signals and systems lecture 4
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Signals and Systems Lecture 4. Representation of signals Convolution Examples. Chapter 2 LTI Systems. 1. 1. L. L. 0 2 t. 0 1 2 t. 1. 1. - PowerPoint PPT PresentationTRANSCRIPT
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2006 Fall
Signals and SystemsSignals and SystemsLecture 4Lecture 4
Representation of signalsConvolutionExamples
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2006 Fall
Chapter 2 LTI Systems
Example 1 an LTI system
0 2 t
tf11
0 1 2 t
ty1
1L
211 tftf 211 tyty
0 2 4 t
1
-10 2 4 t
tf2
1 L
ty2
1
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2006 Fall
Chapter 2 LTI Systems
§2.1 Discrete-time LTI Systems : The Convolution Sum(卷积和)
§2.1.1 The Representation of Discrete-Time Signalsin Terms of impulses
11011 nxnxnxnx
knkxnxk
knkx
Example 2
1 0 1 2
1
2
3 nx
n
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2006 Fall
Representing DT Signals with Sums of Unit Samples
Property of Unit Sample
Examples][][][][ 000 nnnxnnnx
10
1]1[]1[]1[
n
nxnx
00
0]0[][]0[
n
nxnx
10
1]1[]1[]1[
n
nxnx
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2006 Fall
Written Analytically
Coefficients Basic Signals
]1[]1[][]0[]1[]1[]2[]2[
][][
nxnxnxnx
inxnxi
k
knkxnx ][][][
Note the Sifting Property of the Unit Sample
Important to note
the “-” sign
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2006 Fall
Chapter 2 LTI Systems
§2.1.2 The Discrete-Time Unit Impulse Responses and the Convolution-Sum Representation of LTI Systems
1. The Unit Impulse Responses单位冲激响应
0 ,h n L n
2. Convolution-Sum (卷积和)
knhkxnyk
系统在 n时刻的输出包含所有时刻输入脉冲的影响
k时刻的脉冲在 n时刻的响应
nhnx
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2006 Fall
Derivation of Superposition SumDerivation of Superposition Sum
Now suppose the system is LTI, and define the unit sample response h[n]:
– From Time-Invariance:
– From Linearity:
][][ nhn
][][ knhkn
][*][][][][][][][ nhnxknhkxnyknkxnxkk
convolution sum
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2006 Fall
The Superposition Sum for DT SystemsThe Superposition Sum for DT Systems
Graphic View of Superposition Sum
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2006 Fall
Hence a Very Important Property of LTI Systems
The output of any DT LTI System is a convolution of the input signal with the unit-sample response, i.e.
As a result, any DT LTI Systems are completely characterized by its unit sample response
k
knhkx
nhnxnyLTIDTAny
][][
][*][][
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2006 Fall
Calculation of Convolution SumCalculation of Convolution SumChoose the value of n and consider it fixed
k
knhkxny ][][][
View as functions of k with n fixedFrom x[n] and h[n] to x[k] and h[n-k]Note, h[n-k]–k is the mirror image of h[n]–n with the origin shifted to n
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2006 Fall
Calculating Successive Values: Shift, Multiply,
Sum
y[n] = 0 for n <y[-1] =y[0] =y[1] =y[2] =y[3] =y[4] =y[n] = 0 for n >
k
knhkxny ][][][-1
1
2
-2
-3
1
1
4
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2006 Fall
Calculation of Convolution SumCalculation of Convolution SumUse of Analytical FormUse of Analytical Form
Suppose that and
then
][][ nuanx n ][][ nubnh n][*][][ nhnxny
k
knhkx ][][
k
knk knubkua ][][
n
k
knkba0
ba
ba
nuab
abnuna
nn
n
][
][)1(11
knknu
kku
][
,0][
00
,0
时为
n
n
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2006 Fall
Calculation of Convolution SumCalculation of Convolution SumUse of Array MethodUse of Array Method
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2006 Fall
Calculation of Convolution SumCalculation of Convolution SumExample of Array MethodExample of Array Method
If x[n]<0 for n<-1,x[-1]=1,x[0]=2,x[1]=3, x[4]=5,…,and h[n]<0 for n<-2,h[-2]=-1, h[-1]=5,h[0]=3,h[1]=-2,h[2]=1,….In this case ,N=-1,M=-2,and the array is as follows
So,y[-3]=-1, y[-2]=3, y[-1]=10, y[0]=15, y[1]=21,…, and y[n]=0 for n<-3
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2006 Fall
Calculation of Convolution SumCalculation of Convolution SumUsing MatlabUsing Matlab
The convolution of two discrete-time signals can be carried out using the Matlab M-file conv.
