Butterworth filter

Figure 3.3 Breadboard configuration of fourth-order Butterworth filter.

The Chebyshev filter offers higher attenuation and a steeper roll-off near the cutoff frequency than the Butterworth filter. There is a tradeoff to achieve the higher attenuation. The cost of utilizing a Chebyshev filter is higher values of Q, which leads to difficulties in hardware realization, and nonlinear phase characteristics, which can result in difficulties in predicting circuit performance. 

Figure 3.7. Unit-step responses of an instrument (a) and of a high-frequency filter (b) in the same frequency range 1 - no filter 2 - with a filter 3 - ideal filter characteristic 4 - approximation by a second order Butterworth filter.

An important low-pass filter that is most commonly used in nuclear medicine is the Butterworth filter (Fig. 4.7). This filter has two parameters the critical frequency (fc) and the order or power (n). The critical frequency is the frequency at which the filter attenuates the amplitude by 0.707, but not the frequency at which it is reduced to zero as with other filters. The parameter, order n, determines how rapidly the attenuation of amplitudes occurs with increasing frequencies. The higher the order, the sharper the fall. Lowering the critical frequency, while maintaining the order, results in more smoothing of the image. 

Figure 4.7. Butterworth filters with different orders and cutoff frequencies.

Fig. 3 Transmission electron micrographs of different nanosheets.(A to C) Low resolution TEM images of flakes of BN, M0S2, and WS2, respectively. (D to F) High-resolution TEM images of BN, M0S2, and WS2 monolayers. (Insets) Fast Fourier transforms of the images. (G to I) Butterworth-filtered images of sections of the images in (D) to (F). (Reproduced with permission).

The thick line in Fig. 5.9 is the result of filtering the raw data in the forward and reverse directions using a fourth-order, low-pass Butterworth filter set at a cutoff frequency of 6 Hz (note that the front and back ends of the raw data were padded). This raises an interesting question, that is, how do we identify an appropriate cutoff frequency for the filter There are a number of methods that can be used to help select an appropriate cutoff frequency. A fast Fourier transform (FFT) can be used to... 

In order to test the merit of this numerical module, we carried out a comparison test with low-pass Butterworth filter. Fig. 5 shows the signal filtered by digital Butterworth filter with corner frequency 200 Hz. 

Passband ripple The passband of a filter should ideally be flat. As we have seen, the Chebyshev filter sacrifices a flat passband for faster roll-off. The Butterworth filter, on the other hand, has been designed to have the shortest possible transition band while still maintaining a flat passband i.e., there is no ripple. The Bessel filter also has a flat passband however, it has the worst roll-off of the three designs. 

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. 

The principal advantage of the Butterworth alignment is the flatness of the ampHtude response within the pass band. The principal disadvantage is that the rolloff is not as steep in the vicinity of cutoff as that of some other filter designs of the same complexity (number of poles). The Chebyshev filter has a frequency response with a steeper rolloff in the vicinity of cutoff than that of the Butterworth filter, but ripples are present in the pass band. The Chebyshev filter is sometimes called an equiripple filter because the ripples have the same peak-to-peak amplitude throughout the pass band. 

The mathematicalbasis for frequency mapping is omitted here however, the technique involves replacing the LP transfer function variable (s or ) with the mapped variable, as presented in Table 4.13. The mapped amplitude response for a Butterworth filter or a Chebyshev filter is obtained by performing the appropriate operation from column two of Table 4.13 on Eq. (4.60) or (4.61), respectively. The mapped phase response... 

The transfer function T(s) of a fifth-order low-pass Butterworth filter with (3.01-dB) cutoff frequency coc = 1 rad/s is... 

FIGURE 8.107 Magnitude responses of low-pass analog filters (a) Butterworth filter, (b) Chebyshev filter, (c) inverse Chebyshev filter. 

The transfer function of an Nth-order Butterworth filter is given by... 

In comparison, the Butterworth filter requires a higher order than both types of Chebyshev filters to satisfy the same specification. There is another type of filters called elliptic filters (Cauer filters) that have equiripples in the pass band as well as in the stop band. Because of the lengthy expressions, this type of filters is not given here (see the references). The Butterworth filter and the inverse Chebyshev filter have better (closer to linear) phase characteristics in the pass band than Chebyshev and elliptic filters. Elliptic filters require a smaller order than Chebyshev filters to satisfy the same specification. 

A filter allows ac of some frequencies to pass through it with the amplitude (voltage or current) essentially unchanged, but the amplitudes of other frequencies are decreased. (An "active filter," involving transistors, can increase certain frequencies, as well as decreasing some.) A simple example, using just one resistor and one capacitor, which is called a "low-pass RC" filter, is shown in Fig. 11.1. It is a type of "first order Butterworth" filter. 

---