Signal and Systems Chapter 2: LTI Systems Representation of DT - PowerPoint PPT Presentation

Signal and Systems Chapter 2: LTI Systems Representation of DT signals in terms of shifted unit samples 1) System properties and examples Convolution sum representation of DT LTI systems 2) Examples 3) The unit sample response and properties of DT LTI systems 4) Representation of CT Signals in terms of shifted unit impulses 5) Convolution integral representation of CT LTI systems 6) Properties and Examples 7) The unit impulse as an idealized pulse that is “ short enough ” : 8) The operational definition of δ(t)

Book Chapter#: Section# Exploiting Superposition and Time- Invariance 𝑀𝑗𝑜𝑓𝑏𝑠𝑇𝑧𝑡𝑢𝑓𝑛 𝑧[𝑜] = 𝑙 ]  𝑦[𝑜] = 𝑏 𝑙 𝑦 𝑙 [𝑜] 𝑏 𝑙 𝑧 𝑙 [𝑜 𝑙  Question: Are there sets of “ basic ” signals so that:  We can represent rich classes of signals as linear combinations of these building block signals.  The response of LTI Systems to these basic signals are both simple and insightful.  Fact: For LTI Systems (CT or DT) there are two natural choices for these building blocks  Focus for now: DT Shifted unit samples CT Shifted unit impulses Computer Engineering Department, Signal and Systems 2

Book Chapter#: Section# Representation of DT Signals Using Unit Samples Computer Engineering Department, Signal and Systems 3

Book Chapter#: Section#  That is … 𝑦[𝑜] =. . . +𝑦[−2]𝜀[𝑜 + 2] + 𝑦[−1]𝜀[𝑜 + 1] + 𝑦[0]𝜀[𝑜] + 𝑦[1]𝜀[𝑜 − 1]+. . . ∞ ] => 𝑦[𝑜] = 𝑦[𝑙]𝜀[𝑜 − 𝑙 𝑙=−∞ Coefficients Basic Signals The Shifting Property of the Unit Sample Computer Engineering Department, Signal and Systems 4

Book Chapter#: Section#  Suppose the system is linear, and define ℎ 𝑙 [𝑜 as the ] response to 𝜀[𝑜 − 𝑙 : ] ] 𝜀[𝑜 − 𝑙] → ℎ 𝑙 [𝑜  From superposition: ∞ ∞ ] 𝑦[𝑜] = 𝑦[𝑙]𝜀[𝑜 − 𝑙] → 𝑧[𝑜] = 𝑦[𝑙]ℎ 𝑙 [𝑜 𝑙→−∞ 𝑙→−∞ Computer Engineering Department, Signal and Systems 5

Book Chapter#: Section#  Now suppose the system is LTI, and define the unit sample response ℎ[𝑜 : ] ] 𝜀[𝑜] → ℎ[𝑜 From TI: ] 𝜀[𝑜 − 𝑙] → ℎ[𝑜 − 𝑙 From LTI: ∞ ∞ ] 𝑦[𝑜] = 𝑦[𝑙]𝜀[𝑜 − 𝑙] → 𝑧[𝑜] = 𝑦[𝑙]ℎ[𝑜 − 𝑙 𝑙→−∞ 𝑙→−∞ convolution sum Computer Engineering Department, Signal and Systems 6

Book Chapter#: Section# Convolution Sum Representation of Response of LTI Systems ∞ ] 𝑧[𝑜] = 𝑦[𝑜] ∗ ℎ[𝑜] = 𝑦[𝑙]ℎ[𝑜 − 𝑙 𝑙→−∞ Interpretation: Computer Engineering Department, Signal and Systems 7

Book Chapter#: Section# Visualizing the calculation of ] 𝑧[𝑜] = 𝑦[𝑜] ∗ ℎ[𝑜  Choose value of n and consider it fixed ∞ ] 𝑧[𝑜] = 𝑦[𝑙]ℎ[𝑜 − 𝑙 𝑙→−∞ View as functions of k with n fixed prod of overlap for prod of overlap for Computer Engineering Department, Signal and Systems 8

Book Chapter#: Section# Calculating Successive Values: Shift, Multiply, Sum 𝑧[𝑜] = 0 𝑜 < −1 𝑧[−1] = 1 × 1 = 1 𝑧[0] = 0 × 1 + 1 × 2 = 2 