8. CONVOLATION

Convolation is an operation performed on two signals which involves multiplying one signal by a delayed or shifted version of another signal, integrating or averaging the product, and repeating the process for different delays. Convolution is a useful process because it accurately describes some effects that occur widely in scientific measurements, such as the influence of a low-pass filter on an electrical signal or of the spectral bandpass of a spectrometer on the shape of a spectrum. 

Figure 11. Convolution is used here to determine how the atomic line spectrum in Window 1 (top left) will appear when scanned with a spectrometer whose slit function (spectral resolution) is described by the Gaussian function in Window 2 (top right). The Gaussian function has already been rotated so that its maximum falls at x=0. The resulting convoluted spectrum (bottom center) shows that the two lines near x=110 and 120 will not be resolved but the line at x=40 will be partly resolved.

In practice the calculation is usually performed by multiplication of the two signals in the Fourier domain. First, the Fourier transform of each signal is obtained. Then the two Fourier transforms are multiplied by the rules for complex multiplication^* and the result is then inverse Fourier transformed. Although this seems to be a round-about method, it turns out to be faster then the shift-and-multiply algorithm when the number of points in the signal is large. Convolution can be used as a very powerful and general algorithm for smoothing and differentiation. The example of Figure 11 shows how it can be used to predict the broadening effect of a spectrometer on an atomic line spectrum.