Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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 ) 》

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

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. f=1/T: sampling frequency T: sampling period

4 4.1 Periodic Sampling Continuous- time signal T: sampling period impulse train sampling Sampling sequence Unit impulse train

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

6 4.2 Frequency-Domain Representation of Sampling T ： sample period; fs=1/T:sample rate Ωs=2π/T: sample rate Representation of in terms of

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

8 DTFT Representation of in terms of, Continuous FT

9 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.

10 aliasing frequency No aliasing aliasing frequency spectrum of ideal sample signal

Compare the continuous-time and discrete-time FTs for sampled signal 11 Example 4.1: Sampling and Reconstruction of a sinusoidal signal Solution:

12 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

Compare the continuous-time and discrete-time FTs for sampled signal 14 Example 4.2: Aliasing in the Reconstruction of an Undersampled sinusoidal signal Solution:

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

Discrete-Time Processing of Continuous-Time signals

17 C/D Converter Output of C/D Converter

18 D/C Converter Output of D/C Converter

Linear Time-Invariant Discrete-Time Systems Is the system Linear Time-Invariant ?

20 Linear and Time-Invariant Linear and time-invariant behavior of the system of Fig.4.11 depends on two factors: 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 )

21 effective frequency response of the overall LTI continuous-time system

Impulse Invariance Given: Design: impulse-invariant version of the continuous-time system

Impulse Invariance  Two constraints The discrete-time system is called an impulse- invariant version of the continuous-time system 截止频率

Continuous-time Processing of Discrete-Time Signal

Continuous-time Processing of Discrete-Time Signal

Continuous-time Processing of Discrete-Time Signal Figure 4.18 Illustration of moving-average filtering. (a) Input signal x[n] = cos(0.25 π n). (b) Corresponding output of six-point moving- average filter. Errata

27  The Nyquist rate is two times the bandwidth of a bandlimited signal.  The Nyquist frequency is half the sampling frequency of a discrete signal processing system.( The Nyquist frequency is one-half the Nyquist rate)  What is Nyquist rate ？  What is Nyquist frequency ？ Review

28  DTFT derived from the equation.  impulse train sampling x s (t) and x[n] have the same frequency component. Review  What is the physical meaning for the equation: DTFT of a discrete-time signal is equal to the FT of a impulse train sampling.

29  How many factors does the linear and time-invariant behavior of the system of Fig.4.11 depends on ? Review  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 )

30  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(e jw )? Review

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