Abstract:
Method and apparatus to determine scattering coefficients for a device under test (DUT) using a vector network analyzer (VNA) is disclosed. Traditionally, for a DUT having P ports, all combinations of reflective and transmission coefficients are measured and calculated. This is true even for reciprocal devices where S ijA =S jiA  because, during the measurement, the source and load matches vary. However, the present invention teaches that, for reciprocal devices, only one of the two transmission coefficients between a first port and a second port need be measured. Under the inventive technique, error terms are removed from the measured scattering coefficients. Then, the source and the load matches may be normalized to a normalization match value. The normalization process removes the differences of the source and the load matches. Accordingly, for reciprocal devices, only one of two reciprocal transmission coefficients need be measured to determine the transmission coefficients for both directions between the first and the second port. The reduction of the measurement requirement reduces the amount of hardware required for a VNA, time required to characterize a DUT, or both.

Description:
BACKGROUND 
     The present invention relates to vector network analyzers. More specifically, the present invention relates to the method and apparatus for efficiently measuring scattering parameters of multiport devices in vector network analysis. 
     Vector network analyzers (VNAs) are used to determine the transmission and reflection characteristics of various devices under test (DUTs). A transmission or a reflection characteristic is usually referred to as a scattering parameter S. In order to fully characterize a particular DUT, transmission characteristics and reflection characteristics for all combination of two ports are made. In general, for a P port device, this requires P 2  scattering parameters, or S ij  for each i and j where i goes from 1 to P and j goes from 1 to P. A scattering coefficient is represented as S ij  where S ij  represents reflection coefficient for port i when i=j and transmission coefficient from port j to port i when i≠j. For convenience, additional subscript M  is used to indicate a measured value, subscript C  for partial error corrected value (corrected for isolation and tracking errors), subscript N  for normalized value, and subscript A  for actual value. For example, S ijM  indicates measured transmission coefficient from port j to port i. 
     In order to fully characterize a device, the DUT, having P ports, the VNA performing the tests needs to either (1) have P or more receivers and make P sweeps of the DUT (hardware-intensive approach), or (2) have less than P receivers and make more than P sweeps (time-intensive approach). Here, a sweep may be defined as the process of sending out a source signal (the incident signal) from a given port and measuring detected signals from one or more ports which may include the source port. For example, to characterize a device, the DUT, having three ports (P=3), a VNA having 4 or more receivers, may make the sweeps listed in TABLE 1A. One receiver, Receiver  0 , to measure the incident signal, and the other three (Receivers  1 ,  2 , and  3 ) to measure the signal at each of the three ports. This would be the fastest approach but requires a VNA having the most hardware. 
     
       
         
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 1A 
               
             
             
               
                   
                   
               
               
                   
                 Scattering parameter measured 
               
               
                   
                 (Receiver 0 measure the incident signal) 
               
             
          
           
               
                   
                 Where P = 3 
                 Receiver 1 
                 Receiver 2 
                 Receiver 3 
               
               
                   
                   
               
               
                   
                 Sweep 1 
                 S 11   
                 S 21   
                 S 31   
               
               
                   
                 Sweep 2 
                 S 12   
                 S 22   
                 S 32   
               
               
                   
                 Sweep 3 
                 S 13   
                 S 23   
                 S 33   
               
               
                   
                   
               
             
          
         
       
     
     For any VNA, the minimum number of receivers needed is two—to first to the measure the incident signal and the other to measure the signal at a port. For a two-port VNA measuring a DUT having three ports, the sweeps listed in TABLE 1B are required. This would require the least amount of hardware within the VNA, but would be the slowest approach. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 1B 
               
               
                   
                   
               
               
                   
                   
                 Scattering parameter measured 
               
               
                   
                   
                 (Receiver 0 measure the incident signal) 
               
               
                   
                 Where P = 3 
                 Receiver 1 
               
               
                   
                   
               
             
             
               
                   
                 Sweep 1 
                 S 11   
               
               
                   
