Dual parallel Mach-Zehnder-modulator device with pre-distorted driving voltages

Abstract

Disclosed herein is a dual parallel Mach-Zehnder-modulator (DPMZM) device comprising a DPMZM 10 having first and second inner MZMs arranged parallel to each other. The first inner MZM generates an in-phase component E.sub.I of an optical signal in response to a first driving voltage V.sub.I, and the second inner MZM generates a quadrature component E.sub.Q of said optical signal in response to a second driving voltage V.sub.Q. Further disclosed is a calculation unit 52 configured for receiving an in-phase component y.sub.I and a quadrature component y.sub.Q of a desired base-band signal, and for calculating pre-distorted first and second driving voltages V.sub.I, V.sub.Q. The calculation of the pre-distorted first and second driving voltages V.sub.I, V.sub.Q is based on a model of said DPMZM 10 accounting for I-Q cross-talk, and using an algorithm that determines said first and second driving voltages V.sub.I, V.sub.Q each as a function of both of said in-phase and quadrature components y.sub.I, y.sub.Q of said base-band signal.

Claims

1. A dual parallel Mach-Zehnder-modulator (DPMZM) device for generating a modulated optical signal, comprising: a first inner Mach-Zehnder-modulator (MZM) operable to generate an in-phase component E.sub.I of the modulated optical signal in response to a first driving voltage V.sub.I, the first driving voltage V.sub.I comprising at least a first AC component; a second inner MZM operable to generate a quadrature component E.sub.Q of the modulated optical signal in response to a second driving voltage V.sub.Q, the second driving voltage V.sub.Q comprising at least a second AC component; and a calculation unit comprising at least one electrical component, the calculation unit configured for receiving an in-phase component y.sub.I and a quadrature component y.sub.Q of a base-band signal, determining at least the first AC component of the first driving voltage V.sub.I as a function of both the in-phase component y.sub.I and the quadrature component y.sub.Q of the base-band signal, and determining at least the second AC component of the second driving voltage V.sub.Q as a function of both the in-phase component y.sub.I and the quadrature component y.sub.Q of the base-band signal.

2. The DPMZM device of claim 1, wherein in the calculation, the calculation unit employs at least one model parameter relating to a finite inner extinction ratio of at least one of the first inner MZM and the second inner MZM.

3. The DPMZM device of claim further comprising a parameter calculating unit that comprises an electronic component, the parameter calculating unit operative to: receive a quality indicator relating to a quality of an optical signal received by a receiver, and modify, in response to the received quality indicator, the at least one model parameter to improve a quality of the modulated optical signal generated using the DPMZM device.

4. The DPMZM device of claim 3, wherein the quality indicator is relates at least in part to at least one of: a bit-error rate related to the optical signal received by the receiver, and an estimate of a bit-error-rate related to the optical signal received by the receiver.

5. The DPMZM device of claim 3, wherein the quality indicator is relates at least in part to at least one of: a power of a residual carrier, and an estimate of a power of a residual carrier.

6. The DPMZM device of claim 3, wherein the quality indicator relates at least in part to a mean square error between a target transmit signal and an actual transmit signal.

7. The DPMZM device of one of claim 3, wherein the receiver is one of the following: a coherent far-end receiver, and a coherent monitoring receiver associated with local to the DPMZM device.

8. The DPMZM device of claim 1, wherein the calculation unit is further configured for determining at least the first AC component of the first driving voltage V.sub.I and at least the second AC component of the second driving voltage V.sub.Q to mitigate an adverse effect of I-Q cross talk introduced by at least one non-ideal characteristic of the DPMZM device.

9. The DPMZM device of claim 1, wherein the DPMZM device further comprises an AC-coupling for coupling the first driving voltage V.sub.I and the second driving voltage V.sub.Q as, and wherein a first and second biasing electrodes are biasing electrode is associated with the first inner MZM and a second biasing electrode is associated with the second inner MZM, and wherein the first biasing electrode is operative to apply a first bias component to the first inner MZM, the first bias component corresponding to a first DC component of the first driving voltage V.sub.I, and wherein the second biasing electrode is operative to apply a second bias component to the second inner MZM, the second bias component corresponding to a second DC component of the second driving voltage V.sub.Q.

10. The DPMZM device of claim 9, further comprising a bias component control unit operative to adjust, in response to at least one input to the bias component control unit, at least one of the first bias component and the second bias component, the at least one input comprising at least one of: an error-indicating signal communicated by a far-end receiver, or a quality indicator, relating to a quality of an optical signal received by a far-end receiver.

11. The DPMZM device of claim 10, wherein the bias component control unit is configured to adjust the first bias component and the second bias component using a gradient descent algorithm minimizing an error indicated by at least one of: the error indicating signal, and the quality indicator.

12. The DPMZM device of claim 1, wherein each of the first inner MZM and the second inner MZM are independently biased, wherein the first inner MZM delivers a minimum possible output power when the first driving voltage V.sub.I is zero, and wherein the second inner MZM delivers a minimum possible output power when the second driving voltage V.sub.Q is zero.

13. The DPMZM device of claim 1, wherein the calculation unit is configured for solving two coupled non-linear equations relating the in-phase component y.sub.I and the quadrature component y.sub.Q of the desired base-band signal to the first driving voltage V.sub.I and the second driving voltage V.sub.Q.

14. The DPMZM device of claim 13, wherein the two coupled non-linear equations are represented as follows: $\begin{array}{c}{y}_I=\sin \left(\frac{\pi }{2}.Math.\frac{V_I}{V_{\pi }}\right)+{\gamma }_Q.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_Q}{V_{\pi }}\right)\\ y_Q=\sin \left(\frac{\pi }{2}.Math.\frac{V_Q}{V_{\pi }}\right)-{\gamma }_I.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_I}{V_{\pi }}\right)\end{array}$ wherein Vπ, γ.sub.Q and γ.sub.I are positive characteristic constants of the DPMZM device.

