7. FOURIER ANALYSIS AND DATA PROCESSING

Transcription

1 7. FOURIER ANALYSIS AND DATA PROCESSING Fourier 1 analysis plays a dominant role in the treatment of vibrations of mechanical systems responding to deterministic or stochastic excitation, and, as has already been seen, it forms the basis of spectral representation of stochastic time series and random signals. The Fourier transformation as a mathematical tool offers a clear and simple method for treating complex analytic problems by converting equations involving the basic variable to corresponding equations in the frequency domain where solutions are often more easily accessible. In probability theory, the Fourier transform is the key to understanding certain probability distributions through their characteristic functions. Finally, the discrete Fourier series form the basis of signal processing and data manipulation, which has turned into a 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 for application elsewhere in the text readily available. Fourier series representation of periodic functions is introduced and the continous Fourier transform is derived for aperiodic functions. Various convenient relations 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 or discretizing of bandlimited realizations of stochastic processes or any other time signals is briefly discussed. Thereafter, the discrete and inverse discrete Fourier transforms will be introduced as a natural extension of the continuous Fourier transform and discrete Fourier series analysis. The Fast Fourier Transform Algorithm is briefly discussed together with various aspects of the analysis of digital signals. Signal convolution and aliasing, and the ramifications of signal distortion due to truncation is presented in some detail. Inverse filtering and windowing of signals is also presented with examples of standard digital filters. This is a highly specialized field, which is undergoing rapid expansion and development in connection with the ever increasing electronic gadgetry and instruments in use in everyday 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 recent developments in earthquake signal analysis, especially for generation of synthetic earthquake accelerations. Some exciting new topics in system analysis and signal processing will not be treated for the sake of space and limitation of topics. Notably, new concepts such as fuzzy systems and neural 1 ) After the French Mathematician Jean Baptiste Joseph Fourier, , [42].

2 368 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 theory of Fourier series and integral transforms. The following presentation is therefore a refresher text to clarify the concepts and the tools to be used. The most basic relations and definitions will be presented together with some practical examples. Formal rigorous mathematical proof of the various statements has 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 a variety of standard textbooks available. For example, Weaver, [169] and Papoulis, [116], offer a thorough and detailed version of Fourier analysis and its many applications with the latter text concentrating on the Fourier integral transform 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). By induction, f(t+nt)=f(t) where n is any integer number. Therefore, the function repeats itself in any number of intervals of length T. The well known cosine and sine functions are perfect examples of periodic functions with the period 2B. In Fig. 7.1, more general examples of such Fig. 7.1 Periodic Functions

3 369 functions are shown. The question now arises, can any reasonably well behaved function f(t), where t is a real variable (for instance the time), that repeats itself with the period T>0, be expanded into trigonometric series, that is, broken down into its harmonic cosine and sine components. The function f(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 circular frequency T=2B/T by which the function repeats itself. Now, formally write f(t) as the trigonometric series, f(t) = 1/2 a 0 +a 1 costt+b 1 sintt+a 2 cos2tt+b 2 sin2tt+aaa +a n cosntt+b n sinntt+aaa (7.1) in which a i and b i 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 coefficients a i 's are obtained and in the same manner, by multiplying with the term sin(ktt), the Fourier coefficients b i 's are found. Thus, (7.2)

4 370 Thus it has been shown without formal mathematical proof that any reasonably well behaved function f(t) can be expressed as Fourier series of the form (7.1) where the Fourier coefficients {a i, b i } are given by the relations Eq. (7.2). It may be asked under what conditions the series Eq. (7.1) converge, that is, reproduce the function 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 a continuous function in (0,T) and fn(t) is piecewise continuous in (0,T), that is, is continous at all except at most finite number of points in (0,T), where it can have a jump discontinuity. The uniform convergence is of course strongest whereas the other two forms assure a certain convergence under a weaker set of assumptions. The statements of existence and convergence of Fourier series will not be addressed further here as the purpose is more to furnish tools to analyse functions known to have a frequency content, i.e. possess harmonic components. Usually, it is just stated that any function, which satisfies the so-called Dirichlet s conditions, can be expressed as Fourier series. These conditions are the 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

