Patent Publication Number: US-8121493-B2

Title: Distortion compensation circuit and method based on orders of time dependent series of distortion signal

Description:
TECHNICAL FIELD 
     The present invention relates to distortion compensation and more particularly, to a distortion compensation circuit and method based on orders of time dependent series of distortion signal. 
     BACKGROUND INFORMATION 
     A directly modulated laser may be used as an optical transmitter that transmits light at a given wavelength. The power (i.e., amplitude) of the laser light is modulated by corresponding modulation of the current used to drive the laser. For example, the optical transmitter may be modulated to carry a wide-band RF signal. In this case, the electrical current that drives or pumps the laser is modulated with the wide-band RF signal. 
     The use of a directly-modulated laser to carry a wide-band RF signal may result in distortion due to the multiple carrier frequencies of the multichannel RF signal modulating the laser and/or the harmonics produced by the non-linear nature of the laser device. Intermodulation distortion may be produced when two or more signals mix together to form distortion products. Discrete distortion may be produced from only one carrier. Distortion may include even-order distortion and odd-order distortion. In a CATV system, the most significant types of even-order and odd-order distortion products are second-order distortion products and third-order distortion products, respectively. Second-order intermodulation (IM 2 ) distortion products may include, for example, intermodulation products formed by combining signals at frequencies A and B, such as A±B. In a CATV system, the sum of second-order intermodulation products that are present in a particular channel is commonly referred to as composite second order (CSO) distortion. Third-order intermodulation (IM 3 ) distortion products may include, for example, intermodulation products formed by combining signals at frequencies A, B, and C, such as A±B±C, 2A±B. In a CATV system, the sum of third-order intermodulation products that are present in a particular channel is commonly referred to as composite triple beat (CTB) distortion. 
     The non-linearities of a time independent non-linear element, such as an amplifier, may be modeled as Taylor series expansions or power series expansions of an input signal. For example, the output y of a non-linear amplifier may be described as a Taylor series expansion of an input x:
 
 y ( x )= C   0   +C   1   x+C   2   x   2   +C   3   x   3   +C   4   x   4   + . . . C   k   x   k    Eq. 1
 
where C 0 , C 1 , C 2 , C 3 , C 4 , . . . C k  are constants representative of the behavior of the non-linear amplifier. The order within the series is determined by the highest power of x in the expansion. The even order (x 2n , where n=1, 2, 3 . . . ) terms in the series (e.g., C 2 x 2 , C 2 x 4 , C 2 x 6 , . . . ) represent even order distortion and the odd order (x 2n+1  where n=1, 2, 3 . . . ) terms in the series (e.g., C 2 x 3 , C 2 x 5 , C 2 x 7 , . . . ) represent odd order distortion. For example, C 2 x 2  is the second-order term and represents distortion from the first of the even order terms and C 3 x 3  is the third-order term and represents distortion from the first of the odd order terms. When the input x is an RF input, both x and y are time-varying quantities. With an input having two angular frequencies (ω 1  and ω 2 ) represented as x=a sin(ω 1 t)+b sin(ω 2 t), the second order term C 2 x 2  creates second order distortion products at frequencies 2ω 1 , 2ω 2 , ω 1 −ω 2 , and ω 1 +ω 2 . Because the non-linear element in this case is time independent, the magnitude and phase of these distortion products are not dependent upon the modulation frequency. However, some non-linear elements, such as lasers, have time dependence and thus have distortion characteristics dependent on the frequency of the modulating signal.
 
