Abstract:
A high speed signal generator comprises a digital signal processing (DSP) block configured to process an input digital signal to generate in parallel a first digital sub-band signal having frequency components within a first spectral range and a second digital sub-band signal having frequency components within the first spectral range. A first Digital-to-Analog Converter (DAC) is configured to process the first digital sub-band signal to generate a first analog sub-band signal and a second DAC is configured to process the second digital sub-band signal to generate a second analog sub-band signal. A combiner is to combine the first analog sub-band signal and the second analog sub-band signal to generate an output analog signal having frequency components covering a substantially continuous spectral range from a lower frequency f 1  to a higher frequency f 2 . The substantially continuous spectral range is substantially wider than a width of the first spectral range, and the lower frequency f 1  is less than or equal to one-third of the higher frequency f 2 .

Description:
CROSS-REFERENCE TO PRIOR APPLICATIONS 
       [0001]    This application is a continuation of U.S. patent application Ser. No. 14/846,848 filed Sep. 7, 2015, which is a divisional of U.S. patent application Ser. No. 14/102,856, now U.S. Pat. No. 9,130,678, which is a divisional of U.S. patent application Ser. No. 12/692,065, now U.S. Pat. No. 8,693,876, which is based on, and claims benefit of, U.S. provisional patent application 61/146,901 filed Jan. 23, 2009, the entire contents of all of which are hereby incorporated herein by reference. 
     
