Abstract:
A method of calculating a time difference is disclosed. The method includes receiving a first ultrasonic signal (r 21 ) having a first frequency from a first transducer (UT 2 ) at a first time and receiving a second ultrasonic signal (r 12 ) having the first frequency from a second ultrasonic transducer (UT 2 ) at a second time. The first ultrasonic signal and the second ultrasonic signal are sampled at a second frequency ( 302 ). The first sampled ultrasonic frequency is interpolated ( 306 ). The difference in travel time between the first and second ultrasonic signals is calculated in response to the interpolated first sampled ultrasonic signal and the sampled second ultrasonic signal (equation [43]).

Description:
BACKGROUND OF THE INVENTION 
       [0001]    This application claims the benefit of U.S. Provisional Application No. 61/713,385 (TI-72924PS), filed Oct. 12, 2012, and incorporated by reference herein in its entirety. 
     
    
       [0002]    Embodiments of the present invention relate to band pass analog-to-digital (ADC) sampling of ultrasonic signals to determine fluid velocity. 
         [0003]    Ultrasound technology has been developed for measuring fluid velocity in a pipe of known dimensions. Typically, these measurement solutions use only analog processing and limit the accuracy and flexibility of the solution. Ultrasound velocity meters may be attached externally to pipes, or ultrasound transducers may be places within the pipes. Fluid flow may be measured by multiplying fluid velocity by the interior area of the pipe. Cumulative fluid volume may be measured by integrating fluid flow over time. 
         [0004]      FIG. 1  illustrates an example of positioning ultrasonic transducers for fluid velocity measurement. There are many alternative configurations, and  FIG. 1  is just an example for the purpose of illustrating some basic equations for ultrasound measurement of fluid velocity. Two ultrasonic transducers UT 1  and UT 2  are mounted inside a pipe  100 , and a fluid is flowing through the pipe with velocity V. L is the distance between the ultrasonic transducers UT 1  and UT 2  and θ is the angle between the dashed line connecting the transducers and the wall of the pipe. Propagation time t 12  or time of flight (TOF) is the time for an ultrasonic signal to travel from UT 1  to UT 2  within the fluid Likewise, propagation time t 21  is the TOF for an ultrasonic signal to travel from UT 2  to UT 1  within the fluid. If C is the velocity of the ultrasonic signal in the fluid and V is the velocity of the fluid in pipe  100 , these propagation times are given by equations [1] and [2]. 
         [0000]    
       
         
           
             
               
                 
                   
                     t 
                     12 
                   
                   = 
                   
                     L 
                     
                       C 
                       + 
                       
                         V 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   1 
                   ] 
                 
               
             
             
               
                 
                   
                     t 
                     21 
                   
                   = 
                   
                     L 
                     
                       C 
                       - 
                       
                         V 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   2 
                   ] 
                 
               
             
           
         
       
     
         [0005]    The angle θ and the distance L are known, and the objective is to measure the fluid velocity V. If the velocity C of the ultrasonic signal in the fluid is known, then only the difference between propagation times t 12  and t 21  is needed. However, the velocity C is a function of temperature, and a temperature sensor may or may not be included based on the target cost of the measurement system. In addition, a flow meter may be used for different fluids such as water, heating oil, and gas. Measuring two different propagation times (t 12  and t 21 ) cancels the variability of C. Combining equations [1] and [2] yields equation [3] for the fluid velocity V. 
         [0000]    
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       L 
                       2 
                     
                     * 
                     
                       
                         
                           t 
                           21 
                         
                         - 
                         
                           t 
                           12 
                         
                       
                       
                         
                           t 
                           21 
                         
                          
                         
                           t 
                           12 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   3 
                   ] 
                 
               
             
           
         
       
     
         [0006]    Therefore, to determine fluid velocity without knowing the velocity of an ultrasonic signal in the fluid, measurement of two ultrasonic propagation times (t 12  and t 21 ) are needed. The present inventors have realized a need to improve measurement techniques in terms of cost and accuracy. Accordingly, the preferred embodiments described below are directed toward improving upon the prior art. 
       BRIEF SUMMARY OF THE INVENTION 
       [0007]    In a preferred embodiment of the present invention, a method of calculating a time difference is disclosed. The method includes receiving a first ultrasonic signal having a first frequency from a first transducer at a first time and receiving a second ultrasonic signal having the first frequency from a second ultrasonic transducer at a second time. The first and second ultrasonic signals are sampled at a second frequency. The first sampled ultrasonic signal is interpolated so it is aligned in time with the first sampled ultrasonic signal. A difference in travel time of the first and second ultrasonic signals is calculated in response to the interpolated first sampled ultrasonic signal and the sampled second ultrasonic signal. 
     
