信号与系统（英文版）_2 Linear Time-Invariant Systems

2 Linear Time-Invariant Systems 2. Linear Time-Invariant Systems 2.1 Discrete-time lti system: The convolution sum 2.1.1 The Representation of Discrete-time Signals in Terms of Impulses xn]=…+x-2]6n+2]+x[-16[n+1]+x[0j1[n]+x(1]n-1]+x{216[n-2]+ x{k16[n-k] If xIn]=u[n], then x[n]=>8[n-k

2 Linear Time-Invariant Systems 2.1 Discrete-time LTI system: The convolution sum 2.1.1 The Representation of Discrete-time Signals in Terms of Impulses 2. Linear Time-Invariant Systems  + =− = − = + − + + − + + + − + − + k x k n k x n x n x n x n x n x n [ ] [ ] [ ] [ 2] [ 2] [ 1] [ 1] [0] [ ] [1] [ 1] [2] [ 2]         If x[n]=u[n], then  + = = − 0 [ ] [ ] k x n  n k

2 Linear Time-Invariant Systems 32 xl-218[n,21 a-2:8:2 [-11Dn+1 …-3-2I525…

2 Linear Time-Invariant Systems

2 Linear Time-Invariant Systems 2.1.2 The Discrete-time Unit Impulse response and the Convolution Sum Representation of Lti Systems (1)Unit Impulse(Sample) Response xn=8n yIn=hn LTI Unit Impulse Response: hnI

2 Linear Time-Invariant Systems 2.1.2 The Discrete-time Unit Impulse Response and the Convolution Sum Representation of LTI Systems (1) Unit Impulse(Sample) Response LTI x[n]=[n] y[n]=h[n] Unit Impulse Response: h[n]

2 Linear Time-Invariant Systems 2) Convolution Sum of Lti System Question n LTI yin/=? Solution n]—>hn] i[n-k]—>hn-K] k]n-K]—>xk]h[n-k xn=∑xk6n-k]-→yn=∑xkhn-k

2 Linear Time-Invariant Systems (2) Convolution Sum of LTI System LTI x[n] y[n]=? Solution: Question: [n] ⎯→ h[n] [n-k] ⎯→ h[n-k] x[k][n-k] ⎯→x[k] h[n-k]   + =− + =− = − − − → = − k k x[n] x[k][n k] y[n] x[k]h[n k]

2 Linear Time-Invariant Systems ho回o h1回

2 Linear Time-Invariant Systems

2 Linear Time-Invariant Systems x-1]b[n+1 x[o] 8(n] x[O] hoIn] l18n-1] x[1h;[o] d

2 Linear Time-Invariant Systems

2 Linear Time-Invariant Systems So yn]=2x[k]hn-k]( Convolution Sum or yn=xn]*hn 3)Calculation of Convolution Sum Time Inversal: h[k]->h[-k Time shift:h[-k]—>hn-k Multiplication: xk]hn-k Summing:y]=∑xk]n-k k=-0 Example2.1222.32.42.5

2 Linear Time-Invariant Systems So  ( Convolution Sum ) + =− = − k y[n] x[k]h[n k] or y[n] = x[n] * h[n] (3) Calculation of Convolution Sum Time Inversal: h[k] ⎯→ h[-k] Time Shift: h[-k] ⎯→ h[n-k] Multiplication: x[k]h[n-k] Summing:  + =− = − k y[n] x[k]h[n k] Example 2.1 2.2 2.3 2.4 2.5

2 Linear Time-Invariant Systems 2.2 Continuous-time LTi system The convolution integral 2.2.1 The Representation of Continuous-time Signals in Terms of Impulses Defineδ(1)=)4.0≤t≤△ otherwise We have the expression: ()=∑x(k△)△A(t-k△) Therefore x()=m∑x(A△)△δ(t-k△)

2 Linear Time-Invariant Systems 2.2 Continuous-time LTI system: The convolution integral 2.2.1 The Representation of Continuous-time Signals in Terms of Impulses         =  otherwise t t 0, , 0 1 Define  ( ) We have the expression:  + =− =    −  k xˆ(t) x(k )  (t k ) Therefore:  + =−  → =   −  k x(t) lim x(k ) (t k ) 0 

2 Linear Time-Invariant Systems x(0) △D△2△ 2△)△(t+2△)△ x《-2△ 2A-△ (b) x-△)b△(+△ (c)

2 Linear Time-Invariant Systems

2 Linear Time-Invariant Systems or x(t)= x(r8(t-r)dr 8(t-) (b) x()8(t-)=×(t)b(t- (c)

2 Linear Time-Invariant Systems or  + − x(t) = x( ) (t − )d