1. 4. The Continuous time Fourier Transform
4.1 Representation of Aperiodic signals:
The Continuous time Fourier Transform
4.1.1 Development of the Fourier transform
representation of the continuous time Fourier
transform

2. (1) Example 4 The continuous time Fourier transform
( From Fourier series to Fourier transform )

3. (2) Fourier transform representation of Aperiodic signal
4 The continuous time Fourier transform
(2) Fourier transform representation of Aperiodic
signal
For periodic signal :
For aperiodic signal x(t) :

4. 4 The continuous time Fourier transform

5. 4 The continuous time Fourier transform
When T , So

6. Relation between Fourier series and Fourier transform:
4 The continuous time Fourier transform
Fourier transform: or
Relation between Fourier series and
Fourier transform:

7. 4 The continuous time Fourier transform

8. 4.1.2 Convergence of Fourier transform
4 The continuous time Fourier transform
4.1.2 Convergence of Fourier transform
Dirichlet conditions:
(1) x(t) is absolutely integrable.
(2) x(t) have a finite number of maxima and minima
within any finite interval.
(3) x(t) have a finite number of discontinuity within
any finite interval. Furthermore, each of these
discontinuities must be finite.

9. 4.1.3 Examples of Continuous time Fourier Transform
4 The continuous time Fourier transform
4.1.3 Examples of Continuous time Fourier Transform
Example
Example (1)
Example (2)

10. 4.2 The Fourier Transform for Periodic Signal
4 The continuous time Fourier transform
4.2 The Fourier Transform for Periodic Signal
Periodic signal:
thus
Example

11. 4.3 Properties of the Continuous time Fourier Transform
4.3.1 Linearity If
then

12. 4.3.2 Time Shifting If then 4 The continuous time Fourier transform
Example 4.9

13. 4.3.3 Conjugation and Conjugate Symmetry
4 The continuous time Fourier transform
4.3.3 Conjugation and Conjugate Symmetry
(1) If
then
(2) If
then

14. 4 The continuous time Fourier transform
(3) If
then

15. 4.3.4 Differentiation and Integration
4 The continuous time Fourier transform
4.3.4 Differentiation and Integration
(1) If
then
(2) If
then
Example 4.12

16. 4.3.5 Time and Frequency Scaling
4 The continuous time Fourier transform
4.3.5 Time and Frequency Scaling If
then
Especially,

17. 4.3.6 Duality If then 4 The continuous time Fourier transform
Example 4.13

18. 4 The continuous time Fourier transform

19. 4.3.7 Parseval’s Relation If then
4 The continuous time Fourier transform
4.3.7 Parseval’s Relation If
then
Example 4.14

20. h(t) H(j) 4.4 The Convolution Property Consider a LTI system: x(t)
4 The continuous time Fourier transform
4.4 The Convolution Property
Consider a LTI system:
h(t)
H(j)
x(t)
y(t)=x(t)*h(t)
X(j )
Y(j)=X(j)H(j)
4.4.1 Examples
Example

21. 4 The continuous time Fourier transform

22. The multiplication(modulation) property:
4 The continuous time Fourier transform
4.5 The Multiplication Property
The multiplication(modulation) property:
s(t)
p(t)
r(t)
Example

23. 4 The continuous time Fourier transform

24. 4.5.1 Frequency-Selective Filtering with Variable Center Frequency
4 The continuous time Fourier transform
4.5.1 Frequency-Selective Filtering with
Variable Center Frequency
A Bandpass Filter :

25. 4 The continuous time Fourier transform

26. 4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs
4 The continuous time Fourier transform
4.6 Tables of Fourier Properties and
of Basic Fourier Transform Pairs
Table 4.1
Table 4.2

27. 4.7 System Characterized by Linear Constant-
4 The continuous time Fourier transform
4.7 System Characterized by Linear Constant-
Coefficient Differential Equation
Constant-coefficient differential equation:
Fourier transform:
Define:
Example

28. 4 The continuous time Fourier transform
Problems:
(a)
(a)(b)