    
     TECHNICAL FIELD 
       [0002]    The present invention relates generally to optical communication systems, and in particular to a high-speed signal generator. 
       BACKGROUND 
       [0003]    Referring to  FIG. 1 a   , in an optical communications system, a transmitter  2  typically comprises a signal generator  4  for converting a digital signal X(n) to be transmitted into a drive signal S(t) which drives a modulator  6  (such as, for example, an Mach-Zehnder modulator (MZM) so as to modulate a narrow-band optical carrier generated by a laser  8  to generate a corresponding optical channel signal, which may then multiplexed into an optical fiber link for transmission through the optical communications system to a receiver. Typically, the drive signal S(t) is a radio frequency (RF) analog electrical signal. In such cases, the signal generator  4  typically includes a digital signal processor (DSP)  10  cascaded with a digital-to-analog converter (DAC)  12 . The DSP  10  operates to process the digital signal X(n) to generate a corresponding digital drive signal X′(m) which is designed in accordance with the performance and operating requirements of the DAC  12 . The DAC  12  operates in a conventional manner to convert the digital drive signal X′(m) into the required analog RF drive signal S(t) for modulation onto the optical carrier. 
         [0004]    In simple cases, the processing implemented by the DSP  10  will typically include de-serializing a serial digital signal X(n) to generate the digital drive signal X′(m) as a parallel digital sample stream; adjusting a number of bits resolution of the digital drive signal X′(m) to match that of the DAC input; and adjusting a timing of the digital drive signal X′(m) to match the sample rate of the DAC. For example, consider an arrangement in which the signal generator  4  receives a serial digital signal X(n) comprising QPSK encoded data. In this case, the DSP  10  may deserialize the received serial bit stream to generate into a two-bit parallel digital drive signal X′(m) so that the DAC  12  will output a four-level analog drive signal S(t). In additional, the DSP  10  may adjust signal timing so that bits of the serial digital signal X(n) can be latched into the DSP  10  at a timing of the input digital signal X(n), and words of the two-bit parallel digital drive signal X′(m) are latched into DAC  12  at the sample rate Fs of the DAC  12 . 
         [0005]    In more complex arrangements, the DSP  10  may also apply a desired encoding scheme to the digital signal X(n). Various encoding schemes may be used as desired, including but not limited to On-Off Keying (OOK), Phase Shift Keying (PSK), Quadrature Phase Shift Keying (QPSK), Quadrature Amplitude Modulation (QAM) and Forward Error Correction (FEC). The DSP  10  may also apply a compensation function (i.e. pre-distortion) to the digital drive signal X′(m) to compensate impairments of the optical communications link, such as, for example, dispersion. 
         [0006]    As channel bit rates rise to 100 Giga-bits per second (Gb/s) and beyond, high speed signal generators are required to generate the necessary electrical drive signals for modulation. High speed signal generators may also be useful in other fields, such has high speed wireless communications. 
         [0007]    A limitation of known techniques is that, for a DAC having a sample rate of Fs, the frequency spectrum of the output signal S(t) consists of a baseband symmetrically distributed about DC (0 Hz) and extending to ±Fs/2, and corresponding harmonic bands symmetrically distributed about±nFs (where n is an integer). These harmonic bands are duplicates of the baseband, and so do not contribute to the bit-rate of the optical channel. For example, consider a scenario in which the DAC is supplied with a digital drive signal X′(m) defining a pure sinusoid having a frequency of fo. The frequency spectrum of the analog output of the DAC will contain baseband frequency components at±fo, and first harmonic frequency components at −Fs±fo and at Fs±fo, as may be seen in FIG. lb. In order to avoid aliasing, a low-pass filter characteristic  14  having a cut-off frequency of Fs/2 is typically used to attenuate the harmonic bands, leaving only the baseband components in the drive signal S(t) output from the signal generator. 
         [0008]    As is well known in the art, the analog drive signal S(t) output from the signal generator is inherently limited to frequency components between DC (0 Hz) and ½ of the sample rate Fs of the DAC. For example, a DAC having a sample rate Fs=10 Giga-Samples per second (GS/s) can generate an analog output signal S(t) with frequency components between DC and Fs/2=5 GHz. In a system that utilizes Nyquist sampling (2 samples per symbol), such a DAC would be suitable for use in a signal generator that operates at a maximum symbol rate of 5 Giga-baud (Gbaud). In the case of QPSK encoding (2 bits per symbol), this translates into a maximum channel bit rate of 10 Gb/s. In high speed optical communications systems, the sample rate of the DAC used to generate the analog drive signal may be the limiting factor in determining the maximum bit-rate of an optical channel. 
         [0009]    The highest speed commercially available DAC currently known to the inventors has a maximum sample rate of Fs=20 GS/s (available from Tektronix). This allows for the generation of an analog output drive signal with spectral content from DC to 10 GHz, and supports symbol rates of up to 10 Gbaud with Nyquist sampling. A high speed signal source is needed that can generate signals with spectral content from DC to at least 20 GHz, and further increases in spectral width are expected to be needed in the future. 
         [0010]    Techniques and high speed signal sources for generating high bandwidth signals remain highly desirable. 
       SUMMARY 
       [0011]    There are disclosed methods and techniques for generating high-speed signals. 
         [0012]    In one aspect of the invention, there is provided a high-speed signal generator. A digital signal processing (DSP) block generates a set of N (where N is an integer and N≧2) parallel digital sub-band signals, each digital sub-band signal having frequency components within a spectral range between 0 Hz and ±Fs/2. A respective Digital-to-Analog Converter (DAC) processes each digital sub-band signal to generate a corresponding analog sub-band signal, each DAC having a sample rate of Fs. A combiner combines the analog sub-band signals to generate an output analog signal having frequency components within a spectral range between 0 Hz and ±NFs/2. 
         [0013]    The present invention provides a method for generating an analog signal with frequency content from DC to NFs/2 (N≧2), using an arrangement of N identical, lower speed, synchronous DACs, a RF mixing circuit, and a RF summation circuit. The lower speed DACs produce content from DC to±Fs/2. After the arrangement of the mixer and summer, the output content will be from DC to±NFs/2. 
         [0014]    This method is suitable for generating pre-distorted signals in electrical domain compensation of optical dispersion and/or multi-level signals for higher modulation formats (M-ary PSK, M&gt;4, QAM) in high-bit rate optical communication systems (for example, &gt;200 Gb/s). 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]    Representative embodiments of the invention will now be described by way of example only with reference to the accompanying drawings, in which: 
           [0016]      FIGS. 1 a  and 1 b    schematically illustrate principal elements and operations of a signal generator known in the art; 
           [0017]      FIG. 2  schematically illustrates a signal generator in accordance with a representative embodiment of the present invention; 
           [0018]      FIGS. 3 a -3 e    show respective RF spectra at various points of the signal generator of  FIG. 2 . 
           [0019]      FIG. 4  schematically illustrates a signal generator in accordance with a second representative embodiment of the present invention; 
           [0020]      FIGS. 5 a -5 e    show respective RF spectra at various points of the signal generator of  FIG. 4 . 
           [0021]      FIG. 6  is a screen-print of a signal analyser showing a measured eye diagram of a signal generated using a signal generator in accordance with an embodiment of the present invention; 
           [0022]      FIG. 7  is a screen-print of a signal analyser showing a measured eye diagram of another signal generated using a signal generator in accordance with an embodiment of the present invention; 
           [0023]      FIG. 8  is a screen-print of a spectrum analyser showing a measured spectrum of the signal of  FIG. 7 ; 
           [0024]      FIGS. 9 a  and 9 b    are block diagrams schematically illustrating respective representative digital signal processors usable in embodiments of the present invention; 
           [0025]      FIG. 10  is a block diagram schematically illustrating a signal generator in accordance with a third representative embodiment of the present invention; and 
           [0026]      FIG. 11  is a block diagram schematically illustrating a signal generator in accordance with a fourth representative embodiment of the present invention. 
       
    
    
