1) After the French Mathematician Jean Baptiste Joseph Fourier, 1768-1830, [42].

7. FOURIER ANALYSIS AND DATA PROCESSING

Fourier1 analysis plays a dominant role in the treatment of vibrations of mechanical systemsresponding to deterministic or stochastic excitation, and, as has already been seen, it forms the basis ofspectral representation of stochastic time series and random signals. The Fourier transformation as amathematical tool offers a clear and simple method for treating complex analytic problems byconverting equations involving the basic variable to corresponding equations in the frequency domainwhere solutions are often more easily accessible. In probability theory, the Fourier transform is the keyto understanding certain probability distributions through their characteristic functions. Finally, thediscrete Fourier series form the basis of signal processing and data manipulation, which has turned intoa vast field that concerns most if not all scientific studies.

In the following, a brief overview of Fourier analysis is presented to have the background forapplication elsewhere in the text readily available. Fourier series representation of periodic functions isintroduced and the continous Fourier transform is derived for aperiodic functions. Various convenientrelations concerning the Fourier transform are presented and a few examples given to clarify the text. This will be followed by an overview section on signal analysis and data processing. Sampling ordiscretizing of bandlimited realizations of stochastic processes or any other time signals is brieflydiscussed. Thereafter, the discrete and inverse discrete Fourier transforms will be introduced as anatural extension of the continuous Fourier transform and discrete Fourier series analysis. The FastFourier Transform Algorithm is briefly discussed together with various aspects of the analysis of digitalsignals. Signal convolution and aliasing, and the ramifications of signal distortion due to truncation ispresented in some detail. Inverse filtering and windowing of signals is also presented with examples ofstandard digital filters. This is a highly specialized field, which is undergoing rapid expansion anddevelopment in connection with the ever increasing electronic gadgetry and instruments in use ineveryday life. Finally, other related methods, such as the z-transformation and ARMA(AutoRegressive Moving Average) models will be mentioned in order to furnish information on recentdevelopments in earthquake signal analysis, especially for generation of synthetic earthquakeaccelerations.

Some exciting new topics in system analysis and signal processing will not be treated for thesake of space and limitation of topics. Notably, new concepts such as fuzzy systems and neural

368

Fig. 7.1 Periodic Functions

networks should be mentioned in this respect, [170], [171] and [172].

7.1 Fourier Analysis

It is assumed that the reader has already acquired a fundamental knowledge of the basic theoryof Fourier series and integral transforms. The following presentation is therefore a refresher text toclarify the concepts and the tools to be used. The most basic relations and definitions will be presentedtogether with some practical examples. Formal rigorous mathematical proof of the various statementshas been avoided whenever deemed acceptable, and more simple methods of derivation employed. The text is both compact and lacking in details. For more rigorous and detailed presentation, there is avariety of standard textbooks available. For example, Weaver, [169] and Papoulis, [116], offer athorough and detailed version of Fourier analysis and its many applications with the latter textconcentrating on the Fourier integral transform.

7.1.1 Periodic Functions

A function f(t) is called a periodic function with the period T>0 if for all t, f(t+T)=f(t). Byinduction, f(t+nT)=f(t) where n is any integer number. Therefore, the function repeats itself in anynumber of intervals of length T. The well known cosine and sine functions are perfect examples ofperiodic functions with the period 2B. In Fig. 7.1, more general examples of such

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functions are shown. The question now arises, can any reasonably well behaved function f(t), where t isa real variable (for instance the time), that repeats itself with the period T>0, be expanded intotrigonometric series, that is, broken down into its harmonic cosine and sine components. The functionf(t) would then be synthetised by adding the harmonic components together.

Let f(t) be a periodic function with the period T>0. Define the corresponding basic circularfrequency T=2B/T by which the function repeats itself. Now, formally write f(t) as the trigonometricseries,

f(t) = 1/2 a0+a1cosTt+b1sinTt+a2cos2Tt+b2sin2Tt+AAA

+ancosnTt+bnsinnTt+AAA (7.1)

in which ai and bi are real constants-called the Fourier coefficients-to be determined. For this purpose,the orthogonal properties of the trigonometric functions, shown below, will come in handy.

