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

Joseph Fourier(1768 -1830) Joseph Fourier(1768-1830), 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.

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

Fourier Series Periodic Functions

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

Example: Find its period. Fact: smallest T

Example: Find its period. must be a rational number

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

Fourier Series Fourier Series

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

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.

Decomposition

Proof Use the following facts:

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

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

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

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.

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,

In this case the Fourier series is,

Analysis of Periodic Waveforms Fourier Series Analysis of Periodic Waveforms

Waveform Symmetry Even Functions Odd Functions

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

Example Even Part Odd Part

Half-Wave Symmetry and

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

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

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

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)

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]

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

Here

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

Thank You