Figure 12. Deconvolution is used here to remove the distorting influence of an exponential tailing response function from a recorded signal (Window 1, top left) that is the result of an unavoidable RC low-pass filter action in the electronics. The response function (Window 2, top right) is usually either calculated on the basis of some theoretical model or is measured experimentally as the output signal produced by applying an impulse (delta) function to the input of the system. The response function, with its maximum at x=0, is deconvoluted from the original signal . The result (bottom, center) shows a closer approximation to the real shape of the peaks; however, the signal-to-noise ratio is unavoidably degraded.
Note that this process in figure 12 has an effect that is visually similar to resolution enhancement, although the later is done without knowledge of the broadening function that caused the peaks to overlap.
When applying deconvolution to experimental data, to remove the effect of a known broadening or low-pass filter operator caused by the experimental system, a very serious signal-to-noise degradation commonly occurs. Any noise added to the signal by the system after the broadening or low-pass filter operator will be greatly amplified when the Fourier transform of the signal is divided by the Fourier transform of the broadening operator, because the high frequency components of the broadening operator (the denominator in the division of the Fourier transforms) are typically very small, resulting in a great amplification of high frequency noise in the resulting deconvoluted signal. This can be controlled but not completely eliminated by smoothing and by constraining the deconvolution to a frequency region where the signal has a sufficiently high signal-to-noise ratio.
Note: The term "deconvolution" is sometimes also used for the process of resolving or decomposing a set of overlapping peaks into their separate components by the technique of iterative least-squares curve fitting of a putative peak model to the data set. The process is actually conceptually distinct from deconvolution, because in deconvolution the underlying peak shape is unknown but the broadening function is assumed to be known; whereas in iterative least-squares curve fitting the underlying peak shape is assumed to be known but the broadening function is unknown.
* If you know that mx = n, where m and n are known but x is unknown, then x = n/m. Conversely if you know that m convoluted with x = n, where m and n are known but x is unknown, then x = m deconvoluted from n.
** Fourier transforms are usually expressed in terms of complex numbers, with real and imaginary parts. If the Fourier transform of the first signal is a + ib, and the Fourier transform of the second signal is c + id, then the ratio of the two Fourier transforms is