Patent Publication Number: US-2021184775-A1

Title: A method and system for determining a model representation for an optical transmitter

Description:
FIELD OF THE INVENTION 
     The present disclosure relates to transmitters for optical data transmission, in particular to adaptive pre-distortion for optical transmitters. 
     BACKGROUND 
     The main drivers for the evolution of optical transmission systems have constantly been the need to improve the capacity of the telecommunication infrastructure and, at the same time, the urge to reduce the cost per bit. In terms of optical transponder technology, these trends currently translate into the adoption of high-order modulation formats with high-spectral efficiency, increase of the signaling rate and the replacement of traditional discrete optical components by integrated components. 
     Modern transponders make use of phase modulation (for example binary phase-shift keying (BPSK) or quadrature phase-shift keying (QPSK)) or a combination of phase and amplitude modulation (for example quadrature amplitude modulation (QAM)). Commonly, the resulting optical signals are not described by their amplitude and their phase, but rather by two orthogonal components, namely the in-phase component and the quadrature component. However, both representations are equivalent and the representation in one system of coordinates can be transferred into a representation in the other system of coordinates. Additionally, two orthogonal polarizations of the light are used to convey two independent QAM (or phase-modulated) signals or, alternatively, the two-dimensional projections of a fourdimensional signal. 
     For instance, the next generation of optical transponders is expected to support 64-point dual polarization quadrature amplitude modulation (64 QAM) at roughly 64 Gigasymbols per second, employing integrated transmitters and receivers realized, e.g., in silicon photonics technology. 
     The combined use of high signaling rates, high-order modulation formats and integrated photonics makes it a serious challenge to achieve the required transmission quality. On the one hand, high signaling rates may result in bandwidth limitations in the optical and electronic components. On the other hand, high-order modulation formats are very sensitive to any type of linear or nonlinear distortion. In conjunction with the trend towards integrated photonic solutions, the situation becomes even worse due to the fact that integrated optical devices have not yet achieved the performance of their discrete counterparts. 
     State-of-the-art adaptive receivers can effectively compensate transmission impairments, but they can be overwhelmed if the quality of the transmit signal is excessively degraded. It is therefore important to implement high-fidelity transmitters. 
     Device manufacturers have constantly tried to limit the imperfections of the electronic and photonic components of optical transmitters. This is certainly a very reasonable aspiration, but the approach is subject to a price/quality trade-off and can result in unnecessarily expensive components. 
     Static digital pre-distortion based on factory calibration has been proposed by A. Napoli et al., “Novel DAC Digital Pre-Emphasis Algorithm for Next-Generation Flexible Optical Transponders”, published in the Proceedings of the 2015 Optical Fiber Communication Conference (OFC 2015), Los Angeles, Calif., Mar. 22-Mar. 26, 2015. However, calibration is time-consuming and thus impacts heavily the production costs. Further, calibration is unsuitable for systems assembled during installation via pluggable components. 
     In their research paper “Low-Cost Transmitter Self-Calibration of Time Delay and Frequency Response for High Band-Rate QAM Transceivers”, Proceedings of the 2017 Optical Fiber Communication Conference (OFC 2017), Los Angeles, Calif., Mar. 19-Mar. 23, 2017, C. R. Fludger et al. describe a transmitter calibration method using swept frequency tones and a single feedback photodiode. However, this method estimates only transmitter frequency response and skew, and requires the transmission of special signals for calibration purposes. 
     G. Khanna et al. in their paper “A Robust Adaptive Pre-Distortion Method for Optical Communication Transmitters”, IEEE Photonics Technology Letters, Vol. 28, No. 7, pages 752-755 (April 2015) describe an adaptive pre-distortion based on an auxiliary coherent receiver. This solution avoids the need for factory calibration, but requires an expensive and relatively bulky auxiliary receiver, which may impact the transponder cost and size. 
     US 2011/0312290 A1 describes methods and systems for closed-loop pre-distortion using feedback from a distant end of the link. A receiver reconstructs the transmit signal from the received signal and the detected data, respectively. This reconstructed signal is compared with the received signal, and the difference is calculated. In a further step, metrics are derived from this difference signal allowing to calculate the difference signal by using a model. These metrics are communicated to the transmitter at the distant end, which computes an inverse compensation from the metrics received from the distant end. 
     In view of these problems, what is needed is a reliable and cost-efficient technique to improve the transmission quality of an optical transmitter. 
     Overview 
     This objective is achieved with a method and system for determining a model representation for an optical transmitter according to independent claims  1  and  13 , respectively. The dependent claims relate to preferred embodiments. 
     A method for determining a model representation for an optical transmitter according to the disclosure comprises: determining a quantity of a subsequence of output signals of an optical transmitter, the output signals being emitted in response to a sequence of input signals provided to the optical transmitter; and determining a model representation for the optical transmitter based on the determined quantity, wherein the quantity only represents a part of the information encoded in the subsequence of the output signals. 
     The model representation for the optical transmitter may be employed to detect and understand transmission errors of the optical transmitter that may lead to signal distortions. In particular, based on knowledge of the model representation for the optical transmitter, the optical transmitter may be corrected to improve the transmission quality. 
     The quantity of the subsequence of output signals may be any property or characteristic of the subsequence of output signals that is susceptible to physical measurement or detection. 
     However, the quantity may not provide the complete information encoded in the subsequence of the output signals. In particular, the quantity may not allow in itself to decode the output signals. For instance, the quantity may not allow to unambiguously determine the sequence of input signals provided to the optical transmitter. This may require the analysis of further parameters of the subsequence of output signals. 
     The inventors found that the determined quantity, even if it only represents a part of the information encoded in the subsequence of the output signals, may nevertheless allow to reliably determine a model representation for the optical transmitter. Hence, the model representation may be determined without the use of sophisticated equipment. 
     According to an embodiment, the transmitter encodes information in at least one polarization either in phase or in a combination of amplitude and phase. 
     According to an embodiment, the quantity pertains to a power or an intensity of the subsequence of output signals. 
     Determining the model representation for the optical transmitter based on the power or intensity determined from the subsequence of output signals allows for a reliable assessment and representation of the optical transmitter. At the same time, determining the power or intensity of the transmitter output does not require sophisticated equipment. It is a realization of the inventors that the model representation may be determined solely from the determined power or intensity, without relying on a coherent receiver, and may still provide a reliable model representation for the optical transmitter. 
     Intensity, in the sense of the present disclosure, may refer to a power transferred by unit area. The area may be governed by a geometry of an optical fiber, or by a geometry of a detector employed for determining the quantity. The area may be constant, or approximately constant. In this sense, the power and the intensity may be considered equivalent quantities in the context of the present disclosure. 
     The term power, in the sense of the present disclosure, may further comprise quantities equivalent to a power or intensity of the subsequence of output signals. For instance, the power of an optical signal is directly related to an amplitude of its electrical field or magnetic field. Hence, determining the amplitude can be considered determining the power in the sense of the present disclosure, and vice-versa. 
     In an embodiment, the only physical parameter of the subsequence of output signals used for determining the model representation is the determined power or intensity, respectively. 
     In an example, determining the quantity may comprise detecting the quantity of the subsequence of output signals. 
     The quantity, such as the power or intensity may be determined by means of an intensity photo-detector. 
     According to an embodiment, the model representation may be determined based on the determined quantity, and may additionally be based on a model of an intensity photo-detector and/or a feedback receiver. 
     According to an example, the model representation may not be based on determining polarization information and/or phase information of the output signals. 
     In some examples, the subsequence of output signals may comprise all optical signals emitted by the optical transmitter in response to the sequence of input signals. In these instances, the subsequence can be considered an improper subsequence, and may be referred to as the sequence of output signals. 
     However, it is an advantage of the techniques of the present invention that the model representation for the optical transmitter may also be based on the intensity determined only for a (small) portion of the optical transmit signals, or only for a partial signal such as a signal pertaining only to one of a plurality of polarization states. In particular, the techniques of the present invention may not require a full alias-free spectral representation of the transmission intensity, but only sparse samples which can be conveniently acquired by means of a low-speed digitizer. 
     Hence, in an example, the subsequence comprises only a subset of optical signals emitted by the optical transmitter in response to the input signals. 
     In these instances, the subsequence can be considered a proper subsequence, in the sense that it corresponds to a subset or partial set of optical signals emitted in response to the sequence of input signals. 
     In general terms, the subsequence may represent a time-limited subsequence of the output signals emitted by the optical transmitter, and/or may correspond to a single or a plurality of tributaries of the optical signals. 
     In an embodiment, determining the model representation comprises an optimization, in particular a least squares optimization. 
