Chapter 3 Chapter 3 Convolution Representation Convolution - - PDF document

Chapter 3 Convolution Representation Chapter 3 Convolution Representation

• Consider the CT SISO system:
• If the input signal is and the

system has no energy at , the output is called the impulse response impulse response of the system CT Unit-Impulse Response CT Unit-Impulse Response

( ) h t ( ) t δ ( ) ( ) x t t δ = ( ) ( ) y t h t = ( ) y t ( ) x t

System System

t

−

=

• Let x(t) be an arbitrary input signal with

for

• Using the sifting property

sifting property of , we may write

• Exploiting time

time-

• invariance

invariance, it is Exploiting Time-Invariance Exploiting Time-Invariance

( ) 0, x t t = < ( ) t δ ( ) ( ) ( ) , x t x t d t τ δ τ τ

−

∞

= − ≥

∫

( ) h t τ − ( ) t δ τ −

System

Exploiting Time-Invariance Exploiting Time-Invariance

• Exploiting linearity

linearity, , 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 Linearity Exploiting Linearity

( ) ( ) ( ) , y t x h t d t τ τ τ

−

∞

= − ≥

∫

( ) ( ) x h t τ τ − τ = ( ) ( ) ( ) , y t x h t d t τ τ τ

∞

= − ≥

∫

• This particular integration is called the

convolution integral convolution integral

• Equation is called the

convolution representation of the system convolution 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 = ∗ ( ) ( ) ( ) , y t x h t d t τ τ τ

∞ −∞

= − ≥

∫

• Since the impulse response h(t) provides the

complete description of a CT LTI system, we write Block Diagram Representation

• f CT LTI Systems

Block Diagram Representation

• f 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 t T

• In order to compute the convolution integral

we have to consider four cases: Example – Cont’d Example – Cont’d

( ) ( ) ( ) , y t x h t d t τ τ τ

∞ −∞

= − ≥

∫

Example – Cont’d Example – Cont’d

• Case 1:

t ≤ ( ) x τ τ T ( ) h t τ − t T − t ( ) y t =

Example – Cont’d Example – Cont’d

• Case 2: 0

t T ≤ ≤ ( ) x τ τ T ( ) h t τ − t T − t ( )

t

y t d t τ = =

∫

Example – Cont’d Example – Cont’d

• Case 3:

( ) x τ τ T ( ) h t τ − t T − t ( ) ( ) 2

T t T

y t d T t T T t τ

−

= = − − = −

∫

2 t T T T t T ≤ − ≤ → ≤ ≤

Example – Cont’d Example – Cont’d

• Case 4:

( ) y t = ( ) x τ τ T ( ) h t τ − t T − t 2 T t T T t ≤ − → ≤

Example – Cont’d Example – Cont’d

( ) ( ) ( ) y t x t h t = ∗ T t 2T

• Associativity

Associativity

• Commutativity

Commutativity

• Distributivity

Distributivity w.r.t. addition w.r.t. addition Properties of the Convolution Integral Properties 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 impulse

Convolution with the unit impulse

• Convolution with the shifted unit impulse

Convolution 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 q

w t q x t v t x t v t − = ∗ = ∗ ( ) ( )

q

x t x t q = − ( ) ( )

q

v t v t q = −

then

⎧ ⎪ ⎨ ⎪ ⎩ ( ) ( ) ( ) x t t x t δ ∗ = ( ) ( ) ( )

q

x t t x t q δ ∗ = −

• 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 t x t v t v t dt dt ∗ = ∗

[ ]

2 2

( ) ( ) ( ) ( ) d dx t dv t x 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

− − −

∗ = ∗ = ∗

• 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 t = t = ( ) ( ) ( ) 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 t u t h t dt dt dt = ∗ = ∗ ( ) ( ) du t t dt δ = ( ) ( ) ( ) h t h t t δ = ∗ ( ) ( ) dg t h t dt =

and

( ) ( )

t

g t h d τ τ = ∫

• r