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
         [0008]      FIG. 1  is a diagram of a pipe with ultrasonic transducers for fluid flow measurement according to the prior art; 
           [0009]      FIG. 2  is a circuit diagram of an ultrasonic mixer based receiver of the present invention for measuring fluid flow; 
           [0010]      FIG. 3A  is a circuit diagram of an ultrasonic intermediate frequency (IF) sampling based receiver of the present invention for measuring fluid flow; 
           [0011]      FIG. 3B  is a circuit diagram of the signal processing circuit of  FIG. 3A ; 
           [0012]      FIG. 4  is graph illustrating Farrow cubic interpolation that may be used with the circuit of  FIG. 3B ; 
           [0013]      FIG. 5A  is diagram showing the excitation signal applied to a transmitting transducer for transmission to a receiving transducer as 1, 10, or 20 pulses; 
           [0014]      FIG. 5B  is diagram showing the signal at the receiving transducer in response to 1, 10, or 20 pulses at a transmitting transducer; 
           [0015]      FIG. 6  is graph showing accuracy of the measurement circuit of  FIG. 3A  compared to zero crossing and cross correlation measurement techniques; 
           [0016]      FIG. 7  is graph showing accuracy of the measurement circuit of  FIG. 3A  compared to zero crossing and cross correlation measurement techniques with different numbers of ADC bits and different numbers of transducer excitation pulses; 
           [0017]      FIG. 8  is graph showing accuracy of the measurement circuit of  FIG. 3A  for a 12-bit ADC having positive sampling frequency misalignment with respect to the optimal ADC sampling frequency; 
           [0018]      FIG. 9  is graph showing accuracy of the measurement circuit of  FIG. 3A  for a 12-bit ADC having negative sampling frequency misalignment with respect to the optimal ADC sampling frequency; and 
           [0019]      FIG. 10  is a table summarizing accuracy of the measurement circuit of  FIG. 3A  for various conditions. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0020]    The preferred embodiments of the present invention provide significant advantages of ultrasonic differential time of flight (TOF) measurement techniques in a fluid medium over methods of the prior art as will become evident from the following detailed description. 
         [0021]    Referring to  FIG. 2 , there is a mixer based measurement circuit of the present invention for measuring differential time of flight (δt) of ultrasonic signals in a fluid medium. Here and in the following discussion, some circuit functions may be realized in hardware, software, or a combination of hardware and software as will be apparent to one of ordinary skill in the art having access to the instant specification. This circuit advantageously converts ultrasonic transducer signals to a lower intermediate frequency (IF), thereby permitting a lower analog to digital converter (ADC) sampling rate. Referring back to  FIG. 1 , signal r 12  is the ultrasonic signal produced by transducer UT 1  and received from transducer UT 2  as given by equation 4 Likewise, signal r 21  is the ultrasonic signal produced by transducer UT 2  and received from transducer UT 1  as given by equation 5. The center frequency of the transmitting transducer is f C , and f(t) is the envelope of the received signal. 
         [0000]        r   12   =f ( t )sin(2π f   C   t )   [4]
 
         [0000]        r   21   =f ( t+δt )sin(2π f   C ( t+δt ))   [5]
 
         [0022]    The receiver transducer  200  of  FIG. 2  preferably receives the signals of equations [4] and [5] alternately so that transducer UT 1  transmits when transducer UT 2  receives, and transducer UT 2  transmits when transducer UT 1  receives. The received signals are amplified by programmable gain amplifier (PGA)  202  and applied to mixer circuits  204  and  206 . Mixer circuit  204  multiplies the received signal by the modulating signal sin(2π(f C +δf)t) and applies the resulting in phase signal to low pass filter (LPF)  208 . Mixer circuit  206  multiplies the received signal by the modulating signal cos(2π(fC+δf)t) and applies the resulting quadrature signal to low pass filter (LPF)  210 . Here, δf is a frequency error term of the mixer frequency with respect to the transducer center frequency. The output signals from in phase mixer  204  are given by equations [6] and [7]. The output signals from quadrature mixer  206  are given by equations [8] and [9]. 
         [0000]        r   I   12 ( t )= f ( t )sin(2 πf   C   t )sin(2π( f   C   +δf ) t )   [6]
 
