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