     In an embodiment the model representation may model a skew and/or a delay of the input signals as they are passed through the optical transmitter. 
     According to an example, determining the model representation may comprise minimizing an output power of the optical signals. 
     Minimizing the output power of the optical signals requires only relatively modest computational and memory resources, and can still yield a reliable model representation for an optical transmitter, in particular an optical transmitter causing a skew and/or a delay of the input signals. 
     An optical transmitter, in the sense of the present disclosure, may refer to any optical transmission unit in an optical network, such as a transmission unit that converts an input data stream into an optical signal, or a sequence of optical signals. It may be modeled as an optical transmission channel. The output signals of the optical transmitter may be signals transmitted via an optical transmission line, such as through an optical fiber cable. 
     In an example, the output signals are optical analog signals. 
     The input signals, in the sense of the present disclosure, may refer to any kind of input signals of the optical transmitter that trigger the optical transmitter to output the output signals in response to the input signals. 
     In an embodiment, the input signals may be digital data signals or electrical analog signals. 
     In particular, the input signals may be digital data signals that correspond to or represent output signals that a user may want to generate and transmit. In this instance, the optical transmitter should ideally correspond to an ideal digital-to-analog converter channel. However, due to imperfections, the optical transmitter may divert from this ideal, and may introduce signal distortions that may affect the quality of the output signals. The model representation for the optical transmitter may model these imperfections. 
     In an embodiment, determining a model representation may comprise providing an ansatz model representation comprising at least one parameter, and determining the at least one parameter based on the determined quantity. 
     In general, the ansatz may comprise a plurality of parameters, and determining the model representation may comprise determining the plurality of parameters. 
     The inventors found that providing the ansatz allows for a computationally efficient determination of the model representation that allows to reliably capture a large variety of possible signal distortions. 
     In an embodiment, the model representation or ansatz comprises a linear combination of a set of base functions. 
     For instance, the base functions may comprise a set of single-component base functions, wherein each of the single-component base functions has a non-zero projection only on one of a plurality of baseband components of the input signals. 
     The base functions may alternatively or additionally comprise a set of delay functions, wherein the delay functions model linear filters for a plurality of baseband components of the input signals. 
     In an example, the base functions are polynomial functions, and may correspond to a Volterra model for the optical transmitter. 
     In an example, the method may further comprise providing the sequence of input signals to the optical transmitter. 
     The method may also comprise receiving the corresponding subsequence of output signals from the optical transmitter. 
     According to an embodiment, the method may comprise determining a pre-distortion map from the model representation. 
     The pre-distortion map may be employed to correct distortions of the optical transmitter so to enhance the signal quality of the output signal. 
     The pre-distortion map may correspond to a channel upstream of the optical transmitter. 
     In an example, the pre-distortion map is based on an inverse of the model representation. In particular, the pre-distortion map may correspond to an inverse of the model representation or may approximate an inverse of the model representation. 
     According to an embodiment, the method may further comprise submitting the input signals to the pre-distortion map prior to the optical transmitter; and/or submitting output signals of the pre-distortion map to the optical transmitter. 
     According to an embodiment, the method comprises adjusting the model representation and/or the pre-distortion map based on output signals of the optical transmitter. 
     This may allow to adapt the model representation and/or the pre-distortion map during operation of the optical transmitter, such as to accommodate temperature effects or aging of the optical transmitter. 
     In an example, the adjusting comprises iteratively adjusting the model representation and/or the pre-distortion map based on output signals of the optical transmitter. 
     In particular, the model representation and/or the pre-distortion map for a second sequence of input signals for the optical transmitter may be adjusted based on a determined quantity of a subsequence of output signals of the optical transmitter, the output signals corresponding to a first sequence of input signals, the first sequence of input signals preceding the second sequence of input signals. 
     According to an example, the adjusting may comprise comparing output signals of the pre-distortion map with output signals of the optical transmitter. 
     This may provide for an efficient feedback correction mechanism for the optical transmitter. The output signals of the pre-distortion map may undergo a signal processing prior to comparison with the output signals of the optical transmitter. In particular, signal processing may comprise submitting the output signals to a map corresponding to the model representation for the optical transmitter and/or to a map corresponding to an intensity photo-detector and/or a feedback receiver receiving the output signals of the optical transmitter. 
     According to an embodiment, the sequence of input signals may comprise signals of a plurality of optical polarization states. 
     The polarization states may be polarization planes or polarization directions, or may reflect any other property of the polarization. 
     Determining a model representation for the optical transmitter may comprise determining a first model representation corresponding to a first polarization state, and determining a second model representation corresponding to a second polarization state different from the first polarization state. 
     The techniques according to the embodiment may allow for a particularly efficient model generation in configurations in which the optical transmitter can be regarded or can be approximated as the combination of independent component transmitters for different polarization states. 
     In an example, the second model representation may be different from the first model representation. 
     In another example, the second model representation may be equal, or approximately equal to the first model representation. 
     In the latter example, the optical transmitter may transmit simultaneously signals of the first polarization state and the second polarization state, and a feedback receiver may alternatively sample the two polarization states. 
     The optical transmitter may be conveniently modeled in terms of a single polarization transmitter that may be used alternatively to transmit signals of the first and second polarization directions, such as by alternating between signals of the first and second polarization planes. The disclosure further relates to a computer program or to a computer program product comprising computer-readable instructions that are adapted to implement, when processed on a computer, a method with some or all of the features described above. 
     The disclosure further relates to a system for determining a model representation for an optical transmitter, comprising: a determination unit adapted to determine a quantity of a subsequence of output signals of an optical transmitter, the output signals being emitted in response to a sequence of input signals provided to the optical transmitter; and an analysis unit adapted to determine a model representation for the optical transmitter from the determined quantity; wherein the quantity only represents a part of the information encoded in the subsequence of the output signals. 
     In some examples, the determination unit and the analysis unit may be separate units. 
     In other examples, the determination unit and the analysis unit may be implemented as a common unit. 
     According to an embodiment the determination unit may comprise a detection unit adapted to detect the quantity of the subsequence of output signals, and may in particular comprise an intensity photo-detector. 
     The system may further comprise an input unit adapted to provide the sequence of input signals to the optical transmitter. 
     The system may further comprise a receiving unit adapted to receive the corresponding sequence of output signals from the optical transmitter. 
     The receiving unit may be further adapted to provide the sequence of output signals to the determination unit. 
     According to an embodiment, the analysis unit may be adapted to determine a pre-distortion map from the model representation. 
     The system may further comprise a pre-distortion unit upstream of the optical transmitter, wherein the pre-distortion unit implements the pre-distortion map. 
     Hence, the pre-distortion unit may be a signal processing unit or channel that is a realization of the pre-distortion map, or an approximation of the pre-distortion map. 
     The pre-distortion unit may be a data processing unit, in particular a data processing unit processing digital signals. 
     The pre-distortion unit may be a separate unit, or may be incorporated into the optical transmitter. 
     In an example, the analysis unit is adapted to adjust the model representation and/or the pre-distortion unit based on output signals of the optical transmitter. 
     According to an example, the analysis unit may be communicatively coupled to the pre-distortion unit. 
     According to an embodiment, the analysis unit may be adapted to compare output signals of the pre-distortion unit with output signals of the optical transmitter. 
     In an example, the system for determining the model representation may further comprise a model unit for the optical transmitter adapted to receive a subsequence of the output signals of the pre-distortion unit and adapted to submit the subsequence or a subsequence derived therefrom to a model of the optical transmitter. 
     According to an embodiment, the system for determining the model representation may further comprise a model unit for modeling the determination unit. 
     For instance, the model unit for the determination unit may receive output signals from the model unit for the optical transmitter, and may be adapted to submit these signals—preferably after processing them—to a model for the determination unit. 
     Output signals from the model unit for the determination unit may be provided to the analysis unit. 
     The analysis unit may determine the model representation for the optical transmitter based on the output from the model unit for the optical transmitter and/or the output from the model unit for the determination unit, such as by means of a comparison with output signals from said determination unit. 
    
