The Convolution Sum The output of an LTI system is the convolution sum of the input to the system and the impulse response of the system. yIn]=xn]*hin]=>xk]hin-k] k=-00 Derivation:8[n]->h[n](impulse response) 6[n-k]->h[n-k](time invariance) x[k]8[n-k]>x[k]h[n-k](homogeneity) x[n]=∑x[k]n-k]→y[n=∑x[k]Mn-] k=-00 r=-c (superposition)
The Convolution Sum • The output of an LTI system is the convolution sum of the input to the system and the impulse response of the system. [ ] [ ]* [ ] [ ] [ ] Derivation: [ ] [ ] (impulse response) [ ] [ ] (time invariance) [ ] [ ] [ ] [ ] (homogeneity) [ ] [ ] [ ] [ ] [ ] [ ] (superposition) k k k y n x n h n x k h n k n h n n k h n k x k n k x k h n k x n x k n k y n x k h n k
Example Compute the convolution ofx[n]and h[n] hecn对l=trand-y威
Example Compute the convolution of [ ] and [ ], 3 where [ ] [ ] and [ ] [ ]. 4 n x n h n x n u n h n u n
Solution yIn]=x(n]*hin]=x[kJhin-k] k=-0 -立)w-幻 n-k
Solution 1 0 0 [ ] [ ] [ ] [ ] [ ] 3 [ ] [ ] 4 3 3 3 4[1 ] 4 4 4 k n k k n k l n n n k l y n x n h n x k h n k u k u n k
Convolution Sum Evaluation Procedure ·Methods -Convolution Table LTI Form -Reflection and Shift Direct Form yIn]=xIn]*hin]=>x[k]h[n-k] k=-o0
Convolution Sum Evaluation Procedure • Methods – Convolution Table – LTI Form – Reflection and Shift – Direct Form k y[n] x[n] h[n] x[k]h[n k]
Example Given that h[n]=δ[n]+2δ[n-l]+3δ[n-2]and xn]=un-un-3.Find yn=xn *hn
Example Given that [ ] [ ] 2 [ 1] 3 [ 2] and [ ] [ ] [ 3]. Find [ ] [ ] [ ]. h n n n n x n u n u n y n x n h n