Apodisation

Suppose we have a signal like the one in Fig. 21(a), a signal in which we recognise a strong trend that makes it end much higher or lower than it started. 

Another way of looking at it starts from the periodic nature of the basis functions. As far as the sines and cosines are concerned, the signal could be just one period of a cyclic phenomenon. When we plot the concatenated signal (Fig. 22(a)), we see that it is dominated by a triangular oscillation, a saw-tooth as it is called. The Fourier transform of the signal will be equally dominated by the transform of that saw-tooth, which, due to the sharp edges in the saw-tooth, contains a lot of high frequencies. To illustrate this, the saw-tooth and its power spectrum have been plotted in Fig. 22(b) and (c). 

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. 

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. 

When Frequencies Change in Time Towards the Wavelet Transform 

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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. 

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 12.8 Resolution and apodisation. Apodisation A, Rectangle. B. Trapezoid. C. Triangle. D. (Triangle), ...

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 ... 

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).

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

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. 

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