Deconvolution method for emissions measurement

Abstract

Disclosed is a method of correcting a response of an instrument. The method includes determining an inverse convolution function, the inverse convolution function being in the time domain. A response of an instrument to an exhaust sample is recorded as a function of time. The recorded response is then convolved with the inverse convolution function, the result being a convolution corrected instrument response.

What is claimed is: 
     
         1 . A method of correcting a response of an instrument comprising:
 determining an inverse convolution function, the inverse convolution function being in the time domain;   recording a response of an instrument to an exhaust sample as a function of time; and   convolving the recorded response with the inverse convolution function, the result being a convolution corrected instrument response.   
     
     
         2 . The method as recited in  claim 1 , wherein the determining step includes determining an idealized convolution function, the idealized convolution function being in the time domain. 
     
     
         3 . The method as recited in  claim 2 , wherein the idealized convolution function is the first derivative of a response of the instrument to a reference exhaust sample. 
     
     
         4 . The method as recited in  claim 2 , wherein the idealized convolution function is calculated by convolving a Gaussian function with an impulse response function. 
     
     
         5 . The method as recited in  claim 4 , wherein the Gaussian and impulse response functions are based on a scaling factor, the scaling factor determined based on a normalized convolution function and on a response of the instrument to a reference exhaust sample. 
     
     
         6 . The method as recited in  claim 5 , wherein the normalized convolution function is calculated by convolving a normalized Gaussian function with a normalized impulse response function. 
     
     
         7 . The method as recited in  claim 6 , wherein the impulse response function is based on values from the response of the instrument to the reference exhaust sample. 
     
     
         8 . The method as recited in  claim 2 , wherein the determining step includes transforming the idealized convolution function from the time domain to the frequency domain. 
     
     
         9 . The method as recited in  claim 8 , wherein the determining step includes dividing a regularizing filter function by the transformed idealized convolution function, the result being the inverse convolution function in the frequency domain. 
     
     
         10 . The method as recited in  claim 9 , wherein the determining step includes transforming the inverse convolution function from the frequency domain to the time domain. 
     
     
         11 . The method as recited in  claim 9 , wherein the regularizing filter function is based on the transformed idealized convolution function and a positive adjustable filter parameter. 
     
     
         12 . The method as recited in  claim 11 , wherein the positive adjustable filter parameter is a constant value independent of frequency. 
     
     
         13 . The method as recited in  claim 12 , wherein the determining step includes adjusting the positive adjustable filter parameter to adjust overshoots, undershoots, and a dynamic response of the inverse convolution function. 
     
     
         14 . The method as recited in  claim 1 , wherein the instrument is a gas analyzer configured to measure a concentration of a gaseous constituent of the exhaust sample as a function of time. 
     
     
         15 . The method as recited in  claim 1 , further including calculating a derivative corrected instrument response to eliminate noise at step changes in the convolution corrected instrument response. 
     
     
         16 . The method as recited in  claim 1 , wherein a derivative corrected instrument response is calculated by solving for p(t) using the following equation: 
       
         
           
             
               
                 
                   p 
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 + 
                 
                   
                     β 
                     · 
                     
                       ( 
                       
                         
                            
                           p 
                         
                         
                            
                           t 
                         
                       
                       ) 
                     
                   
                   * 
                   
                     k 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
               
               = 
               
                 y 
                  
                 
                   ( 
                   t 
                   ) 
                 
               
             
           
         
         where p(t) is the derivative corrected instrument response, β is a constant, k(t) is the inverse convolution function, and y(t) is the convolution corrected instrument response. 
       
     
     
         17 . A method of determining an inverse convolution function comprising:
 determining an idealized convolution function, the idealized convolution function being in the time domain;   transforming the idealized convolution function from the time domain to the frequency domain;   dividing a regularizing filter function by the transformed idealized convolution function, the result being the inverse convolution function in the frequency domain; and   transforming the inverse convolution function from the frequency domain to the time domain.   
     
     
         18 . The method as recited in  claim 17 , wherein the idealized convolution function is calculated by convolving a Gaussian function with an impulse response function. 
     
     
         19 . The method as recited in  claim 17 , wherein the regularizing filter function is based on the transformed idealized convolution function and a positive adjustable filter parameter, and wherein the positive adjustable filter parameter is a constant value independent of frequency. 
     
     
         20 . The method as recited in  claim 19 , further including adjusting the positive adjustable filter parameter to adjust overshoots, undershoots, and a dynamic response of the inverse convolution function.