chapter 3 convolution representation

1

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 =

3

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= ∗

4

• 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

5

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−

6

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−

7

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∗ + = ∗ + ∗

8

•• 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= ≥

9

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:



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)))

10



[ ] [ ] [ ]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 −=

11

• 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

12

• 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∞

= − ≥∫

13

• 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

14

• 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∞

= − ≥∫


• Case 1: 0t ≤

( )x τ

τT0

( )h t τ−

t T− t

( ) 0y t =

15


• Case 2: 0 t T≤ ≤

( )x τ

τT0

( )h t τ−

t T− t

0

( )t

y t d tτ= =∫


• 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≤ − ≤ → ≤ ≤

16


• Case 4:

( ) 0y t =

( )x τ

τT0

( )h t τ−

t T− t

2T t T T t≤ − → ≤


( ) ( ) ( )y t x t h t= ∗

T0 t2T

17

•• 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


( ) ( ) ( )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∗ = −

18

•• 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



[ ] ( )( ) ( ) ( )d dx tx t v t v tdt dt

∗ = ∗

[ ]2

2( ) ( )( ) ( )d dx t dv tx t v t

dt dt dt∗ = ∗



•• 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− − −∗ = ∗ = ∗

19

• 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

chapter 3 convolution representation

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