New metric for IQ imbalance compensation in optical QPSK coherent systems

Abstract

We report on a simple alternative method for the compensation of quadrature imbalance in optical quadrature phase-shift-keying (QPSK) coherent systems. By introducing a new metric, the phase imbalance can be determined and compensated. The proposed method is theoretically and numerically analyzed. In particular, it is shown that the method exhibits a small bias of estimated phase imbalance value. Thanks to its deterministic property, this bias can be simply compensated by incorporating at the receiver a phase rotator (or phase shift) whose value can be determined based on an analytical analysis. Moreover, the algorithm is also experimentally validated through bit-error-rate and error vector magnitude (EVM) measurements. A good agreement on the performance of the proposed method with that of the Gram–Schmidt orthogonalization procedure is shown in a 20-Gbit/s optical QPSK experiment. The robustness of both methods was verified with up to $30^\circ$ phase imbalance by comparing the signal with and without phase imbalance compensation. A 10% reduction in EVM is achieved with our method for a high phase imbalance of $30^\circ$, while the implementation complexity can be reduced owing to the suppression of the use of square-root operators.

Appendices

Appendix A

Based on the definition of the new metric in (4), the squared modulus of r can be written as

$$\begin{aligned} {| r |^2} = {(U + X)^2} + {(V + Y)^2} , \end{aligned}$$

(A.1)

in which the variables U, X, V and Y are defined by

$$\begin{aligned} \left\{ \begin{array}{l} U = \cos {\phi _S}\\ V = \alpha \cdot \cos {\phi _S} + \beta \cdot \sin {\phi _S}\\ X = {n_C}\\ Y = \alpha \cdot {n_C} + \beta \cdot {n_S}\\ \alpha = \sin {\phi _\mathrm{var}} + \sin {\phi _\mathrm{mis}} \cdot \cos {\phi _\mathrm{var}}\\ \beta = \cos {\phi _\mathrm{mis}} \cdot \cos {\phi _\mathrm{var}} \end{array} \right. . \end{aligned}$$

(A.2)

We first calculate the expected value of ${| r |^2}$. From (A.1), the expectation of $r^2$ is calculated as follows

$$\begin{aligned} \begin{aligned} {\mathbf{E}}\left\{ {{{(U + X)}^2} + {{(V + Y)}^2}} \right\}&= {\mathbf{E}}\left\{ {\left( {{U^2} + {V^2}} \right) } \right\} \\&\quad +\,{\mathbf{E}}\left\{ {\left( {{X^2} + {Y^2}} \right) } \right\} \\&\quad +\, 2{\mathbf{E}}\left( {UX} \right) + 2{\mathbf{E}}\left( {VY} \right) . \end{aligned} \end{aligned}$$

(A.3)

It must be noted that the expected values of U, V, X and Y are zero. On the other hand, U and X, V and Y are uncorrelated so that the last two terms of (A.3) are equal to 0. Moreover, $\phi _S$ is uniformly distributed over the four values $(2k+1)\cdot \pi /4$, $k = 0, 1, 2, 3$. Using the quantities defined in (A.2), the variances of U, V and their covariance can be retrieved as

$$\begin{aligned} {\mathbf{E}}\left( {{U^2}} \right) = \frac{1}{2},\quad {\mathbf{E}}\left( {{V^2}} \right) = \frac{1}{2}\left( {{\alpha ^2} + {\beta ^2}} \right) ,\quad {\mathbf{E}}\left( {UV} \right) = \frac{\alpha }{2}. \end{aligned}$$

(A.4)

As mentioned, $n_C$ and $n_S$ are random Gaussian variables with the same variance $\sigma ^2$. The variances of X, Y and their covariance are

$$\begin{aligned} {\mathbf{E}}\left( {{X^2}} \right) = {\sigma ^2},\quad {\mathbf{E}}\left( {{Y^2}} \right) = {\sigma ^2}\left( {{\alpha ^2} + {\beta ^2}} \right) ,\quad {\mathbf{E}}\left( {XY} \right) = \alpha {\sigma ^2}. \end{aligned}$$