15. The DPMZM device of claim 13, wherein the calculation unit carries out an iterative solution of the two coupled non-linear equations, including at least two iterations of the iterative solution.

16. The DPMZM device of claim 13, wherein the calculation unit is configured to solve the following equations: $V_I=\frac{2V_{\pi }}{\pi }a\sin \left(P_1\left(y_I{y}_Q\right)\right)$ $V_Q=\frac{2V_{\pi }}{\pi }a\sin \left(P_2\left(y_I,y_Q\right)\right)$ wherein P.sub.1(y.sub.I,y.sub.Q) and P.sub.2(y.sub.I,y.sub.Q) are polynomials of y.sub.I,y.sub.Q, wherein P.sub.1(y.sub.I,y.sub.Q) is a first-order polynomial in y.sub.I and a at least a second-order polynomial in y.sub.Q, and wherein P.sub.2(y.sub.I,y.sub.Q) is preferably a first order polynomial in y.sub.Q and a at least a second-order polynomial in y.sub.I.

17. The DPMZM device of claim 1, wherein the calculation unit comprises a look-up table of relating to a function a sin(x), and a look-up table relating to at least one of: a function cos(x), and cos(a sin(x)).

18. A method for generating a modulated optical signal using a dual parallel Mach-Zehnder-modulator (DPMZM) comprising a first inner Mach-Zehnder-modulator (MZM) and a second inner MZM arranged parallel to the first inner MZM, the first inner MZM operable to generate an in-phase component E.sub.I of the modulated optical signal in response to a first driving voltage V.sub.I, the first driving voltage V.sub.I comprising at least a first AC component, and the second inner MZM operable to generate a quadrature component E.sub.Q of the modulated optical signal in response to a second driving voltage V.sub.Q, the second driving voltage V.sub.Q comprising at least a second AC component, the method comprising the following steps: receiving an in-phase component y.sub.I and a quadrature component y.sub.Q of a base-band signal, calculating at least the first AC component of the first driving voltage V.sub.I as a function of both the in-phase component y.sub.I and the quadrature component y.sub.Q of the base-band signal, calculating at least the second AC component of the second driving voltage V.sub.Q as a function of both the in-phase component y.sub.I and the quadrature component y.sub.Q of the base-band signal, applying the first driving voltage V.sub.I to the first inner MZM for generating the in-phase component E.sub.I of the modulated optical signal and applying the second driving voltage V.sub.Q to the second inner MZM for generating the quadrature component E.sub.Q of the modulated optical signal.

19. The method of claim 18, wherein the calculation step employs at least one model parameter relating to a finite inner extinction ratio of at least one of the first inner MZM and the second inner MZM.

20. The method of claim 19, further comprising the steps of receiving a quality indicator relating to a quality of an optical signal received by a receiver, and modifying, in response to the received quality indicator, the at least one model parameter to improve a quality of the modulated optical signal generated using the DPMZM.

21. The method of claim 20, wherein the quality indicator relates at least in part to at least one of: a bit-error rate related to the optical signal received by the receiver, and an estimate of a bit-error-rate related to the optical signal received by the receiver.

22. The method of claim 20, wherein the quality indicator relates at least in part to at least one of: a power of a residual carrier, and an estimate of a power of a residual carrier.

23. The method of claim 20, wherein the quality indicator relates at least in part to a mean square error between a target transmit signal and an actual transmit signal.

24. The method of one of claim 18, wherein the DPMZM further comprises an AC-coupling for coupling at least a portion of the first driving voltage V.sub.I to the first inner MZM and at least a portion of the second driving voltage V.sub.Q to the second inner MZM, and wherein the method further comprises a step of applying a first bias component to the first inner MZM and a second bias component to the second inner MZM for generating the modulated optical signal.

25. The method of claim 24, wherein the first bias component corresponds to a first DC component of the first driving voltage V.sub.I, and the second bias component corresponds to a second DC component of the second driving voltage V.sub.Q.

26. The method of claim 24, wherein the method further comprises a step of adjusting, at least one of the first bias component and the second bias component in response to at least one of: an error-indicating signal communicated by a far-end receiver, and a quality indicator relating to a quality of an optical signal received by a far-end receiver.

27. The method of one of claim 18, wherein the calculation step comprises solving two coupled non-linear equations relating the in-phase component y.sub.I and the quadrature component y.sub.Q of the base-band signal to the first driving voltage V.sub.I and the second driving voltage V.sub.Q.

28. The method of claim 27, wherein the two coupled non-linear equations are represented as follows: $\begin{array}{c}{y}_I=\sin \left(\frac{\pi }{2}.Math.\frac{V_I}{V_{\pi }}\right)+{\gamma }_Q.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_Q}{V_{\pi }}\right)\\ y_Q=\sin \left(\frac{\pi }{2}.Math.\frac{V_Q}{V_{\pi }}\right)-{\gamma }_I.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_I}{V_{\pi }}\right)\end{array}$ wherein Vπ, γ.sub.Q and γ.sub.I are positive characteristic constants of the DPMZM device.

29. The method of claim 28, wherein the calculation step comprises an iterative solution of the two coupled non-linear equations, including at least two iterations of the iterative solution.

30. The method of claim 18, wherein at least the first AC component of the first driving voltage V.sub.I and at least the second AC component of the second driving voltage V.sub.Q are calculated to mitigate an adverse effect of I-Q cross talk introduced by at least one non-ideal characteristic of the DPMZM.

