20.11 Transform Methods

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20.11.1 General Considerations

For linear systems, the Laplace and Fourier transforms (Laplace being more general) have been preeminent

tools with which to study equations of motion (see Appendix 2A and Chapter 10). The author’s

transform experience (like most physicists) is mainly with Fourier transforms (FT). The discrete FT can

be understood in terms of phasors (Peters, 1992). For linear differential equations, transforms are the

means to convert differential forms to an equivalent algebraic form. Unfortunately, they cannot be

directly employed on nonlinear equations due to the failure of superposition. Nevertheless, the linear

approximations continue to be very valuable, so a chapter on damping deserves to mention some of their

properties.

Ideas concerning the FFT were evidently originally treated by Gauss in the early 1800s, but the digital

signal processing (DSP) “explosion” of the 1960s was largely due to the work of Cooley and Tukey (1965).

For an interesting historical account about an “accident” in the publication of their paper, the reader is

referred to Cipra (1993), who says the following about the FFT:

The Fourier transform stands at the center of signal processing, which encompasses everything

from satellite communications to medical imaging, from acoustics to spectroscopy. Fourier

analysis, in the guise of x-ray crystallography, was essential to Watson and Crick’s discovery of

the double helix, and it continues to be important for the study of protein and

viral structures. The Fourier transform is a fundamental tool, both theoretically and

Sensor output (V)

3

2

1

0

−1

−2

−3

Time (S)

residuals from fit

0 100 200 300 400

FIGURE 20.11 Illustration of phase noise in the free-decay of a vertical seismometer.

20-34 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

computationally, in the solution of partial differential equations. As such, it is at the heart of

mathematical physics, from Fourier’s analytic theory of heat to the most modern treatments

of quantum mechanics. Any kind of wave phenomenon — be it seismic, tidal, or

electromagnetic — is a candidate for Fourier analysis. Many statistical processes, such as the

removal of “noise” from data and computing correlations, are also based on working with

Fourier transforms.

Concerning the last statement about noise, this author has used autocorrelation as a powerful means

for identifying short-lived, low-frequency periodic signals in time records that do not readily show up in

power spectra (FFTs). For example, they are the means for studying free-earth oscillations — eigenmodes

excited by rapid relaxations of the Earth under tidal stressing (12 h periodic) (Peters, 2000). The FFT is

used to generate the autocorrelation by means of the Wiener– Khintchine theorem (Press et al., 1986).

The great advantage of the FFT compared with the DFT has to do with degeneracy. The DFT proceeds

to calculate the components of every “vector” in the reciprocal space (frequency reciprocal to time, units

of “second”, or wave number (spatial frequency) reciprocal to displacement, units of “meter”) with

disregard for the fact that many components have the same value, apart from a change of sign.

20.11.2 Bit Reversal

The key to the power of the FFT (central processor unit [CPU] time proportional to n log n) compared

with the discrete Fourier transform (DFT) (CPU time proportional to n2) is the bit reversal scheme of the

Cooley – Tukey algorithm. It is illustrated very simply by the following. Instead of a practically sized

number of samples in the record to be transformed (minimum of n ¼ 1024; typically), consider (for

pedagogy) n ¼ 8; distributed on the unit circle as shown in Figure 20.12.

Observe that the roots of unity in the complex plane, which have been numbered 0 to 7, divide the

“pie” into eight equal pieces. (The algorithm requires that n be expressible as a power of 2). The usual

decimal counting scheme for the eight “vectors” is as indicated, traversing the phasor diagram (circle on

left) sequentially. In the Cooley – Tukey algorithm, a choice is made to reverse the bits of the binary

representation of the vector. Usually, the least significant bit is on the right and the most significant bit on

the left, so that decimal counting is as shown on the right in the table, from 0 to 7. With bit reversal, “lsb”

becomes the leftmost binary digit and the “msb” is the rightmost digit. Thus, for example, binary 110

(usually 6) becomes 3.

Using bit reversal, the phasor diagram is not traversed in the usual phasor (circulatory) sense, but

rather in a “flip-flop” back and forth across the circle. By this means, there is no needless repetition in the

calculation of “vector” components (real and imaginary values of a given term in the transform).

For example, 5 is the simple negative of 1. It is much faster to reverse the sign on 1 to get 5 than to

decimal

(bit-reversed)

binary

number

decimal

2 (usual)

1

0

3

6

5

4

7

0 000 0

4 001 1

2 010 2

6 011 3

1 100 4

5 101 5

3 110 6

7 111 7

FIGURE 20.12 Graphical illustration of why the Cooley – Tukey FFT algorithm is significantly faster than the

original DFT.

Damping Theory 20-35

© 2005 by Taylor & Francis Group, LLC

needlessly calculate values for sine and cosine terms a second time. The saving in time is substantial as n

gets large, since there are then a great number of circulations of the phasor circle. For a 1K record, the FFT

computes the transform 102.4 times faster than does the DFT. Additional details are provided in Peters

(2003a, 2003b, 2003c).

20.11.3 Wavelet Transform

Recent work suggests that the wavelet transform (WT) may in the future replace the FT in some

applications (see Chapter 11). It uses the Haar function, which is orthogonal on [0,1], as opposed to

the orthogonality of the harmonic functions (sine and cosine) corresponding to [0,2p ] (Strang, 1993). It is

claimed that the WT is better able to address features of the Heisenberg uncertainty principle than the FFT.

20.11.4 Heisenberg’s Famous Principle

The heart and soul of quantum mechanics is the Heisenberg uncertainty principle. As noted elsewhere in

this chapter, it has things to say about damping models. According to well-known physicist Hans Bethe

(1992), the principle has received “bad press”:

Many people believe that the uncertainty principle has made everything uncertain. It is quite

the opposite. Without the uncertainty principle there could not exist any atoms, there could

not be any certainty in the behavior of matter. So it is in fact a certainty principle.

Curiously, a failure figured in Heisenberg’s discovery of the principle. During his thesis defense, in

front of great theoretical physicist Arnold Sommerfeld (his director) and the famous experimentalist

Wilhelm Wien, he proved unable to derive the magnifying power of a simple microscope. The scandal

culminated with Professor Wien asking him to explain how a battery works, and he could not answer that

question either. Knowing his extraordinary theoretical giftings, Sommerfeld gave him the highest

possible grade to compensate for Wien’s choice of an F. Thus, Heisenberg was awarded his doctorate.

Later, in an ironic turn of events, Heisenberg chose a microscope to illustrate features of the matrix

quantum mechanics that he originated, and which corrected problems with the Bohr wave mechanics

theory. His greatest source of embarrassment served to make Heisenberg famous!

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