Method for compensating channel distortions by pre-distortion of Mach-Zehnder modulators, based on symmetric imbalance

Abstract

A method for compensating the distortions introduced by impairments of MZMz implementing an optical transmitter, according to which the level of total amplitude and phase distortions caused by the optical transmitter is measured and all impairments in the constellation domain are compensated by pre-distorting the input signal to be transmitted by symmetrically adding imbalance to the voltage applied to the MZM arms. The imbalance is determined by introducing a phase rotation in either I or in the Q path of the optical transmitter, which compensates the total amplitude distortion, and also introducing a phase rotation to both I and Q paths of the optical transmitter, which compensate the total phase distortion and the phase shift caused by compensating the amplitude distortion, until reaching a desired operating point, which corresponds to the level of pre-distortion.

Claims

1. A method for compensating the distortions introduced by impairments of MZMz implementing an optical transmitter, comprising: a) measuring the level of total amplitude and phase distortions caused by said optical transmitter; b) optimizing all impairments in the constellation domain by pre-distorting the input signal to be transmitted by symmetrically adding imbalance to the voltage applied to the MZM arms, said imbalance is determined by: c) introducing a phase rotation in either I or in the Q path of said optical transmitter, which compensates said total amplitude distortion by independently providing different control voltages to the inputs of each MZM, thereby applying a Q-imbalance being a distortion applied in Q MZM or by applying an I-imbalance being a distortion applied in I MZM, to rotate the signal from the distorted constellation point to a new point, which is on the same power equivalent circle as the target point; and d) introducing a phase rotation to both I and Q paths of said optical transmitter by independently providing different control voltages to the inputs of each MZM, thereby, applying a common imbalance on both I MZM and Q MZM, to rotate the signal from said new point to said target point, which compensates said total phase distortion and the phase shift caused by compensating said amplitude distortion, until reaching a desired operating point, which corresponds to the level of pre-distortion.

2. The method according to claim 1, wherein the geometric distance between distorted locations and the target constellation points, being the desired points in the signal space, at which all distortions are compensated, is represented by an Error Vector Magnitude (EVM).

3. The method according to claim 2, wherein the MZM output Error Vector Magnitude (EVM) is bounded by 10-6N [dB].

4. The method according to claim 1, wherein pre-distortion is determined by the number of bits N of digital to analog converter providing voltage to said MZM.

5. The method according to claim 4, wherein the absolute value of the Error Vector Magnitude (EVM) is bounded by 10-6N [dB], where N is the effective number of bits of the digital to analog converter used for driving the signal to the MZM.

6. The method according to claim 1, wherein impairments include one or more of the following: nonlinearity; single MZM gain imbalance; single MZM phase imbalance; dual MZM gain imbalance dual MZM phase imbalance; and any frequency independent impairment.

7. The method according to claim 1, wherein optimization is conducted using means of mean square criterion.

8. The method according to claim 1, wherein pre-distortion is performed by generating a distorted transfer function consisting of a common y.sub.mzmCom(t) signal, and an imbalanced signal y.sub.mzmIm(t) which are defined as $\begin{array}{cc}{y}_{\mathrm{MZM}}\left(t\right)=\underset{\mathrm{YmzmCom}\left(t\right)}{\underbrace{G_{\mathrm{MZM}}.Math.\sin \left(\frac{\pi }{2V_{\pi }}.Math.V\left(t\right)\right)}.Math.}\underset{\underset{\mathrm{YmzmIm}\left(t\right)}{︸}}{e^{-j\left(\frac{\pi }{2V_{\pi }}\mathrm{.Math.}\Delta \left(t\right)\right)}}.& \left(12\right)\end{array}$

9. The method according to claim 1, wherein pre-distortion of each MZM is performed by generating a distorted transfer function consisting of a common y.sub.mzmCom(t) signal, and an imbalanced signal y.sub.mzmIm(t) and defined as $\begin{array}{cc}{y}_{\mathrm{mzmD}}\left(t\right)=\underset{\mathrm{YmzmDcom}\left(t\right)}{\underbrace{G_{\mathrm{MZM}}\frac{1}{2}\left(e^{-j\left(\frac{\pi V\left(t\right)}{2V_{\pi }}\right)}+G_{\mathrm{ab}}.Math.e^{-j\left(\frac{\pi V\left(t\right)}{2V_{\pi }}+p_{a}b\right)}\right)}.Math.}\underset{\underset{\mathrm{YmzmDim}\left(t\right)}{︸}}{e^{-j\left(\frac{\pi }{2V_{\pi }}.Math.\Delta \left(t\right)\right)}}.& \left(13\right)\end{array}$

10. The method according to claim 1, wherein pre-distortion of in-phase and quadrature MZM is performed by generating a distorted transfer function consisting of a common y.sub.mzmIDcom(t)/y.sub.mzmQDcom(t) and imbalance y.sub.mzmIDim(t)/y.sub.mzmQDim(t) components and defined by: $\begin{array}{cc}{y}_{\mathrm{mzmIQD}}\left(t\right)=y_{\mathrm{mzmIDcom}}\left(t\right)\underset{\underset{\mathrm{YmzmIDim}\left(t\right)}{︸}}{.Math.e^{-j\left(\frac{\pi }{2V_{\pi }}.Math.{\Delta }_t\left(t\right)\right)}}+j.Math.G_{\mathrm{IQ}}.Math.e^{j\phi \mathrm{IQ}}.Math.y_{\mathrm{mzmQDcom}}\left(t\right).Math.\underset{\underset{\mathrm{YmzmQDim}\left(t\right)}{︸}}{e^{-j\left(\frac{\pi }{2V_{\pi }}.Math.{\Delta }_Q\left(t\right)\right)}},& \left(14\right)\end{array}$ wherein each component results in a constellation rotation around a point on one of the axes.

