Deconvolution method for emissions measurement

1. A system comprising:

an emissions measurement instrument configured to:

measure a concentration of an emission in a sample of exhaust gas from an engine; and

generate an uncorrected instrument response indicating the measured emission concentration as a function of time;

memory configured to:

record the uncorrected instrument response; and

store an inverse convolution function, wherein the inverse convolution function is in the time domain and represents an inverse of the transient response of the emissions measurement instrument; and

a processor configured to:

retrieve the uncorrected instrument response and the inverse convolution function from the memory;

convolve the uncorrected instrument response with the inverse convolution function to mitigate the effect of the transient response of the emissions measurement instrument on the uncorrected instrument response, the result of the convolution being a convolution corrected instrument response; and

output a signal indicating the convolution corrected instrument response.

2. The system of claim 1 , wherein the processor is configured to determine the inverse convolution function based on an idealized convolution function, the idealized convolution function being in the time domain.

3. The system of claim 2 , wherein the idealized convolution function is the first derivative of a response of the emissions measurement instrument to a reference gas sample.

4. The system of claim 2 , wherein the processor is configured to calculate the idealized convolution function by convolving a Gaussian function with an impulse response function.

5. The system of claim 4 , wherein the processor is configured to:

determine a scaling factor based on a normalized convolution function and a response of the emissions measurement instrument to a reference gas sample; and

determine the Gaussian and impulse response functions based on the scaling factor.

6. The system of claim 5 , wherein the processor is configured to calculate the normalized convolution function by convolving a normalized Gaussian function with a normalized impulse response function.

7. The system of claim 6 , wherein the processor is configured to determine the impulse response function based on values from the response of the emissions measurement instrument to the reference gas sample.

8. The system of claim 2 , wherein the processor is configured to:

transform the idealized convolution function from the time domain to the frequency domain;

determine the inverse convolution function based on the idealized convolution function, the inverse convolution function initially being in the frequency domain; and

transform the idealized convolution function from the frequency domain to the time domain.

9. The system of claim 8 , wherein the processor is configured to divide a regularizing filter function by the transformed idealized convolution function, the result of the division being the inverse convolution function in the frequency domain.

10. The system of claim 9 , wherein the processor is configured to determine the regularizing filter function based on the transformed idealized convolution function and a positive adjustable filter parameter.

11. The system of claim 10 , wherein the positive adjustable filter parameter is a constant value independent of frequency.

12. The system of claim 10 , wherein the processor is configured to adjust the positive adjustable filter parameter to adjust overshoots, undershoots, and a dynamic response of the inverse convolution function.

13. The system of claim 1 , wherein the processor is configured calculate a derivative corrected instrument response to eliminate noise at step changes in the convolution corrected instrument response.

14. The system of claim 1 , wherein the processor is configured to calculate a derivative corrected instrument response by solving for p(t) using the following equation:

p

⁡

(

t

)

+

β

⁢

dp

dt

⁢

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.

15. The system of claim 1 further comprising:

a sample line configured to deliver the exhaust gas sample to the emissions measurement instrument; and

a valve disposed in the sample line and adjustable between a first position and a second position, wherein valve allows a reference gas sample to flow to the emissions measurement instrument when the valve is in the first position, and the valve allows the exhaust gas sample to flow to the emissions measurement instrument when the valve is in the second position.

16. The system of claim 15 , wherein the processor is configured to:

determine the inverse convolution function based on a response of the emissions measurement instrument to the reference gas sample;

adjust the valve to the first position when determining the inverse convolution function; and

adjust the valve to the second position when correcting the response of the emissions measurement instrument to the exhaust gas sample.

17. A system comprising:

an emissions measurement instrument configured to measure a concentration of an emission in a reference gas sample;

memory configured to:

record a response of the emissions measurement instrument to the reference gas sample as a function of time; and

store an idealized convolution function, the idealized convolution function being in the time domain and representing the transient response of an the emissions measurement instrument; and

a processor configured to:

retrieve the idealized convolution function from the memory;

transform the idealized convolution function from the time domain to the frequency domain;

divide a regularizing filter function by the transformed idealized convolution function, the result of the division being an inverse convolution function in the frequency domain;

transform the inverse convolution function from the frequency domain to the time domain using the processor;

output a signal indicating the inverse convolution function in the time domain, wherein the memory is configured to store the inverse convolution function in the time domain;

convolve the recorded response with the transformed inverse convolution function to mitigate the effect of the transient response of the emissions measurement instrument on the recorded response, the result of the convolution being a convolution corrected instrument response indicating the measured emission concentration; and

generate a signal indicating the convolution corrected instrument response.

18. The system of claim 17 , wherein the processor is configured to calculate the idealized convolution function by convolving a Gaussian function with an impulse response function.

19. The system of claim 17 , wherein the processor is configured to:

compare the convolution corrected instrument response signal to a reference gas signal indicating a predetermined concentration of the emission in the reference gas sample;

adjust the inverse convolution function based on the comparison; and

output a signal indicating the inverse convolution function as adjusted.

20. The system of claim 19 , wherein the processor is configured to:

determine the regularizing filter function based on the transformed idealized convolution function and a positive adjustable filter parameter, wherein the positive adjustable filter parameter is a constant value independent of frequency; and

adjust the positive adjustable filter parameter to adjust overshoots, undershoots, and a dynamic response of the convolution corrected instrument response relative to the reference gas signal.