Product Theorem for Gaussian Functions

By | March 27, 2016

It is a well known fact in mathematics that the product of two Gaussian functions is also a Gaussian function. In this post I want to present this result for future reference for me and for anyone who might find it useful.

The product theorem for Gaussian functions states that the product of two overlapping Gaussian functions is also a Gaussian function and determines the center and width of the resulting function in terms of the parameters of the two original functions.

To illustrate the identity, consider a Gaussian function of the form G_i(x)=A_i\exp[-(x-\mu_i)^2/2\sigma_i^2], where \mu_i, \sigma_i and A_i, correspond to its center, width and amplitude, respectively. The result of the product of two Gaussian functions G_1(x) and G_2(x) with different amplitudes, widths and central positions, G_3(x) = G_1(x)G_2(x), is equal to

G_3(x) = A_1 A_2\exp\left(-\frac{(\mu_1-\mu_2)^2}{2(\sigma_1^2+\sigma_2^2)}\right)\exp\left(-\frac{(x-\tilde{\mu})^2}{2\tilde{\sigma}^2}\right)\,,


where the centroid is given by

\tilde{\mu} = \frac{\mu_1\sigma_2^2+\mu_2\sigma_1^2}{\sigma_1^2+\sigma_2^2}\,,


and the width by

\tilde{\sigma}^2=\frac{\sigma_1^2\sigma_2^2}{\sigma_1^2+\sigma_2^2}\,.


The theorem shows that the result of the product of two Gaussian functions is a new Gaussian function of width \tilde{\sigma}, centered in the position \tilde{\mu}, whose amplitude strongly depends on the factor \exp\left(-(\mu_1-\mu_2)^2/2(\sigma_1^2+\sigma_2^2)\right). In addition, the resulting function is narrower than either of the two original Gaussian functions, and its center lies within the interval (\mu_1,\mu_2). Notice that the above result can be further simplified if we consider the scenario where the Gaussian functions are centered in different positions but have the same width. If we define \sigma_1 = \sigma_2 = \sigma, the centroid simplies to \tilde{\mu}= (\mu_1+\mu_2)/2, whereas the width is given by \tilde{\sigma}^2 = \sigma^2/2.

Examples

The next figures show two representative cases where two different Gaussian functions with different and equal widths are multiplied. In both cases the result is a Gaussian function narrower than the original functions centered in between the two original centroids, \mu_1 and \mu_2.

ProductTheoremForGaussianFunctions_001

ProductTheoremForGaussianFunctions_002

Additional information

  • E. Wolf, J. T. Foley and F. Gori, "Frequency shifts of spectral lines produced by scattering from spatially random media," J. Opt. Soc. Am. A, 7, 173, (1990)