Abstract:
A method of digitizing first and second signals in imperfect quadrature for obtaining characteristic parameters of the first signal comprises providing a first signal, the first signal comprising an inphase quasi-sinusoidal analog signal. The method comprises providing a second signal, the second signal comprising a quadrature signal. The method comprises digitizing the first signal at a sampling rate, thereby generating a first plurality of sets of digital signal waveform samples and digitizing the second signal at the sampling rate, thereby generating a second plurality of sets of digital signal waveform samples. The method comprises digitally processing successive first and second sets of digital signal waveform samples to generate continually updated digital characteristic parameters representing a characteristic behavior of the first signal.

Description:
THE FIELD OF THE INVENTION  
       [0001]     This invention relates generally to systems and methods for digitizing the phase of an analog signal. This invention relates more particularly to a system and method for continuously and accurately digitizing the phase progression of quasi-sinusoidal signals in quadrature based on digitals samples of their waveforms.  
       BACKGROUND  
       [0002]     Many existing phase detectors are analog in nature and have a limited dynamic range. Generally, such phase detectors generate an output voltage indicative of the phase difference between two oscillations that are close in frequency. The polarity of the output voltage indicates which oscillation is leading the other. The magnitude of the output voltage tends to be proportional to the phase difference. The dynamic range of such analog phase detectors is typically limited to one cycle in each direction. Digital phase detection is typically preferred for phase detection of dynamic ranges wider than 1 or 2 cycles.  
         [0003]     A prior method of phase digitizing that has very wide dynamic range is described in U.S. Pat. No. 5,663,666, entitled DIGITAL PHASE DETECTOR, by Chu and Sommer. Such a method can be used only on a signal operating within a very narrow frequency band, 100 ppm for example, such as a signal from a crystal oscillator. The method also requires a local oscillator operating at near coherence to the signal.  
         [0004]     Another prior method of phase digitizing involves time-stamping the zero-crossings of a signal, as described in “Phase Digitizing Sharpens Timing Measurements,” David Chu, IEEE Spectrum, July 1988, pp. 28-32. For precise results, such methods usually involve custom time-digitizer circuits, such as described in U.S. Pat. No. 5,166,959, entitled PICOSECOND EVENT TIMER, by Chu and Knotts. Phase digitizing techniques that involve time-stamping the zero-crossings of a signal are better suited for agile signals of high frequencies, where signal frequencies may change radically and suddenly, and many zero-crossings are available to generate time-stamp data. A penalty for such a wide-band approach is noise.  
         [0005]     In an interferometer arrangement, noise is usually generated from fluctuating beam alignment, turbulence, photodiodes, electronic amplification, and the light source itself. In noisy environments, unexpected spurious zero-crossings may occur due to multiple triggering of the same signal edge, causing a catastrophic failure in previous phase digitizing processes.  
         [0006]     In metrology of moving objects, signals are generally quasi-sinusoidal and of limited agility due to the physical inertia of objects being monitored. Frequency of the signal is proportional to the velocity of the object being monitored, and phase of the signal is proportional to the distance of travel. Because physical objects cannot instantaneously jump from one velocity to a much different velocity, the frequency of the signals changes relatively slowly.  
         [0007]     The frequency of the signal, although changing slowly, may traverse a wide range, including very low frequencies where the number of zero-crossings available for measurement may be at a premium. Also, the occurrences of zero-crossings are generally non-uniform. This non-uniformity may pose additional difficulty in ascertaining the “data age”—the time between event occurrence and the presentation of its measurement data. These factors render the zero-crossing approach not an optimum technique for phase digitizing for interferometry.  
         [0008]     A prior method of phase digitizing for interferometry uses block regression as described in U.S. Pat. No. 6,480,126 entitled PHASE DIGITIZER, by Chu, and assigned to Agilent Technologies, Inc. The described method based on linear regression over an entire time segment, and not just at the vicinity of a zero crossing, is effective in averaging out noise. However, the method cannot be used on a signal operating at a frequency within ±100 kHz. This frequency limit effectively places an upper limit on the velocity of the detected object when the object is moving away from the light source to avoid entering this frequency band.  
         [0009]     Therefore, there is a need for a phase digitizing system and method that employs digital signal processing for continuously generating noise-suppressed digital phase data representing the phase of an incoming analog signal, without the disadvantages of previous phase digitizing techniques.  
       SUMMARY  
       [0010]     One aspect of the present invention provides a method of digitizing first and second signals in imperfect quadrature for obtaining characteristic parameters of the first signal. The method comprises providing a first signal, the first signal comprising an inphase quasi-sinusoidal analog signal. The method comprises providing a second signal, the second signal comprising a quadrature signal. The method comprises digitizing the first signal at a sampling rate, thereby generating a first plurality of sets of digital signal waveform samples and digitizing the second signal at the sampling rate, thereby generating a second plurality of sets of digital signal waveform samples. The method comprises digitally processing successive first and second sets of digital signal waveform samples to generate continually updated digital characteristic parameters representing a characteristic behavior of the first signal. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0011]      FIG. 1  is a block diagram illustrating an exemplary embodiment of a heterodyne displacement measuring interferometer system.  
         [0012]      FIG. 2  is a diagram illustrating an exemplary embodiment of optics for generating a quadrature signal for the heterodyne displacement measuring interferometer system.  
         [0013]      FIG. 3  is an electrical block diagram illustrating an exemplary embodiment of a phase digitizer according to the present invention.  
         [0014]      FIG. 4  is an electrical block diagram illustrating the exemplary embodiment of the phase digitizer in greater detail. 
     