                 Sweep 2 
                 S 12   
               
               
                   
                 Sweep 3 
                 S 13   
               
               
                   
                 Sweep 4 
                 S 21   
               
               
                   
                 Sweep 5 
                 S 22   
               
               
                   
                 Sweep 6 
                 S 23   
               
               
                   
                 Sweep 7 
                 S 31   
               
               
                   
                 Sweep 8 
                 S 32   
               
               
                   
                 Sweep 9 
                 S 33   
               
               
                   
                   
               
             
          
         
       
     
     Alternatively, when using a VNA having 3 receivers, the VNA makes the sweeps listed in TABLE 2 to characterize a device having three ports. 
     
       
         
               
               
               
             
               
               
               
             
           
               
                   
                 TABLE 2 
               
             
             
               
                   
                   
               
               
                   
                 Scattering parameter measured 
                   
               
               
                   
                 (Receiver 0 measure the 
               
               
                   
                 incident signal) 
               
             
          
           
               
                 Where P = 3 
                 Receiver 1 
                 Receiver 2 
               
               
                   
               
               
                 Sweep 1 
                 S 11   
                 S 21   
               
               
                 Sweep 2 
                 S 11   
                 S 31   
               
               
                 Sweep 3 
                 S 22   
                 S 12   
               
               
                 Sweep 4 
                 S 22   
                 S 32   
               
               
                 Sweep 5 
                 S 33   
                 S 13   
               
               
                 Sweep 6 
                 S 33   
                 S 23   
               
               
                   
               
             
          
         
       
     
     For practical implementation of VNAs, there is a trade-off between the hardware-intensive solution (higher cost) and the time-intensive solution (slower process). On the one hand, for the hardware-intensive approach, it is costly to build, operate, and maintain additional hardware required for the large number of receivers. On the other hand, as for the time-intensive approach, it takes longer period of time to fully characterize a device because multiple sweeps are made. Further, the number of required sweeps grows very fast as P, the number of ports of the device, grows. In fact, for a VNA having two receivers, the number of sweeps required to fully characterize a device is P 2 . 
     There is a need for a technique and an apparatus to reduce the number of receivers, the number of sweeps, or both to fully characterize a device. 
     SUMMARY 
     These needs are met by the present invention. According to one embodiment of the present invention, a device under test (DUT) may be characterized by measuring transmission coefficient for transmission of signal from a first port to a second port, S ijM . That measurement may be used to determine the actual transmission coefficient of transmission of signal from the second port to the first port, S jiA , without having to measure the transmission coefficient of transmission of signal from the second port to the first port, S jiM . Of course, the measured signal, S ijM , may also be used to determine the actual transmission coefficient of transmission of signal from the first port to the second port 
     According to another aspect of the invention, an apparatus for characterizing a device under test (DUT) is disclosed. The apparatus has a processor and storage connected to the processor. The storage holds instructions for the processor to measure transmission coefficient, S jiM , for transmission of signal from a first port to a second port, and to determine, using that measurement, actual transmission coefficient, S ijA , for transmission of signal from the second port to the first port. 
     Yet another aspect of the invention, a machine readable storage is disclosed. The storage is encoded with instructions causing, when executed by a machine, to measure transmission coefficient, S jiM , for transmission of signal from a first port to a second port; and determine actual transmission coefficient, S ijA , for transmission of signal from the second port to the first port. 
     Other aspects and advantages of the present invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings, illustrating by way of example, the principles of the present invention. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 illustrates a flowchart outlining the technique of the present invention; and 
     FIG. 2 illustrates a device according to one embodiment of the present invention. 
    
