MCRB for timing, phase and frequency estimation in presence of self-phase modulation for low-rate optical communication

Abstract

In this paper we derive the theoretical lower bound, namely Modified Cramér-Rao bound (MCRB) for symbol timing, phase and frequency offset in presence of nonlinear self-phase modulation (SPM) in a dispersion compensated long-haul coherent fiber link. The system model considers multiple span of fiber each associated with optical amplifier. Dual polarization multilevel quadrature amplitude modulation is opted for data transmission to support the data rate lower than 10 Gigabaud. We find that SPM induces underdamped oscillation on the MCRB bounds depending on the pulse shapes (symmetric and asymmetric) utilized. In presence of realistic low-pass filter at the receiver front end, the MCRB degrades significantly due to SPM. We also show the effect of SPM on symbol error rate degradation. Simulation is carried out with symmetric return-to-zero pulse with duty cycles of 33, 67 % and self-generated asymmetric pulse to verify the theoretical results.

Appendices

Appendix 1: Derivation of Fisher information for symmetric pulse without LPF

Derivation of $F_{\theta \tau }=F_{\tau \theta }$: we see that

$$\begin{aligned}&{\mathbb {E}}\left[ \mathfrak {R}\left\{ \frac{\partial \tilde{\mathbf {x}}_{k}^{H}}{\partial \theta } \frac{\partial \tilde{\mathbf {x}}_{k}}{\partial \tau }\right\} \right] \nonumber \\&\quad =-2\beta A^{2}\sum _{n=-(N-1)/2}^{(N-1)/2}E_{4}\left| p(\xi _{n,k})\right| ^{3}\dot{p}(\xi _{n,k}), \end{aligned}$$

(51)

In (51) we used the fact that data symbols are i.i.d. and introduced $E_{l}={\mathbb {E}}\left\{ \left\| \mathbf {S}_{n}\right\| ^{l}\right\}$. The same reasoning will be followed to derive the other components of FIM. Therefore,¹ $F_{\theta \tau }$ is given as

$$\begin{aligned} F_{\theta \tau }=\frac{-4\beta A^{2}}{N_{a}N_{0}/T}{\displaystyle \sum _{k=-\infty }^{+\infty }}\sum _{n=-(N-1)/2}^{(N-1)/2}E_{4}\left| p(\xi _{n,k})\right| ^{3}\dot{p}(\xi _{n,k}) \end{aligned}$$

(52)

Note that $F_{\theta \tau }=F_{\tau \theta }=0$ whenever the system is linear (i.e., $\gamma =0$) or whenever the pulse p(t) is even.

1.1 Derivation of $F_{\theta \theta }$ and $F_{\tau \tau }$

Using similar reasoning, $F_{\theta \theta }$ is given as

$$\begin{aligned} F_{\theta \theta } = \frac{2A}{N_{a}N_{0}/T}{\displaystyle \sum _{k=-\infty }^{+\infty }}{\displaystyle \sum _{n=-(N-1)/2}^{(N-1)/2}E_{2}\left| p(kT-nT_{s}-\tau )\right| ^{2}} \end{aligned}$$

(53)

Finally, it is easily shown that

$$\begin{aligned}&{\mathbb {E}}\left\{ \frac{\partial \tilde{\mathbf {x}}_{k}^{H}}{\partial \tau } \frac{\partial \tilde{\mathbf {x}}_{k}}{\partial \tau }\right\} \nonumber \\&\quad =A{\displaystyle \sum _{n=-(N-1)/2}^{(N-1)/2}} \left| \dot{p}(\xi _{n,k})\right| ^{2}\left( 2+4\beta ^{2}A^{2}E_{6} \left| p(\xi _{n,k})\right| ^{4}\right) .\nonumber \\ \end{aligned}$$

(54)