    
     DETAILED DESCRIPTION  
       [0015]     In the following Detailed Description, reference is made to the accompanying drawings, which form a part hereof, and in which is shown by way of illustration specific embodiments in which the invention may be practiced. In this regard, directional terminology, such as “top,” “bottom,” “front,” “back,” “leading,” “trailing,” etc., is used with reference to the orientation of the Figure(s) being described. Because components of embodiments of the present invention can be positioned in a number of different orientations, the directional terminology is used for purposes of illustration and is in no way limiting. It is to be understood that other embodiments may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The following Detailed Description, therefore, is not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims.  
         [0016]     A displacement measuring interferometry system including phase digitizing is described in this application. Quadrature signal generation from a heterodyne source for use in the interferometry system is also described. In addition, digitized signal processing of the quadrature signal including mathematical treatment of the process and the hardware for performing the process is described. Embodiments of the invention provide a heterodyne interferometer without a low frequency limitation. The quadrature signal allows the frequency to be positive, zero, or negative without hindering phase digitizing.  
         [heading-0017]     I. Displacement Measuring Interferometry System  
         [0018]     The phase digitizing system and method of the present invention is discussed in the context of a displacement measuring interferometry system. However, the phase digitizing techniques disclosed herein are also applicable to any other application in which it is desirable to continuously generate digital phase data representing the phase of an incoming analog signal.  
         [0019]     A typical displacement measuring interferometer system consists of a frequency-stabilized laser light source, interferometer optics and measuring electronics. In metrology based on homodyne interferometry, the phase progression function φ(t) is directly proportional to the object displacement in time, t, usually by the factor λ/4. That is, one unit interval (UI) change represents an object movement of one-quarter of the wavelength of the light wave. One UI represents one cycle of the light interference fringe, or 2π radians. In metrology based on heterodyne interferometry, there are two channels: one Doppler-shifted (Measurement Channel), and the other not shifted (Reference Channel). The difference between the two phase progression functions φ M (t) and φ R (t) of the two channels is proportional to the object displacement to within an arbitrary constant. The phase progression function for the reference channels is monotonically increasing with time. The phase progression function for the measurement channel increases with time only for positive frequencies, but decreases with time for negative frequencies.  
         [0020]      FIG. 1  is a block diagram illustrating a heterodyne displacement measuring interferometer system  100 . Interferometer system  100  includes laser  102 , interferometer  108 , measurement and processing electronics  112 , and quadrature generator  120 . Interferometer  108  includes stationary retroreflector  104 , polarizing beam splitter (PBS)  106 , and movable retroreflector  110 .  
         [0021]     Laser  102  generates a pair of collinear, orthogonally polarized optical beams of equal intensity and of different frequencies f1 and f2, which differ in frequency by F R , which is a reference frequency. The optical beams pass through interferometer  108 . Polarization beam splitter  106  reflects one polarization of the incoming light to stationary retroreflector  104 , and passes the other polarization of light to movable retroreflector  110 . Retroreflectors  104  and  110  return the light to polarization beam splitter  106 , where one beam is transmitted and the other beam is reflected, so that the two beams are again collinear and cobore. Linear motion of movable retroreflector  110  results in a corresponding change in the difference in phase between the two beams. The output beams from interferometer  108  are optically processed in quadrature generator  120  to produce two mixed beams  113  F M  (Inphase) and  114  Q M  (quadrature), both fluctuating in intensity coherently but out of phase. The frequency of fluctuation is in accordance with Doppler shifting of the split-frequency. Both beams are photo-detected and processed in measurement and processing electronics  112 . A third reference fluctuating beam F R    111 , not Doppler shifted, is phase digitized by a processor similar to one described in U.S. Pat. No. 6,480,126. Either mixed beam is referred to as the measurement signal, and the mixing is represented by the following Equation I: 
 
Measurement signal=f1{circle over (X)}f2  Equation I 
        where:     {circle over (X)} indicates a mixing operation; and     the underlining of f1 indicates that the signal is Doppler-shifted.        
 
         [0025]     Measurement and processing electronics  112  contain a photodetector that produces an electrical measurement signals corresponding to the optical measurement signals. The measurement signal has a frequency that is equal to the reference frequency F R  plus the Doppler shift frequency: 
 
F M =F R +nν/λ  Equation II 
        where:     ν is the velocity of the interferometer element whose position is being measured (the sign of ν indicates the direction of travel);     λ is the wavelength of light emitted from laser  102 ; and     n equals 2, 4, etc., depending on the number of passes the light makes through interferometer  108 .        
 
         [0030]     In the example system of  FIG. 1 , the movement of retroreflector  110  produces the Doppler shift and n is equal to 2. The reference signal is produced by mixing the two beams from laser  102  (f1 and f2), which is represented by the following Equation III: 
 