(A.5)

Finally, the expected value of ${| r |^2}$ is derived by [as in Eq. (7)]

$$\begin{aligned} {\mathbf{E}}\left( {{{| r |}^2}} \right) = \left( {\frac{1}{2} + {\sigma ^2}} \right) \cdot \left( {1 + {\alpha ^2} + {\beta ^2}} \right) . \end{aligned}$$

(A.6)

The following part is the derivation of the variance of ${\left| r \right| ^2}$. The variance of ${| r |^2}$ can be written as

$$\begin{aligned} {\mathbf{Var}}\left( {{{| r |}^2}} \right) = {\mathbf{E}}\left( {{{| r |}^4}} \right) - {\left[ {{\mathbf{E}}\left( {{{| r |}^2}} \right) } \right] ^2}. \end{aligned}$$

(A.7)

Due to the assumption of central and uncorrelated variables, some terms cancel and the expectation of ${| r |^4}$ can be written as

$$\begin{aligned} \begin{aligned} {\mathbf{E}}\left( {{{| r |}^4}} \right) =&{} {\mathbf{E}}\left[ {{{\left( {{U^2} + {V^2}} \right) }^2}} \right] \\&+ {\mathbf{E}}\left[ {{{\left( {{X^2} + {Y^2}} \right) }^2}} \right] \\&+ 2{\mathbf{E}}\left[ {\left( {{U^2} + {V^2}} \right) .\left( {{X^2} + {Y^2}} \right) } \right] \\&+ 4{\mathbf{E}}\left[ {{{\left( {UX + VY} \right) }^2}} \right] . \end{aligned} \end{aligned}$$

(A.8)

Based on the definition of the QPSK signal, $(U^2 + V^2)$ is a random discrete variable taking on two equiprobable values, $1/2\cdot (1 + (\alpha + \beta )^2)$ and $1/2\cdot (1 + (\alpha - \beta )^2)$. The first term of (A.8) can then be derived as

$$\begin{aligned} {\mathbf{E}}\left[ {{{\left( {{U^2} + {V^2}} \right) }^2}} \right] = \frac{1}{4}\left[ {{{\left( {1 + {\alpha ^2} + {\beta ^2}} \right) }^2} + 4{\alpha ^2}{\beta ^2}} \right] . \end{aligned}$$

(A.9)

The second term of (A.8) is calculated using the fact that all the moments of a zero-mean Gaussian variable can be derived from its variance [28]. More specifically, the fourth-order moments of (X, Y) can be written as

$$\begin{aligned} \left\{ \begin{array}{l} {\mathbf{E}}\left[ {{X^4}} \right] = 3{\sigma ^4}\\ {\mathbf{E}}\left[ {{Y^4}} \right] = 3{\left( {{\alpha ^2} + {\beta ^2}} \right) ^2}{\sigma ^4}\\ {\mathbf{E}}\left[ {{X^2}{Y^2}} \right] = {\mathbf{E}}\left[ {n_C^2{{\left( {\alpha {n_C} + \beta {n_S}} \right) }^2}} \right] = \left( {3{\alpha ^2} + {\beta ^2}} \right) {\sigma ^4} \end{array} \right. . \end{aligned}$$

(A.10)

Therefore, the second term of (A.8) is given by

$$\begin{aligned} {\mathbf{E}}\left\{ {{{\left( {{X^2} + {Y^2}} \right) }^2}} \right\} = \left[ {3 + 2\left( {3{\alpha ^2} + {\beta ^2}} \right) + 3{{\left( {{\alpha ^2} + {\beta ^2}} \right) }^2}} \right] {\sigma ^4}. \end{aligned}$$

(A.11)