11. The method according to claim 1, wherein the imbalance is added symmetrically to the transmitted symbol.

12. The method according to claim 1, further comprising generating a metric of performance in the form of an Error Vector Magnitude (EVM), being the geometric distance between distorted locations and the target constellation points in the signal space and defined by: $\mathrm{EVM}=10.Math.{\mathrm{Log}}_{10}\left(\frac{{E\left(y_{\mathrm{mzmIDQ}}\left(n\right)-k_{\mathrm{MZM}}{V}_{\mathrm{IQ}}\left(n\right)\right)}^2}{E\left({\left(k_{\mathrm{MZM}}{V}_{\mathrm{IQ}}\left(n\right)\right)}^2\right)}\right).$

13. The method according to claim 1, wherein a push-pull configuration is assumed when Δ.sub.I=Δ.sub.Q=0, said a push-pull configuration is optimal when there are no distortions.

14. The method according to claim 1, wherein the optimal EVM value is achieved with non-zero imbalance.

15. The method according to claim 1, wherein a complex imbalance is used for moving the signal from the distorted constellation point {I.sub.0, Q.sub.0} to the target constellation point {Ic, Qc} by: a) applying either a Q-imbalance being a distortion applied in Q MZM only value or I-imbalance being a distortion applied in I MZM only value, and rotating from the distorted constellation point {I.sub.0, Q.sub.0} to a new point {I.sub.1, Q.sub.1}, which is on the same power equivalent circle as the target point {Ic, Qc}; b) applying a common imbalance on both I and Q rotating the signal from {I.sub.1, Q.sub.1} to {I.sub.c, Q.sub.c}.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) In the drawings:

(2) FIG. 1 illustrates an MZM based I/Q optical modulator structure with a graphical representation of its main impairments;

(3) FIG. 2 illustrates the effect of a complex rotation which is a result of an added complex exponent, when: (a) Δ.sub.Q added to the Q-MZM (b) Δ.sub.I added to the I-MZM;

(4) FIGS. 3a and 3b illustrate EVM as function of the complex corrective Symmetric Imbalance (SI) when applying it to a QAM-16 signal with and without impairments;

(5) FIGS. 4a and 4b illustrate two step complex imbalance based transform applied to a QAM-16 distorted constellation point;

(6) FIG. 5 illustrates optimal and quantized corrective imbalance values Vs V(t)/V.sub.Π for a MZM dominated by nonlinearity;

(7) FIG. 6 illustrates EVM performance of a quantized corrective imbalance in a non-linearity dominated system;

(8) FIG. 7 illustrates the effect of quantization error on pre-distortion performance with 5.5, 6 and 7 bit DAC in a nonlinearity dominated MZM;

(9) FIGS. 8a and 8b illustrate the effect of quantization error with single MZM gain and phase distortion with 5.5, 6 and 7 bit DAC and V(t)/V.sub.Π of −5.5 dB;

(10) FIGS. 9a and 9b illustrate the effect of quantization error with dual (complex) MZM gain and phase distortion with 5.5,6 and 7 bit DAC and a voltage ratio V(t)/V.sub.Π of ˜−5.5 dB;

(11) FIG. 10 illustrates the required OSNR Performance of iDPD for a QAM-16 signal in three cases: when a nonlinear only distortion is applied, with nonlinearity combined with ˜2% IQ Dual MZM gain imbalance (1.02/0.98 gain split), with nonlinearity combined with 2% IQ dual gain imbalance and 1° of single MZM phase offset; and

(12) FIG. 11 illustrates the accuracy of iterative search in respect to the calculated value for a 5.5 bit DAC quantized correction.

DETAILED DESCRIPTION OF THE INVENTION

(13) The method proposed by the present invention provides a generalized closed form for the compensation of nonlinearity, single MZM gain and phase imbalance, and dual MZM gain and phase imbalance. All parameters are compensated for both analytically and by applying an optimization method which requires only two optimization parameters, thereby allowing significant simplification of the MZM pre-distortion problem.

(14) iDPD Optimization Targets

(15) The MZM is targeted to be optimized in Optical Signal to Noise Ratio (OSNR) ranges, where it will have minor effect on the performance of next generation high speed coherent optical modulations.

(16) The MZM target noise floor optimization of the iDPD method is set by the limit for its contribution to the overall noise. For example, if the allowed MZM noise contribution is 2 dB, assuming that the MZM noise floor is flat within the signal BW, and the overall OSNR is set to 20 dB, then it means that OSNR from all noise sources except MZM would be 22 dB and the MZM OSNR should be ˜2.5 dB better, or 24.5 dB. Similarly if the allowed noise contribution of the MZM is 1 dB, and the target OSNR is 30 dB, then the required MZM OSNR would be ˜37 dB. Assuming that the MZM allowed contribution is 1˜2 dB of OSNR degradation, the target for its noise floor optimization for very high rate systems would be in the range of 24 dB to 37 dB OSNR.

(17) Forced Symmetric Imbalance

(18) The proposed solution is to find a parameter, which can be used to optimize all impairments in the constellation domain at once, instead of finding and optimizing each impairment separately.