     Several techniques have been proposed or employed to compensate for distortion by injecting distortion of equal magnitude but opposite phase to the distortion produced by the laser device. For example, a predistortion circuit may be employed to predistort the RF signal being applied to modulate the laser. One such predistortion circuit includes split signal paths—a main or primary signal path and a secondary signal path. A small sample of the RF input is tapped off the main signal path and a distortion generator in the secondary signal path generates distortion (i.e., predistortion). The predistortion is then recombined with the RF signal on the main signal path such that the predistortion is of equal magnitude but opposite sign to the laser-induced distortion. 
     These predistortion circuits have been proposed or employed using frequency independent magnitude adjustments in the secondary path and even magnitude-phase tilt filters to account for the frequency dependent effects in non-linear elements that have time dependence. However, such existing predistortion circuits may not adequately compensate for the frequency-dependent distortion. Further, other non-linear elements may also contribute to distortion. In an optical system, for example, non-linear elements may include at least the laser, optical fiber and amplifier non-linearities. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       These and other features and advantages will be better understood by reading the following detailed description, taken together with the drawings wherein: 
         FIG. 1  is a schematic diagram of an optical transmitter including a predistortion circuit, consistent with one embodiment of the invention. 
         FIG. 2  is a schematic diagram of a distortion compensation circuit including multiple paths based on orders of time dependent series of a distortion signal, consistent with one embodiment of the invention. 
         FIG. 3  is a schematic diagram of a distortion compensation circuit including invertible paths based on orders of time dependent series of a distortion signal, consistent with another embodiment of the invention. 
         FIG. 4  is a schematic diagram of a distortion compensation circuit including multiple paths based on orders of time dependent series of a distortion signal, consistent with another embodiment of the present invention. 
         FIG. 5  is a schematic diagram of a distortion compensation circuit including a frequency independent path and a frequency dependent path, consistent with a further embodiment of the invention. 
         FIG. 6  is a schematic diagram of a distortion compensation circuit including multiple paths based on orders of time dependent series of a distortion signal, consistent with yet another embodiment of the invention. 
         FIG. 7  is a schematic diagram of a distortion compensation circuit including multiple paths based on orders of time dependent series of a distortion signal, consistent with a further embodiment of the invention. 
         FIG. 7A  is a schematic diagram of a distortion compensation circuit including multiple paths in series based on orders of time dependent series of a distortion signal, consistent with a further embodiment of the invention. 
         FIG. 8  is a schematic diagram of an invertible CSO distortion generator for use in a predistortion circuit, consistent with one embodiment of the invention. 
         FIG. 9  is a schematic diagram of a CSO distortion generator for use in a predistortion circuit, consistent with another embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     A distortion compensation circuit, consistent with embodiments of the present invention, may be used with one or more non-linear elements, such as a laser, to compensate for distortion generated by the non-linear element(s), for example, in broadband applications. As will be described in greater detail below, embodiments of the distortion compensation circuit may include a plurality of distortion paths corresponding to different frequency dependent orders of a time dependent series of a distortion signal. The distortion compensation circuit may include, for example, a frequency independent distortion path and one or more frequency dependent distortion paths to produce distortion that compensates for frequency dependent distortion generated by the non-linear element(s). 
     Distortion compensation circuits may include predistortion circuits, which generate compensating distortion before the non-linear element(s), for example, in an optical transmitter. Distortion compensation circuits may also include postdistortion circuits, which generate compensating distortion after the non-linear element(s), for example, in an optical receiver. Although some of the exemplary embodiments may refer specifically to predistortion circuits, the concepts described herein may be used with predistortion compensation, postdistortion compensation, or a combination thereof. Thus, distortion compensation circuits, consistent with the embodiments described herein, may be used to compensate for distortion produced by one or more non-linear elements before and/or after the distortion compensation circuits. 
     Referring to  FIG. 1 , an optical transmitter  100  may include a predistortion circuit  110  to generate predistortion that compensates for distortion produced by one or more non-linear elements, such as a laser  120  and/or an optical fiber  130  when a RF signal modulates the laser  120  to produce a modulated optical output coupled into the optical fiber  130 . As used herein, “compensate” or “compensating” for distortion means reducing distortion to a point that is tolerable in a particular system and does not necessarily require elimination of distortion. To compensate for distortion produced by the laser  120 , the predistortion may be generated by the predistortion circuit  110  with a magnitude substantially equal to the magnitude of the distortion produced by the laser  120  and/or optical fiber  130  and a phase that is substantially opposite the phase of the distortion produced by the laser  120 , optical fiber  130  and/or non-linear elements such as amplifier non-linearities. 
     According to one embodiment, the optical transmitter  100  may also include RF amplifier/anti-clipping circuit  140  to receive and amplify the RF input signal (e.g., a multi-channel carrier multiplex signal) and/or to modify the RF input signal to prevent or reduce clipping in the laser  120 . One example of an anti-clipping circuit is the type described in greater detail in commonly-owned U.S. patent application Ser. No. 11/753,082, which is incorporated herein by reference. The predistortion circuit  110  may then receive the amplified RF signal, generate the predistortion and combine the predistortion with the RF signal that modulates the laser  120 . The laser  120  may be a directly-modulated electrically pumped semiconductor laser, such as a laser diode. 
     One embodiment of the optical transmitter  100  may further include thermo-electric cooler (TEC) controller and laser diode driver circuitry  150  to control the temperature of and to bias the laser  120 . A controller  160 , such as a microprocessor, may be used to control the components and the operation of the optical transmitter  100 . The TEC controller and laser diode driver circuitry  150  and the microcontroller  160  may include components known to those skilled in the art for use in a laser transmitter, such as the type available from Applied Optoelectronics, Inc. 
     One example of an optical transmitter  100  is a laser transmitter designed for forward-path CATV applications. In such “broadband” applications, the optical transmitter  100  and particularly the laser  120  may be designed for high frequency operation, for example, up to about 1 GHz. The distortion compensation circuits and methods described herein may also be used in other applications (e.g., using different or even higher frequencies) and/or with other types of optical transmitters. Embodiments of the distortion compensation circuit may also be used with any non-linear element or device that generates distortion that can be compensated with predistortion or postdistortion. 
     Distortion compensation circuits, such as predistortion circuit  110 , may generate distortion (e.g., predistortion or postdistortion) based on different frequency dependent orders of a time dependent series representative of a distortion signal. As mentioned above, the output y of a non-linear element may be described as a Taylor series expansion of an input x:
 