31. A dual parallel Mach-Zehnder-modulator (DPMZM) device for generating a modulated optical signal, comprising: a first inner Mach-Zehnder-modulator (MZM) operable to generate an in-phase component E.sub.I of the modulated optical signal in response to a first driving voltage V.sub.I, the first driving voltage V.sub.I comprising at least a first AC component; a second inner MZM operable to generate a quadrature component E.sub.Q of the modulated optical signal in response to a second driving voltage V.sub.Q, the second driving voltage V.sub.Q comprising at least a second AC component; and a calculation unit comprising at least one electrical component, the calculation unit operable to: determine at least the first AC component of the first driving voltage V.sub.I as a function of both the in-phase component y.sub.I and the quadrature component y.sub.Q of a base-band signal, and determine at least the second AC component of the second driving voltage V.sub.Q as a function of both the in-phase component y.sub.I and the quadrature component y.sub.Q of the base-band signal.

32. The DPMZM device of claim 31, wherein at least the first AC component of the first driving voltage V.sub.I and at least the second AC component of the second driving voltage V.sub.Q are pre-distorted.

33. The DPMZM device of claim 32, wherein at least the first AC component of the first driving voltage V.sub.I and at least the second AC component of the second driving voltage V.sub.Q are determined based on at least one model accessible to the calculation unit.

34. The DPMZM device of claim 31, wherein the calculation unit is further operable to determine at least the first AC component of the first driving voltage V.sub.I and at least the second AC component of the second driving voltage V.sub.Q to mitigate an adverse effect of I-Q cross talk introduced by at least one non-ideal characteristic of the DPMZM device.

35. The DPMZM device of claim 34, wherein at least the first AC component of the first driving voltage V.sub.I and at least the second AC component of the second driving voltage V.sub.Q are determined based on at least one model accessible to the calculation unit.

Description

SHORT DESCRIPTION OF THE FIGURES

(1) For the purposes of promoting an understanding of the principles of the invention, reference will now be made to the preferred embodiments illustrated in the drawings and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of the invention is thereby intended, such alterations and further modifications in the illustrated device and method and such further applications of the principles of the invention as illustrated therein being contemplated therein as would normally occur now or in the future to one skilled in the art to which the invention relates.

(2) FIG. 1 shows a schematic view of a prior art DPMZM,

(3) FIG. 2 shows a flow diagram of a single iteration of a pre-distortion algorithm according to the invention,

(4) FIG. 3 shows a flow diagram of two consecutive iterations of a pre-distortion algorithm according to the invention,

(5) FIG. 4 shows a flow diagram of three consecutive iterations of a pre-distortion algorithm according to the invention,

(6) FIG. 5 shows a DPMZM device employing a modified version of a generally pre-known bias control, in which error offsets are introduced,

(7) FIG. 6 shows a DPMZM device employing a novel bias control relying on a feedback channel from a far-end receiver,

(8) FIGS. 7 to 9 show the performance of the iterative pre-distortion algorithm as indicated in FIGS. 3 and 4 in simulative investigations,

(9) FIG. 10 shows a DPMZM device including a parameter calculating unit adapted to modify model parameters such as to optimize a quality indicator received from a far-end receiver,

(10) FIG. 11 shows a DPMZM device including a parameter calculating unit employing an indirect learning architecture,

(11) FIG. 12 shows a DPMZM device including a parameter calculating unit, and employing an inverse calculation unit, and

(12) FIG. 13 shows a general setup of a DPMZM and an inverse system providing pre-distorted complex input to the DPMZM.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

(13) Turning back to the DPMZM 10 of FIG. 1, in the following discussion we assume an infinite or a perfectly compensated outer extinction ratio of the outer MZM and concentrate on the inner MZMs 20, 22. Further, for the moment we assume that each inner MZM 20, 22 is biased to deliver the minimum possible output power when the corresponding driving signal V.sub.I, V.sub.Q is zero. Note that from an operation point of view, the latter would actually not be an ideal starting point, as this does not correspond to the biasing that eventually leads to the best signal quality. However, the assumption is made purely for mathematical purposes, as it leads to very simple equations, the solutions of which then lead to DC-components in the pre-distorted first and second driving voltages V.sub.I, V.sub.Q that account for a more proper biasing.

(14) With appropriate normalization of the electric field amplitude by the input electric field amplitudes in the respective first and second arms 16, 18 of the outer MZMs, the input/output relations of the first and second inner MZMs 20, 22 are as follows:

(15) $\begin{array}{cc}{E}_I=\frac{1}{1+{\beta }_I}\left[\exp \left(j.Math.\frac{\pi }{2}.Math.\left(\frac{V_I}{V_{\pi }}-1\right)\right)+{\beta }_I.Math.\exp \left(-j.Math.\frac{\pi }{2}.Math.\left(\frac{V_I}{V_{\pi }}-1\right)\right)\right]\mathrm{and}& \left(2\right)\\ E_Q=\frac{1}{1+{\beta }_Q}\left[\exp \left(j.Math.\frac{\pi }{2}.Math.\left(\frac{V_Q}{V_{\pi }}-1\right)\right)+{\beta }_Q.Math.\exp \left(-j.Math.\frac{\pi }{2}.Math.\left(\frac{V_Q}{V_{\pi }}-1\right)\right)\right],& \left(3\right)\end{array}$

(16) Herein, E.sub.I and E.sub.Q are the normalized in-phase and quadrature components of the optical signal generated in response to the first and second driving voltages V.sub.I, V.sub.Q, respectively. V.sub.π, β.sub.I and β.sub.Q are positive characteristic constants of the DPMZM 10. More particularly, the constants β.sub.I and β.sub.Q represent the ratio of the electric field amplitude in the two arms of the first and second inner MZMs 20, 22, respectively. In other words, a value β.sub.I=1 would resemble a situation where the power is evenly split between the two arms of the first inner MZM 20, while any deviation from this ideal behavior would lead to a value β.sub.I different from 1. The symbol “j” resembles the imaginary part of a complex number in the usual manner.

