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