Measurement noise and signal processing

5.4.1 Passive analogue filters

The very simplest passive filters are circuits that consist only of resistors and capacitors. Unfortunately, these only have a mild filtering effect. This is adequate for circuits like tone controls in radio receivers but unsuitable for the sort of signal processing requirements met in most measurement applications. In such cases, it is normal to use a network of impedances, such as those labelled Z₁ and Z₂ in Figure 5.6(a). Design formulae require the use of a mixture of capacitive and inductive impedances for Z₁ and Z₂. Ideally, these impedances should be either pure capacitances or pure inductances (i.e. components with zero resistance) so that there is no dissipation of energy in the filter. However, this ideal cannot always be achieved in practice since, although capacitors effectively have zero resistance, it is impossible to manufacture resistance-less inductors.

 The detailed design of passive filters is quite complex and the reader is referred to specialist texts (e.g. Blinchikoff, 1976) for full details. The coverage below is therefore only a summary, and filter design formulae are quoted without full derivation.

Each element of the network shown in Figure 5.6(a) can be represented by either a T-section or -section as shown in Figures 5.6(b) and 5.6(c) respectively. To obtain proper matching between filter sections, it is necessary for the input impedance of each section to be equal to the load impedance for that section. This value of impedance is known as the characteristic impedance (Z0). For a T-section of filter, the characteristic impedance is calculated from:

The frequency attenuation characteristics of the filter can be determined by inspecting this expression for Z₀. Frequency values for which Z₀ is real lie in the pass-band of the filter and frequencies for which Z₀ is imaginary lie in its stop-band.

Let Z₁ = jωL and Z2 = 1/jωC, where L is an inductance value, C is a capacitance value and ω is the angular frequency in radian/s, which is related to the frequency 

f according to ω = 2 f. Substituting these values into the expression for Z₀ above gives:

quencies where ω < square root 4/LC, Z₀ is real, and for higher frequencies, Z₀ is imag[1]inary. These values of impedance therefore give a low-pass filter (see Figure 5.7(a)) with cut-off frequency fc given by:

                                               fc = ωc/2π = 1/(2π square root LC)
A high-pass filter (see Figure 5.7(b)) can be synthesized with exactly the same cut-off frequency if the impedance values chosen are:

                                               Z₁ = 1/jωC and Z₂ = jωL

It should be noted in both of these last two examples that the product Z₁Z₂ could be represented by a constant k that is independent of frequency. Because of this, such filters are known by the name of constant-k filters. A constant-k band-pass filter (see Figure 5.7(c)) can be realized with the following choice of impedance values, where a is a constant and the other parameters are as before:

The frequencies f₁ and f₂ defining the end of the pass-band are most easily expressed in terms of a frequency f₀ in the centre of the pass-band. The corresponding equations are:

For a constant-k band-stop filter (see Figure 5.7(d)), the appropriate impedance values are: 

The frequencies defining the ends of the stop-band are again normally defined in terms of the frequency f0 in the centre of the stop-band:

As has already been mentioned, a practical filter does not eliminate frequencies in the stop-band but merely attenuates them by a certain amount. The attenuation coefficient, ˛, at a frequency in the stop band, f, for a single T-section of a low-pass filter is given by:

                                                     α = 2 cosh^-1 (f/fc)
The relatively poor attenuation characteristics are obvious if we evaluate this expression for a value of frequency close to the cut-off frequency given by f = 2fc. Then α = 2 cosh-¹ (2) = 2.64. Further away from the cut-off frequency, for f = 20fc, α = 2 cosh^-1 (20) = 7.38.

 Improved attenuation characteristics can be obtained by putting several T-sections in cascade. If perfect matching is assumed, then two T-sections give twice the attenuation of one section, i.e. at frequencies of 2fc and 20fc, ˛ for two sections would have a value of 5.28 and 14.76 respectively.

 The discussion so far has assumed that the inductances are resistance-less and that there is perfect matching between filter sections. However, it has already been noted that such ideal conditions cannot be achieved in practice, and this has several consequences. Inspection of the expression for the characteristic impedance (5.1) reveals frequency[1]dependent terms. Thus, the condition that the load impedance is equal to the input impedance for a section is only satisfied at one particular frequency. It is usual to match the impedances at zero frequency for a low-pass filter and at infinite frequency for a high-pass filter. This ensures that the frequency where the input and load impedances are matched is comfortably within the pass-band of the filter. Frequency dependency is one of the reasons for the degree of attenuation in the pass-band shown in the practical filter characteristics of Figure 5.5, the other reason being the presence of resistive components in the inductors of the filter. The effect of this in a practical filter is that the value of ˛ at the cut-off frequency is 1.414 whereas the value predicted theoretically for an ideal filter (equation 5.2) is zero. Cascading filter sections together increases this attenuation in the pass-band as well as increasing attenuation of frequencies in the stop-band.

This problem of matching successive sections in a cascaded filter seriously degrades the performance of constant-k filters and this has resulted in the development of other types such as m-derived and n-derived composite filters. These produce less attenuation within the pass-band and greater attenuation outside it than constant-k filters, although this is only achieved at the expense of greater filter complexity and cost. The reader interested in further consideration of these is directed to consult one of the specialist texts recommended in the Further Reading section at the end of this chapter.