(19) The MZM data voltages applied on each arm V.sub.a(t), V.sub.b(t) presented in Eq. 2 can be rewritten with a common value V(t) and imbalance variable Δ(t) so that:

(20) $\begin{array}{cc}\{\begin{array}{c}{V}_a\left(t\right)=V\left(t\right)+\Delta \left(t\right)\\ V_b\left(t\right)=V\left(t\right)-\Delta \left(t\right)\end{array}& \left[\mathrm{Eq}.11\right]\end{array}$

(21) The distorted transfer function, as described in Eq. 2, can be split to two components: the common y.sub.mzmCom(t) signal, and imbalanced signal y.sub.mzmIm(t) which are defined as:

(22) $\begin{array}{cc}{y}_{\mathrm{MZM}}\left(t\right)=\underset{Y_{\mathrm{mzmCom}}\left(t\right)}{\underbrace{G_{\mathrm{MZM}}.Math.\sin \left(\frac{\pi }{2V_{\pi }}.Math.V\left(t\right)\right)}.Math.}\underset{\underset{Y_{\mathrm{mzmIm}}\left(t\right)}{︸}}{e^{-j\left[\frac{\pi }{2V_{\pi }}\mathrm{.Math.}\Delta \left(t\right)\right]}}& \left[\mathrm{Eq}.12\right]\end{array}$

(23) In a similar manner, the separation to common and imbalanced signal can be applied on the generalized MZM definition from Eq. 8 above:

(24) $\begin{array}{cc}{y}_{\mathrm{mzmD}}\left(t\right)=\underset{\mathrm{YmzmDcom}\left(t\right)}{\underbrace{G_{\mathrm{MZM}}\frac{1}{2}\left[e^{-j\left[\frac{\pi V\left(t\right)}{2V_{\pi }}\right]}+G_{\mathrm{ab}}.Math.e^{-j\left[\frac{\pi V\left(t\right)}{2V_{\pi }}+{\phi }_{\mathrm{ab}}\right]}\right]}.Math.}\underset{\underset{\mathrm{YmzmDim}\left(t\right)}{︸}}{e^{-j\left[\frac{\pi }{2V_{\pi }}.Math.\Delta \left(t\right)\right]}}& \left[\mathrm{Eq}.13\right]\end{array}$

(25) The same decomposition can also be applied to in-phase and quadrature MZM, yielding the common y.sub.mzmIDcom(t)/y.sub.mzmQDcom(t) and imbalance signal y.sub.mzmIDim(t)/y.sub.mzmQDim(t) signals. The generalized distorted transfer function from Eq. 10 can be rewritten as an extension of Eq. 13:

(26) $\begin{array}{cc}{y}_{\mathrm{mzmIQD}}\left(t\right)=y_{\mathrm{mzmIDcom}}\left(t\right).Math.\underset{\underset{\mathrm{YmzmIDim}\left(t\right)}{︸}}{e^{-j\left[\frac{\pi }{2V_{\pi }}.Math.{\Delta }_I\left(t\right)\right]}}+j.Math.G_{\mathrm{IQ}}.Math.e^{j\phi \mathrm{IQ}}.Math.y_{\mathrm{mzmQDcom}}\left(t\right).Math.\underset{\underset{\mathrm{YmzmQDim}\left(t\right)}{︸}}{e^{-j\left[\frac{\pi }{2V_{\pi }}.Math.{\Delta }_Q\left(t\right)\right]}}& \left[\mathrm{Eq}.14\right]\end{array}$

(27) Each one of the two components results in a constellation rotation around a point on one of the axes. For example, when applying an imbalance Δ.sub.Q(t) on the quadrature MZM, the constellation (I.sub.0, Q.sub.0) rotates around the point (I.sub.0, 0) on the constellation plane. Equivalently, an imbalance Δ.sub.I (t) applied to the in-phase MZM only, results in a rotation around the point (0, Q.sub.0), as shown in FIG. 2.

(28) If the imbalance is added symmetrically to the transmitted symbol, it can be considered as a method for performing linear mapping in the constellation domain. When the distortion of the signal is known, this reverse operation to the linear mapping can be used to move constellation points from a distorted position to its target one, resulting in a pre-distortion operation, based on the forced imbalance, referred to as iDPD.

(29) Error Vector Magnitude (EVM) as a Function of Complex Imbalance Plane

(30) In the case of I-Q modulation, the complex modulating data stream can be defined by:
V.sub.IQ(n)=V.sub.I(n)+j.Math.V.sub.Q(n)  [Eq. 15]
where V.sub.IQ is normalized to an RMS value of unity. A common measure of MZM distortion can be achieved by calculating the average Mean Square Error (MSE) between the “practical” complex MZM output, as defined in Eq. 14, and the “ideal” complex modulating data stream (Eq. 15). In the constellation symbol space, this metric is referred to the Error Vector Magnitude (EVM), and defined as follows:

(31) $\begin{array}{cc}\mathrm{EVM}=10.Math.{\mathrm{Log}}_{10}\left(\frac{{E\left(y_{\mathrm{mzmIQD}}\left(n\right)-k_{\mathrm{MZM}}{V}_{\mathrm{IQ}}\left(n\right)\right)}^2}{E\left({\left(k_{\mathrm{MZM}}{V}_{\mathrm{IQ}}\left(n\right)\right)}^2\right)}\right)& \left[\mathrm{Eq}.16\right]\end{array}$

(32) The k.sub.MZM value is required in order to normalize constellation voltage (V.sub.IQ) to the MZM input voltage as defined in Eq. 3 above:

(33) $\begin{array}{cc}{k}_{\mathrm{MZM}}=\frac{\pi }{2V_{\pi }}& \left[\mathrm{Eq}.17\right]\end{array}$

(34) Using EVM allows having a joint measure of all impairments, thus gives a unified metric of quality to the MZM. The EVM function can also be used as the target function for optimizing {Δ.sub.I, Δ.sub.Q}.