    
     DETAILED DESCRIPTION 
     As shown in the drawings for purposes of illustration, the present invention is embodied in a technique of measuring transmission coefficient, S jiM , for transmission of signal from a first port to a second port. And, using that measurement to determine the actual coefficients S jiA  as well as for S ijA . This is applicable for reciprocal devices where S ijA =S jiA . For reciprocal devices, it is possible to reduce the number of transmission measurements required by a factor of two. This may be accomplished by using a single transmission measurement, S ijM , to determine two scattering parameters, S ijA  and S jiA , thus reducing the number of measurements required to characterize a DUT. Accordingly, the number of receivers (hardware), the number of required sweeps (time), or some combination of both hardware and time is also reduced. For the reciprocal devices, although the actual coefficients S ijA  and S jiA  are equal, the measured coefficients S ijM  and S jiM  are not equal because match varies at each test port depending upon how the ports are switched as to source match, load match, and the actual VNA port used. However, using the technique of the present invention, S ijM  may be normalized to represent both S ijM  and S jiM  and used in determining all of the DUT S-parameters. 
     Taking the Measurements 
     Referring to FIG. 1, a flowchart  100  outlining the technique of the present invention is illustrated. First, raw measurements are taken. Operation  102 . To determine the transmission coefficient of any two ports i and j where i≠j, only one of the two transmission coefficients need be measured. That is, to fully error correct all S-parameters, only one measurement need be made between each pair of ports i and j—either S ijM  or S jiM . Using the example used for TABLE 2 above (for a three port DUT measured with a VNA having three receivers) only three sweeps are necessary to fully characterize the DUT. The three sweeps are listed in TABLE 3A. This is because once a measurement for a transmission coefficient from a first port to a second port, S ijM , is taken, the measurement for a transmission coefficient from the second port to the first port, S jiM , is not necessary. 
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 3A 
               
             
             
               
                   
               
               
                 (CASE 1) 
               
             
          
           
               
                   
                 Scattering parameter measured 
                   
               
             
          
           
               
                 Where P = 3 
                 Receiver 1 
                 Receiver 2 
               
               
                   
               
               
                 Sweep 1 
                 S 11M   
                 S 21M   
               
               
                 Sweep 2 
                 S 22M   
                 S 32M   
               
               
                 Sweep 3 
                 S 33M   
                 S 13M   
               
               
                   
               
             
          
         
       
     
     Alternative measurements that may be made are listed in TABLE 3B. The alternative measurements exist because, for any port combination i and j, either S ij  or S ji  may be measured. 
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 3B 
               
             
             
               
                   
               
               
                 (CASE 2) 
               
             
          
           
               
                   
                 Scattering parameter measured 
                   
               
             
          
           
               
                 Where P = 3 
                 Receiver 1 
                 Receiver 2 
               
               
                   
               
               
                 Sweep 1 
                 S 11M   
                 S 31M   
               
               
                 Sweep 2 
                 S 22M   
                 S 12M   
               
               
                 Sweep 3 
                 S 33M   
                 S 23M   
               
               
                   
               
             
          
         
       
     
     Error Correction 
     Next, the measured values are partially error corrected to remove isolation errors and frequency response errors introduced by a VNA. Operation  104 . Imperfections in network analyzer hardware degrade measurement accuracy. The effect of some hardware imperfections can be characterized and removed from the measurements via vector error correction. Hardware imperfections corrected by this process include isolation errors (directivity for reflection measurements or crosstalk for transmission measurements), frequency response or tracking errors, and mismatch errors (due to source match or load match). These errors in multiport network analyzers can be characterized using methods established for two-port error correction. Once the errors are characterized they can be removed from the measurements. Here, the isolation error and frequency response errors are removed as follows: 
       S   ijC =( S   ijM   −X   ij )/ T   ij   (Equation 1) 
     where 
     S ijC  is the partial error corrected coefficient; 
     S ijM  is the raw, measured coefficient, 
     X ij  is isolation error; 
     T ij  is frequency response error; and 
     where i goes from 1 to P and j goes from 1 to P where P is the number of ports for the DUT. 
     Techniques to obtain actual values for the isolation error and the frequency response error for any port combination i and j are known in the art. For example, TABLE 4 lists a mapping between the characterized error terms from three two-port calibrations to a three-port error correction. This may be used for the three-port VNA illustrated in FIG.  2 . For other VNA configurations, the correlations and error correction values may vary as known in the art. 
     