Example:– p=[0 ones(1,10) zeros(1,5)]; – x=p; h=p;– y=conv(x,h);– n=-1:14;– subplot(2,1,1),Stem(n, x(1:length(n)))– n=-2:24;– subplot(2,1,2),Stem(n, y(1:length(n)))
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2006 Fall
Calculation of Convolution SumCalculation of Convolution SumUsing Matlab ResultUsing Matlab Result
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2006 Fall
ConclusionConclusionAny DT LTI Systems are completely
characterized by its unit sample response.
Calculation of convolution sum:– Step1:plot x and h vs k, since the convolution sum is
on k;– Step2:Flip h[k] around vertical axis to obtain h[-k];– Step3:Shift h[-k] by n to obtain h[n-k] ;– Step4:Multiply to obtain x[k]h[n-k];– Step5:Sum on k to compute – Step6:Index n and repeat step 3 to 6.
][*][][ nhnxny
k
knhkx ][][
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2006 Fall
ConclusionConclusion
Calculation Methods of Convolution Sum– Using graphical representations;– Compute analytically;– Using an array;– Using Matlab.
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2006 Fall
Chapter 2 LTI Systems
§2.2 Continuous-Time LTI Systems : The Convolution Integral (卷积积分)
§2.2.1 The Representation of Continuous-Time Signalsin Terms of impulses
dtxtx
——Sifting Property
§2.2.2 The Continuous-Time Unit Impulse Response and the Convolution Integral Representation of LTI Systems
y t x t h t x h t d
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2006 Fall
Representation of CT Signals
Approximate any input x(t) as a sum of shifted, scaled pulses (in fact, that is how we do integration)
tkttktkxtx )1(),()(
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2006 Fall
Representation of CT Signals (cont.)
areaunitahast)(
)()( ktkx
k
ktkxtx )()()(
dtxtx )()()(
↓limit as →0
⇓
Sifting
property
of the unit
impulse
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2006 Fall
Response of a CT LTI System
Now suppose the system is LTI, and define the unit impulse response h(t):
(t) →h(t)– From Time-Invariance:
(t −) →h(t −)– From Linearity:
)(*)()()()(
)()()(
thtxdthxty
dtxtx
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2006 Fall
Superposition Integral for CT SystemsSuperposition Integral for CT SystemsGraphic View of Staircase Approximation
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2006 Fall
CT Convolution MechanicsCT Convolution MechanicsTo compute superposition integral
– Step1:plot x and h vs , since the convolution integral is on ;
– Step2:Flip h( around vertical axis to obtain h(-;– Step3:Shift h(-) by n to obtain h(n-) ;– Step4:Multiply to obtain x(h(n-;– Step5:Integral on to compute – Step6:Increase t and repeat step 3 to 6.
dthxthtxty )()()(*)()(
dthx )()(
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2006 Fall
Basic Properties of Convolution
Commutativity: x(t)∗h(t) h(t) ∗x(t)Distributivity :
Associativity:
x(t)∗(t −to ) x(t −to ) (Sifting property: x(t) ∗(t) x(t))
An integrator:
)(*)()(*)()]()([*)( 2121 thtxthtxththtx
)(*)](*)([)](*)([*)( 2121 ththtxththtx
t
dxtutx )()(*)(
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2006 Fall
Convolution with Singularity FunctionsConvolution with Singularity Functions
)()(*)( tfttf
)()(*)( tfttf
t
dftutf )()(*)(
)()(*)( )()( tfttf kk
)()(*)( )()( tfttf kk
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2006 Fall
More about Response of LTI SystemsMore about Response of LTI SystemsHow to get h(t) or h[n]:
– By experiment;– May be computable from some known
mathematical representation of the given system.
Step response:
t
dhts )()(
n
k
khns ][][
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2006 Fall
SummarySummaryWhat we have learned ?
– The representation of DT and CT signals;– Convolution sum and convolution integral
Definition; Mechanics;
– Basic properties of convolution;What was the most important point in the lecture?What was the muddiest point?What would you like to hear more about?
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2006 Fall
ReadlistReadlist
Signals and Systems:– 2.3,2.4– P103~126
Question: The solution of LCCDE
(Linear Constant Coefficient Differential or Difference Equations)
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2006 Fall
Problem SetProblem Set
2.21(a),(c),(d)2.22(a),(b),(c)