 y ( x )= C   0   +C   1   x+C   2   x   2   +C   3   x   3   +C   4   x   4   + . . . C   k   x   k    Eq. 2
 
where C 0 , C 1 , C 2 , C 3 , C 4 , . . . C k  are constants representative of the behavior of the non-linear amplifier. The order within this series is determined by the highest power of x in the expansion. For example, C 2 x 2  is the second-order term. When the non-linear element also has time dependence, such as for lasers, the Taylor series may be further expanded as a time dependent series including time dependent terms as follows:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     When an input having two angular frequencies (ω 1  and ω 2 ) represented as x=a sin(ω 1 t)+b sin(ω 2 t) is applied to the above time dependent non-linear element, the second order distortion at frequencies 2ω 1 , 2ω 2 , ω 1 −ω 2 , and ω 1 +ω 2  will have an amplitude and phase that is dependent on frequency. For the 2ω 1  term, the dependence may be represented as follows: 
     
       
         
           
             
               
                 
                   
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     The first term in the above series represents the frequency independent term. The remaining terms represent frequency dependent terms that are a result of the time dependence upon distortion. A similar dependence can be found for other second order distortion products. 
     When the non-linear element is a laser being modulated by an input current to produce light output, the input is current I(t) and the output is laser output power P(I(t)). The power signal P may thus be represented by the following time dependent series: 
                         P   =       P   ⁡     (     I   ⁡     (   t   )       )       =       ⁢       C   00     +       C   01     ⁢     I   ⁡     (   t   )         +       C   02     ⁢       I   ⁡     (   t   )       2       +     …   ⁢           ⁢     C     0   ⁢   k       ⁢       I   ⁡     (   t   )       k       +                       ⁢         C   11     ⁢       ⅆ   I       ⅆ   t         +       C   12     ⁢   I   ⁢           ⁢       ⅆ   I       ⅆ   t         +     …   ⁢           ⁢     C     1   ⁢   k       ⁢     I     k   -   1       ⁢       ⅆ   I       ⅆ   t         +                     ⁢         C   21     ⁢         ⅆ   2     ⁢   I       ⅆ     t   2           +       C   22     ⁢   I   ⁢         ⅆ   2     ⁢   I       ⅆ     t   2           +     …   ⁢           ⁢     C     2   ⁢   k       ⁢     I     k   -   1       ⁢         ⅆ   2     ⁢   I       ⅆ     t   2           +                     ⁢   ⋮                   ⁢         C     n   ⁢           ⁢   1       ⁢         ⅆ   n     ⁢   I       ⅆ     t   n           +       C     n   ⁢           ⁢   2       ⁢   I   ⁢         ⅆ   n     ⁢   I       ⅆ     t   n           +     …   ⁢           ⁢     C   nk     ⁢     I     k   -   1       ⁢         ⅆ   n     ⁢   I       ⅆ     t   n                           Eq   .           ⁢   5               
In Equation 4, k indicates the order of the series representing distortion (i.e., distortion order) and n indicates the order of the time dependent series (i.e., frequency dependent order). For example, the term C 02 I(t) 2  represents second order distortion and zero order of the time dependent series of the second order distortion.
 
     For a laser that is being modulated by an input current to produce a light output, the input current for a single angular frequency ω may be represented as follows:
 
 I ( t )= I   0   +I   1   e   iωt    Eq. 6
 
Using Equation 5 in the expanded series of Equation 4, the second order distortion terms of the expanded time dependent series of the power signal P may be represented as follows:
 
                           P   2     =       ⁢         C   02     ⁢     I   1   2     ⁢     ⅇ     ⅈ2ω   ⁢           ⁢   t         +                     ⁢         C   12     ⁢     I   1   2     ⁢     ⅇ     ⅈ2ω   ⁢           ⁢   t       ⁢   ⅈω     +                     ⁢         C   22     ⁢     I   1   2     ⁢         ⅇ     ⅈ2ω   ⁢           ⁢   t       ⁡     (   ⅈω   )       2       +                     ⁢         C   32     ⁢     I   1   2     ⁢         ⅇ     ⅈ2ω   ⁢           ⁢   t       ⁡     (   ⅈω   )       3       +                     ⁢     ⋮   +                     ⁢       C     n   ⁢           ⁢   2       ⁢     I   1   2     ⁢         ⅇ     ⅈ2ω   ⁢           ⁢   t       ⁡     (   ⅈω   )       n                     Eq   .           ⁢   7               
In this time dependent series representing the power signal second order distortion P 2  (e.g., CSO distortion), the zero (0) order term C 02 I 1   2   i2ωt  is frequency independent and the higher order (2 . . . n) terms C 12 I 1   2 e i2ωt iω, C 22 I 1   2 e i2ωt (iω) 2 , C 32 I 1   2 e i2ωt (iω) 3 , . . . C n2 I 1   2 e i2ωt (iω) n  are frequency dependent. A similar time dependent series may be used to represent third order distortion (e.g., CTB distortion). Thus, the different distortion orders within the power signal, such as second order distortion (e.g., CSO) and third order distortion (e.g., CTB), may each be represented as a time dependent series having terms with different frequency dependent orders within the time dependent series. The second order intermodulation distortion (i.e., CSO distortion) may be represented more simply as a function of angular frequency (ω) with three (3) frequency dependent orders as follows:
 