       
         
               
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
               
                   
                 Calibration 
                 Calibration 
               
               
                   
                 between ports 1 
                 between ports 2 
               
               
                 Calibration between ports 1 and 2 
                 and 3 
                 and 3 
               
               
                   
               
             
             
               
                 Forward Directivity Error (E DF ) = 
                 E DF  = X 1,1   
                 E DF  = X 2,2 (PORT 3)   
               
               
                 X 1,1   
               
               
                 Forward Source Match Error (E SF ) = 
                 E SF  = G 1S   
                 E SF  = G 2S (PORT 3)   
               
               
                 G 1S   
               
               
                 Forward Reference Tracking Error 
                 E RF  = T 1,1   
                 E RF  = T 2,2 (PORT 3)   
               
               
                 (E RF ) = T 1,1   
               
               
                 Forward Crosstalk Error (E XF ) = X 2,1   
                 E XF  = X 3,1   
                 E XF  = X 3,2   
               
               
                 Forward Load Match Error (E LF ) = 
                 E LF  = G 3L (VIA   
                 E LF  = G 3L (VIA   
               
               
                 G 2L   
                 
                   SWITCH S0) 
                 
                 
                   SWITCH S1) 
                 
               
               
                 Forward Tracking Error (E TF ) = T 2,1   
                 E TF  = T 3,1   
                 E TF  = T 3,2   
               
               
                 Reverse Directivity Error (E DR ) = 
                 E DR  = X 3,3   
                 E DR  = X 3,3   
               
               
                 X 2,2(VIA PORT 1)   
               
               
                 Reverse Source Match Error (E SR ) = 
                 E SR  = G 3S   
                 E SR  = G 3S   
               
               
                 G 2S (PORT 1)   
               
               
                 Reverse Reference Tracking Error 
                 E RR  = T 3,3   
                 E RR  = T 3,3   
               
               
                 (E RR ) = T 2,2 (PORT 1)   
               
               
                 Reverse Crosstalk Error (E XR ) = X 1,2   
                 E XR  = X 1,3   
                 E XR  = X 2,3   
               
               
                 Reverse Load Match Error (E LR ) = 
                 E LR  = G 1L   
                 E LR  = G 2L   
               
               
                 G 1L   
               
               
                 Reverse Tracking Error (E TF ) = T 1,2   
                 E TF  = T 1,3   
                 E TF  = T 2,3   
               
               
                   
               
             
          
         
       
     
     Normalization 
     Then, the partial error corrected coefficients are match normalized. Operation  106 . The normalization modifies effective mismatch at each port to simulate a condition of having a constant mismatch at each port rather than the actual case where the mismatch at each port varies depending upon the S-parameter being measured. The effective port match at any port can be modified by mathematically adding a normalization match term 
     
       
         (G Nk −G Uk ) 
       
     
     where 
     G Nk  is a desired normalized match at port k; and 
     G Uk  is the current match at port k. 
     The description of measured S parameters as a function of actual S parameters and residual errors are established using the Mason&#39;s Rule. See Samuel J. Mason, “Feedback Theory—Further Properties of Signal Flow Graphs,” Proceedings of the IRE, vol. 44, no. 7, p. 920-926, Jul. 1956 (hereinafter the “Mason article”). The Mason article is incorporated here by reference. The Mason&#39;s rule also establishes the relationship between the normalized S parameters as a function of the partially corrected S parameters and the normalization match terms. 
     The normalization technique of adding the normalization match term (G Nk −G Uk , is applicable for DUTs having any number of ports. For the purposes of discussing the present invention, the normalization technique of the present invention is explained herein for a DUT having three ports characterized by a sample VNA  110  of FIG.  2 . The sample VNA and techniques as illustrated by the equations herein below are used herein to illustrate the present invention. 
     Referring to FIG. 2, the VNA  110  has a signal source  112 , a coupler  111  and three ports—Port  1 , Port  2 , and Port  3 . The signal source  112  supplies source signal to any of the three ports via switch S 0  or a combination of S 0  and S 1 . The source signal goes through the coupler  111  to be sampled by a first receiver  114 . Signals received by any of the three ports may be detected by a second receiver  115  or a third receiver  116  via switches S 2 , S 3 , S 4 , or a combination of these switches. 
     Note that each switch has resistors as terminating load paths for the paths not being used as to send or receive signals. When a path is not selected, it is terminated into internal resistors nominally equal to system characteristic impedance. In general, because each port may terminate into different resistors, the match terms will vary depending upon which resistor the line is terminating into. 
     Then, the normalization process requires the determination of the following equations. The normalized coefficients S ijN  are determined as follows: 
     