       [0027]    It will be noted that throughout the appended drawings, like features are identified by like reference numerals. 
       DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
       [0028]    The present invention provides a technique for generating high speed analog signals having a spectral range from DC to an integer multiple of one half of the DAC sample rate. Thus, in a system in which the DAC sample rate Fs is 10 GS/s, the output analog signal can contain frequency components between DC (0 Hz) and NFs/2, where N integer, N≧2. 
         [0029]    In very general terms, the systems in accordance with the present invention include a digital signal processor for generating a digital drive signal X′(m) in the form of N parallel digital sub-band signals v X [m], where x is an index. Each digital sub-band signal contains frequency components lying within a respective sub-band of the digital drive signal X′(m). The width of each sub-band preferably corresponds with one half of the DAC sample rate. Each digital sub-band signal is then converted to a corresponding analog sub-band signal S X [t] by a respective DAC. The resulting set of analog sub-band signals are then up-converted using respective mixers, with one sub-band not upconverted, and then all sub-bands combined together to yield an output analog signal S′(t) having frequency components spanning a desired spectral range. Anti-aliasing filters and phase delays between the respective sub-band signals can be used to mitigate interference between the analog sub-band signals as required by the implementation. The present invention will be described below with reference to embodiments in which it is desired to generate an output analog signal S(t) containing frequency components between DC (0 Hz) and Fs; that is, where N=2. Extensions of these techniques for the case of N&gt;2, will be apparent to those of ordinary skill in the art upon reading the present specification and drawings, and thus will only be briefly mentioned. 
         [0030]      FIG. 2  schematically illustrates a signal generator in accordance with a representative embodiment of the present invention. In the embodiment of  FIG. 2 , the signal generator  16  comprises a DSP  18  cascaded with a parallel pair of analog signal paths  20   A ,  20   B  which are respectively designed to generate a Low sub-band signal S A (t) and a High sub-band signal S B (t). The parallel signal paths  20   A ,  20   B  are connected to respective inputs of an analog signal combiner  22 , which sums the two sub-band signals S A (t) and S B (t) to generate the desired high-speed analog driver signal S′(t) at an output of the signal generator  16 . 
         [0031]    In the illustrated embodiment, the DSP  18  is configured to operate as a signal synthesizer which generates a digital drive signal X′(m) in the form of a pair of parallel digital sub-band signals v A [m] and v B [m]. The low band signal v A [m] contains frequency components of the digital drive signal X′(m) lying within a sub-band between DC (0 Hz) and Fs/2, while the high band signal v B [m] contains frequency components of the digital drive signal X′(m) lying within a sub-band between Fs/2 and Fs. In some embodiments, the digital drive signal X′(m) may be a periodic signal (in time) having a known sequence of bits or symbols, a known bit (or symbol) rate, and a known repetition period. Such a signal is commonly used as a test signal for evaluating performance of optical communications systems, for example. 
         [0032]    The low sub-band signal v A [m] is supplied to a respective low-band signal path  20   A  comprising a DAC  24   A  cascaded with an Anti-Aliasing filter (AAF A )  26   A . Referring to  FIGS. 2 and 3   a , the DAC  24   A  operates in a conventional manner to generate an analog low-band signal S A (t) corresponding to the low sub-band digital signal v A [m]. Accordingly, the analog low-band signal S A (t) will exhibit a baseband spectrum  28  centered on DC (0 Hz), as may be seen in  FIG. 3 a   . An Anti-aliasing filter (AAF A )  28   A , which may be configured as a conventional low-pass filter with a cut-off frequency corresponding to Fs/2, is provided to attenuate (harmonic) frequency components lying outside the desired base-band. Preferably, the AAF A    26   A  is also designed to exhibit a steep roll-off (for example &gt;20 dB/decade) so as to minimize overlap between residual out-of-band signal components of the analog low-band signal S A (t) and the high-band signal S B (t) in the combiner  22 . As may be appreciated, the low-band signal path  20   A  is closely similar to that of a conventional signal generator  4  ( FIG. 1 a   ), and operates in substantially the same manner. As such, the analog low-band signal S A (t) can support a maximum symbol rate corresponding to Fs/2. The High sub-band signal path  20   B  extends the spectral range of the signal generator  16  to support symbol rates of up to Fs. 
         [0033]    The high band signal v B [m] is supplied to a respective high-band signal path  20   B  comprising a DAC  24   B  cascaded with an Anti-Aliasing filter (AAF)  26   B  and a mixer  30 . Referring to  FIGS. 2 and 3   b , the DAC  24   B  operates in a conventional manner to generate an analog signal corresponding to the high sub-band digital signal v B [m]. Accordingly, the analog signal appearing at the output of the DAC  24   B  will exhibit a baseband spectrum  32  centered on DC, as may be seen in  FIG. 3 b   . The Anti-aliasing filter (AAF)  26   B , which may be configured as a conventional low-pass filter with a cut-off frequency corresponding to Fs/2, is provided to attenuate (harmonic) frequency components lying outside the desired base-band. Preferably, the AAF  26   B  is also designed to exhibit a steep roll-off (for example &gt;20 dB/decade) so as to minimize overlap between residual out-of-band signal components of the analog high-band signal S B (t) and the low-band signal S A (t) in the combiner  22 . The resulting filtered base-band signal is then supplied to an input of the mixer  30 . 
         [0034]    In the embodiment of  FIG. 2 , the mixer  30  receives a local oscillator (LO) signal having a frequency corresponding to Fs/2. If desired, the LO signal can be generated by dividing a clock signal by 2, as shown in  FIG. 2 . Alternatively, a conventional oscillator circuit (such as a Phase-Locked Loop) may be used to produce the LO signal having the desired frequency. The mixer  30  operates in a conventional manner to heterodyne the filtered base-band signal ( FIG. 3 b   ) with the LO signal to produce a radio frequency (RF) analog high-band signal S B (t) having a center frequency corresponding to that of the LO signal, which in this case lies at Fs/2. 
         [0035]    In some embodiments, the mixer  30  is provided as an image rejection mixer. As is known in the art, an image rejection mixer operates to reject one side-band from the RF output. In the embodiment of  FIG. 2 , this may be utilized by implementing the image rejection mixer such that only the positive side band of the filtered base-band signal ( FIG. 3 b   ) input to the mixer  30  appears at the mixer output. This yields a high sub-band analog signal S B (t) having a spectrum as illustrated in  FIG. 3 c   , in which the upper side band of the input signal has been up-converted to the frequency range extending from Fs/2 to Fs, and its complex conjugate frequency range extending from −Fs/2 to −Fs. 
         [0036]    In other embodiments, the mixer  30  is provided as a double side-band mixer. In this case, a high-pass filter (not shown) having a cut-off frequency at Fs/2 (and preferably a steep roll-off) can be provided at the output of the mixer to attenuate frequency components lying below Fs/2, and so yield an output spectrum closely similar to that illustrated in  FIG. 3   c.    