    
     
       DETAILED DESCRIPTION 
       The features and numerous advantages of the method and system according to the invention may be best understood from a detailed description of specific embodiments with reference to the accompanying drawings, in which: 
         FIG. 1  is a schematic illustration of a system for determining a model representation for an optical transmitter according to an embodiment; 
         FIG. 2  is a flow diagram illustrating a method for determining a model representation for an optical transmitter according to an embodiment; 
         FIG. 3  is a schematic illustration of a system for determining a model representation for an optical transmitter comprising a pre-distortion unit according to an embodiment; 
         FIG. 4  is a schematic illustration of a system for determining a model representation for an optical transmitter comprising per-polarization, pre-distortion units according to an embodiment; 
         FIG. 5  is a schematic illustration of a system for adaptive pre-distortion according to an embodiment; 
         FIG. 6  is a schematic illustration of a system for adaptive pre-distortion comprising per-polarization processing according to an embodiment; 
         FIG. 7  is a schematic illustration of a system for delay identification based on correlation and corresponding adaptive pre-distortion according to an embodiment; 
         FIG. 8  is a schematic constellation diagram illustrating an optimization technique according to an embodiment; 
         FIGS. 9 a  to 9 e    represent constellation diagrams in the presence of skew to illustrate an optimization technique according to an embodiment; 
         FIG. 10  is a schematic diagram illustrating the detection of minimum instantaneous power in an optimization technique according to an embodiment; 
         FIG. 11 a , 11 b    show a probability density function and cumulative probability distribution of the amplitude values for different skew values and various noise levels according to an embodiment; 
         FIG. 12  illustrates the implementation of an adaptive digital pre-distortion technique in a pluggable module according to an embodiment; and 
         FIGS. 13 a , 13 b    are a schematic illustration of the details of a pluggable module as illustrated in  FIG. 11  according to an embodiment. 
     
    
    