(17) It is further assumed that the outer MZM is biased, by means of the electrodes 28 provided in the second arm 18 thereof, such as to establish a 90° phase shift between the in-phase and quadrature components E.sub.I, E.sub.Q in the combined or total electric field amplitude E, i.e.
E=E.sub.I+j.Math.E.sub.Q.  (4)

(18) Introducing the expressions from Eqs. (2) and (3) above and carrying out a number of arithmetic steps, we obtain the following expression for the combined or total electric field amplitude E:

(19) $\begin{array}{cc}E=\sin \left(\frac{\pi }{2}.Math.\frac{V_I}{V_{\pi }}\right)+{\gamma }_Q.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_Q}{V_{\pi }}\right)+j.Math.\left[\sin \left(\frac{\pi }{2}.Math.\frac{V_Q}{V_{\pi }}\right)-{\gamma }_I.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_I}{V_{\pi }}\right)\right],\mathrm{and}& \left(5\right)\\ {\gamma }_I=\frac{1-{\beta }_I}{1+{\beta }_I}& \left(6\right)\\ {\gamma }_Q=\frac{1-{\beta }_Q}{1+{\beta }_Q}.& \left(7\right)\end{array}$
where we introduced:

(20) Herein, γ.sub.I and γ.sub.Q are the square root of the inverse power extinction ratios ER.sub.I, ER.sub.Q of the first and second inner MZM 20, 22. Usually the extinction ratios ER.sub.I and ER.sub.Q of the first and second inner MZMs 20, 22 are represented as dB values as follows:
ER.sub.I=−20.Math.log.sub.10|γ.sub.I|  (8)
and
ER.sub.Q=−20.Math.log.sub.10|γ.sub.Q|  (9)

(21) From Eq. (5) it can be seen that the DPMZM 10 exhibits an intrinsic sinusoidal non-linearity and, if the inner ERs are finite, additional I-Q-cross-talk.

(22) In a next step, we describe an algorithm for the computation of the driving voltages V.sub.I and V.sub.Q that produce the desire transmit signal if applied to the electrodes 24 and 26 of the non-ideal DPMZM. Assuming the above model of the imperfect DPMZM with non-vanishing inverse extinction ratios γ.sub.I, γ.sub.Q, from Eq. (5) it is seen that the appropriate driving voltages V.sub.I and V.sub.Q to produce the I- and Q-components y.sub.I, y.sub.Q of a desired transmit signal amount to a solution of the following non-linear equation system:

(23) $\begin{array}{cc}\begin{array}{c}{y}_I=\sin \left(\frac{\pi }{2}.Math.\frac{V_I}{V_{\pi }}\right)+{\gamma }_Q.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_Q}{V_{\pi }}\right)\\ y_Q=\sin \left(\frac{\pi }{2}.Math.\frac{V_Q}{V_{\pi }}\right)-{\gamma }_I.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_I}{V_{\pi }}\right).\end{array}& \left(10\right)\end{array}$

(24) For solving the above equation system for V.sub.I and V.sub.Q, the following iterative pre-distortion algorithm may be employed:

(25) $\begin{array}{cc}\begin{array}{c}{V}_I^{\left(n\right)}=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_I-{\gamma }_Q.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_Q^{\left(n-1\right)}}{V_{\pi }}\right)\right)\\ V_Q^{\left(n\right)}=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_Q+{\gamma }_I.Math.\cos \left(\frac{\pi }{2}.Math.\frac{V_I^{\left(n-1\right)}}{V_{\pi }}\right)\right)\end{array}\left(n=1,2,\mathrm{.Math.},K\right),& \left(11\right)\end{array}$
where the positive integer K is a number of iterations, and V.sub.I.sup.(n) and V.sub.Q.sup.(n) are the approximations of the desired driving voltages V.sub.I and V.sub.Q at the n.sup.th iteration. From the above iteration algorithm, the concept of “pre-distortion” becomes particularly apparent. For example, in a perfect DPMZM without cross talk, γ.sub.Q would be zero and V.sub.I.sup.(n) would be dependent on the desired I-component y.sub.I of the transmit signal only. With non-vanishing γ.sub.Q, a “distortion” is introduced to V.sub.I, which distortion is dependent on V.sub.Q and in fact accounts for the cross-talk introduced by non-vanishing parameters γ.sub.I, γ.sub.Q in an anticipatory manner.

(26) The algorithm can be conveniently initialized with
V.sub.1.sup.(0)=0
V.sub.Q.sup.(0)=0,  (12)
which results into

(27) $\begin{array}{cc}{V}_I^{\left(1\right)}=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_I-{\gamma }_Q\right)V_Q^{\left(1\right)}=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_Q+{\gamma }_I\right).& \left(13\right)\end{array}$

(28) If desired, the first iteration can be slightly improved with negligible effort by using the following initialization:

(29) $\begin{array}{cc}{V}_I^{\left(1\right)}=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_I-{\gamma }_Q.Math.E\left\{\cos \left(\frac{\pi }{2}.Math.\frac{y_Q}{V_{\pi }}\right)\right\}\right)V_Q^{\left(1\right)}=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_Q+{\gamma }_I.Math.E\left\{\cos \left(\frac{\pi }{2}.Math.\frac{y_I}{V_{\pi }}\right)\right\}\right).& \left(14\right)\end{array}$

(30) Herein, E{⋅} denotes a stochastic expectation.

(31) FIG. 2 illustrates a single iteration of the pre-distortion algorithm. In case of a digital implementation, the non-linear functions can be conveniently approximated by spline, i.e. piecewise linear interpolation. Since the non-linear functions are completely unaffected by the actual inner extinction ratios, the spline coefficients can be pre-computed offline and implemented via constant look-up tables. The parameters γ.sub.I and γ.sub.Q, on the other hand, need to be calibrated for each individual DPMZM 10, either upon manufacturing or upon operation and can possibly be tracked and adjusted over time in a fashion described in more detail below.