Then, $F_{\tau \tau }$ is given as

$$\begin{aligned} F_{\tau \tau }=\frac{4AN}{N_{a}N_{0}}I_{02} +8\frac{\gamma ^{2}L_{\mathrm {eff}}^{2}N_{a}A^{3}E_{6}N}{N_{0}}I_{42}. \end{aligned}$$

(55)

1.2 Derivation of $F_{\theta \epsilon }=F_{\epsilon \theta }$

Deriving $\frac{\partial \tilde{\mathbf {x}}_{k}^{H}}{\partial \theta }$ from (12) we find that

$$\begin{aligned} F_{\theta \epsilon }= & {} \frac{2}{N_{a}N_{0}/T}{\displaystyle \sum _{k=-\infty }^{+\infty }}\left\{ AkT\sum _{n=-(N-1)/2}^{(N-1)/2}E_{2}\left| p(\xi _{n,k})\right| ^{2}\right\} \nonumber \\= & {} \frac{2ANE_{2}}{N_{a}N_{0}}\intop _{-\infty }^{+\infty }(t+nT_{s}+\tau ) \left| p(t)\right| ^{2}\mathrm {d}t\nonumber \\= & {} \frac{2ANE_{2}}{N_{a}N_{0}}\intop _{-\infty }^{+\infty }(t+\tau ) \left| p(t)\right| ^{2}\mathrm {d}t\nonumber \\= & {} \frac{4AN}{N_{a}N_{0}}\left[ K_{12}+\tau K_{02}\right] . \end{aligned}$$

(56)

In (56) we use the fact that $K_{kl0}=K_{kl}$. Alternatively, in frequency domain $F_{\theta \epsilon }$ can be given as

$$\begin{aligned} F_{\theta \epsilon }= & {} \frac{4AN}{N_{a}N_{0}}\left[ \tau \intop _{-1/T}^{+1/T}\left| P(f)\right| ^{2}\mathrm {d}f\right. \nonumber \\&\left. +\frac{1}{2\pi }\mathrm {Im}\intop _{-1/T}^{+1/T} \left( \frac{\partial }{\partial f}\, P^{*}(f)\right) P(f)\,\mathrm {d}f\right] .\nonumber \\ \end{aligned}$$

(57)

1.3 Derivation of $F_{\epsilon \tau }=F_{\tau \epsilon }$

It can be easily seen from (14) that

$$\begin{aligned}&\frac{\partial \tilde{\mathbf {x}}_{k}^{H}}{\partial \epsilon } =-jkTe^{-j\theta }e^{-j\epsilon kT}\sqrt{A}\nonumber \\&\quad {\displaystyle \sum _{m=-(N-1)/2}^{(N-1)/2}\mathbf {S}_{m}p(\xi _{m,k})} \exp \left[ -j\beta A\left\| {\displaystyle \mathbf {S}}_{m}\right\| ^{2}\left| p(\xi _{m,k})\right| ^{2}\right] .\nonumber \\ \end{aligned}$$

(58)

Hence, using (13) and (58) we obtain

$$\begin{aligned}&{\mathbb {E}}\left\{ \frac{\partial \tilde{\mathbf {x}}_{k}^{H}}{\partial \epsilon } \frac{\partial \tilde{\mathbf {x}}_{k}}{\partial \tau }\right\} =AkT\Biggl [{\displaystyle \sum _{n=-(N-1)/2}^{(N-1)/2}}jE_{2}\dot{\,p}(\xi _{n,k})\,p(\xi _{n,k})\nonumber \\&\quad -2\beta AE_{4}\left| p(\xi _{n,k})\right| ^{3}\dot{\,p}(\xi _{n,k})\Biggr ]. \end{aligned}$$

(59)

Therefore,

$$\begin{aligned}&{\mathbb {E}}\left[ \mathfrak {R}\left\{ \frac{\partial \tilde{\mathbf {x}}_{k}^{H}}{\partial \epsilon } \frac{\partial \tilde{\mathbf {x}}_{k}}{\partial \tau }\right\} \right] \nonumber \\&\quad =-2\beta A^{2}E_{4}kT{\displaystyle \sum _{n=-(N-1)/2}^{(N-1)/2}\left| p(\xi _{n,k})\right| ^{3}\dot{\, p}(\xi _{n,k}).} \end{aligned}$$