 IMD 2(ω)= a+b ( i ω)+ c ( i ω) 2    Eq. 8
 
where a represents the 0 order CSO term, which is frequency independent, b(iω)represents the first order CSO term, which is linear with frequency (i.e., π/2), and c(iω) 2  represents the second order CSO term, which is linear to the second order (i.e., π).
 
     Distortion compensation circuits may thus compensate for multiple distortion terms in the time dependent series representing a distortion signal to improve distortion compensation. In general, the distortion compensation circuit may include multiple distortion generator units corresponding to multiple distortion terms in the time dependent series to independently generate distortion products for those different terms. Various embodiments of such distortion compensation circuits capable of being used with a laser in an optical transmitter are described in greater detail below. Although the exemplary embodiment refers to compensation of distortion generated by a laser in an optical transmitter, the distortion compensation circuits described herein may be used in any type of RF system to compensate for distortion generated by any type of non-linear element. For any such RF system, the distortion profile of the non-linear element may be determined by expanding the non-linear output as a time dependent series, as discussed above. An appropriate number of distortion generator units may be used for different terms in the time dependent series to closely achieve that distortion profile and provide a more linear output from the non-linear element. 
     Referring to  FIG. 2 , a predistortion circuit  200 , consistent with an embodiment of the present invention, is described in greater detail. The predistortion circuit  200  receives a RF input signal at a signal input  202 , generates the predistortion to predistort the RF input signal, and provides a predistorted RF signal at a predistorted signal output  204 . The predistortion circuit  200  may include a primary signal path  210  and a secondary signal path  220  that are coupled together, for example, using directional couplers such as a directional splitter  212  and a directional combiner  214 . At the directional splitter  212 , at least a portion of the RF input signal is received on the primary and secondary signal paths  210 ,  220 , respectively. 
     The secondary signal path  220  produces intermodulation distortion products from the RF input signal received on the secondary signal path  220  and generates those intermodulation distortion products based on the orders of the time dependent series of the distortion described above. At the directional combiner  214 , intermodulation distortion products produced on the secondary signal path  220  are combined with the RF input signal on the primary signal path  210  with a desired magnitude and phase to produce the predistorted RF signal that compensates for distortion generated by the laser and/or other non-linear element (not shown in  FIG. 2 ). The primary signal path  210  may include a delay element  216 , such as a transmission line of a selected length, which delays the RF input signal on the primary signal path  210  to correspond to the delay caused by generating the predistortion on the secondary signal path  220 . Such a delay helps to ensure that the predistortion on the secondary signal path  220  remains in phase with the RF input signal on the primary signal path  210 . 
     According to one embodiment, the secondary signal path  220  may include parallel distortion sub-paths  222 - 0 ,  222 - 1 , . . .  222 -n coupled at one end to splitter  230  and coupled at the other end to a combiner  232 . The distortion sub-paths  222 - 0 ,  222 - 1 , . . .  222 -n correspond to the distortion terms having different frequency dependent orders in the time dependent series representing the distortion. For CSO compensation, for example, the sub-paths  222 - 0 ,  222 - 1 , . . .  222 -n correspond to the 0 order, 1 st  order to n th  order terms C 02 I 1   2 e i2ωt , C 12 I 1   2 e i2ωt iω, . . . C n2 I 1   2 e i2ωt (iω) n , respectively, in the time dependent series representing second order intermodulation distortion. The distortion sub-path  222 - 0  corresponding to the 0 order term is frequency independent and the distortion sub-path(s)  222 - 1  . . .  222 -n corresponding to the 1 st  and higher order terms are frequency dependent. 
     The parallel distortion sub-paths  222 - 0 ,  222 - 1 , . . .  222 -n may include distortion generators  224 - 0 ,  224 - 1 , . . .  224 -n that produce intermodulation distortion products corresponding to the distortion terms with the different respective orders of the time dependent series. For a predistortion circuit  200  that compensates for CSO distortion, for example, each of the distortion generators  224 - 0 ,  224 - 1 , . . .  224 -n generate second order intermodulation distortion products corresponding to the respective CSO terms in the time dependent series representing CSO distortion. For example, the distortion generator  224 - 0  generates second order intermodulation distortion products corresponding to the zero order frequency dependent term C 02 I 1   2 e i2ωt  and the distortion generator  224 - 1  generates second order intermodulation distortion products corresponding to the first order frequency dependent term C 12 I 1   2 e i2ωt iω. If the predistortion circuit  200  is designed to compensate for CTB distortion, the distortion generators  224 - 0 ,  224 - 1 , . . .  224 -n may similarly generate CTB distortion corresponding to CTB terms in a time dependent series representing CTB distortion. 
