1. Notes Assignments Tutorial problems 3.3 3.21
3.22 (a) -> Figs. (b) (d) (f)
3.24
3.25
Tutorial problems
Basic Problems wish Answers 3.8
Basic Problems 3.34
Advanced Problems 3.40

2. Review for Chapter 2 Why to introduce unit impulse response?
Why to introduce convolution?
(DT or CT) Signal can be represented by a linear combination of unit impulse function
When it goes through the system, the output is computed via convolution of input signal and unit impulse response

3. Chapter 3 Fourier Series Representation of Periodic Signals

4. Joseph Fourier

5. A Useful Analogy

6. Example

7. Overview on Frequency Analysis
Fourier Series
(Periodic/Discrete)
Fourier Transform
(Aperiodic/Continuous)
CT Fourier Series
CT Fourier Transform
DT Fourier Series
DT Fourier Transform
Continuous-Time
Domain
Sampling
Discrete-Time
Domain
Special case by using impulse function

8. Two Special Signals for LTI Systems
System Function
System Function

9. System Functions H(s) or H(z)

10. Fourier and Beyond
Observation: if one signal can be written as the linear combination of 𝑒 𝑠𝑡 𝑜𝑟 𝑧 𝑛 , we need NOT to calculate the convolution for the LTI output.
When 𝑠=𝑗𝜔, 𝑧= 𝑒 𝑗𝜔
⇒ 𝑒 𝑗𝜔𝑡 , 𝑒 𝑗𝜔𝑛 : 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑆𝑒𝑟𝑖𝑒𝑠
When s or z is general complex number
⇒𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑇𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 & 𝑍 𝑇𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚

11. A “Special” Class of Periodic Signals
CT Fourier Series: if one CT signal can be written as follows
𝑥 𝑡 = 𝑘=−∞ +∞ 𝑎 𝑘 𝑒 𝑗𝑘 𝜔 0 𝑡
where 𝜔 0 =2𝜋/𝑇.
{ 𝑎 𝑘 }: Fourier series coefficients, which represent the strength of the component 𝑒 𝑗𝑘 𝜔 0 𝑡 .
𝑒 𝑗𝑘 𝜔 0 𝑡 is a signal with pure frequency 𝑘 𝜔 0 ⇒ 𝑥 𝑡 is a periodic signal with period T, it consists of components with different frequencies 𝑘 𝜔 0 and different weights.

12. Convergence of CT Fourier Series
What kind of periodic signals have Fourier series expansion?
Define 𝑒 𝑡 =𝑥 𝑡 − 𝑘=−∞ +∞ 𝑎 𝑘 𝑒 𝑗𝑘 𝜔 0 𝑡 ,
Fourier series expansion exists <=> ∃ {𝑎 𝑘 }, e(t)=0 (D1)
Relaxation:
Fourier series expansion exists <=> ∃ {𝑎 𝑘 }, 𝑇 | ​𝑒 𝑡 2 =0 (D2)
𝐷1⊂𝐷2
Two kind of sufficient conditions for D2
𝑇 | ​𝑥 𝑡 2 𝑑𝑡<∞
Dirichlet condition

13. Dirichlet Conditions

14. Almost all the periodic signal in practice have Fourier series expansion!

18. Inner Product of Exponential Signals
Define inner product as
< 𝒆 𝒋𝒌 𝝎 𝟎 𝒕 ⋅ 𝒆 𝒋𝒏 𝝎 𝟎 𝒕 > = 𝟏 𝑻 𝑻 𝒆 𝒋𝒌 𝝎 𝟎 𝒕 𝒆 𝒋𝒏 𝝎 𝟎 𝒕 ∗ 𝒅𝒕
We have
< 𝒆 𝒋𝒌 𝝎 𝟎 𝒕 ⋅ 𝒆 𝒋𝒏 𝝎 𝟎 𝒕 > = 𝟏 𝐤=𝐧
< 𝒆 𝒋𝒌 𝝎 𝟎 𝒕 ⋅ 𝒆 𝒋𝒏 𝝎 𝟎 𝒕 > = 0 𝐤≠𝐧
{ 𝒆 𝒋𝒌 𝝎 𝟎 𝒕 |∀𝒊𝒏𝒕𝒆𝒈𝒆𝒓 𝒌} is similar to basis of vector space

19. How to Obtain Fourier Coefficients
𝑥 𝑡 = 𝑘=−∞ +∞ 𝑎 𝑘 𝑒 𝑗𝑘 𝜔 0 𝑡 ⇒ 𝐴 = 𝐴 𝑥 𝑥 + 𝐴 𝑦 𝑦 + 𝐴 𝑧 𝑧
Notice
𝐴 𝑥 = 𝐴 ⋅ 𝑥
Similarly, we guess
𝑎 𝑘 =<𝑥 𝑡 ⋅ 𝑒 𝑗𝑘 𝜔 0 𝑡 >
Let’s double-check
<𝑥 𝑡 ⋅ 𝑒 𝑗𝑘 𝜔 0 𝑡 >
= 𝑛=−∞ +∞ 𝑎 𝑛 < 𝑒 𝑗𝑛 𝜔 0 𝑡 ⋅ 𝑒 𝑗𝑘 𝜔 0 𝑡 >
= 𝑎 𝑘

20. CT Fourier Series Pair
Fourier series expansion

21. Example 3.5: Periodic Square WaveT ak

22. Example
Gibbs Phenomenon: The partial sum in the vicinity of the discontinuity exhibits ripples whose amplitude does not seem to decrease with increasing N

23. Periodic Impulse Train

24. (A Few ) Properties of CT Fourier Series
real

25. Time Reversal Time Scaling
the effect of sign change for x(t) and ak are identical
Example: x(t): … a-2 a-1 a0 a1 a2 …
x(-t): … a2 a1 a0 a-1 a-2 …
Time Scaling
positive real number
periodic with period T/α and fundamental frequency αw0
ak unchanged, but x(αt) and each harmonic component are different

27. Power is the same whether measured in the time-domain or the
frequency-domain

28. More Frequency shifting Differentiation Integration
Note:
x(t) is periodic with
fundamental frequency ω0
Frequency shifting
Differentiation
Integration
𝑒 𝑗𝑀 𝜔 0 𝑡 𝑥 𝑡    𝑏 𝑘 = 𝑎 𝑘−𝑀
𝑑𝑥 𝑑𝑡    𝑏 𝑘 =𝑗𝑘 𝜔 0 𝑎 𝑘
−∞ 𝑡 𝑥(𝑡)𝑑𝑡    𝑏 𝑘 =( 1 𝑗𝑘 𝜔 0 ) 𝑎 𝑘