chapter 3 convolution representation
TRANSCRIPT
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Chapter 3Convolution Representation
Chapter 3Convolution Representation
• Consider the DT SISO system:
• If the input signal is and the system has no energy at , the output
is called the impulse responseimpulse response of the system
DT Unit-Impulse ResponseDT Unit-Impulse Response
[ ]y n[ ]x n
[ ]h n[ ]nδ
[ ] [ ]x n nδ=
[ ] [ ]y n h n=
System
System
0n =
2
• Consider the DT system described by
• Its impulse response can be found to be
ExampleExample
[ ] [ 1] [ ]y n ay n bx n+ − =
( ) , 0,1,2,[ ]
0, 1, 2, 3,
na b nh n
n− =⎧
= ⎨= − − −⎩
……
• Let x[n] be an arbitrary input signal to a DT LTI system
• Suppose that for • This signal can be represented as
Representing Signals in Terms ofShifted and Scaled Impulses
Representing Signals in Terms ofShifted and Scaled Impulses
0
[ ] [0] [ ] [1] [ 1] [2] [ 2]
[ ] [ ], 0,1,2,i
x n x n x n x n
x i n i n
δ δ δ
δ∞
=
= + − + − +
= − =∑ …
1, 2,n = − − …[ ] 0x n =
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Exploiting Time-Invariance and Linearity
Exploiting Time-Invariance and Linearity
0[ ] [ ] [ ], 0
iy n x i h n i n
∞
=
= − ≥∑
• This particular summation is called the convolution sumconvolution sum
• Equation is called the convolution representation of the systemconvolution representation of the system
• Remark: a DT LTI system is completely described by its impulse response h[n]
0[ ] [ ] [ ]
iy n x i h n i
∞
=
= −∑
The Convolution Sum The Convolution Sum
[ ] [ ]x n h n∗[ ] [ ] [ ]y n x n h n= ∗
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• Since the impulse response h[n] provides the complete description of a DT LTI system, we write
Block Diagram Representation of DT LTI Systems
Block Diagram Representation of DT LTI Systems
[ ]y n[ ]x n [ ]h n
• Suppose that we have two signals x[n] and v[n] that are not zero for negative times (noncausalnoncausal signalssignals)
• Then, their convolution is expressed by the two-sided series
[ ] [ ] [ ]i
y n x i v n i∞
=−∞
= −∑
The Convolution Sum for Noncausal Signals The Convolution Sum for Noncausal Signals
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Example: Convolution of Two Rectangular Pulses
Example: Convolution of Two Rectangular Pulses
• Suppose that both x[n] and v[n] are equal to the rectangular pulse p[n] (causal signal) depicted below
The Folded PulseThe Folded Pulse
• The signal is equal to the pulse p[i] folded about the vertical axis
[ ]v i−
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Sliding over Sliding over [ ]v n i−[ ]v n i− [ ]x i[ ]x i
Sliding over - Cont’d Sliding over - Cont’d [ ]x i[ ]x i[ ]v n i−[ ]v n i−
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Plot of Plot of [ ] [ ]x n v n∗[ ] [ ]x n v n∗
•• AssociativityAssociativity
•• CommutativityCommutativity
•• DistributivityDistributivity w.r.t. additionw.r.t. addition
Properties of the Convolution Sum Properties of the Convolution Sum
[ ] ( [ ] [ ]) ( [ ] [ ]) [ ]x n v n w n x n v n w n∗ ∗ = ∗ ∗
[ ] [ ] [ ] [ ]x n v n v n x n∗ = ∗
[ ] ( [ ] [ ]) [ ] [ ] [ ] [ ]x n v n w n x n v n x n w n∗ + = ∗ + ∗
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•• Shift property:Shift property: define