By multiplying both sides of Eq. (7.1) with the term cos(kTt) and then integrating over the range (0,T),making use of the above orthogonal properties of the trigonometric functions, the Fourier coefficientsai's are obtained and in the same manner, by multiplying with the term sin(kTt), the Fourier coefficientsbi's are found. Thus,

(7.2)

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Thus it has been shown without formal mathematical proof that any reasonably well behaved functionf(t) can be expressed as Fourier series of the form

(7.1)

where the Fourier coefficients {ai , bi} are given by the relations Eq. (7.2).

It may be asked under what conditions the series Eq. (7.1) converge, that is, reproduce thefunction f(t). There are several modes of convergence, which can be described as follows:

(i) The Fourier series Eq. (7.1) is said to converge uniformly to f(t) in the interval of length T if

a) f(t), fN(t), and fO(t) exist and are continous for 0#t#T

b) f(t) satisfies given boundary conditions

(ii) The Fourier series Eq. (7.1) is said to converge to f(t) in the mean square sense in the interval (0,T)provided only that f(t) is any function for which

(iii) The Fourier series Eq. (7.1) is said to converge point wise to f(t) in the interval of length T if f(t) is acontinuous function in (0,T) and fN(t) is piecewise continuous in (0,T), that is, is continous at all exceptat most finite number of points in (0,T), where it can have a jump discontinuity. The uniformconvergence is of course strongest whereas the other two forms assure a certain convergence under aweaker set of assumptions. The statements of existence and convergence of Fourier series will not beaddressed further here as the purpose is more to furnish tools to analyse functions known to have afrequency content, i.e. possess harmonic components. Usually, it is just stated that any function, whichsatisfies the so-called Dirichlet’s conditions, can be expressed as Fourier series. These conditions arethe following:

a) the function is periodic, that is, f(t+T)=f(t)

b) the function is bounded, that is, *f(t)*<4 for all t

c) the function has at most a finite number of discontinuities within each period

d) the function has at most a finite number of maxima/minima within each period

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The Dirichlet conditions are sufficient for a periodic function to have a Fourier series representation butnot necessary. There are functions which do not satisfy the above conditions but can nevertheless beexpressed as Fourier series.

Consider a function f(t) satisfying the Dirichlet conditions that is either an even or odd functionof t. For an even function, the Fourier coefficients bk vanish (see Eq. (7.2), and for an odd function theak’s vanish. Therefore, the Fourier coefficients are obtained using the simpler form

(7.3)

The Fourier series Eq. (7.1) can be rewritten using a complex mathematical form by applyingthe Euler relations

e±ikTt = coskTt±iAsinkTtor

(7.4)

where

(7.5)

since

(7.6)

Using the Euler relations Eq. (7.3), and comparing with Eq. (7.2), the Fourier coefficients ck, ak and bk

are interrelated as follows:

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(7.7)

The Fourier coefficients ak and bk give information about the strength of the correspondingharmonic component in the function or signal f(t). For instance, if x(t) is the deflection of a certainelastic structure with a representative spring constant k, the strain energy at any time t is ½kx2(t), (alsothe energy carried by an electric signal where x(t) is the current). The strain energy accumulated duringone period T called E (in units of 2/k) is therefore

upon introducing the Fourier series, Eq. (7.4). Interchanging the order of summation and integration,

and applying the orthogonality relations Eq. (7.6),

or

(7.8)

The energy is thus proportional to the sum of the squared amplitudes of the harmonic components in thesignal. The terms within the brackets give the energy density per one complete period T. A energydensity spectrum can therefore be defined by considering the contribution by the k-th frequencycomponent, that is,

F(kT) = F(Tk) = 1/2 (ak2+bk

2) (7.9)

The relation Eq. (7.8) is also a statement of the so-called Parseval's theorem, which relates theaverage energy of the signal over one period to the sum square of the Fourier coefficients, that is,

(7.10)

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Fig. 7.2 The Energy Density Spectrum

In Fig. 7.2 the energy density spectrum F(T) is plotted for a series of imaginative discrete frequencies.

