1. Present by SUBHRANGSU SEKHAR DEY
AMITY UNIVERSITY RAJASHAN
Fourier Series
Present by SUBHRANGSU SEKHAR DEY
M.SC CHEMISTRY
DEPARTMENT OF
ASET

2. Joseph Fourier( )
Joseph Fourier( ), son of a French taylor and friend of nepolean,invented many examples of expressions in trigonometric series in
connection with the problems of conduction heat.His book entitled “Theoric Analytique de le Chaleur”(Analytical theory of heat) published in 1822 is classical is the theory of heat conduction for boundary value problem.the fourier series comes after his name for periodic function to be expanded in trigonometric series.

3. Content Periodic Functions Fourier Series
Analysis of Periodic Waveforms
● Half Range Series

4. Fourier Series
Periodic Functions

5. The Mathematic Formulation
Any function that satisfies
where T is a constant and is called the period of the function.

6. Example:
Find its period.
Fact:
smallest T

7. Example:
Find its period.
must be a rational number

8. Example:
Is this function a periodic one?
not a rational number

9. Fourier Series
Fourier Series

10. Introduction
Decompose a periodic input signal into primitive periodic components.
A periodic sequence T 2T 3T t
f(t)

11. T is a period of all the above signals
Synthesis
T is a period of all the above signals
Even Part
Odd Part
DC Part
Let 0=2/T.

12. Decomposition

13. Proof
Use the following facts:

14. Example (Square Wave)  2 3 4 5 -
-2
-3
-4
-5
-6
f(t) 1

15. Example (Square Wave)  2 3 4 5 -
-2
-3
-4
-5
-6
f(t) 1

16. Example (Square Wave)  2 3 4 5 -
-2
-3
-4
-5
-6
f(t) 1

17. Example Find the Fourier series for
In both cases note that we are integrating an odd function (x is
odd and cosine is even so the product is odd) over the
interval  and so we know that both of these integrals will be zero.

18. Next here is the integral for
In this case we’re integrating an even function (x and sine are both
odd so the product is even) on the interval  and so we can
“simplify” the integral as shown above.  The reason for doing this
here is not actually to simplify the integral however.  It is instead
done so that we can note that we did this integral back in the
Fourier sine series section and so don’t need to redo it in this
section. Using the previous result we get,

19. In this case the Fourier series is,

20. Analysis of Periodic Waveforms
Fourier Series
Analysis of
Periodic Waveforms

21. Waveform Symmetry
Even Functions
Odd Functions

22. Decomposition
Any function f(t) can be expressed as the sum of an even function fe(t) and an odd function fo(t).
Even Part
Odd Part

23. Example
Even Part
Odd Part

24. Half-Wave Symmetry
and

25. Quarter-Wave Symmetry
Even Quarter-Wave Symmetry T
T/2
T/2
Odd Quarter-Wave Symmetry T
T/2
T/2

26. Hidden Symmetry The following is a asymmetry periodic function:
Adding a constant to get symmetry property.
A/2
A/2 T T

27. Fourier Coefficients of Symmetrical Waveforms
The use of symmetry properties simplifies the calculation of Fourier coefficients.
Even Functions
Odd Functions
Half-Wave
Even Quarter-Wave
Odd Quarter-Wave
Hidden

28. HALF RANGE SERIES COSINE SERIES
A function defined in can be expanded as a Fourier series of period containing only cosine terms by extending suitably in
(As an even function)

29. SINE SERIES A function defined in can be expanded
as a Fourier series of period containing
only sine terms by extending suitably in
[As an odd function]

30. Example of Half Range Series SOLUTION Expand in half range
(a) sine Series (b) Cosine series.
SOLUTION
(a)
Extend the definition of given function to that of an odd function of period 4
i.e

32. Here

33. (b)
Extend the definition of given function to that of an even function of period 4

37. Thank You