         [0000]        r   I   21 ( t )= f ( t+δt )sin(2π f   C ( t+δt ))sin(2π( f   C   +δf ) t )   [7]
 
         [0000]        r   Q   12 ( t )= f ( t )sin(2 πf   C   t )cos(2π( f   C   +δf ) t )   [8]
 
         [0000]        r   Q   21 ( t )= f ( t+δt )sin(2 πf   C ( t+δt ))cos(2π( f   C   +δf ) t )   [9]
 
         [0023]    The output signals from LPF  208  are given by equations [10] and [11]. The output signals from LPF  210  are given by equations [12] and [13]. Here, the signal pair of equations [11] and [13] is not a delayed version of the signal pair of equations [10] and [12]. By way of contrast, the received signal of equation [5] is a delayed version of the signal of equation [4]. 
         [0000]        {tilde over (r)}   I   12 ( t )= f ( t )sin(2 πδft )   [10]
 
         [0000]        {tilde over (r)}   I   21 ( t )= f ( t+δt )sin(2π( f   C   δt+δft ))   [11]
 
         [0000]        {tilde over (r)}   Q   12 ( t )= f ( t )cos(2 πδft )   [12]
 
         [0000]        {tilde over (r)}   Q   21 ( t )= f ( t+δt )cos(2π( f   C   δt+δft ))   [13]
 
         [0024]    Analog to digital converter (ADC)  212  converts the analog signals from LPF  208  (equations [10] and [11]) to digital signals and applies them to signal processing circuit  216  Likewise, ADC  214  converts the analog signals from LPF  210  (equations [12] and [13]) to digital signals and applies them to signal processing circuit  216 . Processing circuit  216  is preferably a digital signal processor and estimates the differential TOF (60 according to equation [14]. 
         [0000]    
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     t 
                   
                   = 
                   
                     
                       a 
                        
                       
                           
                       
                        
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 Q 
                                 21 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 I 
                                 21 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       a 
                        
                       
                           
                       
                        
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 Q 
                                 12 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 I 
                                 12 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   14 
                   ] 
                 
               
             
           
         
       
     
         [0025]    Referring now to  FIG. 3A , there is a circuit diagram of an ultrasonic intermediate frequency (IF) sampling based receiver of the present invention for measuring fluid flow differential time of flight (δt) of ultrasonic signals in a fluid medium. Here and in the following discussion, the same reference numerals are used to indicate substantially the same circuit elements. The receiver transducer  200  of  FIG. 3A  preferably receives the signals of equations [4] and [5] alternately so that transducer UT 1  transmits when transducer UT 2  receives, and transducer UT 2  transmits when transducer UT 1  receives. The received signals are amplified by programmable gain amplifier (PGA)  202  and applied to anti aliasing filter  300 . The output signal from anti aliasing filter  300  is then applied to analog-to-digital converter (ADC)  302 . This embodiment of the present invention advantageously samples the received signal from anti aliasing filter  300  at an intermediate frequency (IF) so that ADC  302  may sample at a lower frequency as given by equation [15]. The corresponding sample time is given by equation [16]. Here, K is an integer indicating the sampling frequency, and N is an integer indicating the N th  sample. 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                     SAMP 
                   
                   = 
                   
                     
                       f 
                       C 
                     
                     
                       
                         1 
                         4 
                       
                       + 
                       
                         K 
                         2 
                       
                     
                   
                 
               
               
                 
                   [ 
                   15 
                   ] 
                 
               
             
             
               
                 
                   
                     t 
                     SAMP 
                     N 
                   
                   = 
                   
                     N 
                     
                       f 
                       SAMP 
                     
                   
                 
               
               
                 
                   [ 
                   16 
                   ] 
                 
               
             
           
         
       
     
         [0026]    ADC  302  preferably alternately produces the sample signals of equations [17] and [18] at a sampling rate determined by equations [15] and [16]. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       r 
                       12 
                     
                      
                     
                       ( 
                       N 
                       ) 
                     
                   
                   = 
                   
                     
                       f 
                        
                       
                         ( 
                         
                           t 
                           SAMP 
                           N 
                         
                         ) 
                       
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             
                               f 
                               C 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     N 
                                      
                                     
                                       ( 
                                       
                                         
                                           1 
                                           / 
                                           4 
                                         