Reference Signal=f1{circle over (X)}f2  Equation III 
 
         [0031]     Measurement and processing electronics  112  contain a photodetector that produces an electrical reference signal corresponding to the optical reference signal. The reference signal has a frequency that is equal to the reference frequency F R .  
         [0032]     Measurement and processing electronics  112  measure the phase difference between the reference signal and the measurement signal, and process the difference to provide position and velocity outputs.  
         [0033]     Previous methods for determining and processing phase information employ analog techniques or digital techniques that involved time-stamping the zero-crossings of the signal, or techniques that are of limited frequency range. Embodiments of the present invention provide a more effective technique for generating digitized phase information for interferometry applications such as that shown in  FIG. 1 , as well as any other application where it is desirable to generate digital phase data representing the instantaneous phase of an incoming analog signal.  
         [0034]     One embodiment of the present invention is a method of continuously and accurately digitizing the phase progression of a quasi-sinusoidal signal based on digital samples of its waveform. When the signal comes from a Doppler-shifted light wave reflected from a moving object, possibly down-converted by an interferometer, the signal phase is directly proportional to the position of the object. Therefore, continuous signal phase monitoring is equivalent to continuous position monitoring of the object, accurate to a fraction of the light wavelength.  
         [0035]     Phase digitizing of an analog quasi-sinusoidal signal with a low frequency limit of ±100 kHz is described in U.S. Pat. No. 6,480,126, entitled PHASE DIGITIZER, issued Nov. 12, 2002 to Chu and assigned to Agilent Technologies, Inc., and is incorporated herein by reference. To overcome the low frequency limitation, a quadrature, or substantially quadrature signal is generated from light exiting interferometer  108 .  
         [0036]     In one aspect of the invention, a quasi-sinusoidal inphase signal of unknown and changing frequency, phase, and magnitude is digitized by a first analog-to-digital converter (ADC) at a regular rate greater than twice the bandwidth of the signal. At the same time, a quadrature signal for the inphase signal is digitized by a second ADC at the same rate as the inphase signal. The digitized data from both the inphase and quadrature signals is analyzed in 256-sample segments. For each 256-sample segment, a “best-fit” estimate of the inphase signal is generated of the form V*cos[2π(Freq*i−θ)], and a “best-fit” estimate of the quadrature signal is generated of the form U*sin[2π(Freq*i−θ+Δθ)], where i is an index for identifying consecutive digital signal samples within a segment, V and U represent magnitude estimates of the inphase and quadrature signals respectively, Freq represents a frequency estimate, θ represents a phase-offset estimate, and Δθ represents a phase error estimate.  
         [heading-0037]     II. Quadrature Signal Generation from a Heterodyne Source  
         [0038]     For quadrature detection of the measurement signal, a second, additional heterodyned signal is generated from the light exiting interferometer  108 . The second signal has a phase shift of 90 degrees from the first heterodyned signal. The 90 degree phase shift between heterodyned signals is accomplished by inducing a 90 degree phase shift between the f1 and f2 frequency components in the second beam. This signal is then treated exactly as the first mixed heterodyned signal. The beam is sent through a polarizer and the mixed heterodyned signal is sent to a second detector.  
         [0039]      FIG. 2  is a diagram illustrating the preferred embodiment of quadrature Generator  120 . The optics in quadrature generator  120  includes a non-polarizing beam splitter  132 , quarter wave plate  136 , and two polarizers  140  and  146 .  
         [0040]     The process for developing the two heterodyned signals is as follows. First, the beam exiting interferometer  108 , indicated at  130 , is split spatially into two beams,  134  and  144 , each with approximately equal amounts of f1 and f2 frequency components using non-polarizing beam splitter  132 . The f1 and f2 frequency components in beams  134  and  144  remain orthogonally polarized. In one embodiment, non-polarizing beam splitter  132  is a 50% non-polarizing beam splitter or other suitable non-polarizing beam splitter.  
         [0041]     Second, quarter wave plate  136  is inserted in the path of beam  134  such that its fast axis is located at 45 degrees to the orthogonally polarized f1 and f2 frequency components in beam  134 . Quarter wave plate  136  changes the orthogonally linearly polarized light beam  134  to orthogonally circularly polarized light beam  138 , which is right and left circularly polarized light.  
         [0042]     The preceding two steps have produced two beams,  138  and  144  from beam  130  exiting interferometer  108 . In beam  144 , the f1 and f2 frequency components are in orthogonal and linear polarization states. In beam  138 , the f1 and f2 frequency components are in orthogonal and circular polarization states.  
         [0043]     Third, both beams  138  and  144  pass through polarizers. Beam  138  passes through polarizer  140  and beam  144  passes through polarizer  146 . The polarizer axis of polarizer  146  is oriented at 45 degrees to the orthogonally polarized f1 and f2 frequency components of linearly polarized beam  144 .  
         [0044]     Mathematical treatment of the signal mixing for beam  144  is as follows. The nomenclature E 1  and E 2  is assigned to the two linearly polarized components of optical frequencies f1 and f2 respectively. For simplicity, the electric fields of the polarized beams are written in column vectors (Jones Vectors) as follows:  
               E   1     =     [           A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1       )                   0         ]             Equation   ⁢           ⁢   IV                 E   2     =     [         0             A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2     ⁢   t     +     δ   2       )                 ]             Equation   ⁢           ⁢   V             
 
         [0045]     These signals are projected onto polarizer  146  with its polarizer axis oriented at 45 degrees to the orthogonally polarized f1 and f2 frequency components of linearly polarized beam  144 .  
         [0046]     The Jones Matrix for a polarizer with its polarizer axis oriented at 45 degrees is:  
               P   45     =       (     1   2     )     *     [         1       1           1       1         ]               Equation   ⁢           ⁢   VI             
 
         [0047]     E 1 and E   2  pass through polarizer  146  and become E out  and E 2out , where:  
                     E     1   ⁢   out       =       (     1   2     )     *     [         1       1           1       1         ]     *     [           A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1       )                   0         ]                   =       (     1   2     )     *     [           A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1       )                     A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1       )                 ]                     Equation   ⁢           ⁢   VII                       E     2   ⁢   out       =       (     1   2     )     *     [         1       1           1       1         ]     *     [           A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2     ⁢   t     +     δ   2       )                   0         ]                   =       (     1   2     )     *     [           A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2     ⁢   t     +     δ   2       )                     A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2     ⁢   t     +     δ   2       )                 ]                     Equation   ⁢           ⁢   VIII             
 
         [0048]     The sum of E 1out  and E 2out  exiting polarizer  146  equals signal  148  as follows:  
                 E     1   ⁢   out       +     E     2   ⁢   out         =       (     1   2     )     *     [             A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1       )           +     A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2     ⁢   t     +     δ   2       )                         A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1       )           +     A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2     ⁢   t     +     δ   2       )                   ]               Equation   ⁢           ⁢   IX             
 
