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** 、 **XYXY** input data type.