Example 7.1

A pile hammer produces a rectangular impact load of 1 second duration at the rate of 6per minute as shown in Fig. 7.3. Obtain the Fourier series of the pile driving load function and itsenergy density spectrum.

Solution:

The Fourier coefficients can be obtained directly using Eq. (7.2). Thus,

A typical term in the series will be like

374

Fig. 7.3 Pile Driving Forces

or

The largest contribution to the energy density is due to the first component. The contributions due tothe higher frequency terms rapidly tend to zero, which is reflected in Fig. 7.4.

7.1.2 Aperiodic Functions

Next, consider the type of functions that do not show any pattern of periodicity. Non-periodicfunctions, i.e. functions that do not repeat themselves systematically in any time interval, are calledaperiodic. However, figuratively speaking, one can consider an aperiodic function to have an infiniteperiod, that is, letting T64, the function is periodic with an infinitely

375

Fig. 7.4 A Discrete Fourier Spectrum

long period which corresponds to the entire real axis -4<t<4. For functions f(t) that satisfy theDirichlet's condition in an extended form, i.e. with an infinite period, and are absolutely integrable, thatis,

(7.11)

it is possible to define the Fourier transformation F(T) of f(t). The function F(T) is a function of a realvariable, a variable frequency T, and correponds to the Fourier coefficients cn for the discretefrequencies Tn's in the periodic case, Eq. (7.4). In fact, given an absolutely integrableaperiodic function f(t), which satisfies the Dirichlet condition, (most functions of limited duration do),formally write

(7.12)

and

(7.13)

Now put )T=2B/T as T64. Rewriting Eq. (7.12) using Eq. (7.13) yields

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The expression within the brackets can be given the name F(Tk), and putting Tk = k)T,

(7.14)

By going to the limit (T64), the discrete frequency Tk becomes continous and the infinite sum convertsto the integral

(7.15)

in which the Fourier transformation function, called the Fourier transform of f(t), is

(7.16)

The two functions f(t) and F(T) are usually referred to as Fourier transform pairs, which is indicated asfollows:

f(t) : F(T)

The function f(t) can be said to be the time domain representation of certain time dependent informationor signal. The Fourier transform F(T) on the other hand is the frequency domain representation of thesame information.

The Fourier transform F(T) can be broken up into its real and imaginary parts. Obviously,since f(t) is a real function of time,

and (7.17)

that is, the real part FR(T) is an even function of T whereas the imaginary part FI(T) is an odd functionof T.

A physical interpretation of the Fourier transform may be given as follows. By Eq. (7.14), f(t)is dissembled into a series of harmonic components (N is large) such that

(7.18)

377

Fig. 7.5 The Continuous Fourier Transform

in which

(7.19)

That is, the lumped area under the Fourier transform at selected frequencies Tk with equal frequencyintervals )T adds up to give the amplitude at each selected frequency, Fig. 7.5

Example 7.2

Find the Fourier transform of the function exp(-at2), -4<t<4, a>0.

Solution:

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Fig. 7.6 The Normal Density. Fourier Transformation Pairs

The Fourier transform pairs are shown in Fig. 7.6. Except for the factor 1/%(2B), the time domainfunction corresponds to a normal or Gaussian density function with zero mean and variance 1/(2a). ItsFourier transform is also a Gaussian density function with zero mean, which was already established inthe discussion of characteristic functions, Ex. 1.24.

...........

Just as was the case for discrete Fourier series (see Eq. (7.3)), the Fourier transform issimplified if the function f(t) is either an even or odd function. If f(t) is an even function of t, the Fouriertransform Eq. (7.17) looses its sine component, that is, for f(t) = f(-t)

Similarly, for odd functions f(t) = -f(-t), the cosine component will vanish or

The frequency domain representation of an even respectively odd function of time is thus given by thecosine respectively sine transform of the time function, that is,

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(7.20)

For arbitrary f(t), the cosine and sine transforms correspond to the real and imaginary parts of F(T),Eq. (7.17).