         [0049]     For circularly polarized beam  138 , the amount of phase difference between the f1 and f2 frequency components can be varied by the rotational orientation of the polarizer axis of polarizer  140 . If the axis of polarizer  140  is aligned with the fast axis of quarter wave plate  136 , there is no phase difference between the f1 and f2 frequency components. If the polarizer axis of polarizer  140  is oriented at 45 degrees to the fast axis of quarter wave plate  136 , the phase difference is 90 degrees. The general rule is that for every degree of rotation of the polarizer axis of polarizer  140  off from alignment to the fast axis of quarter wave plate  136 , the phase difference will increase (or decrease) by two degrees.  
         [0050]     Mathematical treatment of the signal mixing for beam  134  is as follows. The nomenclature E 3  and E 4  is assigned to the two linearly polarized components of optical frequencies f1 and f2 respectively. Beam  134  is changed into circularly polarized components of optical frequencies f1 and f2 respectively as follows:  
               E   3     =     [           A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1       )                   0         ]             Equation   ⁢           ⁢   X                 E   4     =     [         0             A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2       +     δ   2       )                 ]             Equation   ⁢             ⁢             ⁢   XI             
 
         [0051]     The Jones Matrix for quarter wave plate  136  with its fast axis set at 45 degrees is:  
               Q   45     =       (     1     2       )     ⁡     [         1       i           i       1         ]               Equation   ⁢           ⁢   XII             
 
         [0052]     The Jones Matrix for polarizer  140  with its polarizer axis set at 90 degrees is:  
               P   90     =     [         0       0           0       1         ]             Equation   ⁢           ⁢   XIII             
 
         [0053]     By multiplying equations X, XII, and XIII and equations XI, XII, and XIII, through transformation E 3  and E 4  become E 3out , and E 4out , as follows.  
         [0054]     For E 3out :  
               E     3   ⁢   out       =       (     1     2       )     *     [         0       0           0       1         ]     *     [         1       i           i       1         ]     *     [           A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1       )                   0         ]               Equation   ⁢           ⁢   XIV             
 
         [0055]     Equation XIV reduces to:  
               E     3   ⁢   out       =       (     1     2       )     *     [         0             A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1     +     π   /   2       )                 ]               Equation   ⁢           ⁢   XV             
 
         [0056]     For E 4out :  
               E     4   ⁢   out       =       (     1     2       )     *     [         0       0           0       1         ]     *     [         1       i           i       1         ]     *     [         0             A   ⁢           ⁢     ⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2     ⁢   t     +     δ   2       )                 ]               Equation   ⁢           ⁢   XVI             
 
         [0057]     Equation XVI reduces to:  
               E     4   ⁢   out       =       (     1     2       )     *     [         0             Aⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2     ⁢   t     +     δ   2       )               ]               Equation   ⁢           ⁢   XVII             
 
         [0058]     The sum of E 3out  and E 4out  exiting polarizer  140  equals signal  142  as follows:  
                 E     3   ⁢   out       +     E     4   ⁢   out         =       (     1     2       )     *     [         0               Aⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   1     ⁢   t     +     δ   1     +     π   /   2       )         +     Aⅇ     ⅈ   ⁡     (       2   ⁢   π   ⁢           ⁢     f   2     ⁢   t     +     δ   2       )                 ]               Equation   ⁢           ⁢   XVIII             
 
         [0059]     Equations IV-IX show that the heterodyne signal generated from combining E 1out  and E 2out  does not add additional phase to the mixed signal, but that combining E 3out  and E 4out  through equations X-XVIII does add a 90 degree phase shift between the f1 and f2 frequency components.  
         [0060]     Signals  142  and  148  are in imperfect quadrature. Signal  148  is referred to as the inphase signal and signal  142  is referred to as the quadrature signal. Quadrature signal  142  has a phase shift of 90 degrees to inphase signal  148 .  
         [heading-0061]     III. Digitized Signal Processing  
         [0062]      FIG. 3  is a block diagram illustrating an exemplary embodiment of phase digitizer  200 . Phase digitizer  200  digitally processes incoming inphase signal V  148  and quadrature signal U  142 . Phase digitizer  200  includes analog-to-digital converters (ADCs)  212  and  214 , digital signal processor  202 , and phase accumulator  208 . Phase digitizer  200  uses a block regression technique for phase digitizing in the steady state.  
         [0063]     Inphase signal V  148  is input to ADC  212  and quadrature signal U  142  is input to ADC  214 . ADC  212  is electrically coupled to digital signal processor  202  through data path  213  and ADC  214  is electrically coupled to digital signal processor  202  through data path  215 . Phase accumulator  208  is electrically coupled to digital signal processor  202  through path  209 . Digital signal processor  202  is electrically coupled to phase-accumulator  208  through path  203  comprising frequency update Freq  220  and phase correction θ cor    218 . Continuous phase output signal Phi(j)  216  is provided by digital signal processor  202 , latched at mid-segment by latch  258 .  
         [0064]     Inphase signal V  148  and quadrature signal U  142  in imperfect quadrature are processed using digital signal processing by phase digitizer  200 . ADC  212  samples inphase signal V  148  and ADC  214  samples quadrature signal U  142  and provides the output samples continuously to digital signal processor  202 . In one embodiment, ADCs  212  and  214  are 12-bit ADCs and sample inphase signal V  148  and quadrature signal U  142  at 80 MHz. In other embodiments, other suitable sampling rates can be used. In this embodiment, the samples are labeled as vectors V, for inphase signal V  148 , and U, for quadrature signal U  142 , each of length  256 . Each segment is therefore 256/80 MHz or 3.2 μs.  
         [0065]     Phase accumulator  208  approximates the signal phase progression φ(t i ) of incoming inphase signal V  148 , where the index “i” indicates an 80-MHz clock count value. Successive t i &#39;s are separated by τ, the period of 80 MHz.  
         [0066]     Phase digitizer  200  accurately digitizes the phase progression (in unit intervals UI) of inphase signal V  148  at the 3.2 μs rate regardless of the inphase signal V  148  frequency. The inphase signal V  148  frequency can be positive, negative, near or at zero, or anywhere within the overall measurement range of approximately ± 40  MHz.  
         [0067]     When the inphase signal V  148  frequency is above approximately +300 kHz or below −300 kHz (referred to as normal frequency range), one vector, V, is used for phase digitizing (as described in U.S. Pat. No. 6,480,126) by digital signal processor  202 . Under this normal frequency range, imperfections of U are measured and calibrated by digital signal processor  202 . Imperfections, including the magnitude ratio r=|V/U| and phase error Δθ (departure from 90° of quadrature signal U  142  to inphase signal V  148 ), are relatively constant. These parameters can be exported by digital signal processor  202  and used later in near-zero or at-zero frequency range to minimize errors.  
         [0068]     When the inphase signal V  148  frequency is between approximately −300 kHz and +300 kHz (referred to as low frequency range), including zero frequency, both vectors U and V are used for phase digitizing by digital signal processor  202 . However, by using calibration data of r and Δθ, exported from previous measurements under normal frequencies, the effect due to imperfections of U are corrected.  
         [0069]     The following is the mathematical formulation for phase digitizer  200 , including digital signal processor  202 . The mathematical model for inphase signal V  148  and quadrature signal U  142  at the 12.5 ns rate (one 80 MHz cycle) is: 
 
