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)》 http://course.sdu.edu.cn/bdsp.html 2018/9/13 1 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

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

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

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

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

4.2 Frequency-Domain Representation of Sampling in terms of

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

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

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

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

No aliasing 满足采样定理条件, 无频率混叠

aliasing frequency aliasing 不满足采样定理条件

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

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

从积分(相同的面积)或冲击函数的定义可证

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

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

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

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

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

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

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

4.4 Discrete-Time Processing of Continuous-Time signals

4.4 Discrete-Time Processing of Continuous-Time signals

C/D Converter Output of C/D Converter

D/C Converter Output of D/C Converter

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

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)

effective frequency response of the overall LTI continuous-time system

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

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

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

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:

Example 4.4: Discrete-Time Implementation of an Ideal Continuous-Time Bandlimited Differentiator effective frequency response : frequency response for DT system :

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

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

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

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 :

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

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:

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

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.

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

4.5 Continuous-time Processing of Discrete-Time Signal

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

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

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

Example 4.7: Non-integer Delay why? DTFT When = n0 is integer,

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

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)

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

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