     Examples of a method and system for determining a model representation for an optical transmitter will now be described with reference to optical transmitters as they may be employed in an optical communication network. These optical transmitters may convert a sequence of data inputs, such as digital data inputs, into a sequence of optical signals, and may feed the optical signals into an optical fiber channel. Although reference is made to a plurality of output signals or optical signals, the invention is of course also applicable to configurations with a single output signal. 
     The techniques of the present disclosure apply generally to any optical transmitter providing a sequence of optical signals. 
     In the following embodiments, sequences of power samples are determined that capture at least partially the evolution of the optical power over time. In order to distinguish the measured power levels from the average power of the optical signal, we will denote the sampled power values as instantaneous power. However, it is not required to capture all features of the optical power. For example, suppressing some frequencies by low pass filtering does not per se restrict the applicability of the technique. Such filtering is not necessarily intended—it might just be caused by the limited bandwidth of the used electrical filter. 
       FIG. 1  is a schematic illustration of an optical transmission system  1   o  in the context of which the techniques according to the present invention may be employed. The system  10  comprises an optical transmitter  12  receiving a sequence of electrical analog or digital input signals  14  and converting them into a sequence of optical output signals  16 . For instance, the input signals  14  may be digital data signals, and the optical output signals  16  may be analog optical signals generated by the optical transmitter  12  in response to the input signals  14  and may be fed into an optical network (not shown). 
     The optical transmitter  12  may comprise data processing equipment adapted to receive and process the input signals  14 , as well as lasers, optical filters or other optical equipment to generate the optical output signals  16  in response to receiving the input signals  14 . Mathematically, the optical transmitter  12  may be modeled as a channel. 
     The optical transmission system  10  further comprises a system  18  for determining a model representation for the optical transmitter  12 . The system  18  comprises a determination unit  20  adapted to receive a test sequence  22  of optical output signals tapped from the optical output signals  16 . The test sequence  22  may correspond to a subsequence of the output signals  16  generated by the optical transmitter  12  in response to receiving a sequence of input signals  14 . 
     The determination unit  20  determines an intensity of the output signals in the test sequence  22 . In particular, the determination unit  20  may comprise a detection unit (not shown in  FIG. 1 ) adapted to detect the intensity of the optical signals in the test sequence  22 , such as an intensity photo-detector. 
     The system  18  further comprises an analysis unit  24  communicatively coupled to the determination unit  20 . The analysis unit  24  is configured to determine a model representation for the optical transmitter  12  based on the intensity detected by the determination unit  20 . The model representation may correspond to a description or mathematical representation of the functionality of the optical transmitter  12 . 
     A method according to an example for determining a model representation for an optical transmitter is schematically illustrated in the flow diagram of  FIG. 2 . 
     In a first step S 10 , an intensity of a subsequence of output signals of an optical transmitter, such as the optical transmitter  12  is determined, wherein the output signals are being emitted in response to a sequence of input signals provided to the optical transmitter. 
     In a subsequent step S 12 , a model representation for the optical transmitter is determined based on the determined intensity. 
     The determined model representation may be employed to correct signal distortions that may result from inaccuracies or malfunctions of the optical transmitter  12 , as will now be described in greater detail with reference to  FIGS. 3 to 13 . 
     For instance, the optical transmitter  12  may be designed to convert digital data input signals  14  to a sequence of optical output signals  16  that are a faithful representation of the digital input signals  14 . However, due to imperfections or detrimental environmental influences, the optical transmitter  12  may introduce signal distortions such as differential signal delays or noise and hence may divert from the ideal channel. The techniques according to the present invention may be employed to correct for these imperfections and thereby enhance the signal quality of the output optical signals  16 . 
     A corresponding configuration is schematically illustrated in  FIG. 3 , which illustrates an optical transmission system  10 ′ and a system  18 ′ for determining a model representation that generally correspond to the optical transmission system  10  and system  18  described above with reference to  FIG. 1 , and the same reference numerals are used to designate corresponding elements. 
     However, as can be taken from  FIG. 3 , the system  10 ′ in addition comprises a pre-distortion unit  26  upstream of the optical transmitter  12 . The pre-distortion unit  26  is implemented based on the model representation determined by the analysis unit  24 , and may correspond to an inverse characteristics of the optical transmitter  12 , or may approximate an inverse of the transfer function of the optical transmitter  12 . Hence, when inserted into the data stream of the input signals  14 , the pre-distortion unit  26  may fully or partially compensate for signal distortions of the optical transmitter  12  so that the signal quality of the optical output signal  16  is enhanced. In other words, the concatenation of the pre-distortion unit  26  correcting distortions caused by the optical transmitter  12  and the optical transmitter  12  may represent or approximate an ideal optical transmitter. 
     The model representation may be adapted to account for parts of the modulator imperfections only. For example, it might be sufficient to take only the most dominant effects into account. In this case, the concatenation of the pre-distortion unit  26  correcting distortions caused by the optical transmitter  12  and the optical transmitter  12  represents a non-ideal optical transmitter providing better performance as compared with the optical transmitter  12  without any compensation of non-ideal characteristics. 
     The schematic illustration of  FIG. 3  shows the pre-distortion unit  26  and the optical transmitter  12  as separate units. However, this is a mere example, and in other embodiments the pre-distortion unit  26  may be integrated into the optical transmitter  12 . 
     In addition, the input signals may undergo usual data processing before entering the pre-distortion unit  26 , such as encoding, bit-to-symbol mapping, and pulse shaping. 
       FIG. 4  is a schematic illustration of the optical transmission system  10 ′ and system  18 ′ for determining a model representation as generally described above with reference to  FIG. 3 , but in a configuration in which the optical input signals are split up into in-phase (I) and quadrature (Q) components of a complex baseband signal for two orthogonal polarization directions X, Y. 
     As can be further taken from  FIG. 4 , the optical transmitter  12  may process the input signals  14   a ,  14   b  corresponding to the X and Y polarization per polarization with corresponding optical transmitter units  12   a ,  12   b  to generate corresponding optical output signals  16   a ,  16   b.    
     A combiner unit (not shown) may combine the optical output signals  16   a ,  16   b  corresponding to the two orthogonal polarization directions into a common output signal. 
     The determination unit  20  may comprise a selector unit  40  that may be represented by an optical switch that receives control signals via a control line  30  from the analysis unit  24  and taps optical output signals  16   a ,  16   b  of both polarization directions as instructed by the analysis unit  24 . 
     The determination unit  20  further comprises a feedback receiver  28  with an intensity photo-detector  32  that receives the tapped output signals from the selector unit  40  and measures an intensity of the subsequence of output signals  16   a ,  16   b.    
     Based on the results from the intensity photo-detector  32 , the analysis unit  24  determines a model representation for the optical transmitters  12   a ,  12   b . As described in detail further below, determining the model representation may involve ansatz and optimization techniques. 
     Based on the determined model representation, a pre-distortion unit  26  may then be implemented in the input path upstream of the optical transmitters  12   a ,  12   b , respectively. In the simplified scenario shown in  FIG. 4  in which the transmitter  12  factorizes into pre-polarization transmitter subunits  12   a ,  12   b , the pre-distortion unit may likewise be implemented as a pair of pre-distortion units  26   a ,  26   b  for the respective polarization directions X, Y. In particular, the pre-distortion units  26   a ,  26   b  may correspond to inverses of the corresponding optical transmitters  12   a ,  12   b , respectively, or approximate inverses of the optical transmitters  12   a ,  12   b . In this way, the undesired signal distortions caused by the optical transmitters  12   a ,  12   b  may be corrected polarization-wise in each of the two polarization branches, similarly to the configuration described above with reference to  FIG. 3 . 
     Before providing more details, a summary of possible implementations for polarization multiplexed signals is provided. Such polarization multiplexed signals (in short form polmux signals) simultaneously transmit independent data streams in both polarization planes. 
     In general form, a pre-distortion unit  26  may receive four tributary signals and may determine—by processing these four tributary signals—four pre-distorted output signals that are provided pairwise to the IQ modulators. The model representation used by the pre-distortion unit  26  can be determined from the power of the polarization multiplex signals, i.e. from the total output power. Therefore, part of the output signal is tapped off after combining the polarization component forming the final output signals for determining the total power. Alternatively, parts of the output signals may be branched off after each IQ modulator. In the following, these parts may be combined either optically or after opto-electronic conversion of both output signals electrically in order to determine the total intensity. 
     In case the model representation does not link output signals assigned to one polarization to input tributary signals of the other polarization, the pre-distortion unit  26  can be structured into two independently acting pre-distortion units  26   a  and  26   b , although the pre-distortion model representations used by each of these independently acting pre-distortion units  26   a  and  26   b  are determined from the total power of the polarization multiplex signal. 
     In a further embodiment, the model representations used by the independently acting pre-distortion units  26   a  and  26   b  are each determined from power levels measured for the corresponding polarization plane. Thus, the determination of the model representations and the processing of the tributary signals are completely independent from each other for the two polarization planes. This is for example a suitable approach for compensating for the skew between the in-phase component and the quadrature component of the same polarization. 
     Finally, it is also possible to determine the power for each polarization independently, i.e. in front of a polarization beam combiner, but by using a pre-distortion unit  26  acting on the tributaries of both polarization planes. 
     Steps related to a single polarization only can be performed in parallel by doubling the required equipment. Alternatively, the steps can also be performed sequentially by sharing the same devices and by providing signals from the two polarizations to the devices by means of electrical or optical switches in an alternating way. 
     In contrast to some of the prior-art solutions, the techniques of the present disclosure do not require complete knowledge of the output of the optical transmitter  12 , or even a segment thereof, which might require a coherent receiver. Instead, the techniques of the present disclosure may make use of solely the detected intensity of the transmitter output, which may not contain direct information about the polarization and phase modulation. Moreover, the required “observable” signal can be obtained by tapping only a small portion of the optical output signals  16 . Further, the techniques of the present disclosure do not require a full alias-free spectral representation of the transmit intensity, but only sparse samples, which can be conveniently acquired via a low-speed digitizer. 
     With reference to  FIGS. 3 and 4 , the digital input signals  14  to the optical transmitter  12  comprising the pair of optical transmitter units  12   a ,  12   b  may be represented by a time sequence of vectors 
     