     The intermodulation distortion products corresponding to the different distortion terms in the time dependent series may then be combined at the combiner  232  to produce predistortion that more closely approximates the distortion profile. The predistortion may then be combined with the RF signal at the combiner  214  to produce the predistorted RF signal. Because the predistortion generated by the multiple sub-paths  222 - 0 ,  222 - 1 , . . .  222 -n in the predistortion circuit  200  more closely approximates the distortion profile, the predistorted RF signal will better compensate for the distortion. 
     Because the intermodulation distortion products generated on one or more of the sub-paths  222 - 0 ,  222 - 1 , . . .  222 -n may be 180° out of phase with respect to other of the sub-paths  222 - 0 ,  222 - 1 , . . .  222 -n, the phase of the intermodulation distortion products on one or more of the sub-paths  222 - 0 ,  222 - 1 , . . .  222 -n may need to be phase inverted. For example, the intermodulation distortion products corresponding to the first order term in the time dependent series are generally 180° out of phase relative to the intermodulation distortion products corresponding to the zero order term in the time dependent series. A respective one of the distortion generators  224 - 0 ,  224 - 1 , . . .  224 -n or the combiner  232  may provide the desired phase inversion such that the intermodulation distortion products on each of the sub-paths  222 - 0 ,  222 - 1 , . . .  222 -n are in phase when combined at the combiner  232 . 
     A more specific embodiment of a predistortion circuit  300  based on orders of a time dependent series of a power distortion signal is shown in  FIG. 3 . The predistortion circuit  300  includes a primary signal path  310  coupled to a secondary signal path  320  with parallel distortion sub-paths  322 - 0 ,  322 - 1 , . . .  322 -n similar to the predistortion circuit  200  described above. According to this embodiment, the parallel distortion sub-paths  322 - 0 ,  322 - 1 , . . .  322 -n each include a distortion generator  324 - 0 ,  324 - 1 , . . .  324 -n that generates the intermodulation distortion products of the desired distortion order. Each of the distortion generators  324 - 0 ,  324 - 1 , . . .  324 -n may be essentially the same, and the intermodulation distortion products generated by the distortion generators  324 - 0 ,  324 - 1 , . . .  324 -n are generally frequency independent. 
     According to the illustrated embodiment, the predistortion circuit  300  compensates for CSO distortion and the distortion generators  324 - 0 ,  324 - 1 , . . .  324 -n are CSO distortion generators. The CSO generators  324 - 0 ,  324 - 1 , . . .  324 -n may include, for example, a square law device that generates second order intermodulation distortion products. A CSO distortion generator (or second order distortion generator) does not necessarily generate only second order distortion. A CSO distortion generator (or second order distortion generator) may include a distortion generator that produces even-order distortion with predominantly second order distortion. Embodiments of CSO distortion generators are shown in  FIGS. 8 and 9  and are described in greater detail below. In a predistortion circuit that compensates for odd orders of distortion, other distortion generators may be used, in addition to or instead of the CSO distortion generators, such as CTB distortion generators to compensate for CTB distortion. A CTB distortion generator may include a distortion generator that produces odd-order distortion with predominantly third order distortion. 
     To produce the distortion products for the respective frequency dependent orders of the time dependent series, the frequency dependent sub-paths  322 - 1 ,  322 -n also include differentiating filters  326 - 1 ,  326 -n that filter the frequency independent distortion products to produce an approximate time derivative of the distortion products corresponding to the desired order of the time dependent series. If the distortion generators  324 - 0 ,  324 - 1 , . . .  324 -n are CSO distortion generators that generate second order distortion products represented by the term C 02 I 1   2 e i2ωt , for example, the d/dt differentiating filter  326 - 1  differentiates the second order distortion products to produce an approximate first time derivative of the second order distortion products corresponding to the first order term C 12 I 1   2 e i2ωt iω. 
     One or more of the distortion sub-paths  322 - 0 ,  322 - 1 , . . .  322 -n may also include a signal controlled phase inverter  325 - 0 ,  325 - 1 , . . .  325 -n to invert the phase of the distortion products in one or more of the distortion sub-paths  322 - 0 ,  322 - 1 , . . .  322 -n. The signal controlled phase inverter(s)  325 - 0 ,  325 - 1 , . . .  325 -n may be responsive to a phase inversion control signal (e.g., provided by controller  160  shown in  FIG. 1 ) to select a phase of either 0° or 180°. Because the d/dt differentiating filter  326 - 1  results in a phase shift of 180° in the frequency dependent distortion sub-path  322 - 1  relative to the frequency independent distortion sub-path  322 - 0 , for example, one or both of the phase inverters  325 - 0 ,  325 - 1  may be used to invert the phase of the distortion products in one or both of the distortion sub-paths  322 - 0 ,  322 - 1  such that the frequency independent distortion products and the frequency dependent distortion products are substantially in phase. Predistortion circuits including signal controlled phase invertible paths are described in greater detail in U.S. patent application Ser. No. 12/026,182, filed concurrently herewith and fully incorporated herein by reference. 
