1. Signals and Systems Lecture 15
The Discrete-Time Fourier Transformation

2. Fourier Series of DT Periodic Signals
Synthesis equation
Analysis equation

3. §5.1 Representation of Aperiodic Signals :
The Discrete-Time Fourier Transform

4. Defining
We can get So
As N→∞,
↓The DTFT Pair
Synthesis equation
DTFT (5.8)
Analysis equation
Inverse DTFT (5.9)

5. DT Fourier Transform Pair
factor
Synthesis equation
DTFT (5.8)
Analysis equation
Inverse DTFT (5.9)
1. A linear combination of complex exponentials.
——Spectrum（频谱） of 2.

6. x[n] is equal to over one period and is zero otherwise
Relationship between DTFS and DTFT
DTFS
ak is samples of X(ejω)
DTFT
x[n] is equal to over one period and is zero otherwise

7. x[n] has a finite interval of integration
Relationship between CTFT and DTFT
CTFT
DTFT
x[n] has a finite interval of integration
is periodic

8. Chapter 4 Fourier Transform
§ Fourier Transforms of Typical Signals
Example 5.1

9. Chapter 4 Fourier Transform
Example 5.2
Example 5.3

10. Chapter 4 Fourier Transform
Excise
5.1(a)
5.1(b)

11. Analysis Equation: 5.1.3 Convergence Issues Synthesis Equation:
None, since integrating over a finite interval
Analysis Equation:
Need conditions analogous to CTFT, e.g.
Example 5.4

12. 5.2 The Fourier Transform for Periodic Signals
Consider a periodic signal
In this case, the Fourier transform is

13. Example 5.5:

14. — sampling function Example 5.8: Periodic impulses train
-N N N
Note in this case, periodic
in both time domain (with
a period N) and frequency
domain (with a period 2π/N)
Same function in the frequency-domain!
-ω ω0 2ω0

15. Chapter 4 Fourier Transform
Excise
5.3(a)
5.3(b)

16. Summary
Synthesis equation
Analysis equation

17. Readlist Signals and Systems: Question: 5.3~5.6
Compare properties of DTFT with of CTFS
Basic Fourier Transformation Pairs

18. Problem Set
5.21(a), (c), (g)
5.22(a), (b), (d)