1. Biomedical Signal processing Chapter 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)》
2018/9/13 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,
discrete-time processing,
and the subsequent reconstruction of a continuous-time signal.
T: sampling period
f=1/T: sampling frequency

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

5. 冲激串的傅立叶变换： T：sample period; fs=1/T:sample rate;Ωs=2π/T:sample rate… …

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

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

8. Representation of in terms of
DTFT
without
Aliasing
Continuous FT
of Sampling
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-time FT 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. Is the system Linear Time-Invariant ?
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 Bandlimited Differentiator
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 Bandlimited Differentiator
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 Two constraints 1. hc(t) is bandlimited
脉冲响应不变法 1.
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
The discrete-time system is called an impulse-invariant version of 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 ω c < π ,
by sampling a continuous-time ideal lowpass filter with cutoff frequency Ωc = ωc /T< π /T defined 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 of this sequence :
Ωc = ωc /T< π /T
满足采样定理条件,无频率混叠

40. Many CT systems have impulse responses of form:
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. CT system have impulse responses of form:
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
存在FT
不满足采样定理条件, 存在频率混叠
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 = n0 is integer,

49. What is Nyquist frequency？
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)

51. Assume that we are given a desired continuous-time system that we wish to implement in the form of the following figure, how to decide h[n] and H(ejw)?
Review

52. Chapter 4 HW
4.5
2018/9/13
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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