                                         + 
                                         
                                           K 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     f 
                                     C 
                                   
                                 
                                 + 
                                 
                                   t 
                                   off 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   17 
                   ] 
                 
               
             
             
               
                 
                   
                     
                       r 
                       21 
                     
                      
                     
                       ( 
                       N 
                       ) 
                     
                   
                   = 
                   
                     
                       f 
                        
                       
                         ( 
                         
                           
                             t 
                             SAMP 
                             N 
                           
                           + 
                           
                             δ 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         ) 
                       
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             
                               f 
                               C 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     N 
                                      
                                     
                                       ( 
                                       
                                         
                                           1 
                                           / 
                                           4 
                                         
                                         + 
                                         
                                           K 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     f 
                                     C 
                                   
                                 
                                 + 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                                 + 
                                 
                                   t 
                                   off 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   18 
                   ] 
                 
               
             
           
         
       
     
         [0027]    Equations [17] and [18] are simplified and rewritten as equations [19] and [20]. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       r 
                       12 
                     
                      
                     
                       ( 
                       N 
                       ) 
                     
                   
                   = 
                   
                     
                       f 
                        
                       
                         ( 
                         
                           
                             t 
                             SAMP 
                             N 
                           
                           + 
                           
                             t 
                             off 
                           
                         
                         ) 
                       
                     
                      
                     
                       sin 
                       ( 
                       
                           
                       
                        
                       
                         
                           π 
                            
                           
                               
                           
                            
                           
                             N 
                              
                             
                               ( 
                               
                                 
                                   1 
                                   2 
                                 
                                 + 
                                 K 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           
                             f 
                             C 
                           
                            
                           
                             t 
                             off 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   17 
                   ] 
                 
               
             
             
               
                 
                   
                     
                       r 
                       21 
                     
                      
                     
                       ( 
                       N 
                       ) 
                     
                   
                   = 
                   
                     
                       f 
                        
                       
                         ( 
                         
                           
                             t 
                             SAMP 
                             N 
                           
                           + 
                           
                             t 
                             off 
                           
                           + 
                           
                             δ 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         ) 
                       
                     
                      
                     
                       sin 
                       ( 
                       
                           
                       
                        
                       
                         
                           π 
                            
                           
                               
                           
                            
                           
                             N 
                              
                             
                               ( 
                               
                                 
                                   1 
                                   2 
                                 
                                 + 
                                 K 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           
                             
                               f 
                               C 
                             
                              
                             
                               ( 
                               
                                 
                                   t 
                                   off 
                                 
                                 + 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   18 
                   ] 
                 
               
             
           
         
       
     
         [0028]    For K even and N=0, 1, 2, 3, r 12 (N) is given by equations [19] through [22], respectively. The pattern of equations [19] through [22] repeats for larger N. 
         [0000]        r   12 (0)= f ( t   SAMP   0   +t   off )sin(2π f   C   t   off )   [19]
 
         [0000]        r   12 (1)= f ( t   SAMP   1   +t   off )cos(2π f   C   t   off )   [20]
 
         [0000]        r   12 (2)=− f ( t   SAMP   2   +t   off )sin(2π f   C   t   off )   [21]
 
         [0000]        r   12 (3)=− f ( t   SAMP   3   +t   off )cos(2π f   C   t   off )   [22]
 
         [0029]    For K odd and N=0, 1, 2, 3, r 12 (N) is given by equations [23] through [26], respectively. The pattern of equations [23] through [26] repeats for larger N. 
         [0000]        r   12 (0)= f ( t   SAMP   0   +t   off )sin(2π f   C   t   off )   [23]
 
         [0000]        r   12 (1)=− f ( t   SAMP   1   +t   off )cos(2π f   C   t   off )   [24]
 
         [0000]        r   12 (2)=− f ( t   SAMP   2   +t   off )sin(2π f   C   t   off )   [25]
 
         [0000]        r   12 (3)= f ( t   SAMP   3   +t   off )cos(2π f   C   t   off )   [26]
 
         [0030]    Similarly, for K even and N=0, 1, 2, 3, r 21 (N) is given by equations [27] through [30], respectively. The pattern of equations [27] through [30] repeats for larger N. 
         [0000]        r   21 (0)= f ( t   SAMP   0   +t   off   +δt )sin(2π f   C ( t   off   +δt ))   [27]
 
         [0000]        r   21 (1)= f ( t   SAMP   1   +t   off   +δt )cos(2π f   C ( t   off   +δt ))   [28]
 