•• Convolution with the unit impulseConvolution with the unit impulse
•• Convolution with the shifted unit impulseConvolution with the shifted unit impulse
Properties of the Convolution Sum - Cont’d Properties of the Convolution Sum - Cont’d
[ ] [ ] [ ]w n x n v n= ∗[ ] [ ] [ ] [ ] [ ]q qw n q x n v n x n v n− = ∗ = ∗
[ ] [ ]qx n x n q= −[ ] [ ]qv n v n q= −
then
⎧⎪⎨⎪⎩
[ ] [ ] [ ]x n n x nδ∗ =
[ ] [ ] [ ]qx n n x n qδ∗ = −
Example: Computing Convolution with Matlab
Example: Computing Convolution with Matlab
• Consider the DT LTI system
• impulse response:
• input signal:
[ ]y n[ ]x n [ ]h n
[ ] sin(0.5 ), 0h n n n= ≥
[ ] sin(0.2 ), 0x n n n= ≥
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Example: Computing Convolution with Matlab – Cont’d
Example: Computing Convolution with Matlab – Cont’d
[ ] sin(0.5 ), 0h n n n= ≥
[ ] sin(0.2 ), 0x n n n= ≥
• Suppose we want to compute y[n] for
• Matlab code:
Example: Computing Convolution with Matlab – Cont’d
Example: Computing Convolution with Matlab – Cont’d
0,1, ,40n = …
n=0:40;
x=sin(0.2*n);
h=sin(0.5*n);
y=conv(x,h);
stem(n,y(1:length(n)))
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Example: Computing Convolution with Matlab – Cont’d
Example: Computing Convolution with Matlab – Cont’d
[ ] [ ] [ ]y n x n h n= ∗
• Consider the CT SISO system:
• If the input signal is and the system has no energy at , the output
is called the impulse responseimpulse response of the system
CT Unit-Impulse ResponseCT Unit-Impulse Response
( )h t( )tδ
( ) ( )x t tδ=
( ) ( )y t h t=
( )y t( )x t System
System
0t −=
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• Let x[n] be an arbitrary input signal withfor
• Using the sifting property sifting property of , we may write
• Exploiting timetime--invarianceinvariance, it is
Exploiting Time-Invariance Exploiting Time-Invariance
( ) 0, 0x t t= <( )tδ
0
( ) ( ) ( ) , 0x t x t d tτ δ τ τ−
∞
= − ≥∫
( )h t τ−( )tδ τ− System
Exploiting Time-Invariance Exploiting Time-Invariance
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• Exploiting linearitylinearity,, it is
• If the integrand does not contain an impulse located at , the lower limit of the integral can be taken to be 0,i.e.,
Exploiting LinearityExploiting Linearity
0
( ) ( ) ( ) , 0y t x h t d tτ τ τ−
∞
= − ≥∫( ) ( )x h tτ τ−
0τ =
0
( ) ( ) ( ) , 0y t x h t d tτ τ τ∞
= − ≥∫
• This particular integration is called the convolution integralconvolution integral
• Equation is called the convolution representation of the systemconvolution representation of the system
• Remark: a CT LTI system is completely described by its impulse response h(t)
The Convolution Integral The Convolution Integral
( ) ( )x t h t∗( ) ( ) ( )y t x t h t= ∗
0
( ) ( ) ( ) , 0y t x h t d tτ τ τ∞
= − ≥∫
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• Since the impulse response h(t) provides the complete description of a CT LTI system, we write
Block Diagram Representation of CT LTI Systems
Block Diagram Representation of CT LTI Systems
( )y t( )x t ( )h t
• Suppose that where p(t) is the rectangular pulse depicted in figure
Example: Analytical Computation of the Convolution Integral
Example: Analytical Computation of the Convolution Integral
