Abstract:
A method for processing data in an optical network element. A multi-carrier signal is linear pre-coded and the linear pre-coded signal is modulated. A corresponding optical network element is also described.

Description:
BACKGROUND OF THE INVENTION 
     Field of the Invention 
     The invention relates to a method and to a device for processing data in an optical network element and to an according optical network element. 
     With broadband Internet connections and mobile data transfer becoming ubiquitous technologies, requirements regarding bitrates over WDM optical channels are increasing. In this context, a spectral efficiency of a modulation format used is of significant relevance. 
     The spectral efficiency of an optical signal can be increased, e.g., by multilevel modulation, polarization multiplexing, orthogonal frequency multiplexing or a combination thereof. However, the complexity of the system noticeably increases with the modulation. 
     When multilevel modulation format and polarization multiplexing were chosen to increase the spectral efficiency (SE) of a transmission system, orthogonal frequency division modulation (OFDM) is a subsequent step that doubles the SE by overlapping the spectra of a multitude of subcarriers. 
     For example, a binary on-off-keying (OOK) signal with a data rate of 100 Gbps uses 200 GHz of optical bandwidth (BW). If OFDM is used with quadrature-phase-shift-keying (QPSK) modulated subcarriers and polarization multiplexing (PolMux) a 100 Gbps line rate signal would roughly use 25 GHz optical bandwidth. 
     However, such an advanced modulation format would require the use of digital coherent detection, and therefore, greatly increase the complexity of the implementation, because of the need for advance digital signal processing (DSP) and an optical local oscillator. 
     A reduced complexity of the implementation could be achieved by using a direct detectable OFDM signal, but a reference carrier has to be sent along the data signal situated at a distance in spectrum equal to the BW of the OFDM signal, which reduces the SE. Another possibility is the use of a Compatible Single Sideband OFDM modulation (CompSSB-OFDM). However, a very high power carrier is required in this case, also compromising an overall performance of the system. 
     In addition, all such cases of enhanced SE require complex DSP algorithms that need to be implemented in the transmitter and the receiver. 
     For high data rates (&gt;10 Gbps), the use of DSP is a challenging problem not only concerning the developments of high-speed electronics, but also with regard to an energy consumption of next generation high-speed systems. 
     BRIEF SUMMARY OF THE INVENTION 
     The problem to be solved is to overcome the disadvantages mentioned above and in particular to provide a high degree of spectral efficiency without a high complexity of digital signal processing in an optical system. 
     This problem is solved according to the features of the independent claims. Further embodiments result from the depending claims. 
     In order to overcome this problem, a method is suggested for processing data in an optical network element,
         wherein a multicarrier signal is linear pre-coded,   wherein the linear pre-coded signal is modulated.       

     The proposed method in particular doubles the spectral efficiency of an optical signal while maintaining a low complexity of the system. This allows for example, the use of high-speed 100 Gbps signals in dense WDM systems without the need of polarization multiplexing or complex digital signal processing at the receiver, therefore allowing for a cost efficient approach. 
     The approach further increases the spectral efficiency of signals and provides compatibility with direct detection receivers. Hence, in particular no local oscillator is required at the receiver. 
     In an embodiment, the pre-coded signal is modulated by a differential phase modulation or by an amplitude modulation. 
     It is noted that direct detection modulation formats can be utilized, in particular OOK, DPSK, DQPSK, D8PSK, Star-D8QAM, Star-D16QAM, PAM, etc. 
     In another embodiment, the multicarrier signal is linear pre-coded, wherein each subcarrier is a linear combination of all other subcarriers. 
     Hence, after direct detection at a receiver, at each k-th sampling point, the electrical signal has a value proportional to the k-th subcarrier. 
     In a further embodiment, the multicarrier signal is linear pre-coded by a matrix T, wherein the coefficients of a linear combination of a k-th subcarrier correspond to the k-th row of an DFT matrix. 
     In a next embodiment, a channel transfer function is pre-compensated by an equalization. 
     This equalization can be conducted at the optical network element after the linear pre-coding. Advantageously, the signal can be sent via a dispersion-unmanaged link. 
     It is also an embodiment that said equalization is a one tap equalization. 
     Pursuant to another embodiment, a subcarrier is discarded by a receiver in case the subcarrier experiences deforming (e.g., due to distortions) and/or attenuating effects above a given threshold. 
     In such case, the receiver may (temporarily) discard at least one subcarrier. The receiver may inform the transmitter of the discarded subcarrier and the transmitter may no longer use this subcarrier towards this receiver. 
     According to an embodiment, a dummy subcarrier is used to provide a guard band. 
     Advantageously, such guard band is useful to reduce (inter-symbol) interference from a previously pre-coded block. 
     According to another embodiment, a zero padding is conducted prior to an inverse discrete Fourier transform. 
     Advantageously, this zero padding simplifies filtering at the receiver. Zero padding can be adjusted to the filtering capability of the respective transmitter. 
     In yet another embodiment, a feedback channel is provided to convey information from a receiver to the optical network element. 
     Such feedback channel can be utilized for various purposes. The optical network element may in particular adjust its signals to increase the efficiency based on the information obtained via the feedback channel. 
     According to a next embodiment, the optical network element is an optical transmitter, e.g., an optical line terminal or an optical network unit. 
     The problem stated above is also solved by an optical network element comprising
         a linear pre-coder processing a multicarrier signal,   a modulator processing the linear pre-coded signal.       