       
         
           
             
               
                 
                   
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                                
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         
                           i 
                           = 
                           0 
                         
                         , 
                         1 
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           
                             N 
                             v 
                           
                           - 
                           1 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     gathers the samples of the real baseband signals transmitted at the i-th time instant over in-phase and quadrature components (I/Q) of the two orthogonal polarizations (X/Y). 
     The tapped samples collected by the feedback receiver  28  correspond to a subsequence of output signals  16  that may be represented as follows: 
       [ r [0] r [1] . . .  r [ N   r −1]].  (3)
 
     The transmitter  12  may be modeled in the equivalent baseband representation by the function f θ :    4×N     ν   →   4×N     ν   , which depends on the unknown parameter vector θ and represents an ansatz for the model representation. Some examples are discussed further below. The output of the transmitter model is represented by 
     
       
         
           
             
               
                 
                   
                     
                       w 
                       θ 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             w 
                             θ 
                           
                            
                           
                             [ 
                             0 
                             ] 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             w 
                             θ 
                           
                            
                           
                             [ 
                             1 
                             ] 
                           
                         
                          
                         
                             
                         
                          
                         ⋯ 
                          
                         
                             
                         
                          
                         
                           
                             w 
                             θ 
                           
                            
                           
                             [ 
                             
                               
                                 N 
                                 v 
                               
                               - 
                               1 
                             
                             ] 
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       w 
                       θ 
                     
                     = 
                     
                       
                         f 
                         θ 
                       
                        
                       
                         ( 
                         v 
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       w 
                       θ 
                     
                      
                     
                       [ 
                       i 
                       ] 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 w 
                                 
                                   XI 
                                   , 
                                   θ 
                                 
                               
                                
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               
                                 w 
                                 
                                   XQ 
                                   , 
                                   θ 
                                 
                               
                                
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               
                                 w 
                                 
                                   YI 
                                   , 
                                   θ 
                                 
                               
                                
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               
                                 w 
                                 
                                   YQ 
                                   , 
                                   θ 
                                 
                               
                                
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                         
                     
                      
                     
                       
                         ( 
                         
                           
                             i 
                             = 
                             0 
                           
                           , 
                           1 
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             
                               N 
                               v 
                             
                             - 
                             1 
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The identification of the model consists in finding the parameter vector θ that results in the “best” estimate of the transmitter characteristics. 
     As already mentioned, the observable is a (“lossy”) function of the transmit signal, obtained through the determination unit  20  comprising the feedback receiver  28  with the intensity photo-detector  32 . Therefore, in order to identify θ, we pass the output w of the transmitter model through the model of the determination unit  20  and we proceed by fitting the resulting signal to the observable r. We may model the determination unit  20  in the equivalent baseband representation by a function g:    4×N     ν   →   N     r   . Thus, the output of the receiver model 
         s   θ [ s   θ [0] s   θ [1] . . .  s   θ [ N   r −1]]  (7)
 
     is expressed as 
         s   θ   =g ( w   θ )= g ( f   θ (ν)).  (8)
 
     The optimal parameter vector in the least squares sense may be determined as follows: 
       {circumflex over (θ)}=arg min θ [ r−g ( f   θ (ν))] 2 .  (9)
 
     Literally speaking, the method selects the parameter vector {circumflex over (θ)} mapping the input time sequence of vectors ν to output values of the receiver model r in such a way that the tapped samples collected from the feedback receiver  28  are approximated in a least mean square sense. The receiver model comprises models of the transmitter and the determination unit. Of course, any known kind of approximation criterion can be used instead of the least man square criterion. 
     Once the model representation is identified, the digital pre-distortion (DPD) unit  26  may be configured to apply the inverse function f θ   −1  or an approximation thereof to the digital representation of the input signals. The result is then passed to the transmitter  12 . 
     The overall organization and identification of the model representation as well as the generation of the pre-distortion as described above with reference to Equations (1) to (9) is illustrated in additional detail in  FIG. 5 . The configuration of the optical transmission system  10 ′ and system  18 ′ for determining a model representation correspond to those described above with reference to  FIGS. 3 and 4 , and hence the same reference numerals are employed. 
     As described above with reference to Equations (4) to (6), the model representation for the optical transmitter  12  may be efficiently determined by means of an ansatz, wherein the ansatz comprises a parameter vector θ and the step of determining the model representation comprises optimizing for θ. In  FIG. 5 , the ansatz for the model representation is schematically illustrated with reference numeral  34 , and the model for the determination unit  20  with reference number  36 . The analysis unit  24  then compares the data signals provided by the determination unit  20 , and hence collected from the optical output signal  16 , with corresponding input signals  14  submitted to a concatenation of the ansatz  34  for the model representation and the model  36  for the determination unit  20  to determine the optimal parameter vector according to Equation (9). 
     As illustrated in  FIG. 5 , the model  36  for the determination unit  20  may comprise a memoryless sample power computer, a delay element and a re-sampler. 
     The sample power computer models the photo-detector, the delay element reproduces the latency of the feedback loop and the resampler implements the ratio between the sampling rates of the digital-to-analog converters (DACs) in the transmitter  12  and the analog-to-digital converter (ADC) in the determination unit  20 . In particular, if the DAC rate is an exact multiple of the ADC rate 
         f   ADC   =M·f   DAC ,  (10)
 
