E210 Lecture Notes Dianguang Ma Spring 2010
EI210 Lecture Notes Dianguang Ma Spring 2010
Chapter 2(Part I) Linear Time-Invariant Systems
Chapter 2 (Part I) Linear Time-Invariant Systems
Introduction In this chapter,we examine several methods for describing the relationship between the input and output signals of linear time-invariant (LTl)systems in time domain. Convolution sum/integral Linear constant-coefficient difference/differential equation
Introduction • In this chapter, we examine several methods for describing the relationship between the input and output signals of linear time-invariant (LTI) systems in time domain. – Convolution sum/integral – Linear constant-coefficient difference/differential equation
Discrete-Time LTI Systems:The Convolution Sum An arbitrary discrete-time signal can be thought of as a weighted superposition of shifted (discrete-time)impulses. x[n]=…+x[-1][n+1]+x[0][n]+x[l][n-1]+… =>xk][n-k] xn:the entire signal []a specific value of the signal x[n]at time k
Discrete-Time LTI Systems: The Convolution Sum • An arbitrary discrete-time signal can be thought of as a weighted superposition of shifted (discrete-time) impulses. [ ] [ 1] [ 1] [0] [ ] [1] [ 1] [ ] [ ] [ ]: the entire signal [ ]: a specific value of the signal [ ] at time . k x n x n x n x n x k n k x n x k x n k
Graphical example illustrating the x-2]dn+2] representation of a signal x[n]as a weighted sum of time-shifted 。。 人 impulses. x-lldn+1】 x-] x[o]dIn -1] xf016 0 xjdn-】 + x1]十 0 x[2]d[n -2] x2I十 ◆一0000—0 x(n]
Graphical example illustrating the representation of a signal x[n] as a weighted sum of time-shifted impulses
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
