1. FOURIER SERIES EVEN AND ODD FUNCTION AND APPLICATION
SIGMA INSTITUTE OF ENGINEERING
FOURIER SERIES EVEN AND ODD FUNCTION AND APPLICATION
PREPARED BY:
VAIBHAV BARIYA:
DIPAK BHARTI:
JAY PATEL:
ARPAN KOLI:
AKASH PATEL:
GUIDED BY:
DIPTI MAM

2. 10.4. Even and Odd Functions. Half-Range Expansions
- If a function is even or odd  more compact form of Fourier series
Even and Odd Functions
Even function y=g(x): g(-x) = g(x) for all x (symmetric w.r.t. y-axis)
Odd function y=g(x): g(-x) = -g(x) for all x
Three Key Facts
(1) For even function, g(x),
(2) For odd function, h(x),
(3) Production of an even and an odd function  odd function
let q(x) = g(x)h(x), then q(-x) = g(-x)h(-x) = -g(x)h(x) = -q(x)
Odd function
Even function x
g(x)
g(x) x

3. In the Fourier series,
f(x) even  f(x)sin(nx/L) odd, then bn=0
f(x) odd  f(x)cos(nx/L) odd, then a0 & an=0
Theorem 1: Fourier cosine series, Fourier sine series
(1) Fourier cosine series for even function with period 2L
(2) Fourier sine series for odd function with period 2L
For even function with period 2
For odd function with period 2

4. Theorem 2: Sum of functions
- Fourier coefficients of a sum of f1 + f2
 sums of the corresponding Fourier coefficients of f1 and f2.
- Fourier coefficients of cf  c times the corresponding coefficients of f.
Ex. 1) Rectangular pulse
Ex. 2) Sawtooth wave: f(x) = x +  (- < x < ) and f(x + ) = f(x)
for f2 = :  
for odd f1 = x  2k -  x
f(x)
+ k
previous result by Ex.1 in 10.2 -  x
f(x)

5. APPLICATION OF FOURIER SERIES
Example 1. Square Wave ——High Frequency
One simple application of Fourier series, the analysis of a “square” wave (Fig. (7.5))
in terms of its Fourier components, may occur in electronic circuits designed to
handle sharply rising pulses. Suppose that our wave is designed by

6. (7.30)
From Eqs. (7.11) and (7.12) we find
(7.31)
(7.32)
(7.33)

7. Example 2 Full Wave Rectifier
The resulting series is
(7.36)
Except for the first term which represents an average of f(x) over the interval
all the cosine terms have vanished. Since
is odd, we have a Fourier sine
series. Although only the odd terms in the sine series occur, they fall only as
This is similar to the convergence (or lack of convergence ) of harmonic series.
Physically this means that our square wave contains a lot of high-frequency
components. If the electronic apparatus will not pass these components, our square
wave input will emerge more or less rounded off, perhaps as an amorphous blob.
Example 2 Full Wave Rectifier
As a second example, let us ask how the output of a full wave rectifier approaches
pure direct current (Fig. 7.6). Our rectifier may be thought of as having passed the
positive peaks of an incoming sine and inverting the negative peaks. This yields

8. Since f(t) defined here is even, no terms of the form will appear.
(7.37)
Since f(t) defined here is even, no terms of the form
will appear.
Again, from Eqs. (7.11) and (7.12), we have
(7.38)

9. is not an orthogonality interval for both sines and cosines
(7.39)
Note carefully that
is not an orthogonality interval for both sines and cosines
together and we do not get zero for even n. The resulting series is
(7.40)
The original frequency
has been eliminated. The lowest frequency oscillation is
The high-frequency components fall off as
, showing that the full wave
rectifier does a fairly good job of approximating direct current. Whether this good
approximation is adequate depends on the particular application. If the remaining ac components are objectionable, they may be further suppressed by appropriate filter circuits.

10. These two examples bring out two features characteristic of Fourier expansion.
1. If f(x) has discontinuities (as in the square wave in Example 7.3.1), we can expect
the nth coefficient to be decreasing as
. Convergence is relatively slow.
If f(x) is continuous (although possibly with discontinuous derivatives as in the
Full wave rectifier of example 7.3.2), we can expect the nth coefficient to be
decreasing as
3. More generally if f and its first r derivatives are continuous, but the r+1 is not
then the nth coefficient will be decreasing as
Example 3 Infinite Series, Riemann Zeta Function
As a final example, we consider the purely mathematical problem of
expanding
(7.41)

11. by symmetry all
For the
’s we have
(7.42)
(7.43)
From this we obtain
(7.44)

12. As it stands, Eq. (7.44) is of no particular importance, but if we set
(7.45)
and Eq. (7.44) becomes
(7.46) or
(7.47)
thus yielding the Riemann zeta function,
, in closed form. From our
expansion of
and expansions of other powers of x numerous other
infinite series can be evaluated.

13. THANK YOU