         [0000]        r   21 (2)=− f ( t   SAMP   2   +t   off   +δt )sin(2π f   C ( t   off   +δt ))   [29]
 
         [0000]        r   21 (3)=− f ( t   SAMP   3   +t   off   +δt )cos(2π f   C ( t   off   +δt ))   [30]
 
         [0031]    Likewise, for K odd and N=0, 1, 2, 3, r 21 (N) is given by equations [31] through [34], respectively. The pattern of equations [31] through [34] repeats for larger N. 
         [0000]        r   21 (0)= f ( t   SAMP   0   +t   off   +δt )sin(2π f   C ( t   off   +δt ))   [31]
 
         [0000]        r   21 (1)=− f ( t   SAMP   1   +t   off   +δt )cos(2π f   C ( t   off   +δt ))   [32]
 
         [0000]        r   21 (2)=− f ( t   SAMP   2   +t   off   +δt )sin(2π f   C ( t   off   +δt ))   [33]
 
         [0000]        r   21 (3)=− f ( t   SAMP   3   +t   off   +δt )cos(2π f   C ( t   off   +δt ))   [34]
 
         [0032]    The sine terms for r 12  and K even are collected from equations [19], [21], and repeated N in equation [35]. The cosine terms for r 12  and K even are collected from equations [20], [22], and repeated N in equation [36]. 
         [0000]        r   Keven   12 ( N )={ f ( t   SAMP   0   +t   off ),− f ( t   SAMP   2   +t   off ), . . . }sin(2π f   C   t   off )   [35]
 
         [0000]        r   Keven   12 ( N )={ f ( t   SAMP   1   +t   off ),− f ( t   SAMP   3   +t   off ), . . . }cos(2π f   C   t   off )   [36]
 
         [0033]    Similarly, the sine terms for r 21  and K even are collected from equations [27], [29], and repeated N in equation [37]. The cosine terms for r 12  and K even are collected from equations [28], [30], and repeated N in equation [38]. 
         [0000]        r   Keven   21 ( N )={ f ( t   SAMP   0   +t   off   δt ),− f ( t   SAMP   2   +t   off   +δt ), . . . }sin(2π f   C ( t   off   +δt ))   [37]
 
         [0000]        r   Keven   21 ( N )={ f ( t   SAMP   1   +t   off   δt ),− f ( t   SAMP   3   +t   off   +δt ), . . . }cos(2π f   C ( t   off   +δt ))   [38]
 
         [0034]    Comparing the sine terms of equation [35] with those of equation [37] and the cosine terms of equation [36] with those of equation [38], the sampling functions differ in time by offset δt. The cosine terms of equations [36] through [38], therefore, are interpolated to match the timing of the sine terms in equations [39] and [42], respectively. 
         [0000]        r   Keven,sin   12 ( N )={ f ( t   SAMP   0   +t   off ),− f ( t   SAMP   2   +t   off ), . . . }sin(2π f   C   t   off )   [39]
 
         [0000]        r   Keven,cos   12,int ( N )={ {tilde over (f)} ( t   SAMP   0   +t   off ),− {tilde over (f)} ( t   SAMP   2   +t   off ), . . . }cos(2π f   C   t   off )   [40]
 
         [0000]        r   Keven,sin   21 ( N )={ f ( t   SAMP   0   +t   off ),− f ( t   SAMP   2   +t   off ), . . . }sin(2π f   C   t   off )   [41]
 
         [0000]        r   Keven,cos   21,int ( N )={ {tilde over (f)} ( t   SAMP   0   +t   off   +δt ),− {tilde over (f)} ( t   SAMP   2   +t   off   +δt ), . . . }cos(2π f   C   t   off )   [42]
 
         [0035]    Referring to  FIG. 3B , there is a signal processing circuit  304  to perform the foregoing calculations in equations [39] through [42]. Sine terms from ADC  302  are applied to sum circuit  310  where they are accumulated and applied to processing circuit  312 . Cosine terms from ADC  302  are interpolated by block  306  and applied to sum circuit  308 . Sum circuit  308  accumulates the cosine terms and applies them to processing circuit  312 . Processing circuit  312  estimates the differential TOF (δt) according to equation [43]. 
         [0000]    
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     t 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               angle 
                                
                               
                                 ( 
                                 
                                   ∑ 
                                   
                                     
                                       r 
                                       
                                         Keven 
                                         , 
                                         cos 
                                       
