Chapter 3 Chapter 3 Convolution Representation Convolution - PDF document

Chapter 3 Chapter 3 Convolution Representation Convolution Representation DT Unit-Impulse Response DT Unit-Impulse Response • Consider the DT SISO system: [ ] y n [ ] x n System = δ • If the input signal is and the x n [ ] [ ] n n = system has no energy at , the output 0 = is called the impulse response impulse response of y n [ ] h n [ ] the system δ [ ] n h n [ ] System 1

Example Example • Consider the DT system described by + − = y n [ ] ay n [ 1] bx n [ ] • Its impulse response can be found to be ⎧ − = … n ( a b ) , n 0,1,2, = ⎨ h n [ ] = − − − … ⎩ 0, n 1, 2, 3, Representing Signals in Terms of Representing Signals in Terms of Shifted and Scaled Impulses Shifted and Scaled Impulses • Let x [ n ] be an arbitrary input signal to a DT LTI system x n = n = − − … • Suppose that for [ ] 0 1, 2, • This signal can be represented as = δ + δ − + δ − + � x n [ ] x [0] [ ] n x [1] [ n 1] x [2] [ n 2] ∞ ∑ = δ − = … x i [ ] [ n i ], n 0,1,2, = i 0 2

Exploiting Time-Invariance Exploiting Time-Invariance and Linearity and Linearity ∞ ∑ = − ≥ y n [ ] x i h n [ ] [ i ], n 0 = 0 i The Convolution Sum The Convolution Sum • This particular summation is called the convolution sum convolution sum ∞ ∑ = − y n [ ] x i h n [ ] [ i ] �� = � ��� � i 0 ∗ x n [ ] h n [ ] = ∗ • Equation is called the y n [ ] x n [ ] h n [ ] convolution representation of the system convolution representation of the system • Remark: a DT LTI system is completely described by its impulse response h [ n ] 3

Block Diagram Representation Block Diagram Representation of DT LTI Systems of DT LTI Systems • Since the impulse response h [ n ] provides the complete description of a DT LTI system, we write y n [ ] x n [ ] h n [ ] The Convolution Sum The Convolution Sum for Noncausal Signals for Noncausal Signals • Suppose that we have two signals x [ n ] and v [ n ] that are not zero for negative times (noncausal noncausal signals signals) • Then, their convolution is expressed by the two-sided series ∞ ∑ = − y n [ ] x i v n [ ] [ i ] =−∞ i 4

Example: Convolution of Two Example: Convolution of Two Rectangular Pulses Rectangular Pulses • Suppose that both x [ n ] and v [ n ] are equal to the rectangular pulse p [ n ] (causal signal) depicted below The Folded Pulse The Folded Pulse − [ ] • The signal is equal to the pulse p [ i ] v i folded about the vertical axis 5

− − Sliding over v n [ i ] x i [ ] Sliding over v n [ i ] x i [ ] − − Sliding over - Cont’d v n [ i ] x i [ ] Sliding over - Cont’d v n [ i ] x i [ ] 6

∗ ∗ x n [ ] v n [ ] Plot of [ ] x n v n [ ] Plot of Properties of the Convolution Sum Properties of the Convolution Sum • Associativity Associativity • ∗ ∗ = ∗ ∗ x n [ ] ( [ ] v n w n [ ]) ( [ ] x n v n [ ]) w n [ ] • Commutativity • Commutativity ∗ = ∗ x n [ ] v n [ ] v n [ ] x n [ ] • Distributivity Distributivity w.r.t. addition w.r.t. addition • ∗ + = ∗ + ∗ x n [ ] ( [ ] v n w n [ ]) x n [ ] v n [ ] x n [ ] w n [ ] 7

Properties of the Convolution Sum - Cont’d Properties of the Convolution Sum - Cont’d = − ⎧ x n [ ] x n [ q ] q ⎪ • Shift property: Shift property: define • = − v n [ ] v n [ q ] ⎨ q ⎪ = ∗ w n [ ] x n [ ] v n [ ] then ⎩ − = ∗ = ∗ w n [ q ] x n [ ] v n [ ] x n [ ] v n [ ] q q • Convolution with the unit impulse • Convolution with the unit impulse ∗ δ = x n [ ] [ ] n x n [ ] • Convolution with the shifted unit impulse Convolution with the shifted unit impulse • ∗ δ = − [ ] [ ] [ ] x n n x n q q Example: Computing Convolution Example: Computing Convolution with Matlab with Matlab • Consider the DT LTI system [ ] y n [ ] x n h n [ ] • impulse response: = ≥ h n [ ] sin(0.5 ), n n 0 • input signal: = ≥ x n [ ] sin(0.2 ), n n 0 8

Example: Computing Convolution Example: Computing Convolution with Matlab – Cont’d with Matlab – Cont’d = ≥ h n [ ] sin(0.5 ), n n 0 = ≥ x n [ ] sin(0.2 ), n n 0 Example: Computing Convolution Example: Computing Convolution with Matlab – Cont’d with Matlab – Cont’d • Suppose we want to compute y [ n ] for n = … 0,1, ,40 • Matlab code: n=0:40; x=sin(0.2*n); h=sin(0.5*n); y=conv(x,h); stem(n,y(1:length(n))) 9

Example: Computing Convolution Example: Computing Convolution with Matlab – Cont’d with Matlab – Cont’d = ∗ y n [ ] x n [ ] h n [ ] 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 = is called the impulse response impulse response of y t ( ) h t ( ) the system δ ( ) t h t ( ) System 10

Exploiting Time-Invariance Exploiting Time-Invariance • Let x [ n ] be an arbitrary input signal with = < x t ( ) 0, for t 0 δ • Using the sifting property sifting property of , we may ( ) t 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 11

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 = an impulse located at , the lower limit of 0 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 ��� ��� � 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 ) 12

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 = = • Suppose that where p ( t ) x t ( ) h t ( ) p t ( ), is the rectangular pulse depicted in figure p t ( ) 0 T t 13

Example – Cont’d Example – Cont’d • In order to compute the convolution integral ∞ ∫ = τ − τ τ ≥ y t ( ) x ( ) ( h t ) d , t 0 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 14

Example – Cont’d Example – Cont’d ≤ ≤ • Case 2: 0 t T − τ x τ ( ) h t ( ) − τ t 0 t T T t ∫ = τ = y t ( ) d t 0 Example – Cont’d Example – Cont’d ≤ − ≤ → ≤ ≤ • Case 3: 0 t T T T t 2 T x τ − τ ( ) h t ( ) − τ 0 t t T T T ∫ = τ = − − = − ( ) ( ) 2 y t d T t T T t − t T 15

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 16

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 17