The Ideal Filter

Filtering is a method for the separation of materials. Materials with a particle diameter larger than the pores of the filter are collected as a filter cake, particles, which are smaller than the pores, can pass through the filter. With a fine filter one expects a transmission curve in the shape of a step function. For particles, which are smaller than the pore size, the transmission is 1 for particles, which are larger than the pore size, the transmission is zero, cf. Fig. 20.2. We will refer the diameter of particles that are in the range of the pore size of the filter the critical particle diameter, or consequently also as the critical pore size in the reverse view. 

We inspect once more Fig. 20.2. If we replace the abscissa label wi h frequency, then in electrotechnics, this particular transmission behavior mimics just an ideal low-pass filter. 

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The ideal filter would be one in which the size of the media particles would decrease uniformly in the direction of flow. The coarsest materials would be at the top the finest at the bottom. Fig. 3 illustrates the ideal arrangement. Materials of different densities and sizes would be required to provide the tapered void... 

The transfer characteristics of the high-frequency filter defined by expression (3.9) are ideal. In practice, it is impossible to achieve the ideal filter, so various approximations of the ideal filter must be used. The design of filters approximating the ideal form have been discussed by numerous authors. One simple design uses a filter described by expression (3.9) with the addition of a second-order filter.161... 

A variety of filters with a 0.45 im pore size is available. These filters vary in uniformity of pore size, chemical composition, and flow characteristics. Each of these factors affects the performance of the filter and some filters are better suited for the study of aquatic humic substances than others. The ideal filter would be inert relative to compounds of interest, exhibit good flow characteristics, have a uniform pore size, and be reasonably priced. An evaluation of the suitable filters with respect to pore size, chemical composition, and flow characteristics follows. 

Roll-cff Roll-off describes the width of the transition band in a filter design. The ideal filter has a transition band of zero and is thought of as a brick-wall filter. Such ideal filters are not realizable in practice, but as we have seen, real filters have a roll-off that can be improved through increasing the filter order. 

In Eq. (20.3) a is the constant gain or attenuation and z is the time delay in passing through the system. With this definition, H f), the Fourier transforms of the ideal filter impulse response and Y(f), the filter output are given by, respectively. 

Figure 20.42 shows the frequency-domain representation of an ideal and a nonideal filter and the time-domain representation of the input and output signals. Figures 20.42(a), 20.42(b), and 20.42(c) show, respectively, the frequency response of the ideal filter, the input signal waveform, and the output signal waveform, which is a time delayed version of the input waveform. Both the filter response and the input signal are bandlimited to B Hz. 

In real life, the availability of spectral filters would be an important consideration. Costs and delivery times of the ideal filter might convince you to settle for an off-the-shelf filter instead of your first choice. 

Assume that the accelerometer has the ideal response shown in Figure 4-223, with a measurement range of 2 g (32.2 ft/s ). We want to measure 1 g, but the ambient vibration level is +3 g. In this case, the accelerometer s indications are shaved and the mean value obtained is not 1 g but 0.5 g. The maximum acceleration due to vibrations which are not filtered mechanically, plus the... 

Expertise required to operate One of the objectives for using microprocessor-based predictive maintenance systems is to reduce the expertise required to acquire error-free, useful vibration and process data from a large population of machinery and systems within a plant. The system should not require user input to establish maximum amplitude, measurement bandwidths, filter settings, or allow free-form data input. All of these functions force the user to be a trained analyst and will increase both the cost and time required to routinely acquire data from plant equipment. Many of the microprocessors on the market provide easy, menu-driven measurement routes that lead the user through the process of acquiring accurate data. The ideal system should require a single key input to automatically acquire, analyze, alarm and store all pertinent data from plant equipment. This type of system would enable an unskilled user to quickly and accurately acquire all of the data required for predictive maintenance. 

Equation (3-307) also leads to a very important physical interpretation of the function Sx(f) namely, that Sx(f) describes the way in which the total power of X(t) is distributed in frequency. This comes about if we accept the physically reasonable definition that the power in X(t) contained in the frequency band /x < / < f2 is equal to the total power output of the ideal band-pass filter H(f) shown in Fig. 3-11, when X(t) is the input. Equation (3-307) now yields the result,... 

A dielectrofilter [Lin and Benguigui, Sep. Purif. Methods, 10(1), 53 (1981) Sisson et al., Sep. Sei. Teennol., 30(7-9), 1421 (1995)] is a device which uses the action of an electric field to aid the filtration and removal of particulates from fluid media. A dielectrofilter can have a very obvious advantage over a mechanical filter in that it can remove particles which are much smaller than the flow channels in the filter. In contrast, the ideal mechanical filter must have all its passages smaller than the particles to be removed. The resultant flow resistance can be use-restrictive and energy-consuming unless a phenomenon such as dielectrofiltration is used. 

Clearly the performance of (48) is bounded by the ideal matched filter output SINRideai obtained by letting R = R in (44). This gives... 

Two basic types of filtration processes may be identified, although there are cases where the two types appear to merge. In the first, frequently referred to as cake filtration, the particles from the suspension, which usually has a high proportion of solids, are deposited on the surface of a porous septum which should ideally offer only a small resistance to flow. As the solids build up on the septum, the initial layers form the effective filter medium, preventing the particles from embedding themselves in the filter cloth, and ensuring that a particle-free filtrate is obtained. 

For the very small AVj values under consideration the above expression may be linearized, co-AVg. Due to the cosine modulating filter function the NSE instrument measures the cosine transform of S Q,oo). The detector output of the (ideal) NSE instrument at exact symmetry is ... 

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