Bessel filter function

The purpose of the high- and low-pass filters, shown in Figure 9.2, is to eliminate interference signals such as electrode half-cell potentials and preamplifier offset potentials and to reduce the noise amplitude by the limitation of the amplifier bandwidth. Because the biosignal should not be distorted or attenuated, higher order sharp-cutting linear-phase filters have to be used. Active Bessel filters are preferred filter types because of their smooth transfer function. Separation of biosignal and interference is in most cases incomplete because of the overlap of their spectra. 

Other window functions than the sine window have been proposed as well (see [Bosi et al., 1996b, Fielder et al., 1996]). Using Kaiser-Bessel-Derived window functions, a filter characteristic exhibiting better side-lobe suppression is possible. This is explained in [Fielder et al., 1996],... 

In actual practice, all filters have a distributed cutoff frequency so that none are infinitely sharp, and the way in which the attenuation "rolls off" with frequency affects the attainable S/N. The world of electrical engineering knows of many different filters (such as the Bessel and the Butterworth) which are characterized by different amplitude rolloff and phase characteristics near the cutoff frequency. A commonly used filter is the RC filter because of its ease of implementation. It consists simply of a capacitor C and a resistor R. It has the time constant RC (check it it has the unit of time) and this simply means that it will not respond to signals that change appreciably in times shorter than RC so it is a low pass filter. Its response to a step function in time is exponential so that the rolloff in the frequency domain, i.e., its Fourier transform, is a Lorentzian and the cutoff is very broad. 

There are two demodulation techniques that are used to measure the signal arriving at the detector into and l (2a>). The first uses a lock-in amplifier and a low-pass filter [10, 69], while the second relies on the synchronous sampling demodulator (SSD) to obtain the intensity difference and average signals [65, 66]. In the first case, J2 and Jo are the second-order and zero-order Bessel functions. In the second case they are described by the following expressions ... 

The effect of geometric filtering by the limiting size of optical elements is the vignetting. This means that the power is modulated by the point spread function of the optical elements. In the case of a circular aperture, the modulation function has a sine like profile for squared apertures, the modulation function has a Bessel profile in both x and y directions of the signal. This effect can be simulated and understood, and can be reduced by re-designing the system to accommodate larger optical elements. 

Fig. 7. Holographic filter shape. For wavelengths dose to the central value, X < Xo +/-1.5 nm, the shap>e is Gaussian and like a Bessel function for wavelengths X > Xo +/-1.5 nm...

In general, the two types of filter more widely used are the Bessel and Butterworth filters. The quality of the filter is expressed in the transfer function, and is defined by the number of poles of its transfer function [7]. That is, the higher the number of poles, the sharper the cut-off of high frequencies found. Nevertheless, it is also important to consider the internal filter delay, which is also a function of the number of poles. This delay can influence the quality and interpretation of the recorded signals, especially if they are very fast, and their time courses have to be studied. 

Each filter alignment has a frequency response with a characteristic shape (Fig. 4.32), which provides some particular advantage. For example, filters with Butterworth, Chebyshev, or Bessel alignment are called aU-pole filters because their lowpass transfer functions have no zeros. Table 4.6 summarizes the characteristics of the standard filter alignments. 

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