5 The Dirichlet conditions are sufficient for a periodic function to have a Fourier series representation but not necessary. There are functions which do not satisfy the above conditions but can nevertheless be expressed as Fourier series. Consider a function f(t) satisfying the Dirichlet conditions that is either an even or odd function of t. For an even function, the Fourier coefficients b k vanish (see Eq. (7.2), and for an odd function the a k s vanish. Therefore, the Fourier coefficients are obtained using the simpler form 371 (7.3) The Fourier series Eq. (7.1) can be rewritten using a complex mathematical form by applying the Euler relations or e ±iktt = cosktt±iasinktt (7.4) where (7.5) since (7.6) Using the Euler relations Eq. (7.3), and comparing with Eq. (7.2), the Fourier coefficients c k, a k and b k are interrelated as follows:

6 372 (7.7) The Fourier coefficients a k and b k give information about the strength of the corresponding harmonic component in the function or signal f(t). For instance, if x(t) is the deflection of a certain elastic structure with a representative spring constant k, the strain energy at any time t is ½kx 2 (t), (also the energy carried by an electric signal where x(t) is the current). The strain energy accumulated during one 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 The energy is thus proportional to the sum of the squared amplitudes of the harmonic components in the signal. The terms within the brackets give the energy density per one complete period T. A energy density spectrum can therefore be defined by considering the contribution by the k-th frequency component, that is, (7.8) F(kT) = F(T k ) = 1/2 (a k 2 +b k 2 ) (7.9) The relation Eq. (7.8) is also a statement of the so-called Parseval's theorem, which relates the average energy of the signal over one period to the sum square of the Fourier coefficients, that is, (7.10)

7 In Fig. 7.2 the energy density spectrum F(T) is plotted for a series of imaginative discrete frequencies. 373 Fig. 7.2 The Energy Density Spectrum Example 7.1 A pile hammer produces a rectangular impact load of 1 second duration at the rate of 6 per minute as shown in Fig Obtain the Fourier series of the pile driving load function and its energy density spectrum. Solution: The Fourier coefficients can be obtained directly using Eq. (7.2). Thus, A typical term in the series will be like

8 374 Fig. 7.3 Pile Driving Forces or The largest contribution to the energy density is due to the first component. The contributions due to the higher frequency terms rapidly tend to zero, which is reflected in Fig Aperiodic Functions Next, consider the type of functions that do not show any pattern of periodicity. Non-periodic functions, i.e. functions that do not repeat themselves systematically in any time interval, are called aperiodic. However, figuratively speaking, one can consider an aperiodic function to have an infinite period, that is, letting T64, the function is periodic with an infinitely

9 375 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 the Dirichlet's condition in an extended form, i.e. with an infinite period, and are absolutely integrable, that is, (7.11) it is possible to define the Fourier transformation F(T) of f(t). The function F(T) is a function of a real variable, a variable frequency T, and correponds to the Fourier coefficients c n for the discrete frequencies T n 's in the periodic case, Eq. (7.4). In fact, given an absolutely integrable aperiodic function f(t), which satisfies the Dirichlet condition, (most functions of limited duration do), formally write and (7.12) (7.13) Now put )T=2B/T as T64. Rewriting Eq. (7.12) using Eq. (7.13) yields

10 376 The expression within the brackets can be given the name F(T k ), and putting T k = k)t, (7.14) By going to the limit (T64), the discrete frequency T k becomes continous and the infinite sum converts to 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 as follows: f(t) : F(T) The function f(t) can be said to be the time domain representation of certain time dependent information or signal. The Fourier transform F(T) on the other hand is the frequency domain representation of the same 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 F R (T) is an even function of T whereas the imaginary part F I (T) is an odd function of 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)

11 377 in which (7.19) That is, the lumped area under the Fourier transform at selected frequencies T k with equal frequency intervals )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(-at 2 ), -4<t<4, a>0. Fig. 7.5 The Continuous Fourier Transform Solution:

12 378 The Fourier transform pairs are shown in Fig Except for the factor 1/%(2B), the time domain function corresponds to a normal or Gaussian density function with zero mean and variance 1/(2a). Its Fourier transform is also a Gaussian density function with zero mean, which was already established in the discussion of characteristic functions, Ex Just as was the case for discrete Fourier series (see Eq. (7.3)), the Fourier transform is simplified if the function f(t) is either an even or odd function. If f(t) is an even function of t, the Fourier transform 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 Fig. 7.6 The Normal Density. Fourier Transformation Pairs The frequency domain representation of an even respectively odd function of time is thus given by the cosine respectively sine transform of the time function, that is,