(60)

Then, $F_{\epsilon \tau }$ is given as

$$\begin{aligned} F_{\epsilon \tau }= & {} \frac{-4\beta A^{2}NE_{4}}{N_{a}N_{0}}\intop _{-\infty }^{+\infty }(t+\tau )\left| p(t)\right| ^{3}\dot{\, p}(t)\,\mathrm {d}t \end{aligned}$$

(61)

It is obvious that $F_{\epsilon \tau }=F_{\tau \epsilon }=0$ whenever p(t) is even and the system is linear, i.e., $\gamma =0$. Alternatively, $F_{\epsilon \tau }$ in frequency domain is

$$\begin{aligned} F_{\epsilon \tau }= & {} \frac{-4\beta A^{2}NE_{4}}{N_{a}N_{0}}\Biggl \{\intop _{-1/T}^{1/T}f \left| P(f)\right| ^{3}\left( \frac{\partial }{\partial f}\,P(f)\right) \mathrm {d}f\nonumber \\&+\tau 2\pi \mathrm {Im}\intop _{-\infty }^{+\infty }f \left| P(f)\right| ^{3}P(f)\mathrm {d}f\Biggr \}. \end{aligned}$$

(62)

1.4 Derivation of $F_{\epsilon \epsilon }$

Using (58) and (14) we derive $F_{\epsilon \epsilon }$ as

$$\begin{aligned} F_{\epsilon \epsilon }= & {} \frac{2AE_{2}}{N_{a}N_{0}}\Biggl [\frac{(N^{2}-1)N}{12}T_{s}^{2} \intop _{-\infty }^{+\infty }\left| p(t)\right| ^{2}dt+N\tau ^{2}\nonumber \\&\times \intop _{-\infty }^{+\infty }\left| p(t)\right| ^{2}dt +N\intop _{-\infty }^{+\infty }t^{2}\left| p(t)\right| ^{2}dt\nonumber \\&+\,2\tau N\intop _{-\infty }^{+\infty }t\left| p(t)\right| ^{2}dt\Biggr ] \end{aligned}$$

(63)

An alternative frequency domain interpretation of (63) is

$$\begin{aligned} F_{\epsilon \epsilon }= & {} \frac{2ANE_{2}}{N_{a}N_{0}(2\pi )^{2}} \Biggl [\intop _{-1/T}^{1/T}\left| \frac{\partial }{\partial f}\, P(f)\right| ^{2}\mathrm {d}f\nonumber \\&+\,\frac{(N^{2}-1)}{12}T_{s}^{2}4\pi ^{2} \intop _{-1/T}^{1/T}\left| P(f)\right| ^{2}\mathrm {d}f +4\pi \tau \,\nonumber \\&\times \,\mathrm {Im}\intop _{-1/T}^{+1/T}\left( \frac{\partial }{\partial f}\, P(f)\right) P^{*}(f)\,\mathrm {d}f\nonumber \\&+\,4\pi ^{2}\tau ^{2}\intop _{-1/T}^{1/T}\left| P(f)\right| ^{2}\mathrm {d}f\Biggr ]. \end{aligned}$$

(64)

Appendix 2: Derivation of Fisher information for asymmetric pulse without LPF

We start with derivation of determinant of $\mathbf {F}(\varvec{\Theta })$ (26)

$$\begin{aligned} \mathrm {det}(\mathrm {\mathbf {\mathbf {F}(\varvec{\Theta })})}= & {} F_{\epsilon \epsilon }F_{\theta \theta }F_{\tau \tau }+2F_{\epsilon \tau }F_{\theta \tau } F_{\theta \epsilon }\nonumber \\&-\,F_{\theta \tau }^{2}F_{\epsilon \epsilon }-F_{\theta \epsilon }^{2} F_{\tau \tau }-F_{\epsilon \tau }^{2}F_{\theta \theta }. \end{aligned}$$

