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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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vibrationdata > Filters, Various > Butterworth > Display Transfer Function

No phase correction.

(100 Hz, 0.707)

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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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vibrationdata > Filters, Various > Butterworth > Display Transfer Function

No phase correction.

(100 Hz, 0.707)

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Common -3 dB Point for three order cases

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

L=6L=4L=2

FREQUENCY (rad/sec)

MA

GN

ITU

DE

BUTTERWORTH LOWPASS FILTER L=ORDER c = 1 rad/sec

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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().

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.

Time domain equivalent of H

xk yk

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Time Domain Implementation

y b x a yk n k nn

L

n k nn

L

0 1

where is the input

an & bn are coefficients

L is the order

The filtering equation is implemented as a digital recursive filtering relationship. The response is

xk

yk

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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

Time Reversal

Time Reversal

x k Time domain equivalent of

Time domain equivalent of Yk

An important note about refiltering is that it reduces the transfer function magnitude at the cut-off frequency to –6 dB.

 

H H

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vibrationdata > Filters, Various > Butterworth > Display Transfer Function

Yes phase correction.

(100 Hz, 0.5)

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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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vibrationdata > Filters, Various > Butterworth

with phase correction.

Highpass filter to find onset of P-wave

External file: sm.txt

PS

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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 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.

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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.