1. Signals and Systems Lecture 11
The Continuous-Time Fourier Transformation

2. Chapter 4 Fourier Transform
Fourier Series of CT Periodic Signals
Synthesis equation
Analysis equation

3. Chapter 4 Fourier Transform
§4.1 Representation of Aperiodic Signals :
The Continuous-Time Fourier Transform
-T T/2 –T T1 T/ T t

4. Chapter 4 Fourier Transform
Figure 4.2
谱线变密

5. Chapter 4 Fourier Transform
Consider an aperiodic Signals
–T T t
-T –T T T t

6. Chapter 4 Fourier Transform
Fourier Transform Pair
factor
Synthesis equation
Analysis equation
1. A linear combination of complex exponentials.
——Spectrum（频谱） of 2.

7. Chapter 4 Fourier Transform
Consider a periodic signal
Defining
The Fourier coefficients of are proportional
to samples of the Fourier transform of one period of

8. Chapter 4 Fourier Transform
§ Convergence of Fourier Transforms
is square integrable
2. Dirichlet Conditions:

9. Chapter 4 Fourier Transform
§ Fourier Transforms of Typical Signals
Example 4.1

10. Chapter 4 Fourier Transform
Example 4.2
Example 4.3

11. Chapter 4 Fourier Transform
Example 3:(Continue)
Do not satisfy the condition of finite energy，so can not get from define directly.

12. Chapter 4 Fourier Transform
Example 4.4, 4.5
–T T t F
Useful facts about CTFT’s
Example above:
Example above: F

13. Chapter 4 Fourier Transform
Excise
4.1(a)
4.2(b)
Solution:
Solution:

14. Chapter 4 Fourier Transform
Example 1
Example 2

15. Chapter 4 Fourier Transform
Example 3 F

16. Chapter 4 Fourier Transform
§4.2 The Fourier Transforms for Periodic Signals
More generally, if x(t)=x(t+T),then
Discrete spectra

17. Chapter 4 Fourier Transform
Example 4.6: Periodic square wave

18. — sampling function Chapter 4 Fourier Transform
Example 4.7: Periodic impulses train
— sampling function
-T T T
Note in this case, periodic
in both time domain (with
a period T) and frequency
domain (with a period 2π/T)
Same function in the frequency-domain!
-ω ω0 2ω0

19. Chapter 4 Fourier Transform
Excise
4.3(a)
4.4(a)
Solution:
Solution: ∵ ∴

20. Summary Chapter 4 Fourier Transform Synthesis equation
Analysis equation F

21. Readlist Signals and Systems: Question: 4.3~4.6
Compare properties of CTFT with of CTFS
Basic Fourier Transformation Pairs

22. Problem Set
4.21(a), (c), (g)
4.22(a), (b), (d)