(32) Notably, when cascading multiple pre-distortion stages in the iterative solution according to Eq. (11), the input non-linear functions at each stage can be combined with the output non-linear functions of the previous stage and conveniently implemented in a single operation. Consequently, every iteration requires in fact the computation of only two non-linear functions with one real input and one real output which can be carried out rapidly in real-time under operation of the DPMZM 10.

(33) To appreciate this fact, in FIG. 3 a 2-stage pre-distortion algorithm is displayed, where the constants I.sub.init and Q.sub.init were introduced as the initial values of the iteration. If the initialization of Eq. (12) is adopted, we set
I.sub.init=1
Q.sub.init=1.  (15)

(34) Alternatively, for the initialization of Eq. (14) we set

(35) 0$\begin{array}{cc}{I}_{\mathrm{init}}=E\left\{\cos \left(\frac{\pi }{2}.Math.\frac{y_i}{V_{\pi }}\right)\right\}{Q}_{\mathrm{init}}=E\left\{\cos \left(\frac{\pi }{2}.Math.\frac{y_Q}{V_n}\right)\right\}.& \left(16\right)\end{array}$

(36) For further exemplification purposes, in FIG. 4 an implementation of the 3-stage pre-distortion algorithm is shown. From this, generalizations to more stages will be apparent to the person skilled in the art.

(37) As mentioned above, the recursion of Eq. (11) was derived under the assumption that each of the inner MZMs 20, 22 is independently biased to deliver the minimum possible output power when the corresponding modulating signal V.sub.I and V.sub.Q is zero. However, if the inner extinction ratio is finite, in the absence of pre-distortion, this biasing condition results into imperfect carrier suppression in the optical output of the DPMZM 10. The consequent residual carrier component is detrimental because it impairs the demodulation algorithms, wastes part of the available optical power, and enhances nonlinearities in the fiber. Therefore, the assumed biasing point in the above mathematical derivation, although leading to simpler pre-distortion equations, is actually neither optimal nor desirable in the absence of pre-distortion.

(38) In the presence of pre-distortion, the suboptimal bias is not problematic, since the recursion of Eq. (11) is able to suppress the residual carrier and to produce the desired optical signal. In the model considered so far, this is possible because the pre-distortion as derived by the iteration of Eq. (11) will automatically lead to values of V.sub.I and V.sub.Q including a DC offset representing the optimal biasing condition. In the mathematical model presented, the pre-distortion hence injects a DC-offset into the driving voltages and thereby effectively corrects the biasing point. However, in practical implementations, the driving voltages V.sub.I, V.sub.Q, i.e. the modulating signal, will generally be AC-coupled with the inner MZMs 20, 22 and a bias correction provided through the driving voltages V.sub.I and V.sub.Q would not reach the DPMZM 10.

(39) In the present invention, this difficulty can be overcome by adopting a proper automatic bias control scheme which suppresses the residual carrier. Such a bias scheme, which in fact reintroduces the DC-correction that has been removed from the modulating signal by the AC-coupling, guarantees that the sum of the bias and driving voltage and, thus, the optical output of the DPMZM 10 remain unchanged. Remarkably, the pre-distortion can still be computed according to the simple recursion (11), i.e. under the assumption that each inner MZM 20, 22 is biased for minimum output power, because the resulting offset is filtered out anyway and does not interfere with the bias control.

(40) Several automatic bias control schemes for a DPMZ have been proposed in prior art, see e.g. P. S. Cho, J. B. Khurgin, and I. Shpantzer, “Closed-loop bias control of optical quadrature modulator,” IEEE Photonics Technology Letters, vol. 18, no. 21, pp. 2209-2211, November 2006 and M. Sotoodeh, Y. Beaulieu, J. Harley, and D. L. McGhan, “Modulator bias and optical power control of optical complex E-field modulators”, IEEE Journal of Lightwave Technology, vol. 29, no. 15, pp. 2235-2248, August 2011. These schemes control the bias voltages according to a gradient descent algorithm applied to an algorithm-specific error signal. Since they are designed to work on an ideal DPMZM with infinite ERs, they fail to suppress the residual carrier when the inner MZMs have a finite ER. However, it is possible to adapt such standard bias controls by injecting a proper offset into the error signal. The offset can be determined by factory calibration in such a way that the modified scheme suppresses the residual carrier.

(41) FIG. 5 illustrates a DPMZM device employing a modified version of the standard bias control. As is seen in FIG. 5, digital representations of the driving voltages V.sub.I, V.sub.Q as computed with the recursion of Eq. (11) are inputted at inputs 30 and DA-converted at DA-converters 32. The thus obtained analog signals are AC-coupled to the DPMZM 10. Due to the AC-coupling, a DC-component of the driving voltages V.sub.I and V.sub.Q as obtained from the iteration of Eq. (11) will be lost.

(42) A part of the optical output signal 14 of the DPMZM 10 is branched off and detected with a photo detector 34. The detection signal of the photo detector 34 is coupled to a bias error computer 36 which computes an error signal in a way per se known from the above citations. Two I- and Q-error signals 38, 40 are outputted from the bias error computer 36, to which error offsets are added using adders 42. The error offset can be determined by factory calibration in such a way that the modified scheme suppresses the residual carrier. These additional error offsets for residual carrier suppression are not provided for in known automatic bias control schemes and specifically relate to the operation of the DPMZM device of the present invention accounting for finite inner ERs.

(43) The error signals 38, 40, together with the added offsets, are introduced to a bias voltage computer 44 which in turn computes a bias 46 for the first inner MZM 20 and a bias 47 for the second MZM 22 which are added to the AC-coupled analog driving voltages V.sub.I, V.sub.Q by further adders 42 prior to introducing them to the DPMZM 10.

