Spectral convolution, operation

If b and g are peaked functions (such as in a spectral line), the area under their convolution product is the product of their individual areas. Thus, if b represents instrumental spreading, the area under the spectral line is preserved through the convolution operation. In spectroscopy, we know this phenomenon as the invariance of the equivalent width of a spectral line when it is subjected to instrumental distortion. This property is again referred to in Section II.F of Chapter 2 and used in our discussion of a method to determine the instrument response function (Chapter 2, Section II.G). 

One way to improve the signal-to-noise ratio is through convolution of the spectrum with an appropriate function such as a boxcar, Lorentzian, or Gaussian function. The operation of spectral convolution has been presented in Section 2.3 Such operations tend to distort the spectrum, as the lineshape function is altered. The broader the convolution function, the greater is the distortion of the spectrum. The most common such convolution is the Savitzky-Golay smoothing algorithm [13]. 

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