Transcript webinar_19_digital_filtering

Unit 19
Vibrationdata
Digital Filtering
(plus some seismology)
1
Introduction
Vibrationdata

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
2
Introduction (Continued)

Vibrationdata
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
3
Highpass & Lowpass Filters





Vibrationdata
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
4
Butterworth Filter Characteristics

Vibrationdata
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
5
Butterworth Filter (Continued)
Vibrationdata

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
100 Hz
The corresponding transfer function magnitude is given in the following figure
6
Vibrationdata
(100 Hz, 0.707)
vibrationdata > Time History > Filters, Various > Butterworth > Display Transfer Function
No phase correction.
7
Transfer Function Characteristics


Vibrationdata
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
8
Vibrationdata
(100 Hz, 0.707)
vibrationdata > Time History > Filters, Various > Butterworth > Display Transfer Function
No phase correction.
9
Common -3 dB Point for three order cases
Vibrationdata
c = 1 rad/sec
BUTTERWORTH LOWPASS FILTER L=ORDER
1.1
L=6
L=4
L=2
1.0
0.9
MAGNITUDE
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
FREQUENCY (rad/sec)
10
Frequency Domain Implementation

Vibrationdata
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.
11
Time Domain Implementation
Vibrationdata
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
Time domain
equivalent of H(w)
yk
12
Time Domain Implementation
Vibrationdata
The filtering equation is implemented as a digital recursive filtering
relationship. The response yk is
L
L




 


yk    bnxkn    anykn 



n  1

n  0
 
where
xk is the input
an & bn are coefficients
L is the order
13
Phase Correction
Vibrationdata

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
14
Vibrationdata
Phase Correction
xk
Time
Reversal
Time domain
equivalent of
H(w)
Time
Reversal
Time domain
equivalent of
Yk
H(w)
An important note about refiltering is that it reduces the transfer function
magnitude at the cut-off frequency to –6 dB.
15
Vibrationdata
(100 Hz, 0.5)
vibrationdata > Time History > Filters, Various > Butterworth > Display Transfer Function
Yes phase correction.
16
Filtering Example
Vibrationdata
•
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
17
Homemade Lehman Seismometer
Vibrationdata
18
Non-contact
Vibrationdata
Displacement
Transducer
19
Ballast Mass Partially Submerged in Oil
Vibrationdata
20
Pivot End of the Boom
Vibrationdata
21
Vibrationdata
The seismometer was given an initial displacement and then allowed to
vibrate freely. The period was 14.2 seconds, with 9.8% damping.
22
Vibrationdata
23
Vibrationdata
24
Vibrationdata
Highpass filter to find
onset of P-wave
External file: sm.txt
P
S
vibrationdata > Time History > Filters, Various > Butterworth
with phase correction.
25
Characteristic Seismic Wave Periods
Vibrationdata
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
26
Vibrationdata
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.
27
Vibrationdata
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.
28
Vibrationdata
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.
29
Vibrationdata
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.
30