Chapter 3 Chapter 3 Convolution Representation Convolution - PDF document

Chapter 3 Chapter 3 Convolution Representation Convolution Representation CT Unit-Impulse Response CT Unit-Impulse Response • Consider the CT SISO system: y t ( ) x t ( ) System = δ • If the input signal is and the x t ( ) ( ) t − = system has no energy at , the output t 0 = y t ( ) h t ( ) is called the impulse response impulse response of the system δ ( ) t h t ( ) System

Exploiting Time-Invariance Exploiting Time-Invariance • Let x ( t) be an arbitrary input signal with = < x t ( ) 0, t 0 for δ ( ) t • Using the sifting property sifting property of , we may write ∞ ∫ = τ δ − τ τ ≥ x t ( ) x ( ) ( t ) d , t 0 − 0 • Exploiting time time- -invariance invariance, it is δ − τ − τ ( t ) h t ( ) System Exploiting Time-Invariance Exploiting Time-Invariance

Exploiting Linearity Exploiting Linearity • Exploiting linearity linearity, , it is ∞ ∫ = τ − τ τ ≥ y t ( ) x ( ) ( h t ) d , t 0 − 0 τ − τ x ( ) ( h t ) • If the integrand does not contain τ = 0 an impulse located at , the lower limit of the integral can be taken to be 0,i.e., ∞ ∫ = τ − τ τ ≥ y t ( ) x ( ) ( h t ) d , t 0 0 The Convolution Integral The Convolution Integral • This particular integration is called the convolution integral convolution integral ∞ ∫ = τ − τ τ ≥ y t ( ) x ( ) ( h t ) d , t 0 −∞ ��� ��� � ∗ x t ( ) h t ( ) = ∗ • Equation is called the y t ( ) x t ( ) h t ( ) convolution representation of the system convolution representation of the system • Remark: a CT LTI system is completely described by its impulse response h ( t )

Block Diagram Representation Block Diagram Representation of CT LTI Systems of CT LTI Systems • Since the impulse response h (t) provides the complete description of a CT LTI system, we write y t ( ) x t ( ) h t ( ) Example: Analytical Computation of Example: Analytical Computation of the Convolution Integral the Convolution Integral = = x t ( ) h t ( ) p t ( ), • Suppose that where p ( t ) is the rectangular pulse depicted in figure p t ( ) 0 T t

Example – Cont’d Example – Cont’d • In order to compute the convolution integral ∞ ∫ = τ − τ τ ≥ y t ( ) x ( ) ( h t ) d , t 0 −∞ we have to consider four cases: Example – Cont’d Example – Cont’d t ≤ • Case 1: 0 − τ x τ h t ( ) ( ) − τ t 0 T t T y t = ( ) 0

Example – Cont’d Example – Cont’d ≤ ≤ • Case 2: 0 t T − τ x τ h t ( ) ( ) − τ 0 t t T T t ∫ = τ = y t ( ) d t 0 Example – Cont’d Example – Cont’d ≤ − ≤ → ≤ ≤ 0 t T T T t 2 T • Case 3: x τ − τ ( ) h t ( ) − τ 0 t t T T T ∫ = τ = − − = − y t ( ) d T ( t T ) 2 T t − t T

Example – Cont’d Example – Cont’d ≤ − → ≤ • Case 4: T t T 2 T t x τ − τ ( ) h t ( ) τ − t 0 T t T y t = ( ) 0 Example – Cont’d Example – Cont’d = ∗ y t ( ) x t ( ) h t ( ) t 0 T 2 T

Properties of the Convolution Integral Properties of the Convolution Integral • Associativity Associativity • ∗ ∗ = ∗ ∗ x t ( ) ( ( ) v t w t ( )) ( ( ) x t v t ( )) w t ( ) • Commutativity Commutativity • ∗ = ∗ x t ( ) v t ( ) v t ( ) x t ( ) • Distributivity Distributivity w.r.t. addition w.r.t. addition • ∗ + = ∗ + ∗ x t ( ) ( ( ) v t w t ( )) x t ( ) v t ( ) x t ( ) w t ( ) Properties of the Properties of the Convolution Integral - Cont’d Convolution Integral - Cont’d = − ⎧ x t ( ) x t ( q ) q ⎪ • • Shift property: Shift property: define = − v t ( ) v t ( q ) ⎨ q ⎪ = ∗ w t ( ) x t ( ) v t ( ) ⎩ then − = ∗ = ∗ w t ( q ) x t ( ) v t ( ) x t ( ) v t ( ) q q • Convolution with the unit impulse Convolution with the unit impulse • ∗ δ = x t ( ) ( ) t x t ( ) • Convolution with the shifted unit impulse Convolution with the shifted unit impulse • ∗ δ = − x t ( ) ( ) t x t ( q ) q

Properties of the Properties of the Convolution Integral - Cont’d Convolution Integral - Cont’d • Derivative property: Derivative property: if the signal x ( t ) is • differentiable, then it is d dx t ( ) [ ] ∗ = ∗ x t ( ) v t ( ) v t ( ) dt dt • If both x ( t ) and v ( t ) are differentiable, then it is also 2 d [ ] dx t ( ) dv t ( ) ∗ = ∗ x t ( ) v t ( ) 2 dt dt dt Properties of the Properties of the Convolution Integral - Cont’d Convolution Integral - Cont’d • Integration property: Integration property: define • ⎧ t ∫ − τ τ ( 1) � ⎪ x ( ) t x ( ) d ⎪ −∞ ⎨ t ⎪ ∫ − τ τ � ( 1) v ( ) t v ( ) d ⎪ ⎩ −∞ then − − − ∗ = ∗ = ∗ ( 1) ( 1) ( 1) ( x v ) ( ) t x ( ) t v t ( ) x t ( ) v ( ) t

Representation of a CT LTI System Representation of a CT LTI System in Terms of the Unit-Step Response in Terms of the Unit-Step Response • Let g ( t ) be the response of a system with = impulse response h ( t ) when x t ( ) u t ( ) with t = 0 no initial energy at time , i.e., u t ( ) g t ( ) h t ( ) • Therefore, it is = ∗ g t ( ) h t ( ) u t ( ) Representation of a CT LTI System Representation of a CT LTI System in Terms of the Unit-Step Response in Terms of the Unit-Step Response – Cont’d – Cont’d • Differentiating both sides dg t ( ) dh t ( ) du t ( ) = ∗ = ∗ u t ( ) h t ( ) dt dt dt • Recalling that du t ( ) = δ = ∗ δ ( ) t h t ( ) h t ( ) ( ) t and dt it is t = ∫ dg t ( ) = τ τ h t ( ) g t ( ) h ( ) d or dt 0