1. Vibrationdata 1 Unit 19 Digital Filtering (plus some seismology)

2. Vibrationdata 2 Introduction n Filtering is a tool for resolving signals n Filtering can be performed on either analog or digital signals n Filtering can be used for a number of purposes n For example, analog signals are typically routed through a lowpass filter prior to analog-to-digital conversion n The lowpass filter in this case is designed to prevent an aliasing error n This is an error whereby high frequency spectral components are added to lower frequencies

3. Vibrationdata 3 Introduction (Continued) n Another purpose of filtering is to clarify resonant behavior by attenuating the energy at frequencies away from the resonance n This Unit is concerned with practical application and examples n It covers filtering in the time domain using a digital Butterworth filter n This filter is implemented using a digital recursive equation in the time domain

4. Vibrationdata 4 Highpass & Lowpass Filters n A highpass filter is a filter which allows the high-frequency energy to pass through n It is thus used to remove low-frequency energy from a signal n A lowpass filter is a filter which allows the low-frequency energy to pass through n It is thus used to remove high-frequency energy from a signal n A bandpass filter may be constructed by using a highpass filter and lowpass filter in series

5. Vibrationdata 5 Butterworth Filter Characteristics n A Butterworth filter is one of several common infinite impulse response (IIR) filters n Other filters in this group include Bessel and Chebyshev filters n These filters are classified as feedback filters n The Butterworth filter can be used either for highpass, lowpass, or bandpass filtering n A Butterworth filter is characterized by its cut-off frequency n The cut-off frequency is the frequency at which the corresponding transfer function magnitude is –3 dB, equivalent to 0.707

6. Vibrationdata 6 Butterworth Filter (Continued) n A Butterworth filter is also characterized by its order n A sixth-order Butterworth filter is the filter of choice for this Unit n A property of Butterworth filters is that the transfer magnitude is –3 dB at the cut-off frequency regardless of the order n Other filter types, such as Bessel, do not share this characteristic n Consider a lowpass, sixth-order Butterworth filter with a cut-off frequency of 100 Hz n The corresponding transfer function magnitude is given in the following figure

7. Vibrationdata 7 vibrationdata > Filters, Various > Butterworth > Display Transfer Function No phase correction. (100 Hz, 0.707)

8. Vibrationdata 8 Transfer Function Characteristics n Note that the curve in the previous figure has a gradual roll-off beginning at about 70 Hz n Ideally, the transfer function would have a rectangular shape, with a corner at (100 Hz, 1.00 ) n This ideal is never realized in practice n Thus, a compromise is usually required to select the cut-off frequency n The transfer function could also be represented in terms of a complex function, with real and imaginary components n A transfer function magnitude plot for a sixth-order Butterworth filter with a cut-off frequency of 100 Hz as shown in the next figure

9. Vibrationdata 9 vibrationdata > Filters, Various > Butterworth > Display Transfer Function No phase correction. (100 Hz, 0.707)

10. Vibrationdata 10 Common -3 dB Point for three order cases

11. Vibrationdata 11 Frequency Domain Implementation n The curves in the previous figures suggests that filtering could be achieved as follows: 1. Take the Fourier transform of the input time history 2. Multiply the Fourier transform by the filter transfer function, in complex form 3. Take the inverse Fourier transform of the product n The above frequency domain method is valid n Nevertheless, the filtering algorithm is usually implemented in the time domain for computational efficiency, to avoid leakage error, etc.

12. Vibrationdata 12 Time Domain Implementation The transfer function can be represented by H(  ). Digital filters are based on this transfer function, as shown in the filter block diagram. Note that x k and y k are the time domain input and output, respectively. Time domain equivalent of H    xkxk ykyk

13. Vibrationdata 13 Time Domain Implementation where is the input a n & b n are coefficients L is the order The filtering equation is implemented as a digital recursive filtering relationship. The response is xkxk ykyk

14. Vibrationdata 14 Phase Correction n Ideally, a filter should provide linear phase response n This is particularly desirable if shock response spectra calculations are required n Butterworth filters, however, do not have a linear phase response n Other IIR filters share this problem n A number of methods are available, however, to correct the phase response n One method is based on time reversals and multiple filtering as shown in the next slide

15. Vibrationdata 15 Phase Correction Time Reversal x k Time domain equivalent of YkYk An important note about refiltering is that it reduces the transfer function magnitude at the cut-off frequency to –6 dB. H 

16. Vibrationdata 16 vibrationdata > Filters, Various > Butterworth > Display Transfer Function Yes phase correction. (100 Hz, 0.5)

17. Vibrationdata 17 Filtering Example Use filtering to find onset of P-wave in seismic time history from Solomon Island earthquake, October 8, 2004 Magnitude 6.8 Measured data is from homemade seismometer in Mesa, Arizona

18. Vibrationdata 18 Homemade Lehman Seismometer

19. Vibrationdata 19 Non-contact Displacement Transducer

20. Vibrationdata 20 Ballast Mass Partially Submerged in Oil

21. Vibrationdata 21 Pivot End of the Boom

22. Vibrationdata 22 The seismometer was given an initial displacement and then allowed to vibrate freely. The period was 14.2 seconds, with 9.8% damping.

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25. Vibrationdata 25 vibrationdata > Filters, Various > Butterworth with phase correction. Highpass filter to find onset of P-wave External file: sm.txt P S

26. Vibrationdata 26 Characteristic Seismic Wave Periods Wave TypePeriod (sec) Natural Frequency (Hz) Body0.01 to 500.02 to 100 Surface10 to 3500.003 to 0.1 Reference: Lay and Wallace, Modern Global Seismology

27. Vibrationdata 27 The primary wave, or P-wave, is a body wave that can propagate through the Earth’s core. This wave can also travel through water. The P-wave is also a sound wave. It thus has longitudinal motion. Note that the P-wave is the fastest of the four waveforms.

28. Vibrationdata 28 The secondary wave, or S-wave, is a shear wave. It is a type of body wave. The S-wave produces an amplitude disturbance that is at right angles to the direction of propagation. Note that water cannot withstand a shear force. S-waves thus do not propagate in water.

29. Vibrationdata 29 Love waves are shearing horizontal waves. The motion of a Love wave is similar to the motion of a secondary wave except that Love wave only travel along the surface of the Earth. Love waves do not propagate in water.

30. Vibrationdata 30 Rayleigh waves travel along the surface of the Earth. Rayleigh waves produce retrograde elliptical motion. The ground motion is thus both horizontal and vertical. The motion of Rayleigh waves is similar to the motion of ocean waves except that ocean waves are prograde.