     The signal controlled phase inverters  325 - 0 ,  325 - 1 , . . .  325 -n may also be located in other locations along the sub-paths  322 - 0 ,  322 - 1 , . . .  322 -n other than directly following the distortion generators  324 - 0 ,  324 - 1 , . . .  324 -n. The primary signal path  310  may also include a signal controlled phase inverter (not shown) to provide controllable phase inversion of the RF signal in the primary signal path  310 . 
     A further embodiment of a predistortion circuit  400  is shown in  FIG. 4 . The predistortion circuit  400  includes a primary signal path  410  coupled to a secondary signal path  420  similar to the predistortion circuit  200  described above. According to this embodiment, three parallel distortion sub-paths  422 - 0 ,  422 - 1 ,  422 - 2  include second order distortion generators  424 - 0 ,  424 - 1 ,  424 - 2  that generate frequency independent second order distortion products. The zero order frequency independent sub-path  422 - 0  may be unfiltered to pass the frequency independent second order distortion products. The first order sub-path  422 - 1  and the second order sub-path  422 - 2  may be filtered to produce the frequency dependent distortion products from the frequency independent distortion products generated by the distortion generators  424 - 1 ,  424 - 2 . 
     The first order frequency dependent sub-path  422 - 1  includes a capacitor  440  following the second order distortion generator  424 - 1  to provide the first time derivative (d/dt) filtering of the second order distortion products. The capacitor  440  may have a capacitance capable of providing the desired d/dt function without blocking too much magnitude of the distortion. For example, the capacitor  440  may have a capacitance between 0.5 pF and 5 pF, and more specifically about 2 pF for a secondary path with 50 ohms impedance. This value gives a reasonable approximation of a d/dt filter over a reasonably wide bandwidth. The second order frequency dependent sub-path  422 - 1  includes capacitors  442 ,  444  and resistor  446  to provide the second time derivative (d 2 /dt 2 ) filtering of the second order distortion products. The capacitors  442 ,  444  may have a capacitance between about 0.5 pF and 5 pF and the resistor  446  may have a resistance of about 75 ohm. Other types of components may also be used to provide the desired differentiation, such as an inductor. 
       FIG. 5  shows another embodiment of a predistortion circuit  500  that provides predistortion based on orders of time dependent series of a power distortion signal. The predistortion circuit  500  may include a primary signal path  510  and a secondary signal path  520  including a frequency independent distortion sub-path  522 - 0  and a frequency dependent distortion sub-path  522 - 1 . The sub-paths  522 - 0 ,  522 - 1  may be coupled at each end by a splitter  530 , such as a 3 dB splitter, and a combiner  532 , such as a 3 dB combiner. According to this embodiment, the frequency independent distortion sub-path  522 - 0  produces the frequency independent distortion products corresponding to the zero order term in the time dependent series and the frequency dependent distortion sub-path  522 - 1  produces frequency dependent distortion products corresponding to the first order term in the time dependent series. 
     Each of the sub-paths  522 - 0 ,  522 - 1  may include invertible distortion generators  524 - 0 ,  524 - 1  that generate distortion products, which may be phase inverted in response to a phase inversion control signal. One embodiment of an invertible CSO distortion generator is shown in  FIG. 8  and described in greater detail below. In the frequency dependent sub-path  522 - 1 , a d/dt differentiating filter  526 - 1  following the invertible distortion generator  524 - 1  filters the distortion products to produce the frequency dependent distortion products corresponding to the first order term. 
     One or both of the distortion sub-paths  522 - 0 ,  522 - 1  of the secondary signal path  520  may also include one or more gain control elements, such as a variable attenuator  527 - 0 ,  527 - 1  and/or an amplifier  528 - 0 ,  528 - 1 , before and/or after the distortion generators  524 - 0 ,  524 - 1  to control a magnitude of the predistortion generated by the distortion generators  524 - 0 ,  524 - 1 . Variable gain control elements help to ensure that the magnitude of the predistortion corresponds sufficiently to the magnitude of the distortion being compensated. The variable attenuators  527 - 0 ,  527 - 1  may be PIN attenuators and may receive attenuation control signals from a controller (e.g., controller  160  shown in  FIG. 1 ) to adjust the attenuation as needed. One or both of the distortion sub-paths  522 - 0 ,  522 - 1  of the secondary signal path  520  may also include delay components (not shown) to add small amounts of delay in each sub-path  522 - 0 ,  522 - 1  to “zero” out any path length differences. Other embodiments of predistortion circuits including invertible distortion generators are described in greater detail U.S. patent application Ser. No. 12/026,182, which is filed concurrently herewith and is fully incorporated herein by reference. 
       FIG. 6  shows yet another embodiment of a predistortion circuit  600 . The predistortion circuit  600  includes a primary signal path  610  coupled to a secondary signal path  620  similar to the predistortion circuit  200  described above. According to this embodiment, the secondary signal path  620  includes a single distortion generator  624  located before the splitter  630  to generate the intermodulation distortion products. The parallel distortion sub-paths  622 - 0 ,  622 - 1 , . . .  622 -n then provide the filtering needed to produce the distortion products based on the respective frequency dependent orders of the time dependent series. The frequency dependent sub-paths  622 - 1 , . . .  622 -n, for example, include the differentiating filters  626 - 1 , . . .  626 -n to produce the distortion products for the 1 st  to n th  order terms. The frequency independent path  622 - 0  remains unfiltered to produce the distortion products for the 0 order term. 