                                       
                                         21 
                                         , 
                                         int 
                                       
                                     
                                     / 
                                     
                                       ∑ 
                                       
                                         r 
                                         
                                           Keven 
                                           , 
                                           sin 
                                         
                                         21 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             angle 
                              
                             
                               ( 
                               
                                 ∑ 
                                 
                                   
                                     r 
                                     
                                       Keven 
                                       , 
                                       cos 
                                     
                                     
                                       21 
                                       , 
                                       int 
                                     
                                   
                                   / 
                                   
                                     ∑ 
                                     
                                       r 
                                       
                                         Keven 
                                         , 
                                         sin 
                                       
                                       21 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       
                         f 
                         C 
                       
                     
                   
                 
               
               
                 
                   [ 
                   43 
                   ] 
                 
               
             
           
         
       
     
         [0036]    Estimation accuracy of the differential TOF (δt) of equation [43] relies on the fact that the summed f sampling coefficients of equations [39] through [42] are close to each other in time. Of course, for increasing δt, the estimation error also increases. Transmitting a larger number of pulses from each transducer reduces the variation of the summed f sampling coefficients by increasing signal duration and, therefore, improves accuracy. At least a 1% measurement accuracy is desirable. A most demanding condition for this measurement is assumed for a differential TOF of approximately 3 ns. This corresponds to a 6 cm transducer spacing and a 5 cm/s flow rate. A 1% error for this condition requires an error of less than 30 ps for a 6σ measurement. 
         [0037]      FIG. 4  is graph illustrating Farrow cubic interpolation that may be used with the circuit of  FIG. 3B  as in equations [40] and [42]. This and other interpolation techniques are disclosed by Erup et al., “Interpolation in digital modems—Part II: Implementation and performance”, IEEE Trans. on Communications, Vol. 41, No. 6, 998-1008 (June 1993). The principle of Farrow cubic interpolation is given by equation [44]. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           Y 
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               λ 
                               3 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     x 
                                      
                                     
                                       ( 
                                       
                                         i 
                                         + 
                                         2 
                                       
                                       ) 
                                     
                                   
                                   6 
                                 
                                 - 
                                 
                                   
                                     x 
                                      
                                     
                                       ( 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     x 
                                      
                                     
                                       ( 
                                       i 
                                       ) 
                                     
                                   
                                   2 
                                 
                                 - 
                                 
                                   
                                     x 
                                      
                                     
                                       ( 
                                       
                                         i 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   6 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               λ 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     x 
                                      
                                     
                                       ( 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   2 
                                 
                                 - 
                                 
                                   x 
                                    
                                   
                                     ( 
                                     i 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     x 
                                      
                                     
                                       ( 
                                       
                                         i 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           + 
                           … 
                         
                       
                         
                     
                      
                     
                       λ 
                        
                       
                         ( 
                         
                           
                             - 
                             
                               
                                 x 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     + 
                                     2 
                                   
                                   ) 
                                 
                               
                               6 
                             
                           
                           + 
                           
                             x 
                              
                             
                               ( 
                               
                                 i 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               x 
                                
                               
                                 ( 
                                 i 
                                 ) 
                               
                             
                             2 
                           
                           - 
                           
                             
                               x 
                                
                               
                                 ( 
                                 
                                   i 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             3 
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     x 
                      
                     
                       ( 
                       i 
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   44 
                   ] 
                 
               
             
           
         
       
     
