Fourier Laplace

Fundamentals of Structural Analysis. Chapter 6. Fourier and Laplace Transforms

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6. Fourier series, Fourier and Laplace transforms

The basic theory for the description of periodic signals was formulated by Jean-Baptiste Fourier (1768-1830) in the beginning of the 19th century. Fourier showed that an arbitrary periodic function could be written as a sum of sine and cosine functions. This is the basis for the transformation of time histories into the frequency domain and all kinds of digital frequency analysis.

Jean-Baptiste Fourier (1768-1830)

6.1 Fourier Series

Let x(t) be a periodic function with period T

)t(x)nTt(x =+ (6-1) where n is an integer. The functions


2

=

=

Tt2mcos)t(c

Tt2msin)t(s

m

m

(6-2)

are all periodic for an integer m over the time T. The functions have m whole periods over the time T. Fourier showed that the arbitrary periodic function x(t) may be written as an infinite sum of those sine and cosine functions.

+

= = T

t2msinbT

t2mcosa)t(x m0m

m

(6-3)

To get simpler formulas let =T2 and (6-3) becomes:

)tmsin(b)tmcos(a)t(x0m

mm =

+= (6-4)

To get the coefficients, multiply both sides in (6-4) by cos(nt) with n an integer. Then:

( ) ( )

+++++=

+=

=

t)mnsin(t)mnsin(2

bt)mncos(t)mncos(2

a)t(x)tncos(

or

)tmsin(b)tmcos(a)tncos()t(x)tncos(

mm

0mmm

(6-5)

We now integrate both sides of (6-5) over the time interval (0, T). All trigonometric functions on the right side have a whole number of periods in the interval, and (nearly) all terms in the integral will become zero. But there is one exception. If we choose n = m, cos((n-m)t) = 1 in the whole interval, and we will get from the right hand side:

2Tadt1

2a n

T

0

n = (6-6)


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From this:

T2

with

dt)t(x)tncos(T2a

T

0n

=

= (6-7)

In the same way:

= T0

n dttxtnT2b )()sin( (6-8)

For n = 0, which corresponds to the mean value of the signal, we will get a special case,

and we write this term 20a . We have then arrived at the formula for the Fourier series:

T2

dt)t(x)tmsin(T2b

dt)t(x)tmcos(T2a

with

)tmsin(b)tmcos(a2a)t(x

T

0m

T

0m

1mmm

0

=

=

=

++=

=

(6-9)

We note, that the only thing we have used in the derivation of the formula is the fact that most of the terms in the integrals turn out to be zero. This is called orthogonality, the functions are said to be orthogonal. This is mathematically equivalent with the same concept for vectors, and the scalar product for vectors then corresponds to the integral of the multiplied functions. We may write:


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=

==

=

dt)t(x)t(a

andmn...1mn...0

dt)t()t(

with)t(a)t(x

mm

mn

mm

(6-10)

This is called a generalised Fourier series, and is a widely used concept in many disciplines. In figure 6-1 the successive approximation with Fourier series to a square wave is shown.

Figure 6-1 Successive Fourier series approximation to a square wave by adding terms.

If we let the numbers in the Fourier series get very large, we get a phenomenon, called Gibbs phenomenon, see figure 6-2.


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Figure 6-2 Gibbs phenomenon for Fourier series approximation with many terms.

We now know, that a periodic signal may be written as a Fourier series. In the sum there are terms of the type:

)tsin(b)tcos(a)t(s += (6-11) We may write s(t) in a different way:

ab)tan(

where)tcos(ba)tsin(b)tcos(a)t(s 22

=

+=+=

(6-12)

we introduce

phase

amplitudeba 22

==+

(6-13)

In the Fourier series we find, that the frequencies appear as multiplies of the basic frequency (1/T). The basic frequency is called the fundamental, while the multiples are called harmonics. Fourier analysis is often called harmonic analysis. A periodic signal


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may then be described with its fundamental and harmonics. For each frequency you give the frequency, the amplitude and the phase, alternatively, you give the sine and cosine components.

6.2 Complex Fourier Series

By using the complex notation for sine and cosine functions, given by Euler:

sinicosei += 6-14) or

i2eesin

2eecos

ii

ii

=

+= (6-15)

we may write the formula for the Fourier series in a more compact way:

2bi

2ad

2bi

2ac

edecc

)tmsin(b)tmcos(a2a)t(x

mmm

mmm

1m 1m

timm

timm0

1mmm

0

+=

=

++

=++=

=

=

=

(6-16)

=

=

=T

0

timm

m

tmim

dte)t(xT1c

ec)t(x

(6-17)


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This is called the complex Fourier series. Please note, that the summation now also covers negative indices, we have negative frequencies. We are not yet putting any physical meaning to those frequencies; we just use the compact mathematical notation.

6.3 Fourier Transform

For a function x(t), defined for all time t, we define the Fourier transform F(f) by:

+

= dte)t(x)(X ti (6-18)

F is a complex-valued function of the variable f, frequency, and is defined for all frequencies. As the function is complex, it may be described by a real and an imaginary part or with magnitude and phase, as with any complex number. The definition integral does not converge for all possible functions x(t), see the special literature on the subject. If the Fourier transform exists, there is an inverse transform formula:

+

+= de)(X

21)t(x ti (6-19)

The transform pair is very symmetric, the only difference being the sign of the exponent and the factor 21 in (3-19). We give an example of a Fourier transform:


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Figure 6-3 Time signal and corresponding Fourier transform.

The example given here results in a real Fourier transform, which stems from the fact that x(t) is placed symmetrical around time zero. We give another example:

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Fourier Laplace