1. 2006 Fall Signals and Systems Lecture 2 Complex Exponentials Unit Impulse and Unit Step Signal Singular Functions

2. 2006 Fall Chapter 1 Signals and Systems § 1.3 Exponential and Sinusoidal Signals 复指数信号和正弦信号 § 1.3.1 Continuous-Time Complex Exponential and Sinusoidal Signals 1. Real Exponential Signals a is real Decaying Exponential, when α<0 Growing Exponential, when α>0 Figure 1.19

3. 2006 Fall Chapter 1 Signals and Systems 2.Periodic Complex Exponential and Sinusoidal Signals ① Period ② Euler’ s relation( 欧拉关系 )

4. 2006 Fall Sinusoids and Complex Exponential Probably the most important elemental signal that we will deal with is the real-valued sinusoid. In its continuous-time form, we write the general form as Chapter 1 Signals and Systems Maybe as important as the general sinusoid, the complex exponential function will become a critical part of our study of signals and systems. Its general form is also written as a is complex

5. 2006 Fall Sinusoids and Complex Exponential Decomposition: The complex exponential signal can thus be written in terms of its real and imaginary parts Chapter 1 Signals and Systems (This decomposition of the sinusoid can be traced to Euler's relation) ω 0 :Fundamental Frequency Φ :Phase A :Amplitude

6. 2006 Fall Chapter 1 Signals and Systems ③ Average Power ④ Harmonic relation 3. General Complex Exponential Signals

7. 2006 Fall Chapter 1 Signals and Systems § 1.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals where 1. Real Exponential Signals real

8. 2006 Fall Chapter 1 Signals and Systems 2. Complex Exponential Signals and Sinusoidal Signals ① Average Power ② Euler’ s relation ④ Periodicity Properties ⑤ Harmonic relation ③ Frequency Properties 3. General Complex Exponential Signals

9. 2006 Fall Chapter 1 Signals and Systems ( a) ω 0 =0 N=1 ( b) ω 0 = π /8 N=16 ( c) ω 0 = π /4 N=8 ( d) ω 0 = π /2 N=4 ( e) ω 0 = π N=2 ( f) ω 0 =3 π/2 N=4 ( g) ω 0 =7 π/4 N=8 ( h) ω 0 =15 π/8 N=16 ( i) ω 0 =2 π N=1 Low Frequency High Frequency Figure 1.27 ω 0 =2 kπ, low frequency ω 0 =(2 k+1)π, high frequency Frequency Properties

10. 2006 Fall ω 0 不同, 信号不同. ω 0 相差 2 kπ, 信号相同. ω 0 越大, 频率越高. ω 0 =2 k π 时, 频率低 ; ω 0 =(2 k+1)π 时, 频率高. 对任意的 ω 0, 信号均为周期的. 为有理数时, 信号为周期的. Chapter 1 Signals and Systems Table 1.1 Comparison of the and

11. 2006 Fall Chapter 1 Signals and Systems § 1.4 The Unit Impulse and Unit Step Functions 单位冲激与单位阶跃函数 § 1.4.1 The Discrete-Time Unit Impulse and Unit Step Sequences Unit Impulse 1n=0 0 n ≠ 0 Unit Step 1n ≥ 0 0 n<0

12. 2006 Fall Chapter 1 Signals and Systems 2.Sifting property 筛选特性 1.Sampling property 取样特性 The properties of

13. 2006 Fall Chapter 1 Signals and Systems 1n=0 0 n ≠ 0

14. 2006 Fall Chapter 1 Signals and Systems

15. 2006 Fall Chapter 1 Signals and Systems § 1.4.2 The Continuous- Time Unit Step and Unit Impulse Functions 1. Unit Step Function 1t > 0 0 t < 0 0 t 1 0 △ t 1 2. Unit Impulse Function 1 C +-+- t=0

16. 2006 Fall Chapter 1 Signals and Systems ① ② 0 t ≠ 0 If ③

17. 2006 Fall Engineering Model for Properties of Engineering Model – The value at t=0 is very large – The duration is very short – The area is one Model 1 Demo Model 2 Demo Chapter 1 Signals and Systems

18. 2006 Fall Chapter 1 Signals and Systems § 1.4.3 The Properties of Unit Impulse Functions 1. Sampling and Sifting properties ① If f(t) is continuous at the point of t=0 ② Sampling property Sifting property 2.Scaling property If a is real, a ≠ 0

19. 2006 Fall Chapter 1 Signals and Systems Example2

20. 2006 Fall CT Unit Impulse Function The Dirac delta function is defined by where f(t) is any function that is continuous at t=0 Mathematical Properties :

21. 2006 Fall Chapter 1 Signals and Systems § 1.4.4 信号的计算 1. 信号的加、减、乘、除 2. 信号的基本表示 -τ 0 τ t 0 1 t 1 -1 0 1 t 1 3. 信号的微分、积分运算

22. 2006 Fall Chapter 1 Signals and Systems Example 1.7 x(t) is depicted in Figure 1.40(a),determine the derivative of x(t). 2 1 0 1 2 3 4 t x(t)

23. 2006 Fall Models for Derivation

24. 2006 Fall Properties of

25. 2006 Fall Higher Derivation of

26. 2006 Fall Summary What we have learned? – Complex Exponentials – Unit Impulse and Unit Step Signal – Singular Functions What was the most important point in the lecture? What was the muddiest point? What would you like to hear more about?

27. 2006 Fall Reading List Signals and Systems : – 1.5,1.6, Question : – Classification of Systems

28. 2006 Fall Problem Set 1.21(e),(f) 1.22(e),(f) 1.25(c),(d) 1.26(a),(e)