$(U^2 + V^2)$ and $(Y^2 + X^2)$ are independent random variables. The third term of (A.8) is then obtained

$$\begin{aligned} \begin{aligned}&{\mathbf{E}}\left[ {\left( {{U^2} + {V^2}} \right) .\left( {{X^2} + {Y^2}} \right) } \right] \\&\quad = {\mathbf{E}}\left[ {\left( {{U^2} + {V^2}} \right) } \right] \cdot {\mathbf{E}}\left[ {\left( {{X^2} + {Y^2}} \right) } \right] \\&\quad = \frac{{{\sigma ^2}}}{2}{\left( {1 + {\alpha ^2} + {\beta ^2}} \right) ^2} \end{aligned}. \end{aligned}$$

(A.12)

Due to the independence property between the variables (U, V) and (X, Y), the last term of (A.8) can be reformulated as

$$\begin{aligned} \begin{aligned} {\mathbf{E}}\left[ {{{\left( {UX + VY} \right) }^2}} \right]&= {\mathbf{E}}\left[ {{U^2}} \right] {\mathbf{E}}\left[ {{X^2}} \right] + 2{\mathbf{E}}\left[ {UV} \right] \cdot {\mathbf{E}}\left[ {XY} \right] \\&\quad +\, {\mathbf{E}}\left[ {{V^2}} \right] \cdot {\mathbf{E}}\left[ {{Y^2}} \right] . \end{aligned} \end{aligned}$$

(A.13)

The last term is expressed as

$$\begin{aligned} {\mathbf{E}}\left[ {{{\left( {UX + VY} \right) }^2}} \right] = \left( {{\alpha ^2} + \frac{{1 + {{\left( {{\alpha ^2} + {\beta ^2}} \right) }^2}}}{2}} \right) \cdot {\sigma ^2}. \end{aligned}$$

(A.14)

From (A.9), (A.11), (A.12) and (A.14), the variance of ${| r |^2}$ is finally derived [as in Eq. (8)]

$$\begin{aligned} {\mathbf{Var}}\left( {{{| r |}^2}} \right) = {\alpha ^2}{\beta ^2} + 2\left( {1 + 2{\alpha ^2} + {{\left( {{\alpha ^2} + {\beta ^2}} \right) }^2}} \right) \cdot \left( {1 + {\alpha ^2}} \right) \cdot {\sigma ^2}. \end{aligned}$$

(A.15)

Appendix B

If ${\phi _0} = \arg \mathop {\max }\limits _{{\phi _\mathrm{var}}} \left( {{D_{\left( {wo} \right) }}\left( {{\phi _\mathrm{var}}} \right) } \right)$ or $\tan \phi _0 = -\sin \phi _\mathrm{mis}$, we evaluate the first derivative of $D(\phi )$ at $\phi = \phi _0$. To do that, the expression of D in Eq. (9) is represented as follows

$$\begin{aligned} D = \frac{{Z{{\left( {1 + {\alpha ^2} + {\beta ^2}} \right) }^2}}}{{{\alpha ^2}{\beta ^2} - 2S{\beta ^2} + S{{\left( {1 + {\alpha ^2} + {\beta ^2}} \right) }^2}}} = \frac{Z}{{S + \frac{{{\beta ^2}\left( {{\alpha ^2} - 2S} \right) }}{{{{\left( {1 + {\alpha ^2} + {\beta ^2}} \right) }^2}}}}}. \end{aligned}$$

(B.1)

where $Z = (1/2+\sigma ^2)^2$ and $S = 2\sigma ^2(1+\sigma ^2)$. If we call ${F_S}\left( \phi \right) = \frac{{{\beta ^2}\left( {{\alpha ^2} - 2S} \right) }}{{{{\left( {1 + {\alpha ^2} + {\beta ^2}} \right) }^2}}}$, Eq. (B.1) can be rewritten by

$$\begin{aligned} D = \frac{Z}{{S + {F_S}\left( \phi \right) }}. \end{aligned}$$

(B.2)