     Pursuant to yet an embodiment, the optical network element further comprises a zero padding unit prior to an inverse discrete Fourier transform unit prior to said modulator. 
     It is noted that said modulator may advantageously comprise at least one digital operation. 
     It is also an option that the modulator is a differential modulator or an amplitude modulator 
     According to an embodiment, the optical network element comprises a control unit that is arranged such that the method as described herein can be executed. 
     The problem stated supra is further solved by an optical communication system comprising the optical network element as described herein. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
       Embodiments of the invention are shown and illustrated in the following figures: 
         FIG. 1  shows a transmission scheme comprising a block diagram of a transmitter, a channel and a receiver; 
         FIG. 2  shows another representation of the transmitter; 
         FIG. 3  shows a spectrum of an optical output signal of the transmitter according to  FIG. 2 . 
     
    
    
     DESCRIPTION OF THE INVENTION 
     The solution provided in particular suggests a discrete inverse Fourier transform in the transmitter that increases the spectral efficiency of an optical signal by shaping its spectrum in an OFDM-like fashion. In addition, it makes the signal compatible with direct detection, without the need for digital signal processing and without having to send an optical carrier along the signal. Hence, without such need for complex DSP, the complexity is feasible even for high data rates (e.g., &gt;10 Gbps) and a high spectral efficiency is achieved as no bandwidth is to be reserved for an optical carrier. 
     At the transmitter, an optical OFDM-like signal is generated, wherein each optical subcarrier is a linear combination of all other subcarriers, so that after direct detection, at each k-th sampling point, the electrical signal has a value proportional to the k-th subcarrier. 
     The transmitter may be implemented using a moderate amount of DSP, digital to analog converters (DACs) and one optical IQ modulator. No DSP is needed at the receiver (therefore also no ADCs) to demodulate the OFDM-like signal. The receiver maintains the complexity of its single-carrier counterpart. 
     It is also an option that after the electrical I- and Q-signals are generated, they can be up-mixed to an intermediate frequency by an electrical IQ-modulator. The output thereof can be used to modulate a laser diode via an optical amplitude modulator, e.g., a mach-zehnder modulator. Then, an optical filter can be used to filter one sideband of the optical signal. 
     As increased SE is desired, one attractive application would be to use this technique with DQPSK mapped subcarriers to form a 100 Gbps optical signal without PolMux and without DSP at the receiver. Instead, a simple DQPSK demodulator can be used at the receiver. In this case, the signal shows a 50 GHz bandwidth and therefore would be compatible with DWDM systems. 
     It is also an option to use PolMux and higher level modulation, therefore increasing the SE even further but retaining the complexity of a receiver equal to the case where direct detected single carrier is used. 
     Another advantage is that, as each subcarrier is available at the transmitter separately; simple one tap equalization can be performed for pre-compensating the channel transfer function. This allows, e.g., to send such a signal through a dispersion-unmanaged link (i.e., without dispersion compensation modules). 
     It is a further option of this approach to allow the receiver to perform simple subcarrier selection. Some detrimental effects on the link (i.e., narrowband filtering) may affect some subcarriers in a stronger way than others. Such bad subcarriers can be discarded easily by the receiver by just ignoring certain sample instants. Similar, dummy subcarriers can be used (containing no useful information or no information at all, i.e. always zero amplitude) to form guard bands that help maintaining the performance. 
     It is noted that the approach presented can be used with modulation formats that are in particular compatible with direct detection, e.g., OOK, DPSK, DQPSK, D8PSK, Star-D8QAM, Star-D16QAM, PAM, etc. 
     Here, DQPSK is an example for such modulation format. Other modulation formats may be applicable as well. 
       FIG. 1  shows a transmission scheme comprising a block diagram of a transmitter  101 , a channel  102  and a receiver  103 . 
     Transmitter: 
     The transmitter  101  according to  FIG. 1  will be described in detail.
     (1) A logical binary data sequence is input to an M*N serial to parallel module  104 . M depends on the modulation format: For OOK or BPSK, M equals 1, for QPSK, M amounts to 2, and so on (M-ary modulation).   (2) After parallelization each of the N sequences are mapped independently according to the desired modulation format (see block  105 ).
       The symbol duration (Ts) is equal to the inverse of the total data rate (Br) multiplied by (M*N).   A vector x is provided as an output of the block  105 .   
       (3) At each Ts instant the vector x composed by the N mapped DQPSK symbols is multiplied by a transform matrix T in a block  106  providing a vector Tx.   (4) Then in order to invert the channel response, one tap equalization (see block  107 ) is performed on each element of the vector Tx.   (5) Before performing the inverse DFT at the transmitter, as an option, zero padding (ZP) is used to ease the filtering of the images of the digital signal that causes aliasing (see block  108 ).   (6) After this block  108 , the resulting vector is multiplied by the inverse DFT matrix in a block  109 .   (7) The output vector of the block  109  is serialized in a block  110  and the real part and imaginary part of the sequence are fed to an optical IQ modulator  111 . As an alternative, an electrical IQ-modulator can be used as indicated above.   (8) The output signal of the optical IQ modulator  111  is fed over the channel  102  towards the receiver  103 .
 