 V   i   =V  cos 2π( ft   i −θ)+noise  Equation XIX 
 
 U   i   =U  sin 2π( ft   i −θ+Δθ)+noise  Equation XX 
 
         [0070]     Inphase signal V  148  and quadrature signal U  142  are in imperfect quadrature because U≠V and Δθ≠0. After expansion of cosine and sine equations XIX and XX become: 
 
 V   i   =X   V  cos 2 πft   i   +Y   V  sin 2π ft   i +noise  Equation XXI 
 
 U   i   =X   U  sin 2 πft   i   +Y   U  cos 2 πft   i +noise  Equation XXII 
 
         [0071]     Two 256-length operation vectors E and D are defined, where: 
 
E=(e,e,e, . . . e)  Equation XXIII 
        where e=1 if 0.25≦ft i &lt;0.50; e=−1 if 0.75≦ft i &lt;1.0; else e=0. 
 
D=(d,d,d, . . . ,d)  Equation XXIV 
    where d=1 if 0≦ft i &lt;0.25; d=−1 if 0.50≦ft i &lt;0.75; else d=0.        
 
         [0074]     Four 256-length data vectors V, U, C, and S are defined, where: 
 
V=(V 1 ,V 2 , . . . ,V 256 ), containing the 256 samples of V.  Equation XXV 
 
U=(U 1 ,U 2 , . . . ,U 256 ), containing the 256 samples of U.  Equation XXVI 
 
C=(cos ft 1 , cos ft 2 , . . . , cos ft 256 ), containing cosine table values addressed by ft i .  Equation XXVII 
 
S=(sin ft 1 , sin ft 2 , . . . , sin ft 256 ) containing sine table values addressed by ft i .  Equation XXVIII 
 
         [0075]     At normal frequencies (i.e. outside ±300 kHz), the 512 equations can be reduced to four equations by E and D operating on V, U, C, and S: 
 
( D·V )= X   V ( D·C )+ Y   V ( D·S )  Equation XXIX 
 
( E·V )= X   V ( E·C )+ Y   V ( E·S )  Equation XXX 
 
( D·U )= X   U ( D·S )− Y   U ( D·C )  Equation XXXI 
 
( E·U )= X   U ( E·S )− Y   U ( E·C )  Equation XXXII 
 
         [0076]     At normal frequencies, computation of V, θ, r=|V/U|, and Δθ is based on 512 equations and four unknowns X V , Y V , X U  and Y U . Equations XXIX and XXX and equations XXXI and XXXII are both independent sets and can be used to solve separately for unknowns X V , Y V  and X U , Y U  respectively as follows:  
               [           X   V               Y   V           ]     =       [           (     E   ·   S     )           -     (     D   ·   S     )                 -     (     E   ·   C     )             (     D   ·   C     )           ]     ·     [           (     D   ·   V     )               (     E   ·   V     )           ]               Equation   ⁢           ⁢   XXXIII                 [           X   U               Y   U           ]     =       [           -     (     E   ·   C     )             (     D   ·   C     )               -     (     E   ·   S     )             (     D   ·   S     )           ]     ·     [           (     D   ·   U     )               (     E   ·   U     )           ]               Equation   ⁢           ⁢   XXXIV             
 
         [0077]     Using a four-quadrant actangent function, the parameter θ cor  (in UI) for tracking and phase digitizing signal V is computed as follows:  
               θ   cor     =       (     1     2   ⁢   π       )     ⁢     arctan   ⁡     (       X   V     ,     Y   V       )                 Equation   ⁢           ⁢   XXXV             
 
         [0078]     The calibration parameters r and Δθ (in UI) are computed as follows:  
             r   =            V   U          =           X   V   2     +     Y   V   2           X   U   2     +     Y   U   2                     Equation   ⁢           ⁢   XXXVI                 Δ   ⁢           ⁢   θ     =       (     1     2   ⁢   π       )     ⁢     Arctan   ⁡     (           X   U     ⁢     Y   V       -       X   V     ⁢     Y   U               X   V     ⁢     X   U       +       Y   V     ⁢     Y   U           )                 Equation   XXXVII             
 