(65)

We introduce $A_{1,\beta }=I_{02}+2\beta ^{2}AE_{6}I_{42}$, $A_{2}=K_{22}+2\tau K_{12}+\tau ^{2}K_{02}$, $A_{3}=K_{131}+\tau J_{31}$, and $A_{4}=K_{12}+\tau K_{02}$. Then,

$$\begin{aligned}&F_{\epsilon \epsilon }F_{\theta \theta }F_{\tau \tau }= \left( \frac{4AN}{N_{a}N_{0}}\right) ^{3}A_{1,\beta }A_{2}, \end{aligned}$$

(66)

$$\begin{aligned}&F_{\epsilon \tau }F_{\theta \tau }F_{\theta \epsilon }= \frac{64\beta ^{2}A^{5}N^{3}E_{4}^{2}J_{31}}{(N_{a}N_{0})^{3}}A_{3}A_{4}, \end{aligned}$$

(67)

$$\begin{aligned}&F_{\theta \tau }^{2}F_{\epsilon \epsilon }=\frac{64 \beta ^{2}A^{5}N^{3}E_{4}^{2}J_{31}^{2}}{(N_{a}N_{0})^{3}}A_{2}, \end{aligned}$$

(68)

$$\begin{aligned}&F_{\theta \epsilon }^{2}F_{\tau \tau }=\frac{(4AN)^{3}}{(N_{a}N_{0})^{3}}A_{1,\beta }A_{4}^{2}, \end{aligned}$$

(69)

and

$$\begin{aligned} F_{\epsilon \tau }^{2}F_{\theta \theta }=\frac{64\beta ^{2}A^{5} N^{3}E_{4}^{2}}{(N_{a}N_{0})^{3}}A_{3}^{2}. \end{aligned}$$

(70)

Hence,

$$\begin{aligned} \mathrm {det}(\mathbf {F}(\varvec{\Theta }))= & {} \frac{(4AN)^{3}}{(N_{a}N_{0})^{3}} \Bigl [A_{1,\beta }(A_{2}-A_{4}^{2})\nonumber \\&+\,\beta ^{2}A^{2}E_{4}^{2}\left( J_{31}A_{3}A_{4}-J_{31}^{2}A_{2}-A_{3}^{2}\right) \Bigr ].\nonumber \\ \end{aligned}$$

(71)

The diagonal elements of $\mathrm {\mathbf {F}}^{-1}(\varvec{\Theta })$ in (25) are then obtained as

$$\begin{aligned} B= & {} F_{\theta \theta }F_{\tau \tau }-F_{\theta \tau }^{2}\nonumber \\= & {} \frac{(4AN)^{2}}{(N_{a}N_{0})^{2}}\left[ A_{1,\beta } -\beta ^{2}A^{3}E_{4}^{2}J_{31}^{2}\right] , \end{aligned}$$

(72)

$$\begin{aligned} F= & {} F_{\epsilon \epsilon }F_{\tau \tau }-F_{\epsilon \tau }^{2} \nonumber \\= & {} \frac{(4AN)^{2}}{(N_{a}N_{0})^{2}} \left[ A_{1,\beta }A_{2}-\beta ^{2}A^{3}E_{4}^{2}A_{3}^{2}\right] , \end{aligned}$$

(73)

$$\begin{aligned} O= & {} F_{\epsilon \epsilon }F_{\theta \theta }-F_{\theta \epsilon }^{2} = \frac{(4AN)^{2}}{(N_{a}N_{0})^{2}}\left[ A_{2}-A_{4}^{2}\right] , \end{aligned}$$

(74)

Substituting back into (8) MCRBs for phase, timing and frequency offset are derived.