(35) In typical systems, the minimum EVM is assumed when Δ.sub.I=Δ.sub.Q=0 (push-pull configuration). However, this is not true in the general case. In order to demonstrate this, the pair {Δ.sub.I, Δ.sub.Q} for minimum EVM value is analytically calculated in the case of a non-ideal MZM distorted by I-Q phase imbalance, and is shown to be different than the push-pull configuration.

(36) Considering the MZM transfer function as described above in Eq. 14 with G.sub.ab=1, G.sub.IQ=1 and φ.sub.ab=0, with only a phase I-Q imbalance φ.sub.IQ≠0 when the average transmitted power is normalized to 1 and the MZM gain is normalized so that G.sub.MZM=−1. The maximum EVM is assumed to be achieved when its derivative in respect to {Δ.sub.I, Δ.sub.Q} is equal to zero. The MSE of the transmitted constellation is defined as (V.sub.I, V.sub.Q, Δ.sub.I, Δ.sub.Q are all a function of the discrete time n):

(37) $\begin{array}{cc}\mathrm{MSE}={10}^{\mathrm{EVM}/10}=E(\sin \left(K_{\mathrm{MZM}}.Math.V_I\right)e^{{\mathrm{jK}}_{\mathrm{MZM}}.Math.{\Delta }_I}+{\hspace{1em}j.Math.\sin \left(K_{\mathrm{MZM}}.Math.V_Q\right)e^{{\mathrm{jK}}_{\mathrm{MZM}}.Math.{\Delta }_Q}{e}^{j{\phi }_{\mathrm{IQ}}}-K_{\mathrm{MZM}}.Math.V_{\mathrm{IQ}})}^2& \left[\mathrm{Eq}.18\right]\end{array}$

(38) When the ratio V(t)/V.sub.Π is low, the nonlinearity effect can be assumed as negligible and the function can be approximated by:

(39) 0$\begin{array}{cc}\mathrm{MSE}\approx k_{\mathrm{MZM}}E\left\{\begin{array}{c}\left({\left(V_Q{\phi }_{\mathrm{IQ}}\right)}^2+2V_Q{\phi }_{\mathrm{IQ}}{V}_Q{\Delta }_Q+{\left(V_Q{\Delta }_Q\right)}^2\right)-\\ \left({\left({\Delta }_I{V}_I\right)}^2-2{\Delta }_I{V}_I{V}_Q{\Delta }_Q{\phi }_{\mathrm{IQ}}+{\left(V_Q{\Delta }_Q{\phi }_{\mathrm{IQ}}\right)}^2\right)\end{array}\right\}& \left[\mathrm{Eq}.19\right]\end{array}$

(40) The derivative of the MSE on Δ.sub.I direction would be:

(41) $\begin{array}{cc}\frac{\partial \mathrm{MSE}}{\partial {\Delta }_I}=2k_{\mathrm{MZM}}E\left\{{\Delta }_I^2{V}_Q{\Delta }_Q{\phi }_{\mathrm{IQ}}-{\Delta }_I{V}_I\right\}& \left[\mathrm{Eq}.20\right]\end{array}$

(42) and on Δ.sub.Q direction:

(43) $\begin{array}{cc}\frac{\partial \mathrm{MSE}}{\partial {\Delta }_Q}=2k_{\mathrm{MZM}}E\left\{V_Q^2{\phi }_{\mathrm{IQ}}+V_Q^2{\Delta }_Q+{\Delta }_I{V}_I{V}_Q{\phi }_{\mathrm{IQ}}-\left(V_Q^2{\Delta }_Q{\phi }_{\mathrm{IQ}}^2\right)\right\}& \left[\mathrm{Eq}.21\right]\end{array}$

(44) In the case of push-pull Δ.sub.I=Δ.sub.Q≈0:

(45) $\begin{array}{cc}\frac{\partial \mathrm{MSE}}{\partial {\Delta }_I}=0& \left[\mathrm{Eq}.22\right]\\ \frac{\partial \mathrm{MSE}}{\partial {\Delta }_Q}=2k_{\mathrm{MZM}}{\phi }_{\mathrm{IQ}}E\left\{V_Q^2\left(n\right)\right\}=k_{\mathrm{MZM}}{\phi }_{\mathrm{IQ}}& \end{array}$

(46) When there is no phase imbalance, the push-pull is potentially at its maximum as the gradient is zero. However, when there is phase imbalance, the gradient is non-zero, which implies that the push-pull configuration does not lead to a minimum MSE value.

(47) Based on the above derivation, even if there is only I-Q phase imbalance, the push-pull configuration is suboptimal. This means that there is no reason to assume that the push-pull configuration would be optimal if additional impairments are included. Thus, some forced imbalance might potentially improve performance in the presence of impairments.

(48) EVM as a Function of Symmetric Imbalance

(49) In order to better understand how the forced imbalance affects EVM, a plain of the EVM (in dB) represented in the complex imbalance domain can be defined. The X-axis is the imbalance applied to the in-phase component in degrees, and the Y-axis is the imbalance applied to the quadrature component. The use of radians is applied in order to represent the circular nature of the complex exponential corrective imbalance y.sub.mzmIDim(t) and y.sub.mzmQDim(t), as defined in Eq. 14 above.