7.1.3 Some Important Properties of the Fourier Transform

Given the three Fourier transform pairs

x(t) : X(T), y(t) : Y(T), z(t) : Z(T)

and the frequency domain relation

Z(T) = X(T)Y(T) (7.21)

a corresponding relation between the three functions x(t), y(t), z(t) in the time domain is sought. Now,through Eq. (7.15)

Reversing the order of integration gives

or by Eq. (7.14)

(7.22)

The relations Eqs. (7.21) and (7.22) are usually referred to as the convolution theorem and Eq. (7.22)

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is often presented in the shorthand form z(t) = x(t)Çy(t).

The convolution theorem is "symmetric", that is, given the time domain relation

z(t) = x(t)Çy(t) (7.23)

a relationship corresponding to Eq. (7.20) in the frequency domain can be found. In fact, introducingthe inverse transform Eq. (7.15) into Eq. (7.21),

or

(7.24)

where S is generally a complex variable. Thus, is the convolution in the

frequency domain, also called the complex convolution theorem.

Next, let Y(T) = X*(T) be the complex conjugate of the Fourier transform for x(t). Then byEq. (7.16),

that is, X*(T) and x(-t) are Fourier trandform pairs. By Eqs. (7.15) and (7.21),

Also by Eq. (7.22),

Thus Parseval's theorem for the continuous transform has been established (cf. Eq. (7.9)), that is,

(7.25)

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If the derivatives of the time domain function f(t) exist, satisfy the extended Dirichlet conditionand are absolute integrable, their Fourier transforms are easily obtainable. In fact, by differentiating theinverse transform Eq. (7.15)

AAAAAAAAAAAAAAAAAAAAAAAAAAAAA

therefore the Fourier transform pairs for the derivatives are the following:

(7.26)

Example 7.3

In Fig. 7.7, an one storey elastic frame under the horizontal load f(t) is shown. During the horizontal motion x(t) due to the force f(t), the frame is considered to be massless and purelyelastic with a viscous damping coefficient c. Find the time history of the response when

Solution:

For a static load P, the horizontal deflection *P is given by

382

Fig. 7.7 A Massless, Elastic Frame Under Dynamic Load

whereby the spring constant k is defined. The equation of motion is then given by

(m-0)(x(t)+c0x(t)+kx(t) = f(t)

orc0x(t)+kx(t) = f(t)

A frequency domain solution is first sought. By Fourier transforming the whole equation, that is,multiplying both sides by e-iTt and integrating from -4 to +4, making use of the relations Eq. (7.26),

c(iT)X(T)+kX(T) = F(T)or

Now

and then by the inverse transform Eqs. (7.15), the time history of the response is given by

383

e dz

z z ikc

iiz t T( )

( )

±

−= + ∑∫ 2π

Fig. 7.8 Contour Integration Paths

To recover the time domain representation from the frequency domain representation, the aboveintegral has to be unlocked. For this purpose, elaborate integral tables, [33], may become handy or themore tedious path of Cauchy's integral theorem for analytic functions can be selected (cf. Ex. 3.5). Choosing to go over the wall where it is highest, study the contour integral +f(z)dz, where the complexanalytic function f(z) is being integrated along a closed contour C. Cauchy's integral theorem thenstates that the value of the contour integral is equal to the sum of residues at the poles of f(z) within theclosed contour C times 2Bi. In other words,

(Residues at poles within C)

The integration path is to be taken anticlockwise. Otherwise the residual sum has to be taken with a negative sign.