         [0037]    The analog signal combiner  22  can be provided as a conventional RF signal combiner to sum the two analog sub-band signals S A (t) and S B (t) to yield an analog drive signal S(t) containing frequency components within a spectral band from DC to Fs, as may be seen in  FIG. 3 d   . If desired, a further Anti-Aliasing Filter  34  can be provided at the output of the signal combiner  22  to remove out-of band noise to obtain the final analog drive signal S(t) shown in  FIG. 3   e.    
         [0038]    The above technique yields an analog driver signal S(t) having frequency components within a spectral range between DC (0 Hz) and Fs. Thus the signal generator of  FIG. 2  can be configured to support maximum transmission symbol rates approximately double that of a prior art signal generator that is constructed using exactly the same DAC. 
         [0039]    As may be appreciated, each of the analog sub-band signals S A (t) and S B (t) contain residual frequency components that extend beyond their respective desired spectral range. Accordingly, when the analog sub-band signals S A (t) and S B (t) are combined, these out-of-band frequency components will tend to interfere (particularly in a frequency band around±Fs/2) and produce small errors in the final analog drive signal S(t) ( FIG. 3 e   ). As noted above, this interference may be reduced, but not eliminated by using Anti-aliasing filters having a steep roll-off. Further mitigation of small errors may be obtained by providing a non-zero phase off-set between the analog sub-band signals S A (t) and S B (t) at the combiner input and correction by digital signal processing. In the embodiment of  FIG. 2 , an analog delay circuit  36  is provided in the High sub-band path  20   B  to impose a predetermined phase delay to the high sub-band signal S B (t). A clock delay  38  circuit is provided in the LO signal path to ensure synchronization between the LO signal and the filtered analog signal at the mixer input. If desired, these delay circuits  36 , 38  may be tuneable so that a desired phase relationship between the analog sub-band signals S A (t) and S B (t) can be established. In some embodiments, a phase offset of 90° between the analog sub-band signals S A (t) and S B (t) is established, however other phase offset values may be used, as desired. It will be appreciated that a non-zero the phase offset is beneficial in that it prevents amplification of errors due to constructive interference between the out-of band components of the two sub-band signals S A (t) and S B (t). 
         [0040]    As may be seen in  FIG. 3 , only the upper side-band  40  of the high sub-band signal S H (t) is up-converted and added to the low-sub-band signal S L (t) to obtain the final output signal S(t). This arrangement is advantageous in that the order of the frequency components is preserved through the mixer and combiner, which simplifies the processing required to generate the respective digital sub-band signals v A [m] and v B [m]. 
         [0041]    As may be appreciated, various methods may be used to generate the digital sub-band signals v A [m] and v B [m]. For example, U.S. Publication No. 2006/0127102, now U.S. Pat. No. 7,676,161, teaches methods and systems for controlling a transmitter capable of synthesizing an arbitrary optical E-field waveform. In this system, a complex digital signal processor computes a complex multi-bit digital representation of a desired optical E-field. This multi-bit digital representation is then used to synthesize a set of digital driver signals which, when converted to analog (by respective DACs) and supplied to an electrical-to-optical (E/O) converter such as a Mach-Zehnder modulator, will yield an optical signal at an output of the E/O converter that is a high fidelity reproduction of the desired E-field waveform. An advantage of this system is that the desired optical E-field can have any arbitrary spectrum, within the dynamic range of the analog signal paths and the E/O converter. A further advantage of this arrangement is that the synthesized analog drive signals are automatically adjusted for the electrical properties of the analog signal paths between the outputs of the DACs and the output of the E/O converter. 
         [0042]    These same techniques can be employed in the high speed signal generator of the present invention. In particular, the transmitter of U.S. Publication No. 2006/0127102 may be modified by replacing the analog signal paths and the E/O converter with the signal paths and combiner of the present signal generator. In this case, the complex digital signal processor computes a complex multi-bit digital representation of a desired analog driver signal S′(t) at the output of the combiner. This multi-bit digital representation is then used to synthesize a set of digital driver signals v A [m] and v B [m] which will yield a combiner output signal that is a high fidelity reproduction of the desired analog driver signal, taking into account both the response of the DACs and the electrical properties of the analog signal paths between the DACs and the output of the combiner. 
         [0043]      FIGS. 4 and 5  illustrate an alternative embodiment of the signal generator, in which the LO signal supplied to the mixer has a frequency corresponding to Fs. Referring to  FIG. 4 , the low-band signal path  20   A  is substantially identical to that of  FIG. 2 , and operates to generate an analog low-band signal S A (t) that exhibits a baseband spectrum  28  centered on DC, as may be seen in  FIG. 5 a   . An Anti-aliasing filter (AAF)  26   A , which may be configured as a described above with reference to  FIG. 2 , is provided to attenuate (harmonic) frequency components lying outside the desired base-band. 
         [0044]    The high-band signal path  20   B  is closely similar to that of  FIG. 2 , and supplies an analog filtered base-band signal ( FIG. 5 b   ) to an input of the mixer  30 , which operates in a conventional manner to heterodyne the filtered base-band signal ( FIG. 5 b   ) with the LO signal to produce an RF high-sub-band analog signal S B (t) signal having a center frequency corresponding to that of the LO signal. The arrangement of  FIG. 4  differs from the embodiment of  FIG. 2 , in that, in the embodiment of  FIG. 4  the LO frequency corresponds with Fs. For the case of a double sideband mixer, this operation yields a high-band signal S B (t) having a spectrum of the form illustrated in  FIG. 5 c   . The analog signal combiner  22  operates to sum the two analog sub-band signals S  A (t) and S B (t) to yield an analog drive signal S(t) containing frequency components within a spectral band from DC to 3Fs/2, as may be seen in  FIG. 5 d   . The Anti-Aliasing Filter (AAF)  34  provided at the output of the signal combiner  22  attenuates frequency components lying above Fs, to obtain the final analog drive signal S(t) shown in  FIG. 5   e.    
         [0045]    As may be appreciated, in the embodiment of  FIGS. 4 and 5 , the order of the frequency components within the high-band signal S B (t) are reversed, in that the lower side band  42  of the high-band signal S B (t) is coupled into the upper side band  44  of the drive signal S(t), while the upper side band  46  of the high-band signal S B (t) is coupled into the lower side band  48  of the drive signal S(t). This reversal of frequency components can be compensated by implementing a periodic convolution algorithm to compute the two sub-band digital signals v A [m] and v B [m]. A representative method for accomplishing this is described below. 
         [0046]    Consider the setup shown in  FIG. 4 , and neglecting the effects of the anti-aliasing filters. As described above, the clock signal at Fs is applied to both DACs and the mixer. The output of the signal generator is assumed to be periodic, and all signals in the time domain are considered to be real. 
         [0047]    Consider the low band path. For a pure sinusoid of frequency fo, the low sub-band digital signal v A [m] would be 
         [0000]    
       