( ) ( ) ( ),x t h t p t= =
( )p t
tT0
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• In order to compute the convolution integral
we have to consider four cases:
Example – Cont’dExample – Cont’d
0
( ) ( ) ( ) , 0y t x h t d tτ τ τ∞
= − ≥∫
Example – Cont’dExample – Cont’d
• Case 1: 0t ≤
( )x τ
τT0
( )h t τ−
t T− t
( ) 0y t =
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Example – Cont’dExample – Cont’d
• Case 2: 0 t T≤ ≤
( )x τ
τT0
( )h t τ−
t T− t
0
( )t
y t d tτ= =∫
Example – Cont’dExample – Cont’d
• Case 3:
( )x τ
τT0
( )h t τ−
t T− t
( ) ( ) 2T
t T
y t d T t T T tτ−
= = − − = −∫
0 2t T T T t T≤ − ≤ → ≤ ≤
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Example – Cont’dExample – Cont’d
• Case 4:
( ) 0y t =
( )x τ
τT0
( )h t τ−
t T− t
2T t T T t≤ − → ≤
Example – Cont’dExample – Cont’d
( ) ( ) ( )y t x t h t= ∗
T0 t2T
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•• AssociativityAssociativity
•• CommutativityCommutativity
•• DistributivityDistributivity w.r.t. additionw.r.t. addition
Properties of the Convolution IntegralProperties of the Convolution Integral
( ) ( ( ) ( )) ( ( ) ( )) ( )x t v t w t x t v t w t∗ ∗ = ∗ ∗
( ) ( ) ( ) ( )x t v t v t x t∗ = ∗
( ) ( ( ) ( )) ( ) ( ) ( ) ( )x t v t w t x t v t x t w t∗ + = ∗ + ∗
•• Shift property:Shift property: define
•• Convolution with the unit impulseConvolution with the unit impulse
•• Convolution with the shifted unit impulseConvolution with the shifted unit impulse
Properties of the Convolution Integral - Cont’d
Properties of the Convolution Integral - Cont’d
( ) ( ) ( )w t x t v t= ∗
( ) ( ) ( ) ( ) ( )q qw t q x t v t x t v t− = ∗ = ∗
( ) ( )qx t x t q= −( ) ( )qv t v t q= −
then
⎧⎪⎨⎪⎩
( ) ( ) ( )x t t x tδ∗ =
( ) ( ) ( )qx t t x t qδ∗ = −
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•• Derivative property: Derivative property: if the signal x(t) is differentiable, then it is
• If both x(t) and v(t) are differentiable, then it is also
Properties of the Convolution Integral - Cont’d
Properties of the Convolution Integral - Cont’d
[ ] ( )( ) ( ) ( )d dx tx t v t v tdt dt
∗ = ∗
[ ]2
2( ) ( )( ) ( )d dx t dv tx t v t
dt dt dt∗ = ∗
Properties of the Convolution Integral - Cont’d
Properties of the Convolution Integral - Cont’d
•• Integration property: Integration property: define
( 1)
( 1)
( ) ( )
( ) ( )
t
t
x t x d
v t v d
τ τ
τ τ
−
−∞
−
−∞
⎧⎪⎪⎨⎪⎪⎩
∫
∫
then ( 1) ( 1) ( 1)( ) ( ) ( ) ( ) ( ) ( )x v t x t v t x t v t− − −∗ = ∗ = ∗
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• Let g(t) be the response of a system with impulse response h(t) when with no initial energy at time , i.e.,
• Therefore, it is
Representation of a CT LTI System in Terms of the Unit-Step ResponseRepresentation of a CT LTI System in Terms of the Unit-Step Response
( )g t( )u t
( ) ( )x t u t=0t =
( ) ( ) ( )g t h t u t= ∗
( )h t
• Differentiating both sides
• Recalling that
it is
Representation of a CT LTI System in Terms of the Unit-Step Response
– Cont’d
Representation of a CT LTI System in Terms of the Unit-Step Response
– Cont’d
( ) ( ) ( )( ) ( )dg t dh t du tu t h tdt dt dt
= ∗ = ∗
( ) ( )du t tdt
δ= ( ) ( ) ( )h t h t tδ= ∗
( ) ( )dg t h tdt
=
and
0
( ) ( )t
g t h dτ τ= ∫or