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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 impulse responseresponse of the system
DT Unit-Impulse ResponseDT Unit-Impulse Response
[ ]y n[ ]x n
[ ]h n[ ]n
[ ] [ ]x n n
[ ] [ ]y n h n
System
System
0n
• 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
1, 2,n [ ] 0x n
Exploiting Time-Invariance and Linearity
Exploiting Time-Invariance and Linearity
0
[ ] [ ] [ ], 0i
y 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
[ ] [ ] [ ]i
y n x i h n i
The Convolution Sum The Convolution Sum
[ ] [ ]x n h n[ ] [ ] [ ]y n x n h n
• 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 (noncausal signalsnoncausal signals)
• 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
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
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
Plot of Plot of [ ] [ ]x n v n[ ] [ ]x n v n
• AssociativityAssociativity
• CommutativityCommutativity
• Distributivity w.r.t. additionDistributivity w.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
• 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 n
[ ] [ ] [ ]qx n n x n q
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
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)))
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 impulse responseresponse of the system
CT Unit-Impulse ResponseCT Unit-Impulse Response
( )h t( )t
( ) ( )x t t
( ) ( )y t h t
( )y t( )x t System
System
0t
• Let x[n] be an arbitrary input signal with for
• Using the sifting property sifting property of , we may write
• Exploiting time-invariancetime-invariance, it is
Exploiting Time-Invariance Exploiting Time-Invariance
( ) 0, 0x t t ( )t
0
( ) ( ) ( ) , 0x t x t d t
( )h t ( )t System
Exploiting Time-Invariance Exploiting Time-Invariance
• 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 t
( ) ( )x h t
0
0
( ) ( ) ( ) , 0y t x h t d t
• 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 t
• 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
• 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 t
Example – Cont’dExample – Cont’d
• Case 1: 0t
( )x
T0
( )h t
t T t
( ) 0y t
Example – Cont’dExample – Cont’d
• Case 2: 0 t T
( )x
T0
( )h t
t T t
0
( )t
y t d t
Example – Cont’dExample – Cont’d
• Case 3:
( )x
T0
( )h t
t T t
( ) ( ) 2T
t T
y t d T t T T t
0 2t T T T t T
Example – Cont’dExample – Cont’d
• Case 4:
( ) 0y t
( )x
T0
( )h t
t T t
2T t T T t
Example – Cont’dExample – Cont’d
( ) ( ) ( )y t x t h t
T0 t2T
• AssociativityAssociativity
• CommutativityCommutativity
• Distributivity w.r.t. additionDistributivity w.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 t
( ) ( ) ( )qx t t x t q
• 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 t
dt 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
then ( 1) ( 1) ( 1)( ) ( ) ( ) ( ) ( ) ( )x v t x t v t x t v t
• 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 Response
Representation of a CT LTI System in Terms of the Unit-Step Response
( )g t( )u t
( ) ( )x t u t0t
( ) ( ) ( )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 t
dt dt dt
( )( )
du tt
dt ( ) ( ) ( )h t h t t
( )( )
dg th t
dt
and
0
( ) ( )t
g t h d or
