Fourier series

1. Fourier series

2. FOURIER SERIES
• The discrete time fourier transform in fourier series.
• The process of deriving the weight that describe a
given function is a from of fouries series.
• The analysis and synthesis anlogies are fouries
transform and inverse transform.

3. DEFINITION
• Consider a real value function
s(x)that is integrable on an interval
of length p.
X€[0,1]and p=1
• The analysis process determines
the weight indexed by integer n.

4. COMPLEX VALUED
FUNCTION
•The complex value function of a real
varaible x both componets are real valued
function that can be represented by a
fouries series.
•The two sets of coefficients and the partica
sum.

5. CONVERGENCE
•The fourier series is generally presumed
to converge everywhere except at
discntinuities.
•The fouries series converges to the
function at almost every point.
•The dirichlet’s condition for fourier series
the convergence of fourier series.

6. RIEMANN LEBESQU LEMMA
•If F is integrable.
•There are four riemann lebesqu lemma the
imaginary parts a complex function.
•This result is known as the riemann
lebesgue lemma.

7. SYMMETRY PROPERTIES
• The real and imaginary parts of a
complex function are demposed into
their even and odd parts.
• There are four components denote the
subscripts RE,RO,IE and IO.
Time domain f=f RE
f=f RE

8. DERIVATIVE PROPERTY
.The fouries coefficients ^f’(n) of the
derivative f’ can be expressed in terms of
the fourier coefficients ^f(n) of the
function f.
•The fourier coefficient converge to zero.

11. CONVOLUTIO
THEOREMS
•The first convolution therom states
that if f and g are in <‘([-Π,Π]).
•The fouries series coefficients of the
2Π periodic convolution of f and g.

12. COMPACT GROUPS
•The fouries series on any compact
group it include those classical groups
that are compact.
•The four transform carries convolutions
to pointwise products.
•An alternative extension tocompact is
the peter way theirs.

13. RIEMANNIAN MANIFOLDS
•X is a compact riemannian manifold it
has a laplace beltrami operator.
•The differential operator that
corresponds to palace operator the
riemannian mainfold x.
•Then analogy one can equation x.

15. •LOCALLY COMPACT ABLIAN
GROUPS
•The generalization to compact groups
discussed above does to noncompact.
•If G is compact one also obtains of fouries
series.

16. THANK YOU