Biomedical Signal processing Chapter 4 Sampling of Continuous-Time Signals

1. Biomedical Signal processingChapter 4 Sampling of Continuous-Time Signals Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University 山东省精品课程《生物医学信号处理(双语)》 http://course.sdu.edu.cn/bdsp.html 2019/11/7 1 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. Chapter 4: Sampling of Continuous-Time Signals 4.0 Introduction 4.1 Periodic Sampling 4.2 Frequency-Domain Representation of Sampling 4.3 Reconstruction of a Bandlimited Signal from its Samples 4.4 Discrete-Time Processing of Continuous-Time signals 4.5 Continuous-time Processing of Discrete-Time Signal

3. 4.0 Introduction • Continuous-time signal processing can be implemented through a process of sampling, and the subsequent reconstructionof a continuous-time signal. discrete-time processing, T: sampling period f=1/T: sampling frequency

4. Continuous-time signal 4.1 Periodic Sampling Unit impulse train 冲激串序列 impulse train sampling T: sampling period t Sampling sequence n

5. 1 … … … … -T T 0 0 冲激串的傅立叶变换： T：sample period; fs=1/T:sample rate;Ωs=2π/T:sample rate s(t)为冲激串序列，可展开傅立叶级数

6. Representation of in terms of 4.2 Frequency-Domain Representation of Sampling

7. Representation of in terms of , 数字角频率ω，rad 模拟角频率Ω, rad/s 采样角频率, rad/s DTFT

8. Representation of in terms of DTFT without Aliasing of Sampling Continuous FT DTFT

9. Representation of X(ejω) in terms of Xc(jΩ) Aliasing

10. Nyquist Sampling Theorem • Let be a bandlimited signal with . Then is uniquely determined by its samples , if . • The frequency is commonly referred as the Nyquist frequency. • The frequency is called the Nyquist rate, which is the minimum sampling rate (frequency). without Aliasing

11. No aliasing 满足采样定理条件, 无频率混叠

12. aliasing frequency aliasing 不满足采样定理条件

13. Example 4.1: Sampling and Reconstruction of a sinusoidal signal Compare the continuous-time and discrete-time FTs for sampled signal Solution:

14. Example 4.1: Sampling and Reconstruction of a sinusoidal signal continuous-timeFT of discrete-time FT of

15. 从积分(相同的面积)或冲击函数的定义可证

16. Ex.4.2: Aliasing in sampling an sinusoidal signal Compare the continuous-time and discrete-time FTs for sampled signal Solution:

17. Example 4.2: Aliasing in the Reconstruction of an Undersampled sinusoidal signal continuous-time FT of discrete-time FT of

18. Ex.4.2: Aliasing in sampling an sinusoidal signal continuous-time FT of discrete-time FT of 低通滤波器

19. 4.3 Reconstruction of a Bandlimited Signal from its Samples 低通滤波器Gain: T

20. 4.3 Reconstruction of a Bandlimited Signal from its Samples Gain: T CTFT DTFT

21. 4.3 Reconstruction of a Bandlimited Signal from its Samples the ideal lowpass filter interpolates between the impulses of xs(t). 26

22. 4.3 Reconstruction of a Bandlimited Signal from its Samples CTFT DTFT 27

23. 4.4 Discrete-Time Processing of Continuous-Time signals

24. 4.4 Discrete-Time Processing of Continuous-Time signals

25. C/D Converter • Output of C/D Converter

26. D/C Converter • Output of D/C Converter

27. 4.4.1 LTI DT Systems Is the system Linear Time-Invariant ?

28. Linear and Time-Invariant • Linear and time-invariant behavior of the system of Fig.4.10 depends on two factors: • Firstly, the discrete-time system must be linear and time invariant. • Secondly, the input signal must be bandlimited, and the sampling rate must be high enough to satisfy Nyquist Sampling Theorem.(避免频率混叠)

29. effective frequency response of the overall LTI continuous-time system

30. Example 4.3 Ideal Continuous-Time Lowpass Filtering using a Discrete-Time Filter Given LTI DT System -π π Solution: LTI CT System

31. Example 4.3 Ideal Continuous-Time Lowpass Filtering using a Discrete-Time Filter Figure 4-12 interpretation of how this effective response is achieved.?

