山东大学：《生物医学信号处理 Biomedical Signal Processing》精品课程教学资源（PPT课件讲稿）Chapter 02 Discrete-Time Signals and Systems

Chapter 2 Discrete-Time Signals and Systems ◆2.0 Introduction 2. 1 Discrete-Time Signals: Sequences 2.2 Discrete-Time Systems 2.3 Linear Time-Invariant(LTI Systems 2 4 Properties of LTI Systems 2.5 Linear Constant- Coefficient Difference equations 2 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ

2 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 2 Discrete-Time Signals and Systems u2.0 Introduction u2.1 Discrete-Time Signals: Sequences u2.2 Discrete-Time Systems u2.3 Linear Time-Invariant (LTI) Systems u2.4 Properties of LTI Systems u2.5 Linear Constant-Coefficient Difference Equations

apter 2 Discrete-Time signals and systems 92.6 Frequency-Domain Representation of Discrete- Time Signals and systems 2.7 Representation of Sequences by Fourier transforms 2. 8 Symmetry Properties of the Fourier Transform 2.9 Fourier Transform Theorems 2.10 Discrete-Time Random signals ◆211 Summary 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ

3 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 2 Discrete-Time Signals and Systems u2.6 Frequency-Domain Representation of Discrete-Time Signals and systems u2.7 Representation of Sequences by Fourier Transforms u2.8 Symmetry Properties of the Fourier Transform u2.9 Fourier Transform Theorems u2.10 Discrete-Time Random Signals u2.11 Summary

2.0 Introduction Signal: something conveys information represented mathematically as functions of one or more independent variables classified as Continuous-time analog) signals discrete-time signals, digital signals Signal-processing systems are classified along the same lines as signals Continuous-time(analog) systems, discrete-time systems digital systems 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ

4 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2.0 Introduction uSignal: something conveys information, represented mathematically as functions of one or more independent variables. Classified as: uContinuous-time (analog) signals, discrete-time signals, digital signals uSignal-processing systems are classified along the same lines as signals: Continuous-time (analog) systems, discrete-time systems, digital systems

2.1 Discrete-Time Signals: Sequences Discrete-Time signals are represented as x={xn]-∞<n<∞,n: Integer Cumbersome, so just use xIn In sampling of an analog signal x) xn=xo(nr), T: sampling period 1/(reciprocal of T): sampling frequency 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ

5 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2.1 Discrete-Time Signals: Sequences uDiscrete-Time signals are represented as uIn sampling of an analog signal xa(t): u1/T (reciprocal of T) : sampling frequency x  xn,   n  , n :integer xn x nT  T sampling period a  , : Cumbersome, so just use xn

Figure 2. 1 Graphical representation of a discrete-time signal -2J 2 7891011 9-8-7-6-54-3-2-10123456 Abscissa: continuous line x[n: is defil ned on ly at dis iscrete instants 6 1/30/2021 Zhongguo Liu_ Biomedical Engineering_ Shandong Univ

6 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Figure 2.1 Graphical representation of a discrete-time signal Abscissa: continuous line : is defined only at discrete instants xn

xin]=xa(t)nr=xa(nT) EXAMPLE Sampling the analog waveform △A 32 (a) 256 samples (b) Figure 2.2

7 Figure 2.2 EXAMPLE Sampling the analog waveform x[n] x (t) | x (nT)  a tnT  a

Basic Sequence operations ◆ Sum of two sequences xln+ vln Product of two sequences x{nm]·y{n] Multiplication of a sequence by a number a a·x[n] Delay shift) of a sequence yIn=xn-nol no: integer 8 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ

8 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. uSum of two sequences uProduct of two sequences uMultiplication of a sequence by a number α uDelay (shift) of a sequence Basic Sequence Operations x[n] y[n] [ ] [ ] :integer 0 0 y n  x n  n n x[n] y[n]   x[n]

Basic sequences ◆ Unit sample sequence 0n≠0 (discrete-time impulse, o(n H=0 impulse, Unit impulse) ◆离散时间单位脉冲(样本)序列,区别连续时间单位冲激 EK=(continuous-time unit impulse function &(t)) Unit sample 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ

9 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences uUnit sample sequence (discrete-time impulse, impulse, Unit impulse)         1 0 0 0 n n  n u离散时间单位脉冲(样本)序列, 区别连续时间单位冲激 函数(continuous-time unit impulse function δ(t) )

Basic sequences 4-20134568n 2 A sum of scaled delayed impulses =a3Dn+]+aoz-1]+a6-2]+a ◆ arbitrary sequence xm=∑xk[m-k k 10 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ

10 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences [ ] [ ] [ ] k x n x k  n k      uarbitrary sequence    3  1  2  7 p n  a 3 n   a1 n   a2 n   a7 n  A sum of scaled, delayed impulses

Basic sequences 1n≥0 AUnit step sequence unI 0n<0 nit step 6[k 10. when n <0 n=∑[k ,henn≥0 0k≠0 k=-∞ since s k=0 l]=8团]+8[n-11+6{m-2]+…=∑8[n-k k=0 an=uln-uln-l First backward difference 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ

11 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences uUnit step sequence       0 0 1 0 [ ] n n u n [ ]   n k u n  k    0 [ ] [ ] [ 1] [ 2] [ ] k u n  n  n  n  n k             [n]  u[n]u[n 1] First backward difference       0, 0 , 1, 0 0 0 1 0 since n k when n k when n k k k                