         [0079]     The calibration parameters r and Δθ are averaged over several segments of 3.2 μs and exported. They are used when the signal frequency becomes low, (i.e. within ±300 kHz). To generate these parameters, the usual determinant for the inversion need not be explicitly computed.  
         [0080]     At low frequencies (i.e. within ±300 kHz), equations XXIX-XXXII may not be linearly independent. Both V and U cannot be computed separately with confidence. By using calibration factors r and Δθ, however, the necessary V and θ cor  can be accurately computed. The calibration factor r remains relatively constant even as V and U fluctuate. To equalize the magnitudes, the last two equations (Equations XXXI and XXXII) are multiplied by the calibration factor r exported. To account for the non-ideal skew of the two signals, Δθ is added to the address of the sine and cosine tables by U. These two steps effectively change (X u , Y u ) to (X v , Y v ) and create four sums (D·S) U , (D·C) U , (E·S) U , (E·C) U . Thus modified, equations XXXI and XXXII become: 
 
 r ( D·U )= X   V ( D·S ) u   −Y   V ( D·C ) u   Equation XXXVIII 
 
 r ( E·U )= X   V ( E·S ) u   −Y   V ( E·C ) u   Equation XXXIX 
 
         [0081]     There are two unknowns X V  and Y V  in equations XXIX, XXX, XXXVIII, and XXXIX. The four equations are combined to form two equations XL and XLI as follows for maximum independence: 
 
( D·V )+ r ·( E·U )= X   V [( D·C )+( E−S ) u   ]+Y   V [( D·S )−( E·C ) u ]  Equation XL 
 
( E·V )− r ·( D·U )= X   V [( E·C )−( D·S ) u   ]+Y   V [( E·S )+( D·C ) u ]  Equation XLI 
 
         [0082]     The solution to this 2-by-2-equation set in matrix notation is:  
               [           X   V               Y   V           ]     =       [             (     E   ·   S     )     +       (     D   ·   C     )     u                 (     E   ·   C     )     u     -     (     D   ·   S     )                     (     D   ·   S     )     u     -     (     E   ·   C     )               (     D   ·   C     )     +       (     E   ·   S     )     u             ]     ·     
     ⁢     [             (     D   ·   V     )     +     r   ·     (     E   ·   U     )                     (     E   ·   V     )     -     r   ·     (     D   ·   U     )               ]               Equation   ⁢           ⁢   XLII             
 
         [0083]     Finally, using a four-quadrant arctangent function, the low-frequency phase-digitizing parameter θ cor  (in UI) is:  
               θ   cor     =       1     2   ⁢   π       ⁢     arctan   ⁡     (       X   V     ,     Y   V       )                 Equation   ⁢           ⁢   XLIII             
 
         [0084]      FIG. 4  illustrates phase digitizer  200 , including digital signal processor  202  illustrated in  FIG. 3  in greater detail. Digital signal processor  202  includes arithmetic logic units (ALUs)  204   a - 204   l  (collectively referred to as ALUs  204 ), adder  260 , cosine table U  228 , cosine table V  230 , sine table U  232 , sine table V  234 , inverters  220  and  222 , digital processing block  210 , counter  254 , latch  258 , inverter  252 , and registers  256 .  
         [0085]     ADC  212  is electrically coupled to ALU E·V  204   a  and ALU D·V  204   b . ADC  214  is electrically coupled to ALU D·U  204   g  and ALU E·U  204   h . Cosine table U  228  is electrically coupled to ALU D·C U    204   i  and ALU E·C U    204   j . Cosine table V  230  is electrically coupled to ALU E·C  204   c  and ALU D·C  204   d . Sine table U  232  is electrically coupled to ALU D·S U    204   k  and ALU E·S U    204   l . Sine table V  234  is electrically coupled to ALU E·S  204   e  and ALU D·S  204   f.    
         [0086]     The outputs of ALUs  204  are electrically coupled to registers  256 . The output of adder  260  is electrically coupled to cosine table U  228  and sine table U  232 . The inputs of adder  260  are electrically coupled to digital processing block  210  through path  219  and the output of phase accumulator  210  through path  209 . The output of phase accumulator  208  is electrically coupled to cosine table V  230  and sine table V  234  through path  209 . The output of phase accumulator  208  is electrically coupled to the polarity (SGN) inputs of ALUs  204  through path  209  and inverter  220 . The output of phase accumulator  208  is electrically coupled to and the clock enable (CE) inputs of ALUs  204   a ,  204   c ,  204   e ,  204   h ,  204   j , and  204   l  through path  209  and the CE inputs of ALUs  204   b ,  204   d ,  204   f ,  204   g ,  204   i , and  204   k  through path  209  and inverter  222 . The output of phase accumulator  208  is electrically coupled to Phi(j) latch  258  through high speed phase output path  209 .  
         [0087]     Phi(j) latch  258  is electrically coupled to digital processing block  210  through path  259 . Registers  256  are electrically coupled to digital processing block  210  through path  257 . Clock signal  252  is input to modulo-2 8  counter  254 . Modulo-2 8  counter  254  is electrically coupled to Phi(j) latch  258  through inverter  252  and to registers  256  through path  255 .  
         [0088]     The twelve dot products used in equations XXIX through XLIII, (D·V), (D·U), (D·C), (D·S), (E·V), (E·U), (E·C), (E·S), (D·C) U , (D·S) U , (E·C) U , (E·S) U , are synthesized by hardware at high speed by ALUs  204  by the same names.  
         [0089]     In one embodiment, the digital circuits shown in  FIG. 4  are clocked synchronously at an 80 MHz rate. The clocking circuit is omitted from  FIG. 4  to simplify the illustration of the invention.  
         [0090]     ADCs  212  and  214  are 12-bit ADCs that digitize at 80 MHz the incoming inphase signal  148  from photodetector  270  of unknown magnitude, frequency, and phase and incoming quadrature signal  142  from photodetector  272 , respectively. In alternative embodiments, other sampling rates can be used. The output of ADC  212  is monitored simultaneously by two ALUs  204   a  and  204   b . The output of ADC  212  is represented by V from equation XXV. The output of ADC  214  is monitored simultaneously by two ALUs  204   g  and  204   h . The output of ADC  214  is represented by U from equation XXVI.  
         [0091]     In one embodiment, phase accumulator  208  is a 42-bit phase accumulator and approximates the signal phase progression φ(t i ) of incoming inphase signal  148 , where the index “i” indicates a clock count value. Successive t i &#39;s are separated by π, the period of 80 MHz. The most significant 25 bits of phase accumulator  208  represent the numbers of whole UI in φ(t i ), and the remaining 17 bits represent fractional UI in φ(t i ). The increment value of phase accumulator  208 , Freq, indicated at  220 , is the latest estimate of the signal frequency expressed in UI/π.  
         [0092]     In one embodiment, the most significant 8 bits of the fractional output of phase accumulator  208  are used to address cosine table V  230  and sine table V  234 . Tables  230  and  234  each span one complete period in the 8-bit address space. Therefore, there are 256 entries that span one period in each table  230  and  234 . In one embodiment, each entry in tables  230  and  234  is 10 bits wide. The output of cosine table V  230  is presented to ALU  204   c  and ALU  204   d . The output of cosine table V  230  is represented by C from equation XXVII. The output of sine table V  234  is presented to ALU  204   e  and ALU  204   f . The output of sine table V  234  is represented by S from equation XXVIII.  
         [0093]     The most significant 8 bits of the fractional output of phase accumulator  208  are modified by Δθ in adder  260  to create the dot products with U-suffixes. The output of adder  260  is used to address cosine table U  228  and sine table U  232 . Tables  228  and  232  each span one complete period in the 8-bit address space. Therefore, there are 256 entries that span one period in each table  228  and  232 . Each entry in tables  228  and  232  is 10 bits wide. The output of cosine table U  228  is presented to ALU  204   i  and ALU  204   j . The output of sine table U  232  is presented to ALU  204   k  and ALU  204   l . The output of cosine table U  228  and sine table U  232  is first used in equation XXXVIII.  
         [0094]     The two most significant bits of the fractional part of the output of phase accumulator  208  determine the quadrants and control the operations of the twelve ALUs  204 . The two most significant bits enable or disable the twelve ALUs  204  and assign the polarity of accumulation for the enabled units as shown in the following Table I:  
                       TABLE I                       bits   Name   Action                   00   1 st  quadrant   ALUs 204b, 204d, 204f, 204g, 204i, and 204k are               enabled to increment, ALUs 204a, 204c, 204e,               204h, 204j, and 204l are disabled       01   2 nd  quadrant   ALUs 204a, 204c, 204e, 204h, 204j, and 204l are               enabled to increment, ALUs 204b, 204d, 204f,               204g, 204i, and 204k are disabled       10   3 rd  quadrant   ALUs 204b, 204d, 204f, 204g, 204i, and 204k are               enabled to decrement, ALUs 204a, 204c, 204e,               204h, 204j, and 204l are disabled       11   4 th  quadrant   ALUs 204a, 204c, 204e, 204h, 204j, and 204l are               enabled to decrement, ALUs 204b, 204d, 204f,               204g, 204i, and 204k are disabled                  
 
