【医学英文课件】《生物医学信号处理(双语)》精品课件.ppt

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1、1,Chapter 4:Sampling of Continuous-Time Signals,4.0 Introduction4.1 Periodic Sampling4.2 Frequency-Domain Representation of Sampling4.3 Reconstruction of a Bandlimited Signal from its Samples4.4 Discrete-Time Processing of Continuous-Time signals4.5 Continuous-time Processing of Discrete-Time Signal

2、,and the subsequent reconstruction of a continuous-time signal.,2,4.0 Introduction,Continuous-time signal processing can be implemented through a process of sampling,f=1/T:sampling frequency,T:sampling period,discrete-time processing,3,4.1 Periodic Sampling,Continuous-time signal,Sampling sequence,U

3、nit impulse train,impulse train sampling,t,T:sampling period,n,冲激串序列,4,T：sample period;fs=1/T:sample rate;s=2/T:sample rate,s(t)为冲激串序列，可展开傅立叶级数,冲激串的傅立叶变换：,5,4.2 Frequency-Domain Representation of Sampling,Representation of in terms of,6,4.2 Frequency-Domain Representation of Sampling,Representation

4、of in terms of,7,DTFT,Representation of in terms of,采样角频率,rad/s,连续时间傅里叶变换和离散时间傅里叶变换间的联系,在奥本海姆的信号与系统教材里,在“第7章 采样”内容之前，连续时间傅里叶变换X(j),和离散时间傅里叶变换X(ej)中涉及的频率都用相同的频率符号表示,没有加以区分,各说各话。,8,但要注意频率的单位,一个是rad/s,另一个无单位;另外,频率高低与取值范围的关系：连续时间傅里叶变换X(j)中的值越大,频率越高；但是离散时间傅里叶变换X(ej)中的值越大,频率却未必越高：X(ej)中的的值是的奇数倍的时候,表示频率最高;

5、的值是的偶数倍的时候,表示频率最低。另外注意连续时间和离散时间的傅里叶变换是否具有周期性:X(ej)具有周期性,周期2。X(j)不具有周期性。,9,奥本海姆 信号与系统在“第7章 采样”的“7.4 Discrete-Time Processing of Continuous-time Signals”一节中,因对连续时间信号xc(t)进行采样(得到xdn),在分析频谱时需要同时涉及到连续时间信号的傅里叶变换和离散时间序列的傅里叶变换。,这是两种不同的傅里叶变换,需加以区分(如下所示:是数字频率,是模拟频率)。因为两种傅里叶变换的频谱特性,特别是随频率变化而变化的特性,如上页所述,表现各有特点,

6、有相似的地方,也有截然不同之处。,连续时间傅里叶变换和离散时间傅里叶变换间的联系,(在535页的最后一段中开始)特别将两种傅里叶变换中的频率符号加以区分(仅7.4一节,其他章节没有区分):,10,因为采样,两种不同的傅里叶变换联系起来了,不但两种变换的变换结果可建立起表达式关系,而且其各自变换的自变量频率之间也有表达式关系:=T,是数字频率,是模拟频率。,连续时间傅里叶变换和离散时间傅里叶变换间的联系,也就是说两种变换的频率含义并非完全相同,而是既区别又有联系。也恰恰是因为采样的缘故,建立起了两种变换的频率之间的表达式关系:=T。,信号与系统第7章7.4节中的538页最上面一段中解释了=T的比

7、例关系:用“4.3-5 Time and Frequency Scaling”性质解释。,11,奥本海姆 信号与系统在“第7章 采样”的“7.4 Discrete-Time Processing of Continuous-time Signals”一节中,两种傅里叶变换的表示方法:,这与奥本海姆离散时间信号处理教材中用的频率符号正好相反(该教材中数字频率=T,是模拟频率):,连续时间傅里叶变换和离散时间傅里叶变换间的联系,12,Representation of in terms of,Continuous FT,of Sampling,DTFT,DTFT,without Aliasing,

