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Filters may be described by their effect on various frequency components in the data. Figure 9-6 illustrates the relative attenuation of frequencies for a least-squares quadratic filter. For 25 data points, attenuation is substantial for «> 0.07«ot and negligible for <o<0.05<om. The quantity um=2n/At is the measurement frequency and At is the time interval between measurements.1 Hie number of data points must be carefully selected to avoid removing desired information from the data.

OJO US 0J0

FREQUENCY RATIO <u<unl

Fig. 9-6. Frequency Attenuation (or a Least-Squares Quadratic Filter [Budurka, 1967]

OJO US 0J0

FREQUENCY RATIO <u<unl

Fig. 9-6. Frequency Attenuation (or a Least-Squares Quadratic Filter [Budurka, 1967]

For some applications, the poor frequency cutoff characteristics of the quadratic least-squares filter (manifested by the persistent sinusoidal oscillation at high frequencies) are undesirable. The Butterworth filter [Dennis, 1974; Budurka, 1967; Rabiner and Gold, 1975; Stanley, 1975] has a much sharper cutoff, as shown in Fig. 9-7. Hie coefficients depend on both the order and the cutoff frequency, ioc.

The difference equation for the fifth-order Butterworth is j

fc-i

Fig. 9-7. Magnitude-Frequency Function for Fifth-Order Butterworth Filter [Budurka, 1967]

where the coefficients for uc = 0.044<dm are given in Table 9-2 for equally spaced data. The recursive nature of Eq. (9-25) implies an infinite memory; that is, the improved bandwidth characteristic (e.g., selective frequency attenuation) is achieved by linking together all the measurements. The infinite memory causes an initial transient response in the filter output The Butterworth filter is particularly

Table 9-2. Coefficients for Fifth-Order Butterworth Filter (Budurka, 1967]

*0 -

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