(44) A further DPMZM device according to an embodiment of the invention employs a novel bias control that relies on a feedback channel from a far-end receiver to the transmitter. In the case of coherent transmission, digital demodulation algorithms at the receiver are able to detect the power of the residual carrier generated at the transmitter. The demodulator shall send back the power of the detected residual carrier to the bias control, which uses this information in a gradient descent algorithm to suppress directly the residual carrier.

(45) A corresponding DPMZM device is shown in FIG. 6. The DPMZM device of FIG. 6 is able to carry out a conventional bias control scheme using the photo detector 34, the bias error computer 36 and the bias voltage computer 44 as shown in FIG. 5, but without the error offset introduction of FIG. 5. This standard biasing algorithm, which is based on the standard local error signal defined along the lines of the previously cited references is, however, only provided for as a fallback means in the initial convergence phase. The proper bias control is based on an error signal 51 generated by a residual carrier detector 50 provided in a far-end receiver 48. This error signal 51 may in fact directly correspond to the power of the residual carrier. The bias error computer 44 may then iteratively adapt the bias voltages 46, 47 by minimizing the error indicated by error signal 51. The bias error computer 36 and the bias voltage computer 44 hence in combination form an example of the bias component control unit referred to in the summary of the invention. Note that this bias component control unit provides I- and Q-bias values which “account for” the DC component of the calculated driving voltage, but which also account for the customary bias control. In other words, the contribution corresponding to the DC-components of V.sub.I and V.sub.Q are not separately determined or applied, but are nevertheless automatically accounted for in the embodiment of FIG. 6.

(46) Both DPMZM devices of FIGS. 5 and 6 hence allow for implementing a proper bias control scheme, i.e. a bias control scheme that suppresses the residual carrier and is compatible with the proposed iterative pre-distortion as defined in Eq. (11).

(47) With reference to FIGS. 7 to 9, the performance of the proposed pre-distortion algorithm is demonstrated by means of simulative investigations. In the analysis, a 16-QAM transmission at 31 Gsymbols/s in the presence of root Nyquist spectral shaping with a digital pre-distortion (DPD) running at two samples per symbol was considered. For the sake of simplicity, the quantization noise introduced by the DAC was neglected, and the same inner ERs for the first and second DPMZMs 20, 22 were assumed. The back-to-back performance (i.e. without transmission fiber in between) was evaluated on the basis of the bit error rate (BER) as a function of the optical signal-to-noise ratio (OSNR) and compared with the performance of a reference system without DPD. In the absence of DPD, the biasing points of the inner MZMs 20, 22 were optimized for minimal BER. In the presence of DPD, as discussed above, the biasing point was chosen such that the residual carrier is maximally suppressed. For a fair comparison, the same average optical power with and without DPD was transmitted. In the absence of DPD, the swing of the driving voltages V.sub.I, V.sub.Q is smaller and the system benefits from using only a limited region of the MZM characteristics. In the presence of DPD, which enhances the signal peaks, i.e. the maximum values of V.sub.I and V.sub.Q, a larger non-linear portion of the MZM characteristics is used, but the nonlinearity and the I-Q-cross-talk are pre-compensated.

(48) FIGS. 7 and 8 illustrate the cases of an inner ER of 20 dB and 15 dB, respectively. When the DPD is active, the inner MZMs 20, 22 are fully driven, i.e. the peak-to-peak swings of the driving voltages V.sub.I, V.sub.Q are fixed to 2.Math.V.sub.π. The simulation results show that, remarkably, only two iterations of the iterative pre-distortion algorithm of Eq. (11) are sufficient to achieve the ideal performance corresponding to a hypothetical ideal DPMZM. By contrast, the system without pre-distortion suffers from a significant OSNR penalty.

(49) FIG. 9 illustrates related simulation results for inner ERs as low as 10 dB. In order to avoid clipping in the implementation of the non-linear functions, the power of the modulated signal was backed-off by 1 dB, i.e. the swing of the modulated signal was reduced by 1 dB as compared to the previous two examples. In this case, three iterations of the DPD algorithm were necessary to achieve the ideal performance, whereas the system without DPD exhibits more than 3 dB penalty at a BER of 10.sup.−3.

(50) The iterative solution according to Eq. (11) of the system of coupled non-linear equations (10) turns out to be very attractive. One advantage is the low computation cost, which is both due to the underlying mathematical structure and the fact that in practice very few iterations are necessary. Further, the memory needed for look-up tables (if employed) for the non-linear functions is very moderate. However, the present invention is not limited to this particular algorithm, and the present invention instead also considers calculation units employing other algorithms. An alternative very useful calculation is derived from a 2-stage implementation of the iteration algorithm of Eq. (11). From FIG. 2, and noting that cos(a sin(x))=√{square root over (1−x.sup.2)}, the second iteration of the driving voltages can be obtained as follows:

(51) $\begin{array}{cc}{V}_I^{\left(2\right)}=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_I-{\gamma }_Q.Math.\sqrt{1-(y_Q+{\gamma }_I.Math.{\left(I_{\mathrm{init}}\right)}^2}\right)V_Q^{\left(2\right)}=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_Q+{\gamma }_I.Math.\sqrt{1-{\left(y_I-{\gamma }_Q.Math.Q_{\mathrm{init}}\right)}^2}\right).& \left(17\right)\end{array}$

(52) In the above expressions, the square roots can be approximated by a Taylor expansion. If the square roots are expanded to second order terms in the components y.sub.I, y.sub.Q of the desired transmit signal, the following alternative solution can be obtained:

(53) $\begin{array}{cc}{V}_I=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_I+a_I.Math.y_Q^2+b_I.Math.y_Q+c_I\right)V_Q=\frac{2.Math.V_{\pi }}{\pi }a\sin \left(y_Q+a_Q.Math.y_I^2+b_Q.Math.y_I+c_Q\right).& \left(18\right)\end{array}$
where the coefficients a.sub.I, b.sub.I, c.sub.I, a.sub.Q, b.sub.Q, c.sub.Q depend implicitly on γ.sub.I and γ.sub.Q. This expression suggests an alternative implementation that requires only two evaluations of a single non-linear function besides the computation of the multivariate polynomials in y.sub.I and y.sub.Q. Again, the non-linear functions can be conveniently approximated by spline, i.e. piecewise linear interpolation. In this implementation, only two look-up tables for the function a sin(x), i.e. one for the I- and one for the Q-component, are necessary.