(50) FIG. 3 illustrates an example of contour graphs of EVM for QAM-16 modulation as a function of the complex imbalance in the range of [−Π, Π]. The [−Π, Π] range represents the entire operating input region around the null point. A 5.5 bit DAC is used with quantization full scale to quantization noise floor of ˜38 dB. When operating in a voltage ratio V(t)/V.sub.Π (as defined in Eq. 4) of −8 dB, the maximum EVM that can be achieved is −30 dB. It can be noted that for a signal without distortion (FIG. 3a), the push-pull configuration with zero imbalance achieves the optimal EVM. However, there are additional non-zero imbalance values which lead to the same optimal EVM. For example, when the Q-imbalance is set to 2.5 (0.8Π radians) and the I-imbalance is set to ˜0.8 (0.25Π radians).

(51) In the case of a distorted MZM, analyzing the EVM versus complex imbalance shows that the best EVM value is achieved with non-zero imbalance. In the example shown in FIG. 3b, a distortion of I-Q imbalance gain ratio of 1.3 and I-Q phase offset of 10° (0.17 rad) is chosen, as it reduces the EVM to be worse than −10 dB, which makes it impractical for QAM-16 transmission. The optimal EVM value of −30 dB is achieved when the I-imbalance is set to 0.25 radians and the Q-imbalance is set to 0.8 radians, while the push-pull configuration achieves an EVM value of ˜−10 dB. Based on Eq. 12, the rotation β due to symmetric imbalance would be:

(52) $\begin{array}{cc}\beta =\frac{\pi }{2V_{\pi }}.Math.\Delta & \left[\mathrm{Eq}.23\right]\end{array}$

(53) Which means that in order to achieve the optimal I value of β=0.25 rad, the I-imbalance would be

(54) ${\Delta }_I=\frac{V_{\pi }}{2\pi }\approx 0.16.Math.V_{\pi }.$

(55) Similarly, the optimal Q value of β=0.8 rad can be achieved by applying a Q-Imbalance of

(56) ${\Delta }_Q=\frac{1.6V_{\pi }}{\pi }\approx 0.5.Math.V_{\pi }.$

(57) The goal of the iDPD concept is to propose a closed form and simple generic iterative method for finding the complex imbalance values which would optimize MZM performance in the EVM sense.

(58) Optimal Pre-distortion Values with known impairments

(59) Normally, each MZM operates in a balanced mode where the same voltage is provided to each of its inputs, i.e., V.sub.Ia=V.sub.Ib and V.sub.Qa=V.sub.Qb (see FIG. 1). In order to compensate the impairments, each MZM operates in an imbalanced mode.

(60) In order to determine the level of imbalance, the total distortion of the transmitter is measured (e.g., by transmitting a known test signal and measuring the distorted signal at the output).

(61) The distortion compensation is carried out both in I and Q paths (complex compensation) by first introducing phase rotation in the I path, which compensates the amplitude (gain) distortion. However, compensation of amplitude distortion entails a shift in the phase (in addition to phase distortion), which should be compensated, as well (until reaching the desired operating point). Therefore, at the next step, different phase rotations are concurrently introduced both in I and Q paths, which compensate the total phase distortion. This two-step complex compensation process will be described in detail later on.

(62) It should be understood that this two-step complex compensation process may start by first introducing phase rotation in the Q path, to compensate the amplitude (gain) distortion and then at the next step, required (and different) phase rotations are concurrently introduced both in I and Q paths, which compensate the total phase distortion.

(63) Representation of Pre-Distortion in the Signal Space

(64) The EVM is the geometric distance between distorted locations and the target constellation points in the signal space. The target constellation point is defined as {I.sub.c, Q.sub.c} and the distorted constellation point is defined as {I.sub.0, Q.sub.0}based on the real and imaginary part of the distorted transfer function defined in (14) above such that:

(65) $\begin{array}{cc}\{\begin{array}{c}{I}_0=\mathrm{Re}\left\{y_{\mathrm{mzmIQD}}\left(t\right)\right\}\\ Q_0=\mathrm{Im}\left\{y_{\mathrm{mzmIQD}}\left(t\right)\right\}\end{array}& \left[\mathrm{Eq}.24\right]\end{array}$

(66) The iDPD concept uses the complex imbalance defined in Eq. 14 for moving the signal from the distorted constellation point {I.sub.0, Q.sub.0} to the target constellation point {I.sub.c, Q.sub.c}, which is the desired point, at which all distortions are compensated.

(67) This can be achieved in two steps as follows:

(68) Step 1: Apply a Q-imbalance (a distortion applied in Q MZM only) value Δ.sub.step1, rotating from the distorted constellation point {I.sub.0, Q.sub.0} to a new point {I.sub.1, Q.sub.1}, which is on the same power equivalent circle as the target point {I.sub.c, Q.sub.c}. The operation can mathematically be described as:
I.sub.0=jQ.sub.0e.sup.jΔ.sup.step1=I.sub.1+jQ.sub.1  [Eq. 25]

(69) Step 2: Apply a common imbalance Δ.sub.step2 on both I and Q rotating the signal from {I.sub.1, Q.sub.1} to {I.sub.c, Q.sub.c}. The Operation can mathematically be described as:
(I.sub.1+jQ.sub.1)e.sup.jΔ.sup.step2=I.sub.c+jQ.sub.c  [Eq.26]

(70) FIG. 4 graphically illustrates the two steps. Following the above two steps, the corrective imbalance factors {Δ.sub.step1, Δ.sub.step2} that are required to perform the rotation can be represented as finding the solution to the nonlinear complex equation:
I.sub.0e.sup.j(Δ.sup.step2.sup.)+jQ.sub.0e.sup.j(Δ.sup.step1.sup.+Δ.sup.step2.sup.)=I.sub.c+jQ.sub.c  [Eq. 27]
which can be rewritten as two equations for the real and imaginary parts:

(71) $\begin{array}{cc}\{\begin{array}{c}{I}_0.Math.\cos \left({\Delta }_{\mathrm{step}2}\right)-Q_0.Math.\sin \left({\Delta }_{\mathrm{step}1}+{\Delta }_{\mathrm{step}2}\right)=I_c\\ I_0.Math.\sin \left({\Delta }_{\mathrm{step}2}\right)+Q_0.Math.\cos \left({\Delta }_{\mathrm{step}1}+{\Delta }_{\mathrm{step}2}\right)=Q_c\end{array}& \left[\mathrm{Eq}.28\right]\end{array}$

(72) The imbalance applied to the in-phase and quadrature parts can therefore be written as:

(73) $\begin{array}{cc}\{\begin{array}{c}{\Delta }_I={\Delta }_{\mathrm{step}2}\\ {\Delta }_Q={\Delta }_{\mathrm{step}1}+{\Delta }_{\mathrm{step}2}\end{array}& \left[\mathrm{Eq}.29\right]\end{array}$

(74) Analytic Derivation of the Pre-Distortion Values

(75) When the distortion parameters are known, the constellation point in the signal space {I.sub.0, Q.sub.0} can be calculated using Eq. 10. In turn, {Δ.sub.step1, Δ.sub.step2} can be derived from the two equations in Eq. 28. Therefore, the proposed iDPD can fully compensate for the MZM distortions.

(76) The solution of Eq. 28 can be performed by analyzing the rotation of the two independent steps described in the previous section as follows:

(77) Step 1: Find the phase of rotation from the distorted signal {I.sub.0, Q.sub.0} set by Δ.sub.step1 to the intermediate point {I.sub.1, Q.sub.1}:
∥I.sub.0+jQ.sub.0e.sup.jΔ.sup.step1∥=∥I.sub.1+jQ.sub.1∥=∥I.sub.c+jQ.sub.c∥
(I.sub.0−Q.sub.0.Math.sin(Δ.sub.step1)).sup.2+(Q.sub.0.Math.cos(Δ.sub.step1)).sup.2=I.sub.c.sup.2+Q.sub.c.sup.2
I.sub.0.sup.2+Q.sub.0.sup.2−2I.sub.0Q.sub.0.Math.sin(Δ.sub.step1)=I.sub.c.sup.2+Q.sub.c.sup.2  [Eq. 30]

(78) If the power of each constellation point is defined as P=I.sup.2+Q.sup.2, then Eq. 30 can be rewritten as:

(79) 0$\begin{array}{cc}{\Delta }_{\mathrm{step}1}=\arcsin \left(\frac{P_0-P_c}{2I_0{Q}_0}\right)& \left[\mathrm{Eq}.31\right]\end{array}$
where P.sub.0 is the power of the distorted point {I.sub.0, Q.sub.0} and P.sub.c is the power of the target intermediate point {I.sub.1, Q.sub.1}, which is equal to the power of the target constellation point {I.sub.c, Q.sub.c}.

(80) Step 2: Find the phase of rotation around the axis from the intermediate point to the target constellation point. The rotation angle can be extracted from the phase difference between {I.sub.1, Q.sub.1} and {I.sub.c, Q.sub.c}, which can be derived by:

(81) $\begin{array}{cc}{\Delta }_{\mathrm{step}2}=\arctan \left(\frac{Q_c}{I_c}\right)-\arctan \left(\frac{Q_1}{I_1}\right)& \left[\mathrm{Eq}.32\right]\end{array}$

(82) The corrective imbalance values {Δ.sub.step1, Δ.sub.step2} from Eq. 31 and Eq. 32 solves the nonlinear equation (Eq. 27) and form a full analytic solution for the pre-distortion values that are required to compensate for the MZM impairments. Thus, following (Eq. 29) the values that should be applied to the I-Q MZM for full compensation, are:

(83) $\begin{array}{cc}\{\begin{array}{c}{\Delta }_I=\arctan \left(\frac{Q_c}{I_c}\right)-\arctan \left(\frac{Q_1}{I_1}\right)\\ {\Delta }_Q=\arcsin \left(\frac{P_0-P_{\mathrm{IQ}}}{2I_0{Q}_0}\right)+\arctan \left(\frac{Q_c}{I_c}\right)-\arctan \left(\frac{Q_1}{I_1}\right)\end{array}& \left[\mathrm{Eq}.33\right]\end{array}$

(84) The complex combination of {Δ.sub.I, Δ.sub.Q} is defined as the iDPD compensation parameters.

(85) Performance of Quantized Pre-Distortion Values

(86) Quantization of iDPD Parameters

(87) In a practical system, the symmetric imbalance values are quantized by the DAC used to apply the voltage on the MZM arms. In FIG. 5, the 5.5 bit DAC quantized iDPD parameters {Δ.sub.I, Δ.sub.Q} are compared with non-quantized iDPD parameters in a nonlinearity dominated MZM as a function of the relative voltage ratio V(t)/V.sub.Π in dB. The phase quantization steps of 2Π/45≈0.13.sub.rad are observed around the V(t)/V.sub.Π values of −8 dB and −4 dB, leading to a potential degradation in performance.

(88) Accurate Correction of Quantized Imbalance

(89) When the iDPD parameters are quantized with N bits over the entire correction circle, precise compensation is achieved only when the solutions to the equations at Eq. 33 are integer multiplications of the quantization steps:

(90) $\begin{array}{cc}\{\begin{array}{c}\mathrm{Round}\left\{{\Delta }_I.Math.\frac{2^N}{2\pi }\right\}={\Delta }_I.Math.\frac{2^N}{2\pi }=k_I\\ \mathrm{Round}\left\{{\Delta }_Q.Math.\frac{2^N}{2\pi }\right\}={\Delta }_Q.Math.\frac{2^N}{2\pi }=k_Q\end{array}& \left[\mathrm{Eq}.34\right]\end{array}$
with N, the effective number of DAC bits and k.sub.I, k.sub.Q defined as integer values.