       
           S   ijN   =S   ijC /[1 −S   ijC ( G   jL   −G   jS )]  (Equation 2) 
       
     
     where 
     S ijN  is the normalized coefficient; 
     S ijC  is the partial error corrected coefficient; 
     G jL  is load match at port j; 
     G jS  is source match at port j; and 
     where i goes from 1 to P and j goes from 1 to P where P is the number of ports for the DUT. 
     For the sample DUT having three ports being measured using the sample VNA  110 , the Equation 2 (“Eq. 2”) is applicable for determining the normalized scattering coefficients having ports  2  or  3  as the source port, namely S 12N , S 22N , S 32N , S 13N , S 23N , and S 33N . As for normalized scattering coefficients having port  1  as the source port, the normalized coefficients S i1N  are determined as follows: 
       S   i1N   =[S   i1C +( S   i3N   S   31C   [G   3L(S1)   −G   3L(S0) ])]/[1 −S   11C (G 1L   −G   1S )]  (Eq. 3) 
     where 
     S i1N  is the normalized coefficient having source at port  1 ; 
     S i1C  is the partial error corrected coefficient having source at port  1 ; 
     G 3L(S1)  is load match at port  3  when terminated into Switch S 1 ; 
     G 3L(S0)  is load match at port  3  when terminated into Switch S 0 ; and 
     where i goes from 1 to P where P is the number of ports for the DUT. 
     For normalized scattering coefficients having port  1  as a the source port, Eq. 3 is required because normalizations are required for both ports one and three. This is because port three may be terminated into switch S 0  or switch S 1 . For the illustrated sample VNA of FIG. 2, TABLE 5A is the switching matrix lists all the switching options. For reflection measurements, the port connected to the other receiver is noted in parenthesis in TABLE 5A. 
     
       
         
               
               
               
             
           
               
                 TABLE 5A 
               
               
                   
               
               
                 Source at Port 1 
                 Source at Port 2 
                 Source at Port 3 
               
               
                   
               
             
             
               
                 S 11(2) ; Port 2 in S1, 
                 S 12 ; Port 1 in S0, Port 3 
                 S 13 ; Port 1 in S0, Port 2 
               
               
                 Port 3 in S0 
                 in S1 
                 in S1 
               
               
                 S 11(3) ; Port 2 in S1, 
               
               
                 Port 3 in S0 
                   
                   
               
               
                 S 21 ; Port 2 in S1, 
                 S 22(1) ; Port 1 in S0, 
                 S 23 ; Port 1 in S0, Port 2 
               
               
                 Port 3 in S0 
                 Port 3 in S1 
                 in S1 
               
               
                   
                 S 22(3) ; Port 1 in S0, 
                   
               
               
                   
                 Port 3 in S1 
                   
               
               
                 S 31 ; port 2 in S1, 
                 S 32 ; Port 1 in S0, Port 3 
                 S 33(1) ; Port 1 in S0, 
               
               
                 Port 3 in S0 
                 in S1 
                 Port 2 in S1 
               
               
                   
