Title: Digital Filtering
1Unit 19
Digital Filtering (plus some seismology)
2Introduction
- 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
3Introduction (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
4Highpass 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
5Butterworth 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
6Butterworth 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 100 Hz - The corresponding transfer function magnitude is
given in the following figure
7 (100 Hz, 0.707)
vibrationdata gt Time History gt Filters, Various gt
Butterworth gt Display Transfer Function No phase
correction.
8Transfer 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 gt Time History gt Filters, Various gt
Butterworth gt Display Transfer Function No phase
correction.
10Common -3 dB Point for three order cases
11Frequency 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.
12Time 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)
13Time 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
14Phase 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
15Phase Correction
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 gt Time History gt Filters, Various gt
Butterworth gt Display Transfer Function Yes
phase correction.
17Filtering 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
18Homemade Lehman Seismometer
19Non-contact Displacement Transducer
20Ballast Mass Partially Submerged in Oil
21Pivot End of the Boom
22The seismometer was given an initial displacement
and then allowed to vibrate freely. The
period was 14.2 seconds, with 9.8 damping.
23(No Transcript)
24(No Transcript)
25Highpass filter to find onset of P-wave External
file sm.txt
P
S
vibrationdata gt Time History gt Filters, Various gt
Butterworth with phase correction.
26Characteristic 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
27The primary wave, or P-wave, is a body wave that
can propagate through the Earths 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.
28The 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.
29Love 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.
30Rayleigh 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.