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

apter 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 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

2 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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

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 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

3 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 2 Discrete-Time Signals and Systems ◆2.6 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 ◆2.11 Summary

2.0 Introduction Signal: something conveys information Signals are represented mathematically as functions of one or more independent variables 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 ◆ Discrete-time signal Sampling a continuous-time signal o Generated directly by some discrete-time process 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

4 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2.0 Introduction ◆Signal: something conveys information ◆Signals are represented mathematically as functions of one or more independent variables. ◆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 ◆Discrete-time signal ◆Sampling a continuous-time signal ◆Generated directly by some discrete-time process

2.1 Discrete-Time Signals: Sequences Discrete-Time signals are represented as ins 00<n<∞,n: Integer Cumbersome, so just use xIn ◆ In sampling, xn =xo(nr), T: sampling period ◆1/T(〔 reciprocal of T): sampling frequency 5 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

5 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2.1 Discrete-Time Signals: Sequences ◆Discrete-Time signals are represented as ◆In sampling, ◆1/T (reciprocal of T) : sampling frequency x =xn, −  n  , n:integer xn= xa (nT), T :sampling period Cumbersome, so just use x n 

Figure 2.1 Graphical representation of a discrete-time signa -2 2 7891011 94-7-6-54-3-2-10123456 Abscissa: continuous line x[n: is defined only at discrete instants 6 2/2/2021 Zhongguo Liu_ Biomedical Engineering_Shandong Univ

6 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Figure 2.1 Graphical representation of a discrete-time signal Abscissa: continuous line x n  : is defined only at discrete instants

xin]=xa(tent=xa(nT) EXAMPLE Sampling the analog waveform 32m 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 xn+yln Product of two sequences x{n]·y{n] Multiplication of a sequence by a numbera a·x[m] Delay(shift) of a sequence n=xIn-n 0· Integer 8 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

8 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. ◆Sum of two sequences ◆Product of two sequences ◆Multiplication of a sequence by a numberα ◆Delay (shift) of a sequence Basic Sequence Operations x[n]+ y[n] [ ] [ ] :integer n n0 n0 y n = x − x[n] y[n]   x[n]

Basic sequences ◆ Unit sample sequence 0n≠0 (discrete-time impulse =0 impulse) Unit sample 0 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

9 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences ◆Unit sample sequence (discrete-time impulse, impulse)      =  = 1 0 0 0 n n  n

Basic sequences 4-20134568 2 A sum of scaled delayed impulses p以]=a3n+3]+aoln-1]+a2|n-2]+a, ◆ arbitrary 对=∑k1[m=k] sequence k 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

10 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences   =− = − k x[n] x[k] [n k] ◆arbitrary 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 Unit step sequence un 0n<0 nit step 0.1hen<0 =∑与 1n≥0 k≠0 k since d k=0 lm=8n]+8n-11+6m-2]+…=∑8[n-k1 fn]=u[n]-un-l First backward difference 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

11 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences ◆Unit 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   =−    =         = =   