Presentation on Fourier Transform

1. WELCOME TO
OUR PRESENTATION

2. CREDIT TO…
 1.Md.Shahedujjaman Shahed :152-33-2643
 2.Monsur Ahmed :152-33-2628
 3.A K M Tanvir Rahman : 152-33-2674
 4.Kaimul Islam Ruddro : 152-33-2653
 5.Md Al Tarek : 152-33-2719

3. JOSEPH FOURIER
(Founder of Fourier series)

4. FOURIER SERIES, which is an infinite series representation of
such functions in terms of ‘sine’ and ‘cosine’ terms, is useful here.
Thus, FOURIER SERIES, are in certain sense, more UNIVERSAL
than TAYLOR’s SERIES as it applies to all continuous, periodic
functions and also to the functions which are discontinuous in their
values and derivatives. FOURIER SERIES a very powerful method
to solve ordinary and partial differential equation, particularly with
periodic functions appearing as non-homogenous terms.
As we know that TAYLOR SERIES representation of functions are valid only for
those functions which are continuous and differentiable. But there are many
discontinuous periodic function which requires to express in terms of an infinite
series containing ‘sine’ and ‘cosine’ terms.

6. Fourier series make use of the orthogonality relationships
of the sine and cosine functions.
FOURIER SERIES can be generally written as,
Where,
……… (1.1)
……… (1.2)
……… (1.3)

7. BASIS FORMULAE OF FOURIER SERIES
The Fourier series of a periodic function ƒ(x) with period 2п is
defined as the trigonometric series with the coefficient a0, an
and bn, known as FOURIER COEFFICIENTS, determined by
formulae (1.1), (1.2) and (1.3).
The individual terms in Fourier Series are known as
HARMONICS.
Every function ƒ(x) of period 2п satisfying following conditions
known as DIRICHLET’S CONDITIONS, can be expressed in
the form of Fourier series.

8. EXAMPLE:
sin-1x, we can say that the function sin-1x cant be
expressed as Fourier series as it is not a single valued
function.
tanx, also in the interval (0,2п) cannot be
expressed as a Fourier Series because it is infinite at
x= п/2.
CONDITIONS :-
1. ƒ(x) is bounded and single value.
( A function ƒ(x) is called single valued if each point
in the domain, it has unique value in the range.)
2. ƒ(x) has at most, a finite no. of maxima and minima in the
interval.
3. ƒ(x) has at most, a finite no. of discontinuities in the
interval.

9. Fourier series for EVEN and ODD functions
If function ƒ(x) is an even periodic function with the period
2L (–L ≤ x ≤ L), then ƒ(x)cos(nпx/L) is even while ƒ(x)sin(nпx/L) is
odd.
Thus the Fourier series expansion of an even periodic
function ƒ(x) with period 2L (–L ≤ x ≤ L) is given by,
L
nx
a
a
xf
n
n

cos
2
)(
1
0




dxxf
L
a
L

0
0 )(
2
Where,
,2,1cos)(
2
0
  ndx
L
xn
xf
L
a
L
n

0nb
EVEN FUNCTIONS

10. If function ƒ(x) is an even periodic function with the period
2L (–L ≤ x ≤ L), then ƒ(x)cos(nпx/L) is even while ƒ(x)sin(nпx/L) is
odd.
Thus the Fourier series expansion of an odd periodic function ƒ(x)
with period 2L (–L ≤ x ≤ L) is given by,
)sin()(
1 L
xn
bxf
n
n





Where,
,2,1sin)(
2
0
  ndx
L
xn
xf
L
b
L
n

ODD FUNCTIONS

11. APPLICATIONS
OF
FOURIER SERIES

12. Consider a mass-spring system as before, where we have a
mass m on a spring with spring
constant k, with damping c, and a force F(t) applied to the
mass.
Suppose the forcing function F(t) is 2L-periodic for some
L > 0.
The equation that governs this
particular setup is
The general solution consists of the
complementary solution xc, which
solves the associated
homogeneous equation mx” + cx’ + kx = 0, and a particular
solution of (1) we call xp.
mx”(t) + cx’(t) + kx(t) = F(t)

13. For c > 0,
the complementary solution xc will decay as time goes by.
Therefore, we are mostly interested in a
particular solution xp that does not decay and is periodic with
the same period as F(t). We call this
particular solution the steady periodic solution and we write it
as xsp as before. What will be new in
this section is that we consider an arbitrary forcing function
F(t) instead of a simple cosine.
For simplicity, let us suppose that c = 0. The problem with c >
0 is very similar. The equation
mx” + kx = 0
has the general solution,
x(t) = A cos(ωt) + B sin(ωt);
Where,

14. Heat on an insulated wire
Let us first study the heat equation. Suppose that we have a wire
(or a thin metal rod) of length L that is insulated except at the
endpoints. Let “x” denote the position along the wire and let “t”
denote time. See Figure,

15. Let u(x; t) denote the temperature at point x at time t. The
equation governing this setup is the so-called one-dimensional
heat equation:
where k > 0 is a constant (the thermal conductivity of the
material). That is, the change in heat at a
specific point is proportional to the second derivative of the heat
along the wire. This makes sense; if at a fixed t the graph of the
heat distribution has a maximum (the graph is concave down),
then heat flows away from the maximum. And vice-versa.
Where,
T
x
tA
Q
k




16. The Method of Separation of Variables
Let us divide the partial differential equation shown earlier by the
positive number σ, define κ/σ ≡ α and rename α f(x, t) as f (x, t)
again. Then we have,
We begin with the homogeneous case f(x, t) ≡ 0. To implement
the method of separation of variables we write
T(x, t) = z(t) y(x), thus expressing T(x, t) as the product of a
function of t and a function of x. Using ̇z to denote dz/dt and y’, y”
to denote dy/dx, d2y/dx2, respectively, we obtain,

17. THANKS EVERYONE
HAVE A NICE DAY…..