The Transform Matrix
   

     The output of the transmitter  101  is an OFDM-like signal, each subcarrier is a combination of the data vector x. The matrix T is chosen so that the coefficients of the linear combination of each k-th subcarrier correspond to the k-th row of the DFT matrix. 
     The purpose of this linear combination is that at the k-th sampling instant the value of the optical signal is proportional to the k-th element of the vector x. Hence, squaring of the direct detector will only affect the k-th element. 
     The transform matrix T is dependent on the amount of zero padding (ZP). If the amount of ZP is equal or greater than N, then T is equal to the DFT matrix. If the zero padding is smaller than N then T correspond to a permutation of the rows of the DFT matrix. The permutation is dependent on the amount of ZP. 
     The Receiver 
     Advantageously, a legacy direct detection receiver  103  may be used depending on the modulation format of the subcarriers. In the case of DQPSK, two delay interferometers with balance detection may be used to separate and demodulate the I- and the Q-components of the optical signal. 
     Preferably, the delay time is matched to the symbol duration Ts (that is the duration of the “OFDM-like block”). 
     Sampling is done at a rate equal to the data rate divided by the modulation order (i.e. Br/M). 
     Details Regarding the Processing of Data 
     The following provides further details of the solution provided and explains its impact on the receiving side. 
     x is regarded as a vector comprising N elements, i.e. 
     
       
         
           
             
               
                 
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                   ( 
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     A transform matrix of N×N dimension is defined as 
     
       
         
           
             
               
                 
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     Further, t′ n  is a row of the matrix T, i.e.
 
 t′   n   =[t   n0   t   n1    . . . t   n(N−1) ].  (3)
 
     The transmitter in particular generates a signal as follows: 
     
       
         
           
             
               
                 
                   
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     As a first example, the vector x is an intensity modulated signal (ON-OFF-keying), i.e.
 
 x   n ε[0;1].  (5)
 
     A photodiode may be used at the receiver side to detect and demodulate the signal transmitted. The received sampled signal ŝ is sampled at sampling points 
     
       
         
           
             
               
                 
                   
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     By utilizing said photodiode the square of the magnitude of the optical signal is received, which (prior to the sampling phase) amounts to 
     
       
         
           
             
               
                 
                   
                     
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                   ( 
                   7 
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     After sampling, i.e. 
             t   =             k   ·   Δ     ⁢           ⁢   t     N     ⁢           ⁢   and   ⁢           ⁢     f   n       =     n     Δ   ⁢           ⁢   t               
the signal amounts to
 