(91) FIG. 6a presents the quantization error of the iDPD parameters from FIG. 5 versus the ratio V(t)/V.sub.Π in dB. FIG. 6b shows the EVM as a function of the ratio V(t)/V.sub.Π in dB. It can be noticed that a significant improvement of the EVM is achieved when both quantization errors are close to zero. The V(t)/V.sub.Π values in which the quantized iDPD parameters are optimal, can be found numerically as the minimum value of the multiplication between the quantization error of the iDPD parameters:

(92) $\begin{array}{cc}{M}_{\mathrm{tDPD}}=\arg \min \left\{.Math.\mathrm{Round}\left\{{\Delta }_I.Math.\frac{2^N}{2\pi }\right\}-{\Delta }_I.Math.\frac{2^N}{2\pi }.Math..Math..Math.\mathrm{Round}\left\{{\Delta }_Q.Math.\frac{2^N}{2\pi }\right\}-{\Delta }_Q.Math.\frac{2^N}{2\pi }.Math.\right\}& \left[\mathrm{Eq}.35\right]\end{array}$
where {Δ.sub.I, Δ.sub.Q} are a function of V(t)/V.sub.Π. This operation is only done once: when the DAC resolution and MZM input voltage range are defined, thus it does not have significant impact on system complexity.

(93) Limits on Quantized iDPD Performance

(94) The worse case error due to quantized correction is a result of the quantization of the two phases {Δ.sub.I, Δ.sub.Q} applied on the distorted point {I.sub.0, Q.sub.0}. Each phase error (as shown above in FIG. 2.) generates an error proportional to the radius of rotation. Thus, assuming that the phase error is small enough, the arc equals its supporting chord, and the maximum signal space error due to the quantized phase would be half the chord length, which can be bounded by:

(95) $\begin{array}{cc}{\mathrm{.Math.}}_{\mathrm{Quantized}\mathrm{\_\Delta }}=\frac{1}{2}.Math.\frac{2\pi .Math.\max \left\{I_0,Q_0\right\}}{2^N}& \left[\mathrm{Eq}.36\right]\end{array}$

(96) When the signal RMS is normalized to unity, the metric of MSE is equivalent to the EVM, thus:
EVM=10 log.sub.10(π.sup.2)−10 log.sub.10(2.sup.2N)+20 log.sub.10(max{I.sub.0,Q.sub.0})≈10−6N+20 log.sub.10(max{(I.sub.0,Q.sub.0})  [Eq. 37]

(97) FIG. 7 shows a graph of EVM as a function of V(t)/V.sub.Π for DAC values with 5.5 (which is the typical benchmark), 6 and 7 bits in a nonlinearity dominated MZM. As the initial distorted point {I.sub.0, Q.sub.0} has a lower power than the target constellation point (due the sinusoidal non-linearity), the relative EVM correction factor 20 Log.sub.10(max{I.sub.0,Q.sub.0}) is always negative which improves the overall EVM as function of the ratio V(t)/V.sub.Π decreases. This means that the limit on EVM performance can be estimated by:
EVM.sub.lim≈10−6N  [Eq. 38]

(98) Quantized Gain and Phase Imbalance Correction

(99) The effect of quantized iDPD has been previously analyzed on a nonlinearity dominated MZM. The analysis showed that there are specific input voltages in which the EVM is minimized. For example when setting the V(t)/V.sub.Π to −5.5 dB (found numerically) the nonlinearity is almost completely compensated (as shown in FIG. 7) and the V(t)/V.sub.Π is referred to as optimal. However, even if the system is designed to work in this input voltage, the single MZM and dual MZM gain and phase distortions may significantly affect performance.

(100) FIG. 8 shows the effect of a single MZM gain and phase imbalance on EVM is shown for the optimal V(t)/V.sub.Π. It can be seen that when there is no gain or phase imbalance, the EVM is ˜−80 dB. However as either gain or phase imbalance increases the absolute value of EVM drops fast.

(101) In FIG. 9, the effect of dual (complex) MZM gain and phase imbalance is shown for the optimal V(t)/V.sub.Π. A pattern similar to the single MZM behavior is observed, in which slight changes in gain or phase imbalance causes fast drops in EVM. In both cases, the EVM performances are in good agreement with the quantization limit defined in Eq. 38.

(102) Performance of Compensated QAM-16 MZM Modulation

(103) In order to have a better understanding of the expected MZM performance with the iDPD, an analysis of required OSNR for a PDM-16QAM signal is presented in FIG. 10. The target BER is 2.4×10.sup.−2 and the target OSNR is 19.8 dB. A 5.5 bit ENOB DAC is applied for the iDPD correction.

(104) Three cases are considered: when there is no impairment except nonlinearity, when there is IQ imbalance gain distortion of 2% and when the single MZM phase distortion is 1°.

(105) The analysis shows that applying the iDPD method on QAM-16 signals with a 5.5 bit AC can be used improve the require OSNR by 5˜8 dB, depending on the applied voltage and the level of distortion, mainly due to iDPD ability of maintaining almost the same level of noise contribution for applied voltage dynamic range.