     the resampler is a simple decimator that keeps  1  sample out of M. If needed, more accurate models of the determination unit  20  can be employed, for instance to emulate bandwidth limitations or nonlinear distortions. 
     Using the model of the determination unit  20  shown in  FIG. 5  and assuming that condition (10) holds, Equation (9) becomes 
       {circumflex over (θ)}=arg min θ Σ i=0   N     r   [ r [ i ]−∥ w   θ [ M·i ]∥ 2 ] 2 ],  (11)
 
     where |⋅| denotes the Euclidean norm of a vector and where, for the sake of notation simplicity, we neglect the latency. 
     It is worth observing that the identification relies upon the samples at the output of the pre-distortion unit  26  and therefore is unaffected by the actual configuration of the pre-distortion unit. This enables the implementation of an iterative approach to track dynamic changes in the optical transmitter  12 , such as due to temperature fluctuations or aging, whereby an updated transmitter model is identified at each iteration and used to pre-distort the desired transmit signal in the next time window. 
     In the following we consider two special cases of practical importance:
         1. Delay identification:
           The vector θ represents the delay of the 4 real components of the transmit signal.   
           2. Linear combination:
           The function f 0  is linear in θ.   
               

     1. Delay Identification: 
     This case has two major applications: estimation of the latency of the feedback loop and of the skew among the transmit in-phase and quadrature components I and Q and/or skew among polarizations X and Y. 
     The estimation of the latency may be a preliminary step for the identification of any other parameter. Before identifying the rest of the transmitter model, the estimated latency may be implemented in the delay element of the feedback receiver model to synchronize r and se. 
     The skew, in particular between the in-phase and the quadrature components, is a critical impairment in any optical amplitude/phase modulation and its accurate compensation is often desired to enable the transmission of high-order modulation formats. 
     In both applications, since the summed Euclidean norms, i.e. the power, of we is, apart from negligible boundary effects, essentially independent of the delays θ, Equation (11) simplifies to 
       {circumflex over (θ)}=arg max θ Σ i=0   N     r     r [ i ]·∥ w   θ [ M·i ]∥ 2 .  (12)
 
     Expressed verbally, Equation (12) indicates that the transmitter model is identified by maximizing the inner product between the sequences of the feedback samples and of the samples obtained by applying the cascade of the transmitter model  34  and the determination unit model  36  to the output of the pre-distortion unit  26 . The identification can be implemented as an exhaustive search over a set of candidate parameter vectors. The evaluation of the sum in Equation (12) for each tentative θ can be efficiently performed by means of Fourier-transform techniques. 
     The vector parameter gathers the delays of the four baseband components: 
       θ=[τ XI τ XQ τ YI τ YQ ].
 
     The overall latency is defined as 
     
       
         
           
             
               
                 
                   
                     
                       τ 
                       
                         r 
                         , 
                         s 
                       
                     
                     = 
                     
                       
                         
                           τ 
                           XI 
                         
                         + 
                         
                           τ 
                           XQ 
                         
                         + 
                         
                           τ 
                           YI 
                         
                         + 
                         
                           τ 
                           YQ 
                         
                       
                       4 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     the IQ-skew for the two polarizations is 
       τ IQ,X =τ XI −τ XQ  
 
       τ IQ,Y =τ YI −τ YQ ,  (15)
 
     and the XY-skew is 
     
       
         
           
             
               
                 
                   
                     τ 
                     XY 
                   
                   = 
                   
                     
                       
                         
                           τ 
                           XI 
                         
                         + 
                         
                           τ 
                           XQ 
                         
                         - 
                         
                           τ 
                           YI 
                         
                         - 
                         
                           τ 
                           YQ 
                         
                       
                       2 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     The exhaustive search is performed over a reasonable interval, which depends on the physical properties of transmitter, digital-to-analog converter (DAC), feedback receiver and analog-to-digital converter (ADC), with a reasonable step-size, commensurate with the symbol rate of the modulated signal. 
     2. Linear Combination: 
     In many practical applications the transmitter can be modelled as the linear combination of a given set of base function f j :    4+N     ν   →   4+N     ν    (j=0, 1, . . . , L−1): 
         f   θ (ν)=Σ j=0   L−1 θ j   f   j (ν).  (17)
 
     In this case, the parameter vector consists of the coefficients of the linear combination: 
       θ=[θ 0 θ 1  . . . θ L−1 ].  (18)
 
     It is worth mentioning that the resulting model can be either linear or nonlinear in the argument ν, depending on the adopted base functions. Some notable and practically relevant choices include:
         Single-component base functions, each having non-zero projection on only one of the four baseband components,   Single-component delay functions, which model independent linear filters for the four baseband components,   Polynomial base functions, which result in a Volterra model.       

     Using the model of the feedback receiver shown in 5 and assuming Equation (17), Equation (9) becomes 
       {circumflex over (θ)}=arg min θ   CF   θ (ν),  (19)
 
     where we introduce the cost function 
         CF   θ (ν)=Σ i=0   N     r   [ r [ i ]−∥Σ j=0   L−1 θ j   f   j (ν)[ M·i ]∥ 2 ] 2 .  (20)
 
     The first derivatives of the cost function can be expressed in closed form as 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           CF 
                           θ 
                         
                          
                         
                           ( 
                           v 
                           ) 
                         
                       
                     
                     
                       ∂ 
                       
                         θ 
                         m 
                       
                     
                   
                   = 
                   
                     
                       - 
                       4 
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           N 
                           r 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           [ 
                           
                             
                               r 
                                
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                             - 
                             
                               
                                  
                                 
                                   
                                     ∑ 
                                     
                                       j 
                                       = 
                                       0 
                                     
                                     
                                       L 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       θ 
                                       j 
                                     
                                      
                                     
                                       
                                         
                                           f 
                                           j 
                                         
                                          
                                         
                                           ( 
                                           v 
                                           ) 
                                         
                                       
                                        
                                       
                                         [ 
                                         
                                           M 
                                           · 
                                           i 
                                         
                                         ] 
                                       
                                     
                                   
                                 
                                  
                               
                               2 
                             
                           
                           ] 
                         
                          
                         
                           
                             
                               f 
                               m 
                             
                              
                             
                               ( 
                               v 
                               ) 
                             
                           
                            
                           
                             [ 
                             
                               M 
                               · 
                               i 
                             
                             ] 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               0 
                             
                             
                               L 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               θ 
                               j 
                             
                              
                             
                               
                                 
                                   f 
                                   j 
                                 
                                  
                                 
                                   ( 
                                   v 
                                   ) 
                                 
                               
                                