                              s   ⁡     (         k   ·   Δ     ⁢           ⁢   t     N     )            2     =              ∑     n   =   0       N   -   1       ⁢       t   n   ′     ·   x   ·     ⅇ     j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢       n     Δ   ⁢           ⁢   t       ·         k   ·   Δ     ⁢           ⁢   t     N                    2       ,           (   8   )               
which corresponds to the inverse Fourier transform of T·x. Equation (8) can be rewritten in matrix form as follows:
 
                         s   ^     k     =              s   ⁡     (         k   ·   Δ     ⁢           ⁢   t     N     )            2     =       (       e   k     ·     W     -   1       ·   T   ·   x     )     ·       (       e   k     ·     W     -   1       ·   T   ·   x     )     *           ,           (   9   )               
wherein ( . . . )* denotes a transposed conjugated matrix and e k  identifies a k-th row of the identity matrix, e.g.,
 
 e   2 =[0 1 0 0 . . . 0].
 
     In addition, W −1  refers to an inverse discrete Fourier transform (DFT) matrix. 
     Equation (9) can be rewritten as follows:
 
 ŝ   k   =e   k   ·W   −1   ·T·x·x*·T*·W   −1   *·e*   k .  (10)
 
     The transform matrix T may be equal to the DFT matrix W (transforming with DFT). Thus, equation (10) results in
 
 ŝ   k   =e   k   ·x·x*·e*   k   (11)
 
and thus in
 
 ŝ   k   |x   k | 2 .  (12)
 
     Therefore, at the suitable sampling point 
               t   =         k   ·   Δ     ⁢           ⁢   t     N       ,         
if the transform matrix T equals the DFT matrix W, the k-th sampling value equals the k-th element of the vector x (magnitude square according to equation (12)).
 
     If x k  was intensity-modulated (i.e. x k  ε [0;1]), the signal |x k | 2  will also amount to either 0 or 1. 
     It is noted that the sampling points may be determined based on, e.g., a bit error rate. A suitable sampling point may thus correspond to an optimized or suitable bit error rate. Hence, the sample points may be chosen that allow for such an acceptable (or optimal) bit error rate. 
     It is further noted that the sampling points may be iteratively or dynamically adjusted by tracing the bit error rate. The receiver may hence at a given time check whether an adjustment of the sampling points result in an improved bit error rate and thus adjust the timing accordingly. 
     In addition to the OOK example described above, DQPSK modulation is another example that could be utilized, which in further detail is described hereinafter. 
     As shown with regard to OOK, the transmitter generates a signal s(t) pursuant to equation (4). Now, the vector x is DQPSK modulated, i.e. information is encoded in the phase difference of subsequent signals x k (t) and x k (t−Δt) for k=0, . . . , N−1. 
     A delay-interferometer plus balance detection can be used to obtain signals I(t) and Q(t) from the signal s(t). The signals I(t) and Q(t) received are sampled at sampling points 
                     t   =             k   ·   Δ     ⁢           ⁢   t     N     ⁢           ⁢   with   ⁢           ⁢   k     =   0       ,   …   ⁢           ,     N   -   1             (   13   )               
to obtain signal vectors Î and {circumflex over (Q)}.
 
     Prior to the sampling phase at the receiver, according to the transfer function (in time) of the demodulator, the signal received can be denoted as 
     
       
         
           
             
               
                 
                   
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                   14 
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     Hence, the signal I(t) is derived from the real part            { . . . }, wherein the signal Q(t) is accordingly derived from the imaginary part ¦{ . . . }.
     After the sampling phase, equation (14) results in 
     
       
         
           
             
               
                 
                   
                       
                   
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                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     With 
               φ   =     π   4       ,         
equation (14) can be rewritten in matrix form
 
 Î   k =           {( e   k   ·W   −1   ·T·x ( k ))( e   k   ·W   −1   ·T·x   φ ( k− 1))*}  (17)
 
and further to
 
 Î   k             {e   k   ·W   −1   ·T·x ( k )· x   φ *( k− 1)·T*·W −1   *·e*   k }.  (18)

     The transform matrix T may be equal to the DFT matrix W (transforming with DFT). Thus, equation (18) results in
 
 Î   k               {e   k   ·x ( k )· x   φ *( k− 1)· e*   k }  (19)
 
and thus in
 
 Î   k   =             {x   k ( k )· x*   kφ ( k− 1)},  (20)
 
which can be denoted as
 
 Î   k =cos(Φ k ( k )−Φ k ( k− 1)−φ),  (21)
 
which further equals the conventional DQPSK with

             Φ   =     [           Φ   0               Φ   1             ⋮             Φ     N   -   1             ]           
and
 
φ n  being the phase of {x n }.
 