8、13,DTFT,Representation of in terms of,Continuous FT,14,Representation of X(ej)in terms of Xc(j),Aliasing,15,DTFT,Representation of in terms of,Continuous FT,of Sampling,16,Nyquist Sampling Theorem,Let be a bandlimited signal with.,The frequency is commonly referred as the Nyquist frequency.,The freq

9、uency is called the Nyquist rate,which is the minimum sampling rate(frequency).,Then is uniquely determined by its samples,if.,without Aliasing,17,No aliasing,满足采样定理条件,无频率混叠,18,aliasing,不满足采样定理条件,aliasing frequency,19,No aliasing,aliasing,20,Example 4.1:Sampling and Reconstruction of a sinusoidal si

10、gnal,Solution:,Compare the continuous-time and discrete-time FTs for sampled signal,21,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,23,Ex.4.2:Aliasing

11、 in sampling an sinusoidal signal,Solution:,Example 4.2:Aliasing in the Reconstruction of an Undersampled sinusoidal signal,continuous-time FT of,discrete-time FT of,24,Ex.4.2:Aliasing in sampling an sinusoidal signal,continuous-time FT of,discrete-time FT of,25,低通滤波器,26,Gain:T,4.3 Reconstruction of

12、 a Bandlimited Signal from its Samples,27,低通滤波器Gain:T,4.3 Reconstruction of a Bandlimited Signal from its Samples,28,Gain:T,4.3 Reconstruction of a Bandlimited Signal from its Samples,CTFT,DTFT,29,4.3 Reconstruction of a Bandlimited Signal from its Samples,CTFT,DTFT,重建原信号,只要满足Qs 2QN,30,4.3 Reconstru

13、ction of a Bandlimited Signal from its Samples:关于重建表达式的理解:,the ideal lowpass filter interpolates between the impulses of xs(t).,30,即可完全重建原信号。,只要满足Qs 2QN,采样,重建,重建过程,卷积,滤波,31,4.4 Discrete-Time Processing of Continuous-Time signals,32,4.4 Discrete-Time Processing of Continuous-Time signals,33,C/D Conve

14、rter,Output of C/D Converter,34,D/C Converter,Output of D/C Converter,35,4.4.1 LTI DT Systems,Is the system Linear Time-Invariant?,即采样时无频率混叠.,采样频率,36,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

15、 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),37,This is the effective frequency response of the overall LTI continuous-time system.,等效的连续时间系统必须是带限的.,38,Example 4.3 bulid a Ideal Continu

16、ous-Time Lowpass Filtering using a Discrete-Time Filter,LTI DT System,LTI CT System,-,Given,Solution:,39,Example 4.3 Ideal Continuous-Time Lowpass Filtering using a Discrete-Time Filter,interpretation of how this effective response is achieved.?,Figure,4-12,40,Example 4.3 Ideal Continuous-Time Lowpa

17、ss Filtering using a Discrete-Time Filter,interpretation of how this effective response is achieved.?,Figure,4-12,等效连续低通滤波器频率响应,41,Example 4.4:Discrete-Time Implementation of an Ideal Continuous-Time Bandlimited Differentiator,Solution:Differentiator:Frequency response:The inputs are restricted to b

18、e bandlimited.For bandlimited signals,it is sufficient that:,The corresponding DT system has frequency response:,If inputs are bandlimited,高于/T的频率响应也用不到.,42,Example 4.4:Discrete-Time Implementation of an Ideal Continuous-Time Bandlimited Differentiator,effective frequency response:,frequency respons

19、e for DT system:,43,Example 4.4:Discrete-Time Implementation of an Ideal Continuous-Time Bandlimited Differentiator,frequency response for DT system:,The corresponding impulse response:,or equivalently,44,4.4.2 Impulse Invariance,Given:,Design:,impulse-invariant version of the continuous-time system

