Fit Convolution

When performing curve fitting on experimental data, the experimental data is often the convolution of the input signal and the instrument response. One may want to remove instrument response in the data. Origin supports to define the fitting function as a convolution, and extract the original signal by fitting.

Origin provides several tools to perform convolution fit.

• GaussMod: Built-in fitting function to convolute Gaussian and an exponential decay function.
• Fitting Function Library app: Provides several fitting functions to convolute Gaussian and all kinds of exponential functions, which are to be used in the nonlinear curve fitting tool.
• Fit Convolution app: Perform fitting on convolution of the signal function with a column for instrument response data.

Convolution of Two Functions

If the instrument response function is a Gaussian function \(y=\frac{1}{\sqrt{2\pi}w}e^{-\frac{(x-x_c)^2}{2w^2}}\), the original signal is an exponential function, Origin provides several fitting functions for it.

Function	Source	ID in App	Exponential Form	Description
GaussMod	Built-in		\(\frac{A}{t_0}e^{-\frac{x}{t_0}} \; x \ge 0\)	Convolution of Gaussian and an exponential decay function.
FitConvExp2	app*	2	\(\frac{A_1}{t_1}e^{-\frac{x}{t_1}}+\frac{A_2}{t_2}e^{-\frac{x}{t_2}} \; x \ge 0\)	Convolution of Gaussian and a two-phase exponential decay function.
FitConvExp3	app*	22	\(\frac{A_1}{t_1}e^{-\frac{x}{t_1}}+\frac{A_2}{t_2}e^{-\frac{x}{t_2}}+\frac{A_3}{t_3}e^{-\frac{x}{t_3}} \; x \ge 0\)	Convolution of Gaussian and a three-phase exponential decay function.
GaussModP6	app*	28	\(\frac{A}{t_0}e^{-\frac{x-x_0}{t_0}} \; x \ge x_0\)	Convolution of Gaussian and an exponential decay function with a shift parameter.
FitConvExp2P8	app*	29	\(\frac{A_1}{t_1}e^{-\frac{x-x_0}{t_1}}+\frac{A_2}{t_2}e^{-\frac{x-x_0}{t_2}} \; x \ge x_0\)	Convolution of Gaussian and a two-phase exponential decay function with a shift parameter.
FitConExp3P10	app*	30	\(\frac{A_1}{t_1}e^{-\frac{x-x_0}{t_1}}+\frac{A_2}{t_2}e^{-\frac{x-x_0}{t_2}}+\frac{A_3}{t_3}e^{-\frac{x-x_0}{t_3}} \; x \ge x_0\)	Convolution of Gaussian and a three-phase exponential decay function with a shift parameter.
GaussModTwoSided	app*	31	\(\frac{A}{t_0}e^{-\frac{|x|}{t_0}}\)	Two-sided exponential decay convolved with a Gaussian function.
GaussModLeft	app*	32	\(\frac{A}{t_0}e^{\frac{x}{t_0}} \; x \le 0\)	Left skew GaussMod function.

_{* Fitting Function Library app}

Typical curves for these convolution functions are listed as below:

Function	Curve
GaussMod
FitConvExp2
FitConvExp3
GaussModP6
FitConvExp2P8
FitConExp3P10
GaussModTwoSided
GaussModLeft

Convolution of One Function and a Column

If the instrument response is defined in a column, Origin provides Fit Convolution app to fit. And two functions in the convolution are more flexible.

• The original signal can be any function.
• The instrument response is defined in a column, and can be any function’s data.
• A shift parameter is introduced to allow to shift the instrument response data towards right or left.

Fit Convolution app supports three types of input data in the worksheet.

Input Data	Columns	Description	Does instrument response support to shift?
Simple	XY1Y2	X: X data for convolution and instrument response Y1: Convolution y data Y2: Instrument response y data	No
X Shift	XY1Y2	X: X data for convolution and instrument response Y1: Convolution y data Y2: Instrument response y data	Yes
XYXY	X1Y1X2Y2	X1Y1: XY data for convolution X2Y2: XY data for instrument response	Yes

Fit Convolution app provides a sample project to show how to use X Shift and XYXY input data type.