Digital Filtering

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Unit 19
Digital Filtering (plus some seismology)
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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

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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

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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

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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

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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 100 Hz
• The corresponding transfer function magnitude is
  given in the following figure

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

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(100 Hz, 0.707)
vibrationdata gt Filters, Various gt Butterworth gt
Display Transfer Function No phase correction.
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Common -3 dB Point for three order cases
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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.

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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)
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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
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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

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Phase 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.  
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(100 Hz, 0.5)
vibrationdata gt Filters, Various gt Butterworth gt
Display Transfer Function Yes phase correction.
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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

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Homemade Lehman Seismometer
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Non-contact Displacement Transducer
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Ballast Mass Partially Submerged in Oil
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Pivot End of the Boom
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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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(No Transcript)
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(No Transcript)
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Highpass filter to find onset of P-wave External
file sm.txt
S
P
vibrationdata gt Filters, Various gt
Butterworth with phase correction.
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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
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The 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.
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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.
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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.
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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.