         
           
             
               
                 
                   v 
                   A 
                 
                  
                 
                   [ 
                   m 
                   ] 
                 
               
               = 
               
                 cos 
                  
                 
                   ( 
                   
                     
                       
                         f 
                         0 
                       
                       · 
                       
                         
                           2 
                            
                           π 
                         
                         Fs 
                       
                     
                      
                     m 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
         [0000]    where m is the sample index, and Fs is the sample rate of the DAC. The analog signal output from an ideal DAC would be v A (t)=cos(2πf 0 t), where t is time. The output of a real DAC (i.e., one that does not have a perfect 
         [0000]    
       
         
           
             
               sin 
                
               
                   
               
                
               x 
             
             x 
           
         
       
     
         [0000]    reconstruction filter, is v A,R (t)=Σ n=−∞   ∞ v A [n]h(t−nT) 
         [0048]    where h(t) is the impulse response of the DAC, and Tis the sample time. Neglecting a scaling factor, the representation in the frequency domain of the DAC output will be, 
         [0049]      V   A,R ( f )=[Σ n=−∞   ∞   v   A ( f−F   S   k )] H ( f ) 
         [0050]    which is made up of copies of the ideal output, filtered by the response of the DAC. This is shown in  FIG. 1   b.    
         [0051]    The signal generator is required to generate frequency content from 0 to ±Fs. Within this range, it can be seen from  FIG. 1 b    that for a given frequency of fo, the output of the DAC will have spectral components at −Fs+f 0 , −f 0 , f 0  and Fs−f 0 . 
         [0052]    If the amplitude of the spectral component at f 0  is A, then the respective amplitudes of each of the four spectral components will be, 
         [0000]    
       
         
               
               
               
             
           
               
                   
                   
               
               
                   
                 Frequency 
                 Complex Amplitude 
               
               
                   
                   
               
             
             
               
                   
                 −Fs + f 0   
                 A  
               
               
                   
                 −f 0   
                 A* 
               
               
                   
                   f 0   
                 A  
               
               
                   
                   Fs − f 0   
                 A* 
               
               
                   
                   
               
             
          
         
       
     
         [0053]    where ( )* is the complex conjugate. The spectral components are filtered by the response of the DAC, and other subsequent electrical elements in the signal path, so that the complex amplitude of the corresponding frequency components in the Low-band analog signal S A (t) will be, 
         [0000]    
       