By inspecting Eq. (B.2), it can be observed that the extremum of D corresponds to the extremum of $F_S(\phi )$. Instead of evaluating the first derivative of D at $\phi = \phi _0$, we can equivalently investigate the first derivative of $F_S(\phi )$ at $\phi = \phi _0$. In order to simplify the expression, the following variables and its corresponding derivative at $\phi = \phi _0$ are established

$$\begin{aligned} \left\{ {\begin{array}{ll} {u = {\alpha ^2}} ,&{}\quad {v = {\beta ^2}} \\ {{u_0} = {{\left. {{\alpha ^2}} \right| }_{\phi = {\phi _0}}} = 0} ,&{}\quad {{v_0} = {{\left. {{\beta ^2}} \right| }_{\phi ={\phi _0}}}} \\ {u{'_0} = {{\left. {\left( {\frac{{\mathrm{d}\left( {{\alpha ^2}} \right) }}{{\mathrm{d}\phi }}} \right) } \right| }_{\phi = {\phi _0}}} = 0}, &{}\quad {v{'_0} = {{\left. {\left( {\frac{{\mathrm{d}\left( {{\beta ^2}} \right) }}{{\mathrm{d}\phi }}} \right) } \right| }_{\phi = {\phi _0}}}} \end{array}} \right. . \end{aligned}$$

(B.3)

The expression of $F_S(\phi )$ is now given by

$$\begin{aligned} {F_S}\left( \phi \right) = \frac{{v\left( {u - 2S} \right) }}{{{{\left( {1 + u + v} \right) }^2}}} \end{aligned}$$

(B.4)

and its first derivative at $\phi = \phi _0$ can be established

$$\begin{aligned} {\left. {\frac{{d\left( {{F_S}\left( \phi \right) } \right) }}{{d\phi }}} \right| _{\phi = {\phi _0}}}= & {} \frac{{v{'_0}\left( {{u_0} - 2S} \right) + {v_0}\left( {u{'_0} - 2S} \right) }}{{{{\left( {1 + {u_0} + {v_0}} \right) }^2}}} \nonumber \\&-\,\frac{{2{v_0}\left( {{u_0} - 2S} \right) }}{{{{\left( {1 + {u_0} + {v_0}} \right) }^3}}} \end{aligned}$$

(B.5)

or equivalently,

$$\begin{aligned} {\left. {\frac{{d\left( {{F_S}\left( \phi \right) } \right) }}{{d\phi }}} \right| _{\phi = {\phi _0}}} = \frac{{2S \cdot K}}{{{{\left( {1 + {v_0}} \right) }^2}}} , \end{aligned}$$

(B.6)

in which K is calculated by

$$\begin{aligned} K = \frac{{2{v_0}}}{{1 + {v_0}}} - \left( {{v_0} + v{'_0}} \right) . \end{aligned}$$

(B.7)

Due to the fact that $\tan \phi _0 = -\sin \phi _\mathrm{mis}$ and from the definition of $\beta$ in Eq. (6), the values of $v_0$ and $v'_0$ can be deduced as follows

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{v_0} = \frac{{{{\cos }^2}{\phi _\mathrm{mis}}}}{{1 + {{\sin }^2}{\phi _\mathrm{mis}}}}}\\ {v{'_0} = \frac{{2{{\cos }^2}{\phi _\mathrm{mis}}\sin {\phi _\mathrm{mis}}}}{{1 + {{\sin }^2}{\phi _\mathrm{mis}}}}} \end{array}} \right. . \end{aligned}$$

(B.8)

Substituting (B.8) into (B.7), K can be obtained as

$$\begin{aligned} K = \frac{{{{\cos }^2}{\phi _\mathrm{mis}}\sin {\phi _\mathrm{mis}}\left( {\sin {\phi _\mathrm{mis}} - 2} \right) }}{{1 + {{\sin }^2}{\phi _\mathrm{mis}}}}. \end{aligned}$$

(B.9)

From (B.6) and (B.9), it can be concluded that K is different from 0 and hence the first derivative of $F_S(\phi )$ at $\phi = \phi _0$ is also not equal to 0 whenever $S \ne 0$. As a consequence, the first derivative of $D(\phi )$ at $\phi = \phi _0$ differs from 0, so the dependence of the maximum value of D on the SNR of the received samples is confirmed.