Comparison of GIBBS PHENOMENON based on different basis functions.ppt

This is my presentation from signals and system course at IIT Delhi, India

1. GIBBS PHENOMENON
FOR DIFFERENT BASIS FUNCTIONS
VISHAL VERMA 2017PH10859
SURAJ PUNIA 2017PH10857

2. What is Gibbs Phenomenon?
In 1848, Henry Wilbraham discovered a peculiar manner in which the Fourier series of a piecewise
continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the
Fourier series has large oscillations near the jump. The overshoot does not die out as n increases, but
approaches a finite limit. The Gibbs phenomenon reflects the difficulty inherent in approximating a
discontinuous function by a finite series of continuous functions.
Any global or high order approximation method suffers from the Gibbs phenomenon if the approximant
has a jump discontinuity in the given domain.
In signal processing, the Gibbs phenomenon is undesirable because it causes overshoot and undershoot,
and ringing artifacts from the oscillations.

3. What is a Basis?
We can approximate a periodic function as a weighted summation of basis functions. A basis function is
an element of a particular basis for a function space. Every continuous function in the function space can
be represented as a linear combination of basis functions, just as every vector in a vector space can be
represented as a linear combination of basis vectors.
Any set of functions that form a complete orthogonal system have a corresponding generalized Fourier
series analogous to the Fourier series. Some of the popular bases used in series expansion are :
●Fourier Basis
●Legendre Polynomials
●Bessel functions
●Hermite Polynomials, etc.

4. Gibbs Phenomenon in Fourier Basis
A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of much simpler
sine and cosine functions .

5. Gibbs Phenomenon in Fourier Basis
Example : Let’s approximate a Square Wave as Fourier Series
Amplitude = 1;
Period = 2;

6. Gibbs Phenomenon in Fourier Basis

7. Gibbs Phenomenon in Fourier Basis

8. Gibbs Phenomenon in Fourier Basis

9. Gibbs Phenomenon in Legendre Basis
Legendre polynomials form a complete orthogonal system over the interval [-1,1]. Any function f(x) may
be expanded in terms of them as

10. Gibbs Phenomenon in Legendre Basis
Legendre polynomials form a complete orthogonal system over the interval [-1,1]. Any function f(x) may
be expanded in terms of them as

11. Gibbs Phenomenon in Legendre Basis
Legendre polynomials form a complete orthogonal system over the interval [-1,1]. Any function f(x) may
be expanded in terms of them as

12. Gibbs Phenomenon in Legendre Basis
Legendre polynomials form a complete orthogonal system over the interval [-1,1]. Any function f(x) may
be expanded in terms of them as

13. Gibbs Phenomenon in Legendre Basis
The Legendre polynomials, are solutions to the Legendre differential equation.
The Legendre polynomials P
n(x) are illustrated below for x in [-1,1] and n=1, 2, ..., 5.

14. Gibbs Phenomenon in Legendre Basis
Example : Let’s approximate a Square Wave as Legendre Series Expansion
Amplitude = 1;
Period = 2;

15. Gibbs Phenomenon in Legendre Basis

16. Gibbs Phenomenon in Legendre Basis

17. Gibbs Phenomenon in Legendre Basis

18. Gibbs Phenomenon in Bessel Basis
Bessel functions form a complete orthogonal system in the interval (0,1) . Let n>=0 and alpha
1,
alpha
2, ...be the positive roots of J
n(x)=0, where J
n(z) is a Bessel function of the first kind. An expansion of
a function in the interval (0,1) in terms of Bessel functions of the first kind :

19. Gibbs Phenomenon in Bessel Basis
Bessel functions form a complete orthogonal system in the interval (0,1) . Let n>=0 and alpha
1,
alpha
2, ...be the positive roots of J
n(x)=0, where J
n(z) is a Bessel function of the first kind. An expansion of
a function in the interval (0,1) in terms of Bessel functions of the first kind :

20. Gibbs Phenomenon in Bessel Basis
Bessel functions form a complete orthogonal system in the interval (0,1) . Let n>=0 and alpha
1,
alpha
2, ...be the positive roots of J
n(x)=0, where J
n(z) is a Bessel function of the first kind. An expansion of
a function in the interval (0,1) in terms of Bessel functions of the first kind :

21. Gibbs Phenomenon in Bessel Basis
Bessel functions form a complete orthogonal system in the interval (0,1) . Let n>=0 and alpha
1,
alpha
2, ...be the positive roots of J
n(x)=0, where J
n(z) is a Bessel function of the first kind. An expansion of
a function in the interval (0,1) in terms of Bessel functions of the first kind :

22. Gibbs Phenomenon in Bessel Basis
he Bessel functions of the first kind J
n(x) are defined as the solutions to the Bessel differential equation
The Bessel polynomials J
n(x) are illustrated below for x in [0,10] and n=0,1, 2, ..., 5.

23. Gibbs Phenomenon in Bessel Basis
Example : Let’s approximate a Square Wave as Legendre Series Expansion
Amplitude = 1;
Period = 2;

24. Gibbs Phenomenon in Bessel Basis

25. Gibbs Phenomenon in Bessel Basis

26. Gibbs Phenomenon in Bessel Basis

27. Compare
Fourier Basis Legendre
Basis
Bessel Basis
Approximation Medium (8.949%) Best (8.730%) Worst (9.023%)
Ease Best Medium Worst
Computation Time Best Worst Medium

28. Conclusion
There are many way in which we can approximate a function. Some provide better
result whereas some are easy to implement. The choice of our method (basis
functions) depends upon the application as well as the resources. We observed
gibbs phenomenon in all the considered basis. On comparing the results, we can
conclude that Fourier basis is a better method for general computations whereas
legendre basis is better for high level error sensitive computations.
There are many more basis functions which are used for different purposes. Some
may even be better than Legendre Basis.

29. References
●https://en.wikipedia.org/wiki/Gibbs_phenomenon
●http://mathworld.wolfram.com/Fourier-BesselSeries.html
●http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
●http://mathworld.wolfram.com/LegendrePolynomial.html
●http://mathworld.wolfram.com/LegendreDifferentialEquation.html
●http://mathworld.wolfram.com/Fourier-LegendreSeries.html
●https://in.mathworks.com/help/matlab/ref/integral.html
●https://in.mathworks.com/help/symbolic/legendrep.html
●https://in.mathworks.com/help/matlab/ref/besselj.html
●https://in.mathworks.com/matlabcentral/fileexchange/48403-bessel-zero-solver

30. THANK YOU