信号与系统双语课件chapter2.1

n
0, n k 0 0, k n h[n k ] 1, n k 0 1, k n
0, k 0 x[k ] u[k ] k , k 0
n
n , x[k ]h[n k ] 0,
0k n otherwise
k 0 k 0
[n k ]
This is identical to the expression we derived in section1.4.
2.1.2 The discrete-time impulse response and the convolution-sum representation of LTI system
(b)
From the figure, we can get
1, n 0,1,2 h[n] 0, otherwise
0.5, n 0 x[n] 2, n 1 0, otherwise
y[n] x[n] h[n]
k
x[k ]h[n k ]
x[1], n 1 x[1] [n 1] 0, n 1
1 n 1 ,the sum of the So ,we can deduce that, for ` three sequences equal to x[n]:
x[n] x[1] [n 1] x[0] [n] x[1] [n 1]
More general, by including additional shifted, scaled impulses, we can write:
x[n] x[1] [n 1] x[0] [n] x[1] [n 1] x[k ] [n k ]
CHAPTER 2 Linear Time-Invariant System
LTI system: A system which is linear and time-invariant is
called LTI system. It is an ideal model of a system, in order to simplify a system for analysis. LTI system play an important role in signal and system analysis. ★First ,many physical processes can be modeled as LTI systems. ★In addition, LTI system can be analyzed in considerable detail, providing both insight into their properties and a set of powerful tools that form the core of signal and a system analysis.
y[n]
k
x[k ]h[n k ]

This result is refer to as the convolution sum ,and the right-hand side of equation is known as the convolution of sequence x[n] and h[n]. We will represent the operation convolution symbolically as
k
Example
Consider an LTI system with impulse h[n] and input x[n], as illustrated in figure (a) and (b), find the response y[n].
(a)
(b)
(a) Solution :
y[1] 0.5h[1] 2h[1 1] 0.5h[1] 2h[0] 0.5 1 2 1 2.5 y[3] 0.5h[3] 2h[3 1] 0.5h[3] 2h[2] 0.5 0 2 1 2.5
Example
Consider an input x[n] and a unit impulse response h[n] given by
From the knowledge we learned in 1.4
x[n] [n n0 ] x[n0 ] [n n0 ]
We can get :
x[1], n -1 x[1] [n 1] n -1 0,
x[0], n 0 x[0] [n] n0 0,
Consider the response of a LTI system to an arbitrary input x[n], if we let h[n k ] denote the response of this system to the shifted unit impulse [n k ] ,then the response of the system to the input x[n] is:
Example
Consider x(t ) u(t ), how to repesentit in termsof impulse?
0 u (t ) 1
0, k n h[n k ] 1, k n
n , x[k ]h[n k ] 0,
0k n otherwise
2.2 Continuous-time LTI system: The convolution integral
2.2.2 The representation of continuous-time signals in terms of impulses
Writing this summation in a more compact form, we have
x[n]
k
x[k ] [n k ]
x[n]
k
x[k ] [n k ]

1 This equation corresponds to the representation of an arbitrary signal as a linear combination of shifted unit impulses [n k ] , where the weights in this linear combination are x[k] 2 This equation is called the sifting property of the discretetime unit impulse. Because the sequence [n k ] is nonzero only when n=k, the summation on the right-hand side “sift” through the sequence values x[k] and preserves only the value corresponding to k=n.
For an arbitrary continuous-time signal x(t), it can be represented as follow:
x(t ) x( ) (t )d


We refer to this equation as the sifting property of the continuous-time impulse.
Example
Consider x[n] u[n], how to repesentit in termsof impulse?
0 x[n] u[n] 1
x[n]
0 k k
n0 n0

x[k ] [n k ] 0 [n k ] 1 [n k ]
n , x[k ]h[n k ] 0,
0k n otherwise
y[n]
k n
x[k ]h[n k ]
for n 0

n 1 1 k 1 k 0
0, n 0 h[n] u[n] 1, n 0
n
0, n 0 h[n] u[n] 1, n 0
0, n 0 x[n] u[n] n , n 0
n
0, n 0 h[n] u[n] 1, n 0
0, n 0 h[n] u[n] 1, n 0
0, n 0 x[n] u[n] n , n 0

x[0]h[n] x[1]h[n 1] 0.5h[n] 2h[n 1]
0 .5
Time shifting

2
y[n] 0.5h[n] 2h[n 1]
y[n] 0.5h[n] 2h[n 1]
y[0] 0.5h[0] 2h[0 1] 0.5 1 2 0 0.5
y[n] x[n] h[n]
y[n]
k
x[k ]h[n k ]
Байду номын сангаас
Note that this equation expresses the response of an LTI system to an arbitrary input in terms of the system’s response to the unit impulse. From this, we see that an LTI system is completely characterized by its response to the unit impulse.