NCTU Department of Electrical and Computer Engineering
Senior Course <Dynamic System Analysis and Simulation>
By Prof. Yon-Ping Chen
5.Fourier Transform and Laplace Transform
A.Fourier Series
B.Fourier Transform
C.Laplace Transform
In system enegineering, there are two important transforms which are Fourier transformand Laplace transform. Fourier transform is a tool for signal processing and Laplace transform is mainly applied to controller design.
A. Fourier Series
Before introducing Fourier transform and Laplace transform, let’s consider the so-called Fourier series, which was propsed by French mathematician Jean Baptiste Joseph Fourier (1768-1830) and mainly applied to periodical functions.
In Figure 5-1, there is a periodical function fT(t), ∞t∞, with period T=ba. Based on Fourier series, the period function can be expressed as
for t(∞, ∞)(5-1)
where
(5-2)
(5-3)
(5-4)
It has been proved that Fourier series could represent a periodical function which satisfies the following Dirichlet conditions:
C1- In one period, the number of discontinuous points is finite.
C2- In one period, the number of maximum and minimum points is finite.
C3- In one period, the function is absolutely integrable.
It is interesting to point out that although all the infinite number of sinusoidal functions in (1) are continuous, their sum may be discontinuous just like the function fT(t) depicted in Figure 5-1, which satisfies C1~C3.
Actually, Fourier series (5-1) can be also used to represent a function with finite time duration. For example, the function f(t) in Figure 5-2 is part of fT(t) for t(a, b) and expressed as
for t(a, b)(5-5)
which is the same as (5-1) except that f(t)=0 for ta or tb. However, (5-5) is not a unique expression for the finite time duration function f(t) since it can be also treated as part of the period function shown in Figure 5-3 with period T’=ba’. If for t(∞, ∞) (5-6)
then f(t) can be written as
for t(a, b)(5-7)
From (5-5) and (5-7), we know that the expression of a finite duration function in Fourier series is not unique.
B. Fourier Transform
Now, one question is raised: Is that possible to find a unique expression for any functions satisfying Dirichlet conditions? The answer is “Yes,” if the period in (5-6) is T=∞, from t=∞ to t=∞, as shown in Figure 5-4. The resulted expression is
(5-8)
where is the angular frequency and
(5-9)
is called Fourier transform of f(t). In fact, (5-8) can be further written as
(5-10)
which implies that the function f(t) is composed of the sinusoidal functions of all the frequencies from =∞ to =∞. Moreover, Fourier transform can be represented by the so-called Euler form as
(5-11)
with magnitude and phase.
C. Laplace Transform
In addition to Fourier transform, a function f(t) can be also represented by the following Laplace transform:
(5-12)
where s=+j contains the real part Re(s)= and the imaginary part Im(s)=. The initial time is set as t0 due to the fact that sometimes f(t) may possess sigularity functions at t=0, such as the impulse signal. However, practical functions in engineering are impossible to have any sigularities; therefore, Laplace transform is often given as
(5-13)
It is noticed that if s=j, along the imaginary axis, then (5-13) becomes
(5-14)
Viewing from (5-11), it is known that if f(t) is a function staring from t=0, i.e., f(t)=0 for t<0, then
(5-15)
which is the same as Laplace trasform (5-13) with s=j. This is the relation between Fourier transform and Laplace transform.
For the detail of Fourier transform and Laplace transform, please refer to textbooks of Engineering Mathematics or System Engineering.
Problems
P.5-1Expand the following periodic function with period T=3 into Fourier series:
P.5-2Find the Fourier transform of g(t) and h(t), where
and h(t)=g(t)cos(t)
P.5-3Let Yi(s) be the Laplace transform of yi(t), i=1,2,3. Determine yi(t).
, ,
5-1