Fourier Series
Fourier Integrals
Fourier Transforms
F OURIER T RANSFORMS
G. Ramesh
23rd Sep 2015
Fourier Series
Fourier Integrals
Fourier Transforms
O UTLINE
1
F OURIER S ERIES
2
F OURIER I NTEGRALS
3
F OURIER T RANSFORMS
Fourier Series
Fourier Integrals
Fourier Transforms
T RIGONOMETRIC S ERIES
A series of the form
a0 +
∞ X
an cos(nx) + bn sin(nx)
(a0 , an , bn are constants)
n=1
(1)
is called a trigonometric series.
Let PN (x) = a0 +
N X
an cos(nx) + bn sin(nx) . Then PN is a
n=1
periodic function with period 2π. Hence the trigonometric series
is 2π periodic.
Fourier Series
Fourier Integrals
Fourier Transforms
Let f : R → R be a function such that
1
f is periodic function with period 2π
2
f is integrable over the interval of length 2π
3
f is given by (1).
Then we have
Z π
1
f (x)dx
a0 =
2π −π
Z
1 π
an =
f (x) cos(nx)dx
π −π
Z
1 π
bn =
f (x) sin(nx)dx.
π −π
(2)
(3)
(4)
Fourier Series
Fourier Integrals
Fourier Transforms
O RTHOGONALITY
In finding the above coefficients we use the following relations:
π
Z
cos(mx) cos(nx)dx = 0, (m 6= n)
−π
Z π
sin(mx) sin(nx)dx = 0, (m 6= n)
Z
−π
π
−π
cos(mx) sin(nx)dx = 0, (for all m, n)
Z π
sin2 (nx)dx = π,
Z −π
π
cos2 (nx)dx = π.
−π
Fourier Series
Fourier Integrals
Fourier Transforms
F OURIER S ERIES
The series
f (x) = a0 +
∞ X
an cos(nx) + bn sin(nx)
Z n=1
π
1
a0 =
f (x)dx
2π −π
Z
1 π
f (x) cos(nx)dx
an =
π −π
Z
1 π
bn =
f (x) sin(nx)dx.
π −π
is called the Fourier series of f and the coefficients
a0 , an , bn (n = 1, 2, 3 . . . ) are called as the Fourier coefficients.
Fourier Series
Fourier Integrals
Fourier Transforms
Find theFourier series of the function
−k , if − π < x < 0
f (x) =
k , if 0 < x < π.
Fourier Series
Fourier Integrals
Fourier Transforms
Find theFourier series of the function
−k , if − π < x < 0
f (x) =
k , if 0 < x < π.

 4k , n is odd
Sol: a0 = 0, an = 0, bn = nπ
0, n is even.
Fourier Series
Fourier Integrals
Fourier Transforms
Find theFourier series of the function
−k , if − π < x < 0
f (x) =
k , if 0 < x < π.

 4k , n is odd
Sol: a0 = 0, an = 0, bn = nπ
0, n is even.
Hence f (x) =
4k
1
1
{sin(x) + sin(3x) + sin(5x) + . . .}.
π
3
5
Fourier Series
Fourier Integrals
Fourier Transforms
E XERCISES
Find the Fourier series representation of the following functions:
1
f (t) = t, 0 < t < π, f (t + 2π) = f (t) for all t ∈ R.
2
f (t) = t, −π ≤ t ≤ π, f (t + 2π) = f (t) for all t ∈ R.

−t, if − π ≤ t ≤ 0,
f (t) =
t, if 0 < t ≤ π
3
and f (t + 2π) = f (t) for all t ∈ R.
4
5
f (t) = t 2 , −π ≤ t ≤ π and f (t + 2 π) = f (t) for all t ∈ R.

0, if − π < t ≤ 0,
f (t) =
t, if 0 < t ≤ π
and f (t + 2π) = f (t) for all t ∈ R.
Fourier Series
Fourier Integrals
Fourier Transforms
THANK YOU