       FIG. 7  shows yet another embodiment of a predistortion circuit  700  including a primary signal path  710  and a secondary signal path  720 . According to this embodiment, the secondary signal path  720  includes a plurality of parallel distortion sub-paths  722 - 0 ,  722 - 1 , . . .  722 -n coupled directly to the primary signal path  710  via splitter  712  and combiner  714 . 
       FIG. 7A  shows yet another embodiment of a distortion compensation circuit  700   a.  According to this embodiment, the secondary distortion sub-paths  722 - 0 ,  722 - 1 , . . .  722 -n are arranged in series instead of in parallel. Various other configurations of distortion compensation circuits are also possible based on orders of time dependent series of a distortion signal. 
     One embodiment of an invertible CSO generator  800  is shown in greater detail in  FIG. 8 . The invertible CSO generator  800  may be used in distortion compensation circuits described herein (e.g., the predistortion circuit  500  shown in  FIG. 5 ). The invertible CSO generator  800  receives the RF input signal at an input  802 , generates second order intermodulation products from the RF input signal, and provides the second order intermodulation products (i.e., the CSO distortion) at an output  804 . 
     The invertible CSO generator  800  may include a CSO distortion generator portion  810  including diodes  811 ,  812  connected and arranged to generate the CSO distortion. One example of the diodes  811 ,  812  is a matched series pair of Schottky diodes. The diodes  811 ,  812  are connected and arranged relative to the RF input such that RF voltage drop across the diodes  811 ,  812  are opposite relative to the polarity of the diode. The RF currents through the diodes are then added by use of a balun  830  or other similar devices which block common mode or odd-order signals, but adds differential or even-order signals. Thus, odd order components of the current from the diodes  811 ,  812  are effectively blocked, but even order components are passed. Although the exemplary embodiment shows one arrangement of a series pair of diodes, other arrangements and numbers of diodes are possible such that the diodes are capable of producing distortion corresponding to the distortion to be compensated. 
     The CSO generator portion may also include bias resistors  814 ,  816  coupled in series with the diodes  811 ,  812 . A DC bias voltage coupled to the network of diodes  811 ,  812  and bias resistors  814 ,  816  results in a bias current (I b ) across the diodes  811 ,  812 . In general, the diodes  811 ,  812  are biased to operate in the forward bias region when generating distortion. The bias resistors are chosen along with bias current to provide, among other things, good input impedance match. The diode bias may be set manually by an on-board variable resistor (not shown). In other embodiments, an adjustable bias control may adjust the bias current (I b ) provided to the diodes  811 ,  812  to control, among other things, compensating distortion magnitude, for example, as described in greater detail in U.S. patent application Ser. No. 11/834,873, which is fully incorporated herein by reference. Although the exemplary embodiment shows one configuration and arrangement of the bias resistors together with the diodes, other configurations and bias resistor networks are possible to provide a desired bias current across the diodes. The CSO generator portion  810  may also include DC blocking capacitors  824 ,  826  coupled to the diodes  811 ,  812 , respectively, to isolate the DC bias signals from RF signals. 
     The output of the balun  830  may be connected to an RF switching device  840  to provide phase inversion capabilities. The phase inversion state of the even-order distortion passing through the balun  830  to the output  804  depends on which of the output terminals  836 ,  838  is coupled to the output  804  and which of the output terminals  836 ,  838  is coupled to ground  808   a,    808   b.    
     The switching device  840  is coupled to the output terminals  836 ,  838  of the balun  830  and selects which side of the balun  830  to tap off of in response to a phase inversion control signal received at control signal input  806 . For example, when the switching device  840  invert control signal input  806  is low, the balun terminal  836  may be coupled to output  804  and the balun terminal  838  may be coupled to ground  808   b  providing a phase of 0°. When the switching device  840  invert control signal input  808  is high, the switching device  840  causes the balun terminal  836  to be coupled to ground  808   a  and causes the balun terminal  838  to be coupled to output  804  providing in a phase change of 180°. In other words, the switching device  840  causes the balun  830  to invert the distortion provided to the output  804  in response to an inversion control signal. 
     The switching device  840  may be a solid state RF switch as shown. The switching device  840  may also be implemented using other discrete devices, such as a RF relay or a RF MEMS (microelectromechanical system) switch. The control signal input  806  of the switching device  840  may be coupled to a controller (e.g., controller  160  shown in  FIG. 1 ), which provides the phase inversion control signal as a digital output signal based on various parameters, as described in greater detail below. The controller may include firmware configured to generate the phase inversion control signal in response to various parameters affecting distortion in a system, such as temperature, bias current, and fiber length, as described in greater detail below. Thus, the controller or microprocessor may control distortion compensation in response to the various parameters. 