(106) Finding iDPD Parameters Through Iterative Search

(107) Iterative Search Problem Definition

(108) When the parameters of the impairments are only partially known, or known with low confidence, analytic solution would yield unknown error. On the other hand, applying a full iterative search in which all potential quantized imbalance values are calculated, is typically non feasible from most practical aspects. The approach proposed by the present invention is to define the following optimization problem based on the EVM definition in Eq. 16 above: Maximize ABS{EVM} where EVM=f{Δ.sub.I, Δ.sub.Q}. f{Δ.sub.I, Δ.sub.Q} is defined by Eq. 16 above Δ.sub.I=Δ.sub.step2 Δ.sub.Q=Δ.sub.step1+Δ.sub.step2 Apply the Inequality constraints: 0<Δ.sub.I, Δ.sub.Q<2Π

(109) The inequality constraints imply that the dynamic range of the DAC driving signal allows a full rotation of correction parameters.

(110) The EVM can be calculated by adding a loopback to the system from the transmitter to the receiver. The signal at the output of the loopback is coherently demodulated and compared with the transmitted data. Alternatively, the optimization concept can be applied as part of the manufacturing line, in which the compensation scheme for each symbol constellation is calculated separately. However in this approach, potential changes in impairments, due to temperature changes and aging, are difficult to track.

(111) The EVM function may have several minima, which locations are the solutions of the optimization problem. Thus a method for locating and identifying the global minima is required. The trust region optimization is a known method for finding global minima in non-convex functions.

(112) The method is based on finding an initial point which is an approximation of the target function, and it is assumed that within the ‘trust region’ the function is convex and that the global minima is inside that region around the approximated point. On each iteration the search direction and trust region size are updated, until convergence is achieved.

(113) Initial Conditions for the Trust Region Optimization

(114) Initial conditions can be derived from the analytic solution, however, the use of complex trigonometric functions is typically avoided due to the computational resources required for their calculation. An alternative approach is to use the first order linear approximation of Eq. 28 which is presented as:

(115) $\begin{array}{cc}\{\begin{array}{c}{I}_0-Q_0\left({\Delta }_{\mathrm{step}1}+{\Delta }_{\mathrm{step}2}\right)=I_c\\ I_0{\Delta }_{\mathrm{step}2}+Q_0=Q_c\end{array}& \left[\mathrm{Eq}.39\right]\end{array}$

(116) And its solution:

(117) $\begin{array}{cc}\{\begin{array}{c}{\Delta }_{\mathrm{step}1}=\frac{I_0-I_c}{Q_0}-\frac{Q_c-Q_0}{I_0}\\ {\Delta }_{\mathrm{step}2}=\frac{Q_c-Q_0}{I_0}\end{array}& \left[\mathrm{Eq}.40\right]\end{array}$

(118) As optimizing the EVM plane is equivalent to optimizing the minimum geometric distance in the symbol space, the values found in Eq. 40 can be used as initial conditions for the trust region method for finding the minimum EVM value in the complex imbalance plane:

(119) $\begin{array}{cc}\{\begin{array}{c}{\Delta }_{I\_\mathrm{ini}t}={\Delta }_{\mathrm{step}2}=\frac{Q_c-Q_0}{I_0}\\ {\Delta }_{Q\_\mathrm{init}}={\Delta }_{\mathrm{step}1}+{\Delta }_{\mathrm{step}2}=\frac{I_0-I_c}{Q_0}\end{array}& \left[\mathrm{Eq}.41\right]\end{array}$

(120) Therefore, finding the optimal pre-distortion values requires setting the initial conditions {I.sub.0, Q.sub.0} based on the known MZM distortion parameters and the transmitted data symbol as shown in Eq. 41 above.

(121) FIG. 11a shows a comparison between the iterative trust region search and the analytic solution of the quantized correction in a nonlinearity dominated MZM. In FIG. 11b shows the noise floor which is added to the solution by the trust region algorithm is presented as a function of the relative voltage V(t)/V.sub.Π. It can be seen that the iterative search adds a noise floor of ˜15 dB below the corrected impairment noise floor, which has a negligible effect on overall performance. Thus the trust region method can provide a reliable tool for optimizing the iDPD parameters when only an approximation of the actual distortion is available.

(122) The simulation showed that there are only specific relative voltage V(t)/V.sub.Π values in which the distortion can be fully compensated for, and that for other relative voltages, there is a minimum noise floor contribution that can be assured, limited only by the DAC quantization noise and can be well estimated by [10-6N]dB for a DAC with N effective number of bits.

(123) The method was also tested on various distortions applied to a QAM-16 signal which showed that the required OSNR when applying a 5.5 bits correction DAC to a Polarization Division Multiplexing of 16 QAM (PDM-QAM-16) noise floor with strong FEC and a target of 19.8 dB OSNR can be improved in 5˜8 dB on the extreme high power cases.

(124) When the parameters of the MZM distortion are not known, or can only be approximated, an iterative search based on the trust region algorithm has been proposed. The algorithm uses the preliminary linear approximation to the iDPD basic equations as the initial condition for the search. It was shown that the algorithm has the potential to contribute less than 0.1 dB to the optimal EVM when it fully converges.

(125) The iDPD analysis can be extended to consider the limited bandwidth of MZM driver. Considering the driver BW would result in a linear dependency between symbols, and thus the analytically calculated iDPD values applied on each symbol would be a combination of a linear function of the current and previous symbols.

(126) In order to overcome the limitations of prior art, the present invention proposes a new pre-distortion method, based on adding an imbalance symmetrically to the information applied to the MZM arms. The imbalance pre-distorts the signal in a way that the overall effect of the symmetric imbalance and MZM impairments is minimized by means of mean square error.

(127) Although embodiments of the invention have been described by way of illustration, it will be understood that the invention may be carried out with many variations, modifications, and adaptations, without exceeding the scope of the claims.

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