                               
                                 [ 
                                 
                                   M 
                                   · 
                                   i 
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     and can be used to solve Equation (19) numerically using a gradient-based iterative approach. 
     As explained above with reference to  FIG. 4 , simplifications result in configurations in which no joint polarization processing of the input signals  14  is required. In these cases, the signal processing factors into independent components for the X and Y polarizations, and the optical transmitter  12  may be regarded as the combination of two independent component transmitter units  12   a ,  12   b  for the X and Y polarizations, respectively. 
     In configurations in which the optical transmitters  12   a ,  12   b  for the X and Y polarizations can be assumed to be identical, the ansatz  34  for the model representation may be chosen as a single polarization transmitter and may be used alternatively to identify the two single-polarization transmitters  12   a ,  12   b.    
     The corresponding configuration of an optical transmission system  10 ′ and system  18 ′ for determining a model representation is shown schematically in  FIG. 6 .  FIG. 6  generally corresponds to  FIG. 5 , and hence the same reference numerals are used to denote corresponding components. However, the polarization directions X and Y can be treated independently, and hence the optical transmitter  12  can be represented as a combination of otherwise identical optical transmitters  12   a ,  12   b  for the polarization directions X and Y, respectively. As can be further taken from  FIG. 6 , the optical transmitter  12  in addition comprises a combiner  38  to combine the optical output signals  16   a ,  16   b  of the optical transmitters  12   a ,  12   b  into a common output signal  16 . 
     As can be further taken from  FIG. 6 , the system  18 ′ for determining the model representation comprises an analog X/Y switch  40  to switch between the output polarization branches and alternatively provide the optical output signals  16   a ,  16   b  to the determination unit  20 . 
     The system  18 ′ further comprises a digital X/Y switch  42  that alternatively samples the input signals  14   a ,  14   b  of the X and Y polarizations, respectively. The ansatz  34 ′ for the model representation and model  36 ′ for the determination unit  20  generally correspond to the components  34 ,  36  described above with reference to  FIG. 5 , but are simplified in that they comprise a single polarization direction only. 
     The mathematical representation described above with reference to  FIG. 5  remains essentially unchanged except for the simplification that one of the two polarization directions is dropped. 
     The analysis unit  24  determines the optimal model representation for the optical transmitters  12   a ,  12   b  by comparing the output of the concatenation of the ansatz  34 ′ for the model representation and model  36 ′ for the determination unit with the sample output measured by the determination unit  20  for each polarization sequentially, and adapting the parameters Θ of the ansatz  34 ′ so that the match becomes maximum. The operations for the two polarizations can also be performed in parallel by doubling the units of system  18 ′. If it is justified to assume that both transmitters  12   a  and  12   b  behave almost identically, it is possible to determine the mode representation for one polarization only and to apply this model representation to both polarizations. Preferably, the approach offers the possibility to determine different parameter sets for the different polarizations planes although the same ansatz is used. 
     In the configuration of  FIG. 6  in which the polarization directions are independent, the pre-distortion unit  26  comprises two identical sub-units  26   a ,  26   b , one for each polarization direction X, Y. 
     As illustrated in  FIG. 7 , this parameter optimization technique effectively corresponds to an optical transmission system  10 ″ with a single polarization direction and two quadrature components I and Q. Corresponding components again share corresponding reference signs. Based on the determination of the model representation for the optical transmitter  12 , the pre-distortion unit  26  may be implemented as an inverse or approximate inverse of the optical transmitter  12 . In the configuration represented in  FIG. 7 , the optical transmitter  12  is assumed to introduce only skew/delay, and hence the analysis unit  24  may determine the representation for the pre-distortion unit  26  based on a correlation criterion, as described above with reference to Equation (12). 
     A simplified optimization technique with reduced degree of complexity will now be explained in greater detail with reference to  FIGS. 8 to 1   o  for the special case in which the optical transmitter  12  is essentially limited to delay/skew between the in-phase (I) and quadrature (Q) part of the optical signal. Although the technique is described here with respect to quadrature phase shift keying (QPSK), it should be mentioned here that the technique can also be used for different modulation formats. 
       FIG. 8  schematically illustrates a constellation diagram of a QPSK signal with transitions from the 00 constellation point in an ideal/noiseless case without skew. 
       FIGS. 9 a  to 9 e    show corresponding constellation diagrams in the presence of skew, wherein the magnitude of the skew increases from  FIG. 9 a    (negligible skew) to  FIG. 9 b    (medium level of skew) to  FIG. 9 e    (substantial level of skew). 
     As can be taken from  FIGS. 9 a  to 9 c   , in the presence of skew the transition paths from one symbol to its point symmetric counterpart no longer go through the center point of the constellation diagram. In particular, none of the transitions traverses the shaded circular areas in  FIGS. 9 b  and 9 c   , and the diameter of these areas increases with increasing magnitude of the skew. As a consequence, the power measured over time will never vanish, and the minimum power detected by the determination unit  20  increases with increasing level of skew. 
       FIG. 10  schematically illustrates the minimum instantaneous power detected versus skew of the in-phase and quadrature components. 
     Minimizing the minimum instantaneous power detected by the determination unit  20  can hence be a suitable technique for adapting the model representation for the optical transmitter  12  in case of skew. Advantageously, to detect the minimum power of the transmit signal not only can the determination unit  20  operate at a lower sampling rate, but it can also run asynchronously with respect to the optical transmitter  12 . 
     As an advantage, a combined processing of the input signals  14  and the output signals  16  is not necessary in the technique described with reference to  FIGS. 8 to 10 . The model representation may be solely determined from an analysis of the output intensity of the optical transmitter  12 , which reduces the complexity of the signal processing. 
     In many practical applications, signals may be affected by noise that has been neglected in the above description for the sake of easier understanding. Although the noise level in the configurations considered here is usually quite small and its influence might even be neglected in many configurations, techniques for determining the minimum amplitude or power level in the presence of noise will now be explained. As will become apparent in the following, the term “determining the minimum” should be understood in a broad sense in the presence of noise. 
       FIG. 11  visualizes the probability distribution of the amplitude values for different skew values and for various noise levels. In detail, the upper plot  FIG. 11 a    shows the probability density function (pdf) of the amplitude values for negligible skew, medium-level skew and substantial skew. All curves relating to the same skew level but different noise levels are displayed with the same line style. Solid curves show the pdf for vanishing skew, whereas dash-dotted curves indicate that there is a medium level skew and dashed curves refer to substantial skew. Among the curves relating to the same skew level, the curve with the steepest raising edge illustrates the probability density for negligible noise. Furthermore, the slope of the curves decreases with increasing noise. Bended arrows point in the direction of increasing noise (to be interpreted as a kind of rotation of the curves). The lower plot  FIG. 11 b    shows the corresponding cumulative probability distribution. 
