1. Không gian tín hiệu trực giao
2. Chuỗi Fourier lượng giác
3. Chuỗi Fourier phức
4. Biến đổi Fourier
5. Các tính chất của Biến đổi Fourier
Created and edited by: Nguyen Phuoc Bao Duy
HCMC University of Technology
Chương 5:
Chuỗi Fourier và Biến đổi Fourier
Created and edited by: Nguyen Phuoc Bao Duy
HCMC University of Technology
Component of a Vector:
•The error is minimized when:
•The two vectors are orthogonal when:
: component of along
: error
f cx f x
f cx e e f cx
2
1.c f x
x
0 . 0c f x
Created and edited by: Nguyen Phuoc Bao Duy
HCMC University of Technology
Component of a Signal:
•Approximating a real signal f(t) in terms of other real signal
x(t) over an interval [t1, t2]:
•The error is minimized when its energy is minimum.
( ) . ( ) ( )
( ) . ( ) : component of ( ) in term of ( )
( ) ( ) . ( ) : error
f t c x t e t
f t c x t f t x t
e t f t c x t
2
2 2
1
2
1 1
1
2
2
( ). ( ) 1
( ) min ( ). ( )
( )
t
t t
t
et
t t
x
t
f t x t dt
E e t dt c f t x t dt
E
x t dt
Created and edited by: Nguyen Phuoc Bao Duy
HCMC University of Technology
Orthogonality:
•We define the real signals x1(t) and x2(t) to be orthogonal
over the interval [t1, t2] if:
•Orthogonality in complex signals:
2
1
1 2
( ). ( ) 0
t
t
x t x t dt
2 2
1 1
* *
1 2 1 2
( ). ( ) 0 ( ). ( ) 0
t t
t t
x t x t dt or x t x t dt
Created and edited by: Nguyen Phuoc Bao Duy
HCMC University of Technology
Orthogonality:
•Example 5.01: How many of the following pairs of functions
are orthogonal over [0, 3]?
•Answers: a b c d