

3.5 Group Delay Equalization of CD 57

1.4 1.6 1.8 2 2.2 2.4 2.6

x 10⁵ 14.1

14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 15

CD [ps/nm]

Required OSNR [dB] for BER = 10−3

M=1024, N

t=9

Fig. 3.17. Comparison of basic CD model with extended channel model for widely considered I/FFT length ofM = 1024andN_t= 9.

which is bounded to−1/2 < qk ≤ 1/2. A change in the sign from qk toqk+1 corresponding to an abrupt change in the group delay compensation caused by the change from ℓk to ℓk+1 points to these frequency sub-bands is where the concept of dual-tap is most useful. All those index pairs (k, k+ 1)are denoted by(u, u+ 1)and are collected in the set SD. We exclude the index pair(M/2−1, M/2)which comprises the two frequency bins at the upper and lower end of the baseband spectrum.

The mathematical expression of the delayed dual-tap equalizer is obtained by modifying (3.40) as follows

E_u^D(f) = e^′_u[ℓ_u]exp(−j2πf ℓ_uM T /2) +e^′_u[ℓ_u±1]exp(−j2πf(ℓ_u±1)M T /2), u∈S_D. (3.48) The active tape^′_u[ℓu−1]is placed at positionℓu−1for a positivequwhereas the active tape^′_u[ℓu+1]

is placed at positionℓ_u+ 1for a negativeq_u.

In order to find the value of the two active tapse^′_u[ℓ_u]ande^′_u[ℓ_u±1], we recall that in general the desired equalizer in each sub-banduaims at equalizing the CD channel as given in (3.41). The equalizerE_u^′(f)in (3.41) can be rewritten as follows

E_u^′(f) =H_CD⁻¹(fu)exp(−j2πfuτc)

=aH_CD⁻¹(fu)exp(−j2πfuτc) + (1−a)H_CD⁻¹(fu)exp(−j2πfuτc)

=aH_CD⁻¹(f_u)exp(−j2πf_uℓ_uM T /2)exp(+j2πf_uℓ_uM T /2)exp(−j2πf_uτ_c) + (1−a)H_CD⁻¹(fu)exp(−j2πfuτc)

×exp(−j2πfu(ℓu±1)M T /2)exp(+j2πfu(ℓu±1)M T /2) (3.49) where 0 < a < 1 is an amplitude factor assigned to the tap derived for the delayed single-tap equalizer. By setting (3.49) and (3.48) equal, the value of the two active tapse^′_u[ℓu]ande^′_u[ℓu±1]

58 3. Chromatic Dispersion Compensation in Long-haul Transmission Systems function of the active tapeu[ℓu]of the delayed single-tap equalizer as

e^′_u[ℓu] =aeu[ℓu], (3.52)

e^′_u[ℓu±1] =

( (1−a)×eu[ℓu], for u even,

−(1−a)×eu[ℓu], for u odd. (3.53) In order to keep the same complexity in terms of multiplications for the delayed dual-tap equal-izer and the delayed single-tap equalequal-izer, both active taps are chosen to be equal¹. This is achieved by having

a= 1

2. (3.54)

With this assumption, the delayed dual-tap equalizerE_u^D(z_M/2)in (3.48) can be rewritten as

E_u^D(f) = In general, for evenuthe term

This leads actually to the following two important conclusions for the delayed dual-tap equalizer witha = ¹₂. Firstly, in the sub-bands where it is applied, there are(ℓ_u ± ¹₂)M T /2delay elements represented by the terms (−j2πf T M/2)and(∓j2πf T M/4), respectively. This is illustrated in Fig. 3.18. The delay elements realize a sub-band group delay necessary to compensate the normal-ized group delay of the CD channel τCD(f) and the latency timeτ_c in (3.23). Thus the sub-band group delay is now smoothed where the delayed dual-tap equalizer is applied. The second conclu-sion is that the delayed dual-tap equalizer is not an all-pass filter since its amplitude is not equal to one; it has a sinusoidal shape represented by the termscos(2πf T M/4)andsin(2πf T M/4)for even and oddu, respectively.

1For oddu, both taps have equal magnitudes but opposite signs.

3.5 Group Delay Equalization of CD 59

Fig. 3.18. Delayed dual-tap equalizer with taps of same amplitude: an integer and a fractional group delay

−3 −2 −1 0 1 2 3 dis-continuity of the quantization errorqk

Fig. 3.19. Delayed dual-tap equalizer for CD= 60,000 ps/nm,M = 256,N_t= 7: the frequencies at which it is applied

3.5.3.2 Performance Analysis

The performance of the system with delayed dual-tap equalizer and with delayed single-tap equal-izer is evaluated for compensating different CD values.

In a first set of simulations, in Fig. 3.19 we plot ℓk and and the quantization errorqk for CD value of 60,000 ps/nm, M = 256 andNt = 7(correspondingly optimum τc). The discontinuity (as marked in the figures) in ℓk and qk highly depends on the quantization method (rounding in our case) chosen to getℓk for a given CD value,M andNt. There is a total of twelve sub-bands where the discontinuity occurs. This defines the elements of the set SD which has the following indexes SD = {10,11,32,33,54,55,201,202,223,224,245,246}. In these frequency sub-bands, the delayed dual-tap equalizer is applied to smoothℓk.

For the same settings as in Fig. 3.19, the smoothing of ℓk is shown in Fig. 3.20 by applying the delayed dual-tap equalizer at the corresponding sub-bands. We show just the frequency range f ∈ [0· · ·10] GHz. At the even indexes of the set SD, the delayed dual-tap equalizer has the additional active tap at position ℓk −1. At the odd indexes of the set SD, the delayed dual-tap equalizer has the additional active tap at positionℓ_k+ 1. Fractional values ofℓ_u=k are noticed in the sub-bands ofSD.

In a second set of simulations, the required OSNR is plotted in Fig. 3.21 forM = 256,Nt= 7 (and optimumτc) anda = ¹₂ for different CD values. The performance of the system with a delayed dual-tap equalizer further improves as compared to the system with delayed single-tap equalizer.

This is due to the finer quantization of ℓk by smoothing it in the sub-bands where it is discon-tinuous. The improvement in performance in terms of the required OSNR can be observed when

60 3. Chromatic Dispersion Compensation in Long-haul Transmission Systems

0 2 4 6 8 10

x 10⁹ 1

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8