20、,脉冲响应不变型,关系？,45,4.4.2 Impulse Invariance,Two constraints,The discrete-time system is called an impulse-invariant version of the continuous-time system.,截止频率,1.,2.hc(t)is bandlimited,采样不产生频率混叠,脉冲响应不变型,If,Let,且与采样频率关系？,脉冲响应不变法,这是前面推出的结果要求,46,Example 4.5:A Discrete-Time Lowpass Filter Obtained by Impul

21、se Invariance,Suppose that we wish to obtain an ideal lowpass discrete-time filter with cutoff frequency c,by sampling a continuous-time ideal lowpass filter with cutoff frequency c=c/T/T defined by:,The impulse response of CT system:,Nyquist frequency c 采样频率/2满足采样定理条件,Solution:,find,47,Example 4.5:

22、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:,The DTFT of this sequence:,满足采样定理条件,无频率混叠,check the frequency response,c=c/T/T,48,Example 4.6:Impulse Invariance Applied to Continuous-Time Systems with

23、 Rational System Functions,Many CT systems have impulse responses of form:,Its Laplace transform:,apply impulse invariance concept to CT system,we obtain the hn of DT system:,which has z-transform system function:,频带无限,不满足采样定理条件,49,Example 4.6:Impulse Invariance Applied to CT Systems with Rational S

24、ystem Functions,CT system have impulse responses of form:,Sampling:,z-transform:,assuming Re(s0)0,the frequency response:,L,单位圆在收敛域,不满足采样定理条件,存在频率混叠,但对高阶系统,频率混叠可忽略,所以脉冲响应不变法可用于设计滤波器。,存在FT,50,Example 4.6:Impulse Invariance Applied to CT Systems with Rational System Functions,CT system have impulse re

25、sponses of form:,Sampling:,z-transform:,assuming Re(s0)0,the frequency response:,L,单位圆在收敛域,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,In this case,Eq.(4.55)does not hold exactly,because the original continuous-time system did not have a strictly bandlimited frequency response,and therefore,the

26、resulting discrete-time frequency response is an aliased version.Even though aliasing occurs in such a case as this,the effect may be small.Higher-order systems whose impulse responses are sums of complex exponentials may in fact have frequency responses that fall off rapidly at high frequencies,so

27、that aliasing is minimal if the sampling rate is high enough.Thus,one approach to the discrete-time simulation of continuous-time systems and also to the design of digital filters is through sampling of the impulse response of a corresponding analog filter.,51,4.5 Continuous-time Processing of Discr

28、ete-Time Signal,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.,Figure 4.15,52,4.5 Continuous-time Processing of Discrete-Time Signal,Sam

29、pling without aliasing,53,4.5 Continuous-time Processing of Discrete-Time Signal,Sampling without aliasing,54,4.5 Continuous-time Processing of Discrete-Time Signal,When is integer,55,Example 4.7:Non-integer Delay,The frequency response of a discrete-time system,When(=0.5)is not an integer,it has no

30、 formal meaning we cannot shift the sequence xn by(=0.5).,It can be interpreted by system,integer,如何理解？,56,Let,It represents a time delay of T seconds.Therefore,and,Example 4.7:Non-integer Delay,输出yn是将xn先变换成xc(t),将其延迟T(0.5T)时刻之后再采样得到的离散时间序列.,对连续时间系统,57,Ex.4.7:Non-integer Delay,For=1/2,用图形表示该过程,重建,58

31、,Example 4.7:Non-integer Delay,When=n0 is integer,for LTI system,proof?,DTFT,59,4.5 Continuous-time Processing of Discrete-Time Signal,Figure 4.18 Illustration of moving-average filtering.(a)Input signalxn=cos(0.25n).(b)Corresponding output of six-point moving-average filter.,第二版Errata,60,The Nyquis

32、t 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.),What is Nyquist rate？What is Nyquist frequency？,Review,minimum,max frequency,Sampling without aliasing,61,How many factors does the lin

33、ear 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),62,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 hn and H(ejw)?,Review,