13 379 (7.20) For arbitrary f(t), the cosine and sine transforms correspond to the real and imaginary parts of F(T), Eq. (7.17) 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)

14 380 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, introducing the 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 by Eq. (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)

15 If the derivatives of the time domain function f(t) exist, satisfy the extended Dirichlet condition and are absolute integrable, their Fourier transforms are easily obtainable. In fact, by differentiating the inverse transform Eq. (7.15) 381 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 purely elastic 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

16 382 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) Fig. 7.7 A Massless, Elastic Frame Under Dynamic Load or c0x(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), or c(it)x(t)+kx(t) = F(T) Now and then by the inverse transform Eqs. (7.15), the time history of the response is given by

17 To recover the time domain representation from the frequency domain representation, the above integral has to be unlocked. For this purpose, elaborate integral tables, [33], may become handy or the more 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 complex analytic function f(z) is being integrated along a closed contour C. Cauchy's integral theorem then states that the value of the contour integral is equal to the sum of residues at the poles of f(z) within the closed contour C times 2Bi. In other words, 383 e iz( t ± T ) dz z( z i k ) c = + 2πi (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 are shown. Within the circle of radius R, there are two poles, z = 0 and z = ik/c. The residues at these poles are: Fig. 7.8 Contour Integration Paths Now along the T-axis, the contour integral is equivalent to the sought after integral, that is, Along the great circle contour, path 1: (I)

18 384 and path 2: (II) Along the small circle contour (about the origo, z =,e iv ), 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 includes both 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 and B. Since the final value of the time history x(t) is based on the difference A-B, the small circel contribution cancels out. Now to evaluate the two integrals A and B for the time domain solution, the integration paths are chosen such that the integrals (II) and (III) vanish in all four cases. That is, for A: For t>t, use path 1, sinv>0 For t<t, use path 2, sinv<0

19 385 B: For t>-t, use path 1, sinv>0 For t<-t, use path 2, sinv<0 Therefore, and The response x(t) is shown graphically in Fig Example 7.5 The delta function (*(t) = 4 for t=0, *(t)=0 for t 0) can be defined as a Gaussian probability density 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 behaved function g(t) that is smooth in the vicinity of t = t 0. Then, (7.27)

20 386 Fig. 7.9 Time History of the Response x(t) where the last of Eqs. (7.27), corresponding to a non-zero mean value t 0, portrays the shifting property of the delta function. Does the delta function have a Fourier transform? Obviously the function f(t-t 0 ), (:=t 0 ), does, and it is given by the characteristic function n(t) = exp[it 0 T-F 2 T 2 /2], (see Ex. 1.24). Therefore, at least formally, so The time shifting property of the delta function can be used more directly since (7.28) Next, let F(T) = *(T-T 0 ). What is the time domain representation? By the inverse transform, so (7.29)

21 387 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 a constant value component (a d.c. component) will give rise to an impulse in the Fourier transform at zero frequency. A periodic component in the signal of the type exp(±it 0 t) with a period T 0 = 2B/T 0, will on the other hand give rise to an impulse in the Fourier transform at the controversial frequencies T = ±T 0. From Eq. (7.30), it can be seen that the delta function can be interpreted as the sum of sinusoids of all frequencies. These are in-phase only at time t=0 or at time t=t 0 in the case of Eq. (7.28), giving a very large amplitude, and are out-of-phase at all other frequencies giving zero amplitudes. Mathematically speaking, periodic and d.c. components do not belong to the class of absolutely integrable functions for which the Fourier transform is defined. However, by introduction of the delta function such difficulties have been overcome, and discrete Fourier series can now also be handled as continuous Fourier transforms. The analogy with the treatment of discrete and continuous random variables in Chapter 1 is also evident. Example 7.6 Consider the signal or sample function where a 0 is a constant (a d.c. component), and to a smooth aperiodic function g(t) is added a sum of periodic components. The Fourier transform F(T) is sought. Since cos (T j t+2 j ) =½ (exp(+i(t j t+2 j )+exp(-(t j t+2 j )), the Fourier transform consists of a spike at zero frequency and symmetrical spikes at the controversial frequencies ±T j added to the Fourier transform of the aperiodic function g(t) called G(T). That is, where a j = a -j, 2 -j = -2 j and T -j = -T j. The Fourier transform is plotted in Fig to illustrate the behaviour 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

22 388 continuous Fourier transform is where the coefficients c k are given by Eq. (7.5). Fig Composition of the Fourier Transform