       FIG. 9  shows another embodiment of a distortion generator  900 , which is not invertible. Similar to the invertible distortion generator  800  described above, the distortion generator  900  includes a RF input  902 , a distortion output  904 , and a distortion generator portion  910  coupled to a balun  930 . The distortion generator portion  910  includes a pair of diodes  911 ,  912  configured to generate second order intermodulation distortion products, as described above, although other configurations may be used to provide second order distortion products or higher order distortion products. The balun  930  includes input terminals  932 ,  934  coupled to the respective diodes  911 ,  912  to combine the output from the diodes  911 ,  912  to a single distortion output. In this embodiment, the balun  930  includes an output terminal  936  coupled to the distortion output  904  and an output terminal  938  coupled to ground  908  via resistor  922 . Although two different exemplary distortion generators  800 ,  900  are shown, other types of distortion generators may also be used in the embodiments of the distortion compensation circuit described herein. 
     Accordingly, distortion compensation circuits based on orders of time dependent series of a distortion signal may improve distortion compensation by producing predistortion and/or postdistortion that more closely approximates the distortion profile of the distortion being compensated. In particular, the distortion compensation circuits described herein provide improved compensation of the frequency dependent components of the distortion. 
     Consistent with one embodiment, a distortion compensation circuit is provided for compensating for distortion of a distortion order produced by at least one non-linear element. The distortion compensation circuit includes a primary signal path configured to receive at least a portion of an input signal and a secondary signal path coupled to the primary signal path. The secondary signal path is configured to receive at least a portion of the input signal and to generate distortion of the distortion order from the input signal. The secondary signal path includes a plurality of distortion sub-paths with each of the distortion sub-paths configured to produce intermodulation distortion products of the distortion order but for different frequency dependent orders of a time dependent series representative of the distortion produced by the non-linear element. 
     Consistent with another embodiment, an optical transmitter includes a RF signal input configured to provide a RF input signal and a predistortion circuit configured to receive the RF input signal and to generate a predistorted RF input signal. The predistortion circuit includes a primary signal path configured to receive at least a portion of the RF input signal and a secondary signal path coupled to the primary signal path and configured to receive at least a portion of the RF input signal and to generate distortion from the RF input signal. The secondary signal path including a plurality of parallel distortion sub-paths with each of the distortion sub-paths configured to produce intermodulation distortion products of the same distortion order but for different frequency dependent orders of a time dependent series representative of the distortion produced by the non-linear amplifier. The optical transmitter also includes a laser configured to receive the predistorted RF input signal and to generate a modulated optical output, wherein the predistorted RF input signal compensates for distortion generated by at least the laser. 
     Consistent with a further embodiment, a method is provided for compensating for distortion of a distortion order produced by at least one non-linear element. The method includes: providing a distortion compensation circuit including a primary signal path configured to receive at least a portion of an input signal and a secondary signal path coupled to the primary signal path and configured to receive at least a portion of the input signal, the secondary signal path comprising at least one distortion generator configured to generate intermodulation distortion products of a distortion order from the input signal; and receiving a portion of an RF signal on the primary signal path; receiving a portion of the RF signal on the secondary signal path; generating frequency independent intermodulation distortion products of the distortion order from the RF signal on a first distortion sub-path of the secondary signal path; generating frequency dependent intermodulation distortion products of the distortion order from the RF signal on at least a second distortion sub-path of the secondary signal path; combining the frequency independent intermodulation distortion products and the frequency dependent intermodulation distortion products on the secondary signal path to produce compensating distortion of the distortion order; and combining the compensating distortion with the RF signal. 
     While the principles of the invention have been described herein, it is to be understood by those skilled in the art that this description is made only by way of example and not as a limitation as to the scope of the invention. Other embodiments are contemplated within the scope of the present invention in addition to the exemplary embodiments shown and described herein. Modifications and substitutions by one of ordinary skill in the art are considered to be within the scope of the present invention, which is not to be limited except by the following claims.