In Fig. 7.8, the selected integration paths areshown. Within the circle of radius R, there are twopoles, z = 0 and z = ik/c. The residues at these polesare:

Now along the T-axis, the contour integral is equivalentto the sought after integral, that is,

Along the great circle contour, path 1:

(I)

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and path 2:

(II)

Along the small circle contour (about the origo, z = ,eiv), for path 1 (below the T-axis) and path 2(above the axis) replace R by , in the two above integrals, now called (III) and (IV). Path 1 includesboth poles, whereas path 2 does not include any poles. Therefore,

Path 1,

Path 2, +f(z)dz = 2BiA0 = 0

Then let R 6 0 and , 6 0 and see what happens. There are four different possibilities:

1) t>T, 2) t<T, 3) t>-T, 4) t<-T

The integrand in the great circle integrals (I) and (II) is evaluated as:

The integrand in the small circle integrals (III) and (IV) is evaluated as:

Thus the two small circle integrals III and IV both give a contribution iBc/k to the two integrals A andB. Since the final value of the time history x(t) is based on the difference A-B, the small circelcontribution cancels out. Now to evaluate the two integrals A and B for the time domain solution, theintegration paths are chosen such that the integrals (II) and (III) vanish in all four cases. That is, for

For t>T, use path 1 , sinv>0A:

For t<T, use path 2 , sinv<0

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For t>-T, use path 1 , sinv>0B:

For t<-T, use path 2 , sinv<0

Therefore,

and

The response x(t) is shown graphically in Fig. 7.9.

Example 7.5

The delta function (*(t) = 4 for t=0, *(t)=0 for t…0) can be defined as a Gaussian probabilitydensity with a zero mean value and a standard deviation F which tends to zero. That is,

Going to the limit, the delta function has the following properties. Take any reasonably well behavedfunction g(t) that is smooth in the vicinity of t = t0. Then,

(7.27)

386

Fig. 7.9 Time History of the Response x(t)

where the last of Eqs. (7.27), corresponding to a non-zero mean value t0, portrays the shifting propertyof the delta function.

Does the delta function have a Fourier transform? Obviously the function f(t-t0), (:=t0), does,and it is given by the characteristic function n(T) = exp[it0T-F2T2/2], (see Ex. 1.24). Therefore, at leastformally,

so

(7.28)

The time shifting property of the delta function can be used more directly since

Next, let F(T) = *(T-T0). What is the time domain representation? By the inverse transform,

so

(7.29)

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Obviously

*(t) ] 1 and 1 ] 2B*(T) (7.30)

The above relations can be given the following physical explanation. A time signal x(t) with aconstant value component (a d.c. component) will give rise to an impulse in the Fourier transform atzero frequency. A periodic component in the signal of the type exp(±iT0t) with a period T0 = 2B/T0,will on the other hand give rise to an impulse in the Fourier transform at the controversial frequencies T= ±T0. From Eq. (7.30), it can be seen that the delta function can be interpreted as the sum ofsinusoids of all frequencies. These are in-phase only at time t=0 or at time t=t0 in the case of Eq.(7.28), giving a very large amplitude, and are out-of-phase at all other frequencies giving zeroamplitudes.

Mathematically speaking, periodic and d.c. components do not belong to the class of absolutelyintegrable functions for which the Fourier transform is defined. However, by introduction of the deltafunction such difficulties have been overcome, and discrete Fourier series can now also be handled ascontinuous Fourier transforms. The analogy with the treatment of discrete and continuous randomvariables in Chapter 1 is also evident.

Example 7.6

Consider the signal or sample function

where a0 is a constant (a d.c. component), and to a smooth aperiodic function g(t) is added a sum ofperiodic components. The Fourier transform F(T) is sought.

Since cos (Tjt+2j) =½ (exp(+i(Tjt+2j)+exp(-(Tjt+2j)), the Fourier transform consists of aspike at zero frequency and symmetrical spikes at the controversial frequencies ±Tj added to theFourier transform of the aperiodic function g(t) called G(T). That is,

where aj = a-j , 2-j = -2j and T-j = -Tj. The Fourier transform is plotted in Fig. 7.10 to illustrate thebehaviour of the separate components.

Finally, if the function f(t) were periodic with a period T, then by Eqs. (7.4) and (7.29), the

388

Fig. 7.10 Composition of the Fourier Transform

continuous Fourier transform is

where the coefficients ck are given by Eq. (7.5).