                   
                 S 33(2) ; Port 1 in S0, 
               
               
                   
                   
                 Port 2 in S1 
               
               
                   
               
             
          
         
       
     
     Equation 3 above is applicable for all devices. For reciprocal devices, not all transmission coefficients are required to be measured. Therefore, not all of the measured coefficients are available for application by the equations Eq. 2 and Eq. 3. This is because only one of reciprocal pairs, S ij  and S ji , is measured. Thus, depending on which of the reciprocal pairs are measured, and in the present example, also depending on which switch is used for the measurement, the match terms are taken into account. Reviewing two sample cases, TABLES 3A and 3B may be combined as 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 5B 
               
             
             
               
                   
                   
               
               
                   
                 Scattering parameters 
                   
                 Scattering parameter 
                   
               
               
                   
                 measured 
                   
                 measured 
               
               
                   
                 (CASE 1) 
                   
                 (CASE 2) 
               
             
          
           
               
                   
                 Receiver 1 
                 Receiver 2 
                 Receiver 1 
                 Receiver 2 
               
               
                   
                   
               
             
          
           
               
                 Sweep 
                 S 11M   
                 S 21M   
                 S 11M   
                 S 31M   
               
               
                 1 
               
               
                 Sweep 
                 S 22M   
                 S 32M   
                 S 22M   
                 S 12M   
               
               
                 2 
               
               
                 Sweep 
                 S 33M   
                 S 13M   
                 S 33M   
                 S 23M   
               
               
                 3 
               
               
                   
               
             
          
         
       
     
     For CASE 1, normalization equation Eq. 3 is applicable except for S 11  and S 21 . These are as follows: 
     
       
           S   11N   =[S   11C /(1 −S   11C   [G   1L   −G   1S ])]+[([ S   13N ] 2   [G   3L)S1)   G   3L(S0)])/( 1 +S   33N   [G   3L(S1)   −G   3L(S0) ])]  (Eq. 4) 
       
     
     and 
     
       
           S   21N   =[S   21C /(1 −S   11C   [G   1L   −G   1S ]) ]+[( S   32N   S   13N   [G   3L(S1)   −G   3L(S0) ])/(1 +S   33N   [G   3L(S1)   −G   3L(S0) ])]  (Eq. 5) 
       
     
     where 
     S 11N  is the normalized reflection coefficient for port  1 ; 
     S 11C  is the partial error corrected reflection coefficient for port  1 ; 
     G 1L  is load match at port  1 ; 
     G 1S  is source match at port  1 ; 
     S 13N  is the normalized transmission coefficient from port  3  to port  1 ; 
     S 33N  is the normalized reflection coefficient for port  3 ; 
     G 3L(S1)  is load match at port  3  when terminated at switch S 1 ; 
     G 3L(S0)  is load match at port  3  when terminated at switch S 0 ; 
     S 21N  is the normalized transmission coefficient from port  1  to port  2 ; 
     S 21C  is the partial error corrected transmission coefficient from port  1  to port  2 ; and 
     S 32N  is the normalized transmission coefficient from port  2  to port  3 . 
     Using the values gained from equations Eq. 2, 3, 4, and 5, the following matrix S N−1  of normalized scattering coefficients may be formed in Case 1:          S   NCase1     =     [           S     11      N             S     21      N             S     13      N                 S     21      N             S     22      N             S     32      N                 S     13      N             S     32      N             S     33      N             ]                            
     For CASE 2, normalization equation Eq. 3 is applicable except for S 11  and S 21 . These are as follows: 
     