32. Example 4.3 Ideal Continuous-Time Lowpass Filtering using a Discrete-Time Filter Figure 4-12 interpretation of how this effective response is achieved.?

33. Example 4.4: Discrete-Time Implementation of an Ideal Continuous-Time BandlimitedDifferentiator If inputs are bandlimited, Solution: Differentiator: • Frequency response: • The inputs are restricted to be bandlimited. • For bandlimited signals, it is sufficient that: • The corresponding DT system has frequency response:

34. Example 4.4: Discrete-Time Implementation of an Ideal Continuous-Time BandlimitedDifferentiator • effective frequency response : • frequency response for DT system :

35. Example 4.4: Discrete-Time Implementation of an Ideal Continuous-Time Bandlimited Differentiator • frequency response for DT system : • The corresponding impulse response : • or equivalently,

36. 4.4.2 Impulse Invariance Given: Design: impulse-invariant version of the continuous-time system 脉冲响应不变型

37. 4.4.2 Impulse Invariance 脉冲响应不变法 1. • Two constraints hc(t) is bandlimited 与采样频率关系 2. 截止频率 采样不产生频率混叠 脉冲响应不变型 Thediscrete-time systemis called an impulse-invariant versionof the continuous-time system If Let

38. Example 4.5: A Discrete-Time Lowpass Filter Obtained by Impulse Invariance • Suppose that we wish to obtain an ideal lowpass discrete-time filter with cutoff fre­quency ωc< π , • by sampling a continuous-time ideal lowpass filter with cutoff frequencyΩc= ωc/T< π/Tdefined by : Nyquist frequency < 采样频率/2 满足采样定理条件 find Solution: • The impulse response of CT system :

39. Example 4.5: A Discrete-Time Lowpass Filter Obtained by Impulse Invariance • The impulse response of CT system : • so define the impulse response of DT system to be: check the frequency response • The DTFT ofthis sequence : Ωc = ωc/T< π/T 满足采样定理条件,无频率混叠

40. Example 4.6: Impulse Invariance Applied to Continuous-Time Systems with Rational System Functions • Many CT systems have impulse responses of form: 频带无限 • Its Laplace transform: • apply impulse invariance concept to CT system, we obtain the h[n] of DT system: 不满足采样定理条件 • which has z-transform system function:

41. Example 4.6: Impulse Invariance Applied to CT Systems with Rational System Functions • CT system have impulse responses of form: L • Sampling: • z-transform : • assuming Re(s0) < 0, the frequency response: 单位圆在收敛域 不满足采样定理条件, 存在频率混叠 存在FT 高阶系统, 频率混叠可忽略, 所以脉冲响应不变法可用于设计滤波器

42. 4.5 Continuous-time Processing of Discrete-Time Signal Fig­ure 4.15 • the system of Fig­ure 4.15 is not typically used to implement discrete-time systems, • it provides a useful interpretation of certain discrete-time systems that have no simple interpretation in the discrete domain.

43. 4.5 Continuous-time Processing of Discrete-Time Signal Sampling without aliasing

44. 4.5 Continuous-time Processing of Discrete-Time Signal

45. Example 4.7: Non-integer Delay • The frequency response of a discrete-time system • When is integer, • When is not an integer, it has no formal meaning we cannot shift the sequence x[n] by . • It can be interpreted by system

46. Example 4.7: Non-integer Delay • Let • It represents a time delay of T seconds.Therefore, • and

47. Ex. 4.7: Non-integer Delay • For example, if =1/2

48. Example 4.7: Non-integer Delay why? DTFT • When =n0is integer,

49. Review • What is Nyquist rate？ • What is Nyquist frequency？ • The Nyquist rate is two times the bandwidth of a bandlimited signal. • The Nyquist frequency is one-half the Nyquist rate. • (The Nyquist frequency is half the sampling frequency.) max frequency minimum

50. Review How many factors does the linear and time-invariant behavior of the system of Fig.4.11 depends on ? • First, the discrete-time system must be linear and time invariant. • Second, the input signal must be bandlimited, and the sampling rate must be high enough to satisfy Nyquist Sampling Theorem.(避免频率混叠)