DECONVOLUTION METHOD FOR EMISSIONS MEASUREMENT
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.
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.