Analog interferogram

Interferograms are sampled at equal intervals of retardation x, which mathematically is the multiplication of an analog interferogram by a repetitive impulse function, the Dirac comb, given by... 

It is a property of Fourier transform mathematics that multiplication in one domain is equivalent to convolution in the other. (Convolution has already been introduced with regard to apodization in Section 2.3.) If we sample an analog interferogram at constant intervals of retardation, we have in effect multiplied the interferogram by a repetitive impulse function. The repetitive impulse function is in actuality an infinite series of Dirac delta functions spaced at an interval 1 jx. That is,... 

Figure 3Ab illustrates the Dirac delta comb as a function of wavenumber, v. The function in Eq. 3.2 is an inbnite series. We multiply the analog interferogram with the Dirac delta comb of Eq. 3.1, and consequently, the Eourier transform of the interferogram (i.e., the spectrum) is convolved with the transformed comb (see Section 2.3). The effect of this convolution is to repeat the spectrum ad infinitum. If the spectrum covers the bandwidth 0 to v ax, the transformed Dirac delta comb must have a period of at least 2Vniax otherwise, the spectra will overlap as a result of the convolution. In other words. 

The interferogram is in the form of an analog signal at the detector, which then has to be digitised in order that Fourier transformation into a conventional spectrum can be carried out. There are two particular sources of error in transforming the digitised information on the interferogram into a spectrum. 

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