1. (plus some seismology)
Unit 18
Digital Filtering
(plus some seismology)

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

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

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

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

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

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

8. Transfer Function Characteristics
Note that the curve in the previous figure has a gradual roll-off beginning at about 70 Hz
Ideally, the transfer function would have a rectangular shape, with a corner at (100 Hz, 1.00 )
This ideal is never realized in practice
Thus, a compromise is usually required to select the cut-off frequency
The transfer function could also be represented in terms of a complex function, with real and imaginary components
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. (100 Hz, 0.707)
vibrationdata > Time History > Filters, Various > Butterworth > Display Transfer Function
No phase correction.

10. Common -3 dB Point for three order cases

11. Frequency Domain Implementation
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
The above frequency domain method is valid  
Nevertheless, the filtering algorithm is usually implemented in the time domain for computational efficiency, to avoid leakage error, etc.

12. Time Domain Implementation
The transfer function can be represented by H(w).
Digital filters are based on this transfer function, as shown in the filter block diagram.
Note that xk and yk are the time domain input and output, respectively. xk yk
Time domain equivalent of H(w)

13. Time Domain Implementation
The filtering equation is implemented as a digital recursive filtering relationship. The response is yk
where
is the input
an & bn are coefficients
L is the order xk

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

15. Phase Correction H(w) H(w)
Time Reversal
x k
Time domain equivalent of Yk
H(w)
H(w)
An important note about refiltering is that it reduces the transfer function magnitude at the cut-off frequency to –6 dB.

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

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. Homemade Lehman Seismometer

19. Non-contact Displacement Transducer

20. Ballast Mass Partially Submerged in Oil

21. Pivot End of the Boom

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

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

26. Characteristic Seismic Wave Periods
Wave Type
Period
(sec)
Natural Frequency
(Hz)
Body
0.01 to 50
0.02 to 100
Surface
10 to 350
0.003 to 0.1
Reference: Lay and Wallace, Modern Global Seismology

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. The secondary wave, or S-wave, is a shear wave
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. Love waves are shearing horizontal waves
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. 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.