       
           S   11N   [S   11C /(1 −S   11C   [G   1L   −G   1S ])]+[([ S   31C ] 2   [G   3L(S1)   −G   3L(S0) ][1 +S   33N ( G   3L(S1)   −G   3L(S0) )])/(1 −S   11C   [G   1L   −G   1S ]) 2 ]  (Eq. 6) 
       
     
     and 
     
       
           S   31N   =[S   31C (1 +S   33N   [G   3L(S1)   −G   3L(S0) ])]/[1 −S   11C ( G   1L   −G   1S )]  (Eq. 7) 
       
     
     where 
     S 11N  is the normalized reflection coefficient for port  1 ; 
     S 11C  is the partial error corrected reflection coefficient for port  1 ; 
     G 1L  is load match at port  1 ; 
     G 1S  is source match at port 1; 
     S 31C  is the partial error corrected transmission coefficient from port  1  to port  3 ; 
     S 31N  is the normalized transmission coefficient from port  1  to port  3 ; 
     S 31N  is the normalized transmission coefficient from port  1  to port  3 ; 
     S 33N  is the normalized reflection coefficient for port  3 ; 
     G 3L(S1)  is load match at port  3  when terminated at switch S 1 ; 
     G 3L(SO)  is load match at port  3  when terminated at switch S 0 ; 
     S 21N  is the normalized transmission coefficient from port  1  to port  2 ; 
     S 21C  is the partial error corrected transmission coefficient from port  1  to port  2 ; and 
     S 23N  is the normalized transmission coefficient from port  3  to port  2 . 
     Using the values gained from equations Eq. 2, 3, 6, and 7, the following matrix S N−2  of normalized scattering coefficients may be formed in Case 2:          S   NCase2     =     [           S     11      N             S     12      N             S     31      N                 S     12      N             S     22      N             S     23      N                 S     31      N             S     23      N             S     33      N             ]                            
     Using the normalized scattering coefficient matrix, the actual scattering coefficients may be determined as 
     
       
           S   A   =S   N ( I+G S   N ) −1   (Eq. 8) 
       
     
     where 
     S A  is a matrix of the actual scattering coefficients          [           S     11      A             S     12      A             S     13      A                 S     21      A             S     22      A             S     23      A                 S     31      A             S     32      A             S     33      A             ]     ;                          
     S N  is either S NCase1  or S NCase2  as described above for the sample VNA of FIG. 2, or, in general, a normalization matrix formed using Equations 2 and 3; is the identity matrix          [         1       0       0           0       1       0           0       0       1         ]     ;                          
     and 
     G is a normalized match matrix          [           G   N1         0       0           0         G   N2         0           0       0         G   N3           ]     .                          
     Equation Eq. 8 shows that even for reciprocal devices, S ijM  cannot be merely substituted as S ijM  to find S ijA  and S jiA . Rather, each S parameter is influenced by all the other S parameters. The normalization step allows the use of S ijM  to find both S ijA  and S jiA  thus take advantage of the reciprocity of the device under test. 
     Continuing to refer to FIG. 2, the VNA  110  may also include a processor  118  connected to the receivers  114 ,  115 , and  116  for reading the measured scattered values. The processor is preferably connected to storage  120 . Preferably the storage  120  is a programmable read only memory (PROM); however, the storage  120  may be any suitable machine readable device including, without limitation, random access memory (RAM), magnetic drive (a harddrive device), optical drive (CD-ROM), or a combination of these. The storage  120  may be encoded with instructions that, when executed by the processor  118 , cause the processor  118  to measure S-parameter S ijM , for of signal from a first port to a second port, and to determine actual coefficient, S ijA  from the second port to the first port. The first port may be the same or may be different from the second port. Preferably, the instructions implement the technique and the processes discussed herein above. The processor  120  may also be connected to an output device  122  to display the scattering coefficients. 
     From the foregoing it will be appreciated that the DUT characterization technique and apparatus of the present invention reduces hardware requirements for a VNA, reduces characterization time, or both. Although specific embodiments of the present invention have been described and illustrated, the invention is not to be limited to the specific forms or arrangements of parts so described and illustrated. For example, the DUT may have many ports, or the VNA may include any number of receiver circuits. Moreover, the technique, including without limitation the equations, as illustrated herein above may be modified for application for VNAs having varying number of receiver circuits. The invention is limited only by the claims.