(54) In practice, the effectiveness of the pre-distortion requires an accurate characterization of the imperfect DPMZM. In particular, the recursive implementation according to Eq. (11) depends directly on γ.sub.I and γ.sub.Q which need to be precisely determined. Also, the polynomial implementation according to Eq. (18) depends on six real coefficients, which themselves are implicitly dependent on γ.sub.I and γ.sub.Q and can be individually tuned. In both cases, the parameters of the DPD can be set during factory calibration. In addition or alternatively, they can be continuously adapted at run time.

(55) FIG. 10 is an example of a DPMZM device according to a preferred embodiment of the present invention. The DPMZM device comprises, in addition to the DPMZM 10 itself, a calculation unit 52 receiving desired in-phase and quadrature components y.sub.I, y.sub.Q for the transmit signal to be generated. The calculation unit 52 is designated as “MZM.sup.−1” in the figure, because effectively, it represents the inverse operation of the DPMZM 10 to the extent that the underlying model captures the true characteristics and deficiencies of the DPMZM 10, and to the extent that the algorithm provides an exact solution of the set of coupled equations, depending on the number of iterations and the like.

(56) The calculation unit 52 outputs the first and second driving voltages V.sub.I, V.sub.Q, which are converted to analog signals by a DAC 32 and are applied to the first and second inner MZMs 20, 22 (not shown in FIG. 10) of the DPMZM 10. Again, the DC-components of the first and second driving voltages V.sub.I, V.sub.Q are lost due to AC-coupling (not shown in FIG. 10) which is compensated by adding corresponding bias components using adders 42 in a manner described with reference to FIGS. 5 and 6. Note that for simplicity the bias control units are not shown in FIGS. 10 to 12.

(57) Further shown in FIG. 10 is a far-end receiver 48 receiving the optical output signal 14 transmitted by the DPMZM 10. The far-end receiver 48 returns, through a feedback channel 54, a quality indicator, in the present example an estimated BER, to a parameter calculation unit 56. The parameter calculation unit 56 adjusts the parameters of the model employed by the calculation unit 52 such as to minimize the estimated BER. The parameters calculated by the parameter calculating unit 52 can for example be the values γ.sub.I and γ.sub.Q characterizing the finite ERs of the first and second inner MZMs 20, 22, or the coefficients a.sub.I, b.sub.I, c.sub.I and a.sub.Q, b.sub.Q, c.sub.Q in the polynomial approximation according to Eq. (18), which is why in the embodiment shown in FIG. 10, the parameter calculation unit 52 is referred to as “coefficient calculation”. Note, however, that the model parameters employed by the models of the invention and determined by the parameter calculation unit 52 generally do not need to be coefficients but could also be other types of parameters.

(58) FIG. 11 shows an alternative DPMZM device allowing for adapting the parameters or coefficients by means of what is referred to as an “indirect learning architecture” introduced by C. Eun and E. J. Powers in “A new Volterra predistorter based on the indirect learning architecture”, IEEE Transactions on Signal Processing, pp. 223-227, January 1997.

(59) In FIG. 11, the calculation unit receives frequency dependent target values Y.sub.I.sup.tgt(f), (short noted as “e(f)”) and in response calculates corresponding driving voltages referred to as “z(f)” for short in FIG. 11.

(60) A local coherent monitoring receiver 58 receives part of the output signal 14 and provides hence the actual in-phase and quadrature components Y.sub.I.sup.act(f) and Y.sub.Q.sup.act(f). The better the pre-distortion, or, in other words, the closer the model employed by the calculating unit 52 is to the true MZM 10, the more similar Y.sub.I.sup.act(f) Y.sub.Q.sup.act(f) should be to the target values Y.sub.I.sup.tgt(f), Y.sub.Q.sup.tgt(f), respectively.

(61) In FIG. 11, Y.sub.I.sup.tgt (Y.sub.Q.sup.tgt) and Y.sub.I.sup.act (Y.sub.Q.sup.act) are not directly compared. Instead, in the indirect learning architecture of FIG. 11, Y.sub.I.sup.act and Y.sub.Q.sup.act are applied to a copy of the calculating unit 52 designated at 60. The output of this copy 60 are hence driving voltages referred to as “z′(f)” which, if the models underlying the calculation unit 52 and its copy 60 were precisely reflecting the true MZM 10, should be identical with the applied driving voltages z(f). At a subtracting unit 62, the difference between z(f) and z′(f) is inputted as a frequency dependent error signal into a parameter calculating unit 64 which adapts the parameter (coefficients) such as to minimize the error.

(62) Note that the parameter calculation unit 64 allows for introducing artificial offsets y.sub.I.sup.off, y.sub.Q.sup.off in y.sub.I, y.sub.Q using a further subtractor 62, meaning that the locations of the QAM states are purposefully shifted in the two-dimensional plane. This can be desirable to better exploit the capabilities of the actual DPMZM.