         [0038]    Here, λ is the interpolation factor and k is the output sample index. To is the time between output samples and Ti is the time between input samples. In a preferred embodiment of the present invention, λ is ½. If the ADC sampling frequency is taken as 4/3 (1.733 MHz) or ⅘ (1.04 MHz) of the transducer excitation frequency, it is only necessary to interpolate the summed cosine (Q channel) terms. According to a preferred embodiment of the present invention, this may be accomplished by dividing a 5.2 MHz clock by 4 to produce the 1.3 MHz excitation frequency. The same 5.2 MHz clock may be divided by 3 to produce a 1.733 MHz sampling frequency or by 5 to produce a 1.04 MHz sampling frequency. Both the excitation frequency and the sampling frequency, therefore, are advantageously synchronized. Although some variation of sampling frequency with respect to excitation frequency is possible, the sampling frequency is preferably constrained to +/−5% of the target sampling frequency. Each x(.) term of equation [44] represents a sampled input f(.) from equations [36] and [38]. These input terms are used to interpolate output terms Y(k) or {tilde over (f)}(.) terms of equations [40] and [42], respectively. These interpolated terms are time shifted so that they are aligned with the summed sine (I channel) terms. Thus, the superscripts of the interpolated terms are changed to match the summed cosine terms. 
         [0039]      FIG. 5A  is diagram showing the pulse excitation signal applied to a transmitting transducer as 1, 10, or 20 pulses for transmission to receiving transducer.  FIG. 5B  is graph showing the signal at the receiving transducer in response to 1, 10, and 20 pulses at a transmitting transducer. Preferred embodiments of the present invention utilize transducers with a center frequency (f C ) of 1.3 MHz. However, other transducers and different center frequencies may also be used. The diagram illustrates the resonance of a transmitting transducer in response to 1, 10, and 20 input pulses. In each case, the envelope of the resonant signal rises to a peak amplitude corresponding to the duration of the pulses. After this the envelope of each signal decays as in a typical RLC circuit. For example, 20 pulses at 1.3 MHz have a duration of t=20/1.3e6=15.4 μs. A 10 pulse excitation signal has a duration of t=10/1.3e6=7.7 μs. Several simulations are given at  FIGS. 6-9  for various numbers of input pulses to demonstrate the accuracy of the measurement system of  FIG. 3A  for a transducer center frequency (f C ) of 1.3 MHz. Results for 15 different conditions are summarized at the table in  FIG. 10 . 
         [0040]    Referring now to  FIG. 6 , there is graph showing accuracy of the measurement circuit of  FIG. 3A  compared to zero crossing and cross correlation measurement techniques. The vertical scale is root mean squared (RMS) error of the differential TOF estimation as a function of noise. Simulations show less than 1% RMS for all IF sampling except 1.04 MHz sampling and 20 pulse transmission at 5 ns δt and 1.733 MHz sampling and 1 pulse transmission at 5 ns δt. Other simulations, however, show low tolerance to noise. IF sampling at 1.04 MHz sampling and 20 pulse transmission at 100 ns δt and 1.733 MHz sampling and 20 pulse transmission at 100 ns δt maintain less than 1% RMS error to 40 dBnV/sqrt(Hz). 
         [0041]    Referring now to  FIG. 7 , there is graph showing accuracy of the measurement circuit of  FIG. 3A  compared to zero crossing and cross correlation measurement techniques with different numbers of ADC bits and different numbers of transducer excitation pulses. The vertical scale is root mean squared (RMS) error of the differential TOF estimation as a function of noise. All IF sampling simulations are at 1.733 MHz. The greatest noise tolerance is 20 pulse transmission at either 5 ns or 100 ns δt. Both maintain less than 1% RMS error to 40 dBnV/sqrt(Hz). 
         [0042]    Referring now to  FIG. 8 , there is graph showing accuracy of the measurement circuit of  FIG. 3A  for a 12-bit ADC having positive sampling frequency misalignment with respect to the optimal ADC sampling frequency accuracy of the measurement circuit of  FIG. 3A  compared to zero crossing and cross correlation measurement techniques. The vertical scale is root mean squared (RMS) error of the differential TOF estimation as a function of noise. All IF sampling simulations compare results of a desired 1.733 MHz sampling frequency with positive sampling frequency errors of 0.2%, 1%, and 5%. 
         [0043]    Referring now to  FIG. 9 , there is graph showing accuracy of the measurement circuit of  FIG. 3A  for a 12-bit ADC having negative sampling frequency misalignment with respect to the optimal ADC sampling frequency accuracy of the measurement circuit of  FIG. 3A  compared to zero crossing and cross correlation measurement techniques. The vertical scale is root mean squared (RMS) error of the differential TOF estimation as a function of noise. All IF sampling simulations compare results of a desired 1.733 MHz sampling frequency with negative sampling frequency errors of −0.2% and −1%. 
         [0044]      FIG. 10  is a table summarizing accuracy of the measurement circuit of  FIG. 3A  for various conditions. These conditions include variation of ADC sampling frequency, number of ADC bits, differential TOF (δt), and number of transducer pulses. 
         [0045]    Still further, while numerous examples have thus been provided, one skilled in the art should recognize that various modifications, substitutions, or alterations may be made to the described embodiments while still falling within the inventive scope as defined by the following claims. Other combinations will be readily apparent to one of ordinary skill in the art having access to the instant specification.