         [0095]     The operation of Table I is represented by the operational vectors E and D, defined in equations XXIII and XXIV. The CE inputs of ALUs  204  are enabled when there is a need to add or to subtract and the CE inputs of ALUs  204  are disabled when E or D should do nothing (i.e. when e=0 or d=0).  
         [0096]     In one embodiment, the digitized data from ADCs  212  and  214  is analyzed in 256-sample segments. Modulo-2 8  counter  254  sequences the events in each 256-clock segment. At the negative transition of counter  254 , halfway into a segment, 16 bits of output of phase accumulator  208  (6 bits of whole UI and 10 bits of fractional UI) are latched by Phi(j) latch  258 . The latched value represents a temporary mid-segment value, Phi(j), which is held in reserve to be modified at the end of the segment. The letter “j” is an index for identifying segments.  
         [0097]     At the positive transition of counter  254  at the end of a segment, the outputs of the twelve ALUs  204  are latched into twelve registers  256 , omitting the 4 least significant bits. The latched values are E·V, D·V, E·C, D·C, E·S, D·S, D·U, E·U, D·Cu, E·Cu, D·Su, and E·Su, which are associated with ALUs  204   a ,  204   b ,  204   c ,  204   d ,  204   e ,  204   f ,  204   g ,  204   h ,  204   i ,  204   j ,  204   k , and  204   l , respectively. Immediately after the values are latched, all twelve ALUs  204  are reset to zero (reset circuit not shown for clarity purposes) so that ALUs  204  are ready for the next segment.  
         [0098]     The latched values of the twelve ALUs  204  are digitally processed by digital processing block  210  as shown in the following Equations XLIV through LVIII:  
         [0099]     The temporary mid-segment value Phi(j) latched by Phi(j) latch  258  is now corrected by a computed parameter θ cor  as follows: 
 
 Phi ( j )= Phi ( j )−θ cor   Equation XLIV 
 
         [0100]     Simultaneously, the current value φ of the phase accumulator  208  is also corrected by computed parameter θ cor  as follows: 
 
φ=φ−θ cor   Equation XLV 
 
         [0101]     The corrected Phi(j), together with 320 values from past segments, are stored in memory and exported as measured phase progression values. An updated frequency value, Freq, under steady state, is derived from the current value Phi(j) and one historical value Phi(j−1) recorded one segment ago. One embodiment of the formulation for the new steady-state Freq is: 
 