     With noise and an infinitely long measurement period, there is no range of amplitude values wherein the pdf vanishes completely. Defining a threshold and looking for amplitude values at which the probability density exceeds the threshold value is a suitable way for determining the minimum value. In particular, the “minimum value” should be understood as the smallest amplitude value at which the probability density exceeds a predefined threshold. As illustrated in  FIG. 11 , the threshold value is preferably a function of the noise level. The noise level can be determined from amplitude values determined for the same transmitted symbol or from amplitude values determined for symbols with identical amplitude. Suitable symbols might either be taken from a stream of data to be transmitted or from a sequence of symbols generated for this purpose. In another embodiment, the raising edge is approximated by a linear function and the skew is then determined from the slope and/or the intersection of this linear function with the y-axis or any other intersection. 
     It is an advantage of the techniques according to the present disclosure that they can be implemented and integrated into a pluggable module of an optical transmitter card, as will now be described with reference to  FIGS. 12 and 13 . 
     An example for an optical transceiver card  44  in which the techniques according to the present disclosure may be employed is schematically illustrated in  FIG. 12 . 
     The optical transceiver card  44  comprises a digital signal processing unit  46  communicatively connected to a pluggable optical unit  48 . 
     The digital signal processing unit  46  supplies electrical data signals  14  corresponding to in-phase (I) and quadrature (Q) components of optical signals of two orthogonal polarization directions X, Y to the pluggable optical unit  48 . The pluggable optical unit  48  comprises a controller unit  50  for setting and controlling internal parameters. By means of optical modulators and an optical light source  52 , such as a laser light source or a LED, the received input signals  14  are converted into analog optical signals  16 . The pluggable optical unit  48  may hence implement an optical transmitter  12  as described above with reference to  FIGS. 1 to 7 , and may generate optical output signals  16  that can be supplied to an optical network (not shown). 
     The optical transceiver card  44  also acts as an optical signal receiver. In particular, the pluggable optical unit  48  comprises an optical receiver unit  54  for receiving analog optical signals  56 , such as signals from an optical network node. The pluggable optical unit  48  may coherently detect the incoming optical signals  56 , such as by means of a light source  52 ′ that generally corresponds to the light source  52 . This light source  52 ′ may serve as a local oscillator that is mixed with the received optical signals  56  in order to retrieve two orthogonal (in-phase and quadrature) signal components. The resulting electrical output signals may correspond to in-phase (I) and quadrature (Q) components in two orthogonal polarization directions X and Y, and may be provided as input signals  58  to the digital signal processing unit  46  for analysis and/or signal processing. Instead of using two independent light sources  52  and  52 ′ for the transmit and the receive direction, the pluggable optical unit  48  may generate transmit and receive light sources by splitting a single source using a suitable splitting ratio. 
     An example for an optical output path that may be implemented in the pluggable optical unit  48  is shown in additional detail in the schematic illustration of  FIG. 13 a    for one of the two polarizations. The second polarization may be processed in the same way. 
     As can be taken from  FIG. 13 a   , the pluggable optical unit  48  comprises an IQ modulation unit  60  to generate the optical output signals  16  from the incoming electrical I/Q components employing the laser diode  52 . As further illustrated in  FIG. 13 a   , the pluggable optical unit  48  may further incorporate an intensity photo-detector  32  to tap a subsequence of the optical output signals  16  and determine an intensity of the subsequence of output signals  16 . For instance, the I/Q modulation unit  6   o  may comprise a Mach-Zehnder modulator, and the intensity photo-detector  32  may monitor the output power of the Mach-Zehnder modulator. 
     Based on the determined intensity, the analysis unit  24  may determine a model representation for the optical transmitter  12  as described above with reference to  FIGS. 1 to 10 . The analysis unit  24  may be incorporated into the controller unit  50  of the pluggable optical unit  48 . 
     In particular, in order to implement the techniques described above with reference to  FIGS. 8 to 10  a combined processing of the input signals  14  and the feedback signals  16  is not necessary. In this case, the controller unit  50  may conveniently incorporate the logic for the analysis unit  24 . In other examples, the analysis unit  24  may be incorporated into the digital signal processing unit  46 , or may be a separate unit. 
     Based on the model representation determined by the analysis unit  24 , a pre-distortion unit may be implemented as described above with reference to  FIGS. 1 to 10 , and may be incorporated into the pluggable optical unit  48  or digital signal processing unit  46 . 
       FIG. 13 b    is a schematic illustration of an optical input path of the pluggable optical unit  48  for a single polarization direction. The input path for the second (orthogonal) polarization direction may be identical. 
     The analog optical input signals  56  are split into two parts that are mixed with the light emitted by a light source  52 ′ acting as a local oscillator. In one of the branches, the light from light source  52 ′ is shifted by a phase shifter  64 . Thus, in-phase (I) and quadrature (Q) components are obtained from each optical signal. The resulting I and Q components are each submitted to balanced receivers  66 ,  66 ′ for optical-to-electrical conversion, and the resulting electrical signals are amplified by electrical amplifiers  68 ,  68 ′ to provide the IQ electrical input signals  58  that are subsequently submitted to the digital signal processing unit  46 . 
     In summary, information transmitted in each polarization is encoded in phase or in phase and amplitude (intensity/power) in the presented embodiments. Just knowing the amplitude does not allow to decode the optical signals and to derive the complete information. In particular with respect to digital data signals, a detected value of said quantity does not allow to unambiguously determine the symbol transmitted by the transmitter when the intensity value is determined. Nevertheless, it has been found that knowledge of the amplitude is sufficient for determining a model representation of the transmitter. In other words, the model representation is determined from a quantity that is not sufficient to retrieve the complete information encoded in an optical signal. 
     The description of the embodiments and the Figures merely serve to illustrate the invention and the beneficial effects associated therewith, but should not be understood to imply any limitation. The scope of the invention is to be determined by means of the appended claims. 
     REFERENCE SIGNS 
     
         
           10 ,  10 ′,  10 ″ optical transmission system 
           12  optical transmitter 
           12   a ,  12   b  optical transmitters 
           14  input signals 
           14   a ,  14   b  input polarization directions 
           16  optical output signals 
           16   a ,  16   b  optical output signals 
           18 ,  18 ′,  18 ″ system for determining a model representation 
           20  determination unit 
           22  test sequence 
           24  analysis unit 
           26  pre-distortion unit 
           26   a ,  26   b  pre-distortion units 
           28  feedback receiver 
           30  control line 
           32  intensity photo-detector 
           34 ,  34 ′ ansatz for the model representation 
           36 ,  36 ′ model for determination unit  20   
           38  combiner 
           40  selector unit, analog X/Y switch 
           42  digital X/Y switch 
           44  optical transceiver card 
           46  digital signal processing unit 
           48  pluggable optical unit 
           50  controller unit 
           52 ,  52 ′ light source 
           54  optical receiver unit 
           56  analog optical input signals 
           58  electrical input signals 
           60  IQ modulation unit 
           62  optical splitter 
           64  phase shifter 
           66 ,  66 ′ balanced receivers 
           68 ,  68 ′ electrical amplifiers