中山大学 信号 Chapter+2+LTI+Systems

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Denote
the response of the linear system to the shifted unit impulse [n k ] Consider an input:
hk [n]
x[n]
The response of the above system to this input can be expressed as
u(t ) u( ) (t )d (t )d
0


As for an arbitrary signal x(t)



x( ) (t )d x(t ) (t )d


x(t ) (t )d x(t )
can be expressed as
x(t ) x( ) (t )d


Sifting Property
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It is easy to represent unit step function in terms of integral of impulse function.
y[n] x[n] h[n]
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Example
Consider an LTI system with impulse response h[n] and input x[n], as illustrated in below. What will the response of this system be with input x[n]? What does h[n] mean?
[n] h[n] [n 1] h[n 1]

What does x[n] tell us?
x[n] x[0] [n 0] x[1] [n 1] 0.5 [n 0] 2 [n 1]
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y[n] x[0]h[n 0] x[1]h[n 1] 0.5h[n] 2h[n 1]
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2.2 Continuous-Time LTI Systems: The Convolution Integral
• Think of the unit impulse as the idealization of a pulse which is so short that its duration is inconsequential for any real, physical system. • A representation for arbitrary continuous-time signals in terms of these idealized pulse with vanishingly small duration.
The Representation of ContinuousTime Signals in Terms of Impulse
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Represent x(t) as a series of weighted rectangular pulse
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x(t ) lim x(k) (t k)
0 k
x(k )


(t k )
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As Δ approach 0
k x(k) x( ) (t k) (t ) d
So, x(t ) lim x(k)
0 k

(t k)
y[n]
k
x[k ]h[n k ]

THIS IS REFERRED TO AS THE CONVOLUTION SUM OR SUPERPOSITION SUM. （卷积和）
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y[n]
k
x[k ]h[n k ]

Always be represented symbolically as
• Causal LTI systems described by differential and difference equations
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2.1 Discrete-Time LTI Systems: The Convolution Sum
• The representation of discrete-time signals in terms of impulse
Chapter II Linear Time-Invariant Systems
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• Discrete-Time LTI Systems
– The representation of discrete-time signals in terms of impulse – The discrete-time unit impulse response and the convolution sum representation of LTI systems
– the responses of a time-invariant system to the time-shifted unit impulses are simply time-shifted versions of one another.
• The convolution-sum（卷积和） representation for a LTI discrete-time system results from putting the above two basic facts together.
k
x[k ] [n k ]


y[n]
k
x[k ]h [n]
k
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Expla
x[k ]h [n]
k

8
Continue
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Consider the effect of the time invariance property
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The Discrete-Time unit Impulse Response and the Convolution-sum Representation of LTI Systems
• The sifting property （sift v.筛选）
– it represents x[n] as a superposition （叠加） of scaled versions of a very simple of elementary functions, shifted unit impulses.
ˆ x(k)h

k
(t )
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As Δapproaches 0,
ˆ (t ) ˆ (t ) lim x(k)h y(t ) lim y k
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• How do you accurately represent an arbitrary signal by a series ‘simple’ signals?
– Simple??? – ‘Small’ duration – Easy to deal with
accurate easy
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hk [n] h0 [n k ]
DO YOU REMEMBER THE TIME INVARIANCE PROPERTY INTRODUCED IN CHAPTER ONE?
e(t )
r (t )
e(t t0 )
r (t t0 )
What would happen if the system is a Linear Time Invariant one?
– According to the sampling weigh property of impulse signal, discrete-time signals can be represented in terms of t locatio n – Regard a discrete-time signal as a sequence of individual impulses.
• Continuous-Time LTI Systems
– The representation of continuous-time signals in terms of impulse – The continuous-Time unit impulse response and the convolution integral representation
10
Denote the input to a LTI system as
x[n]
k
x[k ] [n k ]

Define the unit impulse response
h[n] h0 [n]
The response of this LTI system to the input can be expressed as
0 k
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Define
1 0t (t ) 0 others (t ) 1
Then
ˆ (t ) x
k
x(k)



(t k)
The approximation becomes better and better as Δ approach 0, and in the limit equals x(t).
• The response of a linear system to x[n]
– the superposition of the scaled responses of the system to each of these shifted impulses.
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• From the property of time invariance
k
x[n]
x[k ] [n k ]
Do you remember the property of unit impulse ? x[n] [n k ] x[k ] [n k ]
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Any discrete-time signal can be represented as the sum of several weighted impulse.
x(t ) lim x(k) (t k)
0 k
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From another viewpoint …
x(t ) lim
0 k
x(k ) u t k u t k

u t k u t k lim x(k ) 0 k lim
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x(t ) lim x(k) (t k)
0 k

Define
ˆ (t ) as the response of an LTI system to the input (t k) h k
Then, we can see
ˆ (t ) y
k
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The Continuous-Time Unit Impulse Response and the Convolution Integral Representation of LTI System
• An arbitrary continuous-time signal can be viewed as the superposition of scaled and shifted pulses. • The response of a linear system to this signal will be the superposition of the response to the scaled and shifted versions of pulses.