APODISATION FUNCTION

To eliminate this ringing", apodisation functions are used. The most commonly used is the triangle function which reduces the ringing effect but also the spectral resolution (Fig. I2.8C). 

Figure 3.17. Carbon spectra often display distortions when transformed directly (a) which appear to be phase errors but which actually arise from a short acquisition time. Applying a line-broadening apodisation function prior to the transform removes these distortions (b, 1 Hz line-broadening).

Without apodisation, the basis functions of the Fourier set correspond to sharp pulses in the frequency domain. With apodisation, these pulses become convoluted with the transform of the apodisation function, e.g. a Gaussian. The convolution of a pulse with some shape moves this shape to the position of the pulse. As a consequence, the frequency domain is not cut up in disjoint frequencies, but in a series of overlapping Gaussians. Note that this is no more than a different view of the filtering effect of the transform of the apodisation function. 

Each Fourier coefficient in a transform with apodisation represents a band of frequencies. The width of that band is controlled via the length of the signal that is transformed and the shape of the apodisation function. We can introduce the notion of frequency localisation as an extension of the previously introduced frequency resolution and in analogy to localisation in time. When the bands are wide, the frequency information returned by the transform is less localised than when the bands are narrow. In other words, when the time localisation is good, the frequency localisation is poor. 

Figure 6.5 Spectral features as a function of the apodisation function in mid-IR microspectroscopy of tissues (colon tissue cryo-section, same sample position as in Figure 6.4). Transmission type spectra were acquired using Bruker s IRScope II microscope and an IFS28/B spectrometer. Further measurement parameters aperture diameter 900 pm, Cassegrain objective (36 x, NA 0.5), 128 scans, optical substrate CaF2 pm of 1 mm thickness. Spectral resolution 6 cm zero-filling factor (ZFF) 4. Transmission spectra were processed with a first derivative Savitzky-Golay filter with nine smoothing points.

Simulated spectra, as well as measured ones, are apodised with the Norton-Beer strong function [10], in order to reduce the interference of nearby lines. 

The raw bandpass of an AOTF has a sine squared function line shape with sidebands, which if ignored may amount to 10% of the pass optical energy in off-centre wavelengths. This apodisation issue is normally addressed by careful control of the transducer coupling to the crystal. 

Window functions. The experimental signal S(t) is multiplied by the window function in a process termed apodisation to produce a new weighted signal A(t) which is Fourier transformed. Useful window functions commonly used are ... 

We have introduced apodisation as a weighting of the signal, but we can just as well view it as a weighting of the Fourier basis functions. The sines and cosines become squeezed down at the ends, as illustrated by Fig. 24. To the left it shows a sine base function, a gaussian apodisation that is chosen narrow in order to amplify its effect, and the resulting apodised base function that has the shape of a ripple. 

Fig. 5 The first apodised sine basis functions of the Fourier basis (left) and the short-time...

A Lorentzian LBF was used in the deconvolution. A deconvolution factor of 100 was chosen in order to achieve a net amplification of 3.4. This value was chosen after trials with deconvolution factors of 50,100 and 1000 in the manner recommended by Kauppinen et al. (52). With a factor of 50, underdeconvolution is observed, while a factor of 1000 results in over-deconvolution and negative side lobes. The deconvolution function amplifies the noise as well, and in order to reduce the noise, apodisation with a Blackman function is performed in conjunction with the deconvolution over the fraction of the interferogram specified by a noise reduction factor. A noise reduction factor of 0.5 was chosen. While the intensities of the deconvolved peaks are higher than those in the original spectra, the relative peak intensities in the deconvolved spectra remain the same as in the original (57). 

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