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 xn, n , n :integer xn x nT T sampling period a , : Cumbersome, so just use xn
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 xn
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 tnT 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