         
               
               
             
           
               
                   
               
               
                 Frequency 
                 Complex Amplitude 
               
               
                   
               
             
             
               
                 −Fs + f 0   
                 A H A (−Fs + f 0 ) 
               
               
                 −f 0   
                 A* H A (−f 0 ) 
               
               
                   f 0   
                 A H A (f 0 ) 
               
               
                   Fs − f 0   
                 A* H A (Fs − f 0 ) 
               
               
                   
               
             
          
         
       
     
         [0054]    where H A (f) is the total equivalent response of the low sub-band signal path. 
         [0055]    Up-stream of the mixer, the high sub-band signal path operates in a similar manner, so that the complex amplitude of the corresponding frequency components in analog signal input to the mixer will be, 
         [0000]    
       
         
               
               
             
           
               
                   
               
               
                 Frequency 
                 Complex Amplitude 
               
               
                   
               
             
             
               
                 −Fs + f 0   
                 B H B1 (−Fs + f 0 ) 
               
               
                 −f 0   
                 B* H B1 (−f 0 ) 
               
               
                   f 0   
                 B H B1 (f 0 ) 
               
               
                   Fs − f 0   
                 B* H B1 (Fs − f 0 ) 
               
               
                   
               
             
          
         
       
     
         [0056]    where B is the amplitude of the spectral component at f 0 , and H B1 (f) is the Fourier transform of the DAC impulse response and other components in the High sub-band signal path between the DAC and the input to the mixer. The mixer can be modeled as an ideal mixer, followed by filtering, H M (f). Therefore, the output of the mixer will be, 
         [0000]    
       
         
               
               
             
           
               
                   
               
               
                 Frequency 
                 Complex Amplitude 
               
               
                   
               
             
             
               
                 −Fs + f 0   
                 B H B (−f 0 ) H M (−Fs + f 0 ) 
               
               
                 −f 0   
                 B* H B (Fs − f 0 ) H M (−f 0 ) 
               
               
                   f 0   
                 B H B (−Fs + f 0 ) H M (f 0 ) 
               
               
                   Fs − f 0   
                 B* H B (−f 0 ) H M (Fs − f 0 ) 
               
               
                   
               
             
          
         
       
     
         [0057]    The important result from the mixing is that the order of the original amplitudes (i.e., B, B*) is maintained. Therefore, the filtering from the DAC impulse response, mixer, and other components in the high sub-band signal path can be replaced by a total equivalent filter, which for convenience will be denoted H B (f). Thus: 
         [0000]    
       
         
               
               
             
           
               
                   
               
               
                 Frequency 
                 Complex Amplitude 
               
               
                   
               
             
             
               
                 −Fs + f 0   
                 B H B (−Fs + f 0 ) 
               
               
                 −f 0   
                 B* H B (−f 0 ) 
               
               
                   f 0   
                 B H B (f 0 ) 
               
               
                   Fs − f 0   
                 B* H B (Fs − f 0 ) 
               
               
                   
               
             
          
         
       
     
         [0058]    Calculation of the Signals 
         [0059]    As all signals are real, the Fourier transforms are all conjugate symmetric, so the following discussion will reference only the positive frequencies. 
         [0060]    To calculate the required sub-band digital signals v A [m] and v B [m] for input to the two DACs, the following steps are carried out: 
         [0061]    The responses, H A (f) and H B (f) can be measured, using methods known in the art. In some embodiments, these responses are measured at each one of a plurality of frequencies where i is an index. These frequencies may correspond with respective tap frequencies of a Fast Fourier Transform (FFT) block. Thus, for example, the responses may be measured at each of 256 frequencies f i , i=0 . . . 255, corresponding with the tap frequencies of a 256-tap FFT block. 
         [0062]    As the desired output signal S(t) is periodic, the Fourier transform can be found easily using the FFT. In the frequency domain, the amplitude spectral pairs of the desired output signal S(f), are specified as follows (for the range f i =0 . . . Fs/2): 
         [0000]    
       
         
               
               
               
             
           
               
                   
                   
               
               
                   
                 Frequency 
                 Complex Amplitude 
               
               
                   
                   
               
             
             
               
                   
                 f i   
                 C LOW, i   
               
               
                   
                 Fs − f i   
                 C HIGH, i   
               
               
                   
                   
               
             
          
         
       
     
         [0063]    where C LOW,i  or C HIGH,i  are the complex amplitudes of the respective spectral components. 
         [0064]    The responses are found at the frequency pair, 
         [0000]    
       
         
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Frequency 
                 Response 
                 Response 
               
               
                   
                   
               
             
             
               
                   
                 f i   
                 H A (f i ) 
                 H B (f i ) 
               
               
                   
                 Fs − f i   
                 H A (Fs − f i ) 
                 H B (Fs − f i ) 
               
               
                   
                   
               
             
          
         
       
     
         [0065]    The total output is the sum of the two DAC output signal paths, 
         [0000]        C   LOW,i   =H   A ( f   i ) A   i   +H   B ( f   i ) B   i    
         [0000]        C   HIGH,i   =H   A ( Fs−f   i ) A*   i   +H   B ( Fs−f   i ) B*   i    
         [0066]    These two equations can be cast into matrix form, 
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         
                           H 
                           A 
                         
                          
                         
                           ( 
                           
                             f 
                             i 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           H 
                           B 
                         
                          
                         
                           ( 
                           
                             f 
                             i 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           H 
                           A 
                           * 
                         
                          
                         
                           ( 
                           
                             Fs 
                             - 
                             
                               f 
                               i 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           H 
                           B 
                           * 
                         
                          
                         
                           ( 
                           
                             Fs 
                             - 
                             
                               f 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
                
               
                 [ 
                 
                   
                     
                       
                         A 
                         i 
                       
                     
                   
                   
                     
                       
                         B 
                         i 
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       C 
                       
                         LOW 
                         , 
                         i 
                       
                     
                   
                 
                 
                   
                     
                       C 
                       
                         HIGH 
                         , 
                         i 
                       
                       * 
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0067]    So long as the 2×2 response matrix is invertible, then DAC controlled spectral amplitudes, A i  and B i  can be found as 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       A 
                       i 
                     
                   
                 
                 
                   
                     
                       B 
                       i 
                     
                   
                 
               
               ] 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             H 
                             A 
                           
                            
                           
                             ( 
                             
                               f 
                               i 
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             H 
                             B 
                           
                            
                           
                             ( 
                             
                               f 
                               i 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             H 
                             A 
                             * 
                           
                            
                           
                             ( 
                             
                               Fs 
                               - 
                               
                                 f 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             H 
                             B 
                             * 
                           
                            
                           
                             ( 
                             
                               Fs 
                               - 
                               
                                 f 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 
                   - 
                   1 
                 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         C 
                         
                           LOW 
                           , 
                           i 
                         
                       
                     
                   
                   
                     
                       
                         C 
                         
                           HIGH 
                           , 
                           i 
                         
                         * 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0068]    and the required digital sub-band signals v A [m] and v B [m] found by computing the Inverse Fast Fourier Transform (IFFT) of A i  and B i , respectively. For embodiments in which the desired output signal S(t) is a periodic signal having a known bit (or symbol) sequence and periodicity, these calculations can be implemented in the DSP using methods known in the art. 
         [0069]    It has been found that some values of the respective phase delays for the LO of the mixer, and the high sub-band path to the input of the mixer can cause small determinants for the 2×2 response matrix. The best scheme appears to be setting the time domain impulse response maximum for the high sub-band path 
         [0000]    
       
         
           
             1 
             
               2 
                
               
                   
               
                
               Fs 
             
           
         
       
     
         [0000]    after the maximum for the impulse response for the low sub-band path. 
         [0070]    Experimental Results 
         [0071]    A 30 GBaud (giga-symbol per second) periodic analog signal was generated and measured. This signal has four analog levels. A wideband RF amplifier with bandwidth suitable to 30 GBaud was used as an anti-aliasing filter.  FIG. 6  is a screen-print of a signal analyser showing the eye diagram measured at a receiver. 
         [0072]    A 28 GBaud periodic analog signal was generated and measured.  FIG. 7  is a screen-print of a signal analyser showing the eye diagram measured at a receiver. The spectral shape of the signal has been adjusted to give the best performance in the particular system under test, and so the eye appears more square than rounded.  FIG. 8  is a screen-print of a spectrum analyser showing the measured optical signal spectrum for the upper side band. As may be seen in  FIG. 8 , the spectrum exhibits a small harmonic at 20 GHz, which is due to LO feed-through from the mixer. The signal rolls-off to a minimum at 20 GHz, by design, and the low-pass filter removes aliased spectral content above 20 GHz. 
         [0073]    In the forgoing description, the desired output signal S(t) is a periodic signal having a known bit (or symbol) sequence and periodicity. Such a signal is commonly used for evaluating performance of network components and systems, for example by generating the eye diagrams of  FIGS. 6 and 7 . In cases where it is desired to transmit subscriber data traffic, the output signal S(t) is not periodic, and the sequence of symbols is not known in advance.  FIGS. 9 a  and 9 b    illustrate representative DSPs  18  for generating the digital sub-band signals v A [m] and v B [m] based on a subscriber data signal x[n] using the methods of the present invention. 
         [0074]    In the embodiment of  FIG. 9 a   , the DSP  18  comprises an encoding block  50  which receives and processes the subscriber data signal x[n] to generate a digital symbol stream X[m] to be transmitted. In some embodiments, the subscriber data signal x[n] may be a serial bit stream, but it could also be any type of digital signal such as, but not limited to a quantized signal. The encoding block  50  may implement any of a variety of algorithms including, but not limited to: encoding the subscriber data signal x[n] using a desired encoding scheme such as M-ary PSK or QAM; applying Forward Error Correction (FEC); and pre-distortion to compensate link impairments such as dispersion. In some embodiments, the symbols of the digital symbol stream X[m] may be complex valued symbols. 
         [0075]    During each clock cycle, a set of M/2 successive symbols output from the encoder block  50  are deserialized (at  52 ) to generate a parallel input vector {r NEW }. This input vector is combined with the input vector of the previous cycle {r OLD }  54 , and the resulting M-valued input array supplied to an FFT block  56 , which computes an array {R} representing the spectrum of the M-valued input array. The FFT output array {R} is then supplied to a frequency-domain processor (FDP)  58 , which implements the periodic convolution algorithm described above to generate corresponding sub-band arrays {A} and {B} containing the respective complex amplitudes of the spectral components for each digital sub-band signal. Each of the sub-band arrays {A} and {B} is processed using a respective IFFT block  60   A , 60   B  to generate corresponding M-valued output vectors {v A } and {v B }  62   A , 62   B . The low-band output vector {v A } can be divided into a pair of M/2-valued low sub-band vectors {v A   OLD } and {v A   NEW } respectively representing the sub-band signal v A [m] for the current and previous clock cycles. Similarly, the high-band output vector {v B } can be divided into a pair of M/2-valued high sub-band vectors {v B   OLD } and {v B   NEW } respectively representing the sub-band signal v B [m] for the current and previous clock cycles. Accordingly, the respective sub-band signals v A [m] and v B [m] for the current clock cycle can be obtained by serializing the respective sub-band vectors {v A   NEW } and {v B   NEW }, and discarding the vectors {v A   OLD } and {v B   OLD } for the previous clock cycle. 
         [0076]    If desired the resulting sub-band signals v A [m] and v B [m] can be retimed, for example by using a decimation function (not shown), to match the DAC symbol rate. 
         [0077]    An advantage of the arrangement of  FIG. 9 a    is that each input vector {r} is processed through the signal generator, between the input of the FFT and the output of the IFFT blocks, in each of two successive clock cycles. This effectively converts the digital symbol stream X[m] into a periodic signal, so that the cyclic convolution algorithm described above can be used. 
         [0078]    The DSP  18  of  FIG. 9 a    may be used in a signal generator of a high-speed optical transmitter. In this case, the output analog signal S(t) is used to drive an electrical-to-optical converter (not shown), and the encoding block  50  may also implement a pre-distortion to compensate non-linearities of the entire signal path between the DSP output and the output of the electrical-to-optical converter. In embodiments in which the desired output analog signal S(t) is a periodic signal having a known bit (or symbol) sequence and periodicity, an emulation of the DSP  18  of  FIG. 9 a    may be used to process the known bit (or symbol) sequence, represented as input signal x[n], to obtain the digital sub-band signals v A [m] and v B [m], which can then be saved in a memory. In this case, a signal generator  16  can be implemented using a simplified DSP  18  of a type shown in  FIG. 9 b   , in which a high-speed digital controller  64  operates to retrieve successive pre-computed samples of the digital sub-band signals v A [m] and v B [m] from memory  66  at the required sample rate Fs, and output the samples to the sub-band path DACs  24 . 
         [0079]    In the embodiments illustrated in  FIGS. 2 and 4 , the signal generator  16  has a low sub-band path  20   A  and a single high-sub-band path  20   B , which operate together to generate an output analog signal S(t) containing frequency components between DC (0 Hz) and Fs; that is, where N=2. Those of ordinary skill in the art will recognise that this architecture can be extended for N&gt;2, by adding additional high-band signal paths as needed.  FIG. 10  illustrates a generalized example, in which N sub-band paths  20  are provided. Each of the high sub-band paths  20   B - 20   N  includes a respective mixer  30  driven with an LO signal having a suitable frequency for up-converting its respective filtered baseband signal to an appropriate RF band. Such an arrangement can operate substantially as described above, but in this case, all of the analog sub-band signals are combined to obtain an output analog signal S(t) having frequency components within a spectral range extending between DC and 
         [0000]    
       
         
           
             
               
                 N 
                 · 
                 Fs 
               
               2 
             
             . 
           
         
       
     
         [0080]      FIGS. 9 a  and 9 b    show specific embodiments of the DSP  18 , which are configured to generate N=2 digital sub-band signals v A [m] and v B [m]. Both of these DSPs  18  can be modified to facilitate generation of N&gt;2 digital sub-band signals v x [m]. In view of the foregoing teaching, such modifications are considered to be within the purview of those of ordinary skill in the art, and thus will not be discussed in detail herein. 
         [0081]    In the forgoing embodiments, an RF combiner  22  is used to combine the analog sub-band signals, in the electrical domain, to obtain the output analog signal S(t). It will be appreciated, however, that other methods may be used to combine the analog sub-band signals. For example, the sub-band signals may be combined optically, as shown in  FIG. 11 . In this case, each analog sub-band signal is used to drive a respective optical modulator  64  (such as, for example, a Mach-Zehnder modulator). The desired combination of spectral ranges is obtained by separately modulating the optical carrier signal through the optical modulators in parallel, and then combining the resulting modulated optical signals. Naturally, the arrangement of  FIG. 11  may be readily extended, in a manner directly analogous to that described above with reference to  FIG. 10 , to the general case of a signal generator having N sub-band paths  20 . 
         [0082]    Although the invention has been described with reference to certain specific embodiments, various modifications thereof will be apparent to those skilled in the art without departing from the spirit and scope of the invention as outlined in the claims appended hereto.