 Freq=[Phi ( j )− Phi ( j− 1)]/256  Equation XLVI 
 
         [0102]     Exporting Phi(j) completes the tracking and phase digitizing process, which is the same for all frequencies, including normal, low, positive, zero or negative frequency. However, how computed parameter θ cor  is computed depends on the signal frequency.  
         [0103]     As previously described, there are two basic modes of operation, including operation at normal frequencies and operation at low frequencies. No special consideration is necessary to handle positive and negative frequencies. The generation of dot products by ALUs  204  and phase tracking and correction by computed parameter θ cor  are the same for either mode. The following equations XLVII through LVIII show how the dot products from ALUs  204  are used to generate the phase correction computed parameter θ cor  in each mode during digital processing in digital processing block  210 .  
         [0104]     Normal Frequencies: Outside ±300 kHz (i.e. Freq Outside ±0.00375)  
         [0105]     In this mode, four dot products D·C U  from ALU  204   i , D·Su from ALU  204   k , E·Cu from ALU  204   j , and E·Su from ALU  204   l  are not used.  
         [0106]     Four intermediate parameters X v , Y v , X u , Y u  are derived from the remaining eight dot products as follows: 
 
 X   V =( E·S )( D·V )−( D·S )( E·V )  Equation XLVII 
 
 Y   V =( D·C )( E·V )−( E·C )( D·V )  Equation XLVIII 
 
 X   U =( D·C )( D·U )−( E·C )( E·U )  Equation XLIX 
 
 Y   U =( D·S )( E·U )−( E·S )( D·U)   Equation L 
 
         [0107]     The two quadrature calibration factors Δθ(in UI) and r (using the principal arctangent function) are computed from these intermediate parameters as follows:  
               Δ   ⁢           ⁢   θ     =       (     1     2   ⁢   π       )     ⁢     Arctan   [     (           X   U     ⁢     Y   V       -       X   V     ⁢     Y   U               X   U     ⁢     X   V       +       Y   V     ⁢     Y   U           )     ]               Equation   LI               r   =       [     V   U     ]     =           X   V   2     +     Y   V   2           X   U   2     +     Y   U   2                     Equation   LII             
 
         [0108]     Low Frequencies: Inside ±300 kHz (i.e. Freq Inside ±0.00375)  
         [0109]     At low frequencies, no new quadrature calibration factors r and AO are generated. Instead, the factors last produced are used to improve accuracy. The factor Δθ is added to the phase value φ by adder  260 . The sum (fractional part) is used to address cosine table U  228  and sine table U  232 . The results of these tables provide input to ALUs D·C U    204   i , E·C U    204   j  and D·S U    204   k , E·S U    204   l , respectively, as shown in  FIG. 4 . Previously generated parameter r is used, in conjunction with all 12 dot products, to produce two intermediate parameters X V  and Y V  as follows: 
 
 X   V =( E·S+D·C   U )( D·V+rE·U )+( E·C   U   −D·S )( E·V−rD·U )  Equation LIII 
 
 Y   V =( D·S   U   −E·C )( D·V+rE·U )+( D·C+E·S   U )( E·V−rD·U)   Equation LIV 
 
         [0110]     All Frequencies: Inside ±39.7 MHz (i.e. Freq Inside ±0.49625)  
         [0111]     Regardless of the computation of intermediate parameters X V  and Y V  in either the normal or low frequency mode, the phase correction computerized parameter θ cor  is computed using the four-quadrant arctangent function as follows:  
               θ   cor     =       (     1     2   ⁢   π       )     ⁢     arctan   ⁡     (       X   V     ,     Y   V       )                 Equation   ⁢           ⁢   LV             
 
         [0112]     In either normal or low frequency mode, the magnitude estimate of inphase signal V  148  is formally given by: 
 
 V ={square root}{square root over ( X   V   2   +   V   2 )}/ det   Equation LVI 
 
         [0113]     However, the computation for X V  and Y V , is signal frequency dependent as previously described. The computation of the determinant is also signal frequency dependent.  
         [0114]     For normal frequencies, the determinant is: 
 
 det =( E·S )( D·C )−( D·S )( E·C )  Equation LVII 
 
         [0115]     For low frequencies, the determinant is: 
 
 det =( D·C+E·S   U )( E·S+D·C   U )+( E·C   U   −D·S )( E·C−D·S   U )  Equation LVIII 
 
         [0116]     In one form of the invention, digital processing block  210  is implemented as a field programmable gate array (FPGA). In an alternative embodiment, digital processing block  210  is implemented as a DSP processor.  
         [0117]     In one embodiment, as soon as computation of θ cor  and Freq values is completed by digital processing block  210 , the increment value of phase accumulator  208  is modified by digital processing block  210  to Freq−θ cor  for one clock cycle, then to Freq for the next 255 clock cycles. The value of Freq, which is no larger than ½, should be carried to a precision of 17 bits.  
         [0118]     The above process is then repeated for the next 256-sample segment. Throughout the process, the clocked phase accumulator  208  output φ(t i ) serves as a good digitized representation of the phase progression of the incoming inphase signal  148  to 25 bits of whole numbers and 10 bits of fractional numbers.  
         [0119]     The embodiments of the invention described herein, including generating a quadrature signal and phase digitizing the quadrature signal, provide a heterodyne interferometer without a low frequency limitation. The quadrature signal allows the frequency to be positive, zero, or negative without hindering phase digitizing. At normal frequencies, the inphase signal is used to determine the phase progression and parameters representing the imperfections of the quadrature signal are measured and exported. At low frequencies, both the inphase and quadrature signals waveform samples and the exported parameters are used to determine the phase progression.  
         [0120]     Although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that a variety of alternate and/or equivalent implementations may be substituted for the specific embodiments shown and described without departing from the scope of the present invention. This application is intended to cover any adaptations or variations of the specific embodiments discussed herein. Therefore, it is intended that this invention be limited only by the claims and the equivalents thereof.