(63) FIG. 12 indicates yet an alternative DPMZM device which is similar to that of FIG. 11. The main difference is that instead of a copy of the calculation unit 52, which essentially reflects an inverse model of the DPMZM 10, this version employs an “inverse” calculation unit 66 which performs the inverse calculation of the calculation unit 52 and can hence be regarded as the direct model of the DPMZM 10. Note in this regard that of course the calculation units 52, 60 and 66 all are based on the same model of the true DPMZM but differ by the corresponding algorithm receiving y.sub.I, y.sub.Q as inputs and yielding V.sub.I and V.sub.Q as outputs or vice versa.

(64) The inverse calculation unit 66 receives the driving voltages (referred to as “z(f)” in FIG. 12) and calculates target signals based thereon, from which actual signals as picked up by the coherent monitoring receiver 58 are subtracted by the subtracting unit 62.

(65) The parameter calculation or coefficient adaptation as employed in FIGS. 11 and 12 can be based on a minimization of the mean square error (MSE) between the desired and actual transmit signals. The MSE can be expressed either in the frequency or in the time domain. A general frequency formulation is
MSE.sub.f=∫w(f)E[(Y.sub.I.sup.act(f)−Y.sub.I.sup.tgt(f)).sup.2+(Y.sub.Q.sup.act(f)−Y.sub.Q.sup.tgt(f)).sup.2]df,  (19)
where w(f) is the desired weighting function, E[⋅] denotes stochastic expectation, Y.sub.I.sup.tgt(f)+j.Math.Y.sub.Q.sup.tgt(f) is the desired transmit signal at frequency f and Y.sub.I.sup.act(f)+j.Math.Y.sub.Q.sup.act(f) is the actual transmit signal at frequency f, as captured by the monitor receiver 58. A possible formulation in the time domain is
MSE.sub.t=E[y.sub.I.sup.act−y.sub.I.sup.tgt).sup.2+(y.sub.Q.sup.act−y.sub.Q.sup.tgt).sup.2].  (20)

(66) To provide the DPD with additional degrees of freedom, it may be advantageous allowing a DC offset on the transmit constellation. In this case

(67) $\begin{array}{cc}{\mathrm{MSE}}_{\mathrm{off}}=E\left[{\left(y_I^{\mathrm{act}}-y_I^{\mathrm{tgt}}-y_I^{\mathrm{off}}\right)}^2+{\left(y_Q^{\mathrm{act}}-y_Q^{\mathrm{tgt}}-y_Q^{\mathrm{off}}\right)}^2\right]+\lambda \frac{{\left(y_I^{\mathrm{off}}\right)}^2+{\left(y_Q^{\mathrm{off}}\right)}^2}{{\left(y_I^{\mathrm{tgt}}\right)}^2+{\left(y_Q^{\mathrm{tgt}}\right)}^2},& \left(21\right)\end{array}$

(68) Note that in FIGS. 11 and 12, the calculation units 52 are referred to as “MZM.sup.−1”, because essentially the calculation unit establishes a model of the actual, imperfect DPMZM 10 but carries out a calculation that inverts its operation. From a more general viewpoint, a DPMZM 10 can be modelled as a memory-free non-linear system with complex input and complex output. To pre-compensate the DPMZM 10, according to the invention one generally synthesizes the inverse system and then inserts it between the source of the baseband modulating signal and the DPMZM. The desired base-band signals are provided to the inverse system, which then delivers the corresponding pre-distorted complex input to the DPMZM. This generic setup is illustrated in FIG. 13. If the inverse system is modelled with sufficient accuracy, the DPMZM 10 returns a close approximation of the desired signal as a response to the pre-distorted input.

(69) Generally, a memory-free non-linear system can be represented by a look-up table. Accordingly, one could think of synthesizing the inverse DPMZM simply by a look-up table with complex input and complex output. However, such a more straight-forward approach has significant implementation drawbacks. If real and imaginary parts of the complex signals are represented with n bits, respectively, the required look-up table would have 2n input bits and 2n output bits and would require a local storage of (2n)2.sup.2n bits. This can quickly become a prohibitive size, especially if circuit parallelization is also taken into account, as required to support optical data rates on integrated digital circuits. Instead, employing a model as described above and solving a corresponding set of coupled non-linear equations iteratively or in some other approximated way is a much more efficient way of handling this problem.

(70) The embodiments described above and the accompanying figures merely serve to illustrate the method according to the present invention, and should not be taken to indicate any limitation of the method. The scope of the patent is solely determined by the following claims.

LIST OF REFERENCE SIGNS

(71) 10 dual parallel Mach-Zehnder-modulator (DPMZM) 12 optical input 14 optical output 16 first arm 18 second arm 20 first inner MZM 22 second inner MZM 24 set of electrodes 26 set of electrodes 28 set of electrodes 30 input 32 DA-converter 34 photo detector 36 bias error computer 38 error signal 40 error signal 42 adder 44 bias voltage computer 46 bias 47 bias 48 far-end receiver 50 residual carrier detector 51 error signal 52 calculation unit 54 feedback channel 56 parameter calculation unit 58 local coherent monitoring receiver 60 copy of the calculating unit 52 62 subtracting unit 64 parameter calculating unit 66 inverse calculation unit

(72) TABLE-US-00001 LIST OF ABBREVIATIONS AC Alternating Current ADC Analog-to-Digital Converter BER Bit Error Rate CMOS Complementary Metal-Oxide-Semiconductor DAC Digital-to-Analog Converter DC Direct Current DPD Digital Pre-Distorsion DPMZM Dual Parallel Mach-Zehnder Modulator ER Extinction Ratio MSE Mean Square Error I In-phase component MZM Mach-Zehnder Modulator OFDM Orthogonal Frequency Division Multiplexing OSNR Optical Signal-to-Noise Ratio Q Quadrature component QAM Quadrature Amplitude Modulation