     It is in particular noted that the transform matrix may consider channel characteristics that could be determined in advance to or during data processing. In this case, the transform matrix allows precoding of the data to be conveyed across such a channel in a way that the channel&#39;s distortions are at least partially compensated. Hence, noise and/or interference imposed on the channel, e.g., near end and/or far end cross talk, can (at least partially) be compensated. In addition, dispersion of an optical fiber could be compensated by the transform matrix. 
     In order to consider channel characteristics, the transform matrix may be
 
 T=H   D   −1   ·W,  
 
wherein H D   −1  denotes a diagonal matrix comprising the channel&#39;s characteristics.
 
     Common channel estimation techniques could be utilized to determine the characteristics of the channel. One example is a receiver that conveys information regarding the channel quality back to the transmitter (e.g., via a physical or logical feedback channel  113 ). In addition, loops can be used at the transmitter to determine crosstalk from adjacent fibers (channels). 
     However, such predistortion based on channel properties is an option and not necessarily required for the approach presented herein. Insofar, predistortion in the context of this document also comprises a mere transformation utilized by the transform matrix as described and does not require consideration of particular channel characteristics. 
       FIG. 2  shows another representation of the transmitter. A vector x  201  is generated by a modulator MOD with guard bands (GB) being inserted (see blocks  216 ,  217  and  218 ), wherein the vector x  201  pursuant to equation (1) is fed to a processing unit  206 , where it is transformed with the matrix T to {circumflex over (x)}=T·x. Furthermore, zero padding is conducted at a processing unit  207  with a vector z of dimension M×1. An output  202  amounts to 
     
       
         
           
             
               
                 
                   
                     x 
                     ~ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               x 
                               ^ 
                             
                           
                         
                         
                           
                             z 
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   x 
                                   ~ 
                                 
                                 0 
                               
                             
                           
                           
                             
                               
                                 
                                   x 
                                   ~ 
                                 
                                 1 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   x 
                                   ~ 
                                 
                                 
                                   N 
                                   + 
                                   M 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     This vector {tilde over (x)} as output  202  is input to an iDFT  208  as indicated by the square matrix W −1  of dimension (N+M)×(N+M). An output  203  of the block  208  can be denoted as 
     
       
         
           
             
               
                 
                   y 
                   = 
                   
                     
                       W 
                       
                         - 
                         1 
                       
                     
                     · 
                     
                       x 
                       ~ 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     with 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     y 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               0 
                             
                           
                         
                         
                           
                             
                               y 
                               1 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               y 
                               
                                 N 
                                 + 
                                 M 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Each element of the vector y can be written as 
                       y   k     =       ∑     n   =   0       N   +   M   -   1       ⁢         x   ~     n     ·     ⅇ     j   ⁢           ⁢   2   ⁢   π   ⁢     nk     N   +   M                 ,           (   25   )               
which stems from the fact that the matrix W −1  comprises elements
 
     
       
         
           
             
               
                 
                   
                     w 
                     ij 
                   
                   = 
                   
                     
                       ⅇ 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                           ij 
                           
                             N 
                             + 
                             M 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     The elements of the vector y can be converted into serial signals by a block  209  and further the imaginary part and the real part can each be serially input to a DAC  210 ,  211  at time intervals 
     
       
         
           
             
               
                 
                   t 
                   = 
                   
                     
                       
                         k 
                         · 
                         Δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                     
                       N 
                       + 
                       M 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
         
         
           
             with 0&lt;k&lt;N+M−1. 
           
         
       
    
     With 
                 f   n     =     n     Δ   ⁢           ⁢   t         ,         
the output signal  204  of the DAC  211  after a low pass filter  213  amounts to
 
     
       
         
           
             
               
                 
                   
                     
                       y 
                       ?? 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     ?? 
                     ⁢ 
                     
                       
                         { 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               + 
                               M 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 x 
                                 ~ 
                               
                               n 
                             
                             · 
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   f 
                                   n 
                                 
                                 ⁢ 
                                 t 
                               
                             
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     Thus, signal  204  comprises the imaginary part of the complex baseband (BB) representation of the signal that is going to be upconverted at an IQ modulator  214  to the optical carrier frequency provided by a laser diode LD  215 . 
     The baseband (BB) signal can be rewritten as follows: 
     
       
         
           
             
               
                 
                   
                     
                       y 
                       BB 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             x 
                             ~ 
                           
                           n 
                         
                         · 
                         
                           ⅇ 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               f 
                               n 
                             
                             ⁢ 
                             t 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           N 
                         
                         
                           N 
                           + 
                           M 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             x 
                             ~ 
                           
                           n 
                         
                         · 
                         
                           
                             ⅇ 
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 f 
                                 n 
                               
                               ⁢ 
                               t 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     If
 
 z   n =0∀ nε               *| 0&lt; n&lt;M− 1,  (30)
 
hence

             z   =     [         0           0           ⋮           0         ]           
is a vector comprising zeros. Accordingly,
 
 {tilde over (x)}   n =0 for  N&lt;n&lt;N+M.   (31)
 
     Therefore, 
                       ∑     n   =   N       N   +   M   -   1       ⁢         x   ~     n     ·     ⅇ     j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     f   n     ⁢   t           =   0           (   32   )               
and equation (29) results in
 
                         y   BB     ⁡     (   t   )       =       ∑     n   =   0       N   -   1       ⁢         x   ~     n     ·     ⅇ     j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     f   n     ⁢   t             ,           (   33   )               
and based on
 
 {tilde over (x)}   n   ={circumflex over (x)}   n  for 0&lt; n&lt;N− 1  (34)
 
equation (33) can be denoted as
 
                         y   BB     ⁡     (   t   )       =       ∑     n   =   0       N   -   1       ⁢         x   ^     n     ·     ⅇ     j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     f   n     ⁢   t             ,           (   35   )               
which equals
 
                         y   BB     ⁡     (   t   )       =       ∑     n   =   0       N   -   1       ⁢       T   n     ·   x   ·     ⅇ     j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     f   n     ⁢   t             ,           (   36   )               
being the signal to be upconverted by the modulator  214 .
 
     After being processed by the modulator  214  with an optical carrier of a frequency f, an optical output signal  205  amounts to 
     
       
         
           
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         T 
                         n 
                       
                       · 
                       x 
                       · 
                       
                         
                           
                             ⅇ 
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       f 
                                       c 
                                     
                                     + 
                                     
                                       f 
                                       
                                         n 
                                         ⁢ 
                                         
                                             
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           t 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     The spectrum of this optical output signal  205  is visualized by  FIG. 3 . 
     Further Advantages 
     The approach provided in particular bears the following advantages:
     (a) The spectral efficiency can be significantly increased utilizing an OFDM-like approach to be compatible with direct detection.   (b) No power is required for sending an optical carrier signal.   (c) A one tap pre-compensation is possible, permitting the transmission over dispersion uncompensated links.   (d) A simple subcarrier selection is possible in the receiver, e.g., by discarding samples, therefore allowing a selection of samples that suffered less from detrimental effects (i.e. use “good subcarriers”).   (e) Side information can be sent multiplexed in some subcarriers (pilot tone, training symbols, coding data, protocol data, etc.).   (f) There is no need for complex digital signal processing to be supplied with the receiver.   (g) The approach allows for inexpensive highly spectral efficient systems, e.g., 100 Gbps systems with direct detection, without polarization multiplexing and without DSP for DWDM systems.   (h) There is no need for a local oscillator in the receiver and no need to waste energy transmitting a pilot signal.   

     LIST OF ABBREVIATIONS 
     
         
         ADC Analog-to-Digital Converter 
         BER Bit Error Rate 
         Br Bit Rate 
         BW Bandwidth 
         CompSSB-OFDM Compatible Single Sideband OFDM modulation 
         DAC Digital-to-Analog Converter 
         DFT Discrete Fourier Transform 
         DOSM Digital Orthogonal Subcarrier Multiplexing 
         DPSK Differential Phase Shift Keying 
         DQPSK Differential Quaternary Phase Shift Keying 
         DSP Digital Signal Processing 
         DWDM Dense WDM 
         IDFT Inverse DFT 
         OFDM Orthogonal Frequency-Division Multiplexing 
         OOK On Off Keying 
         PolMux Polarization Multiplexing 
         QPSK Quaternary Phase Shift Keying 
         SE Spectral Efficiency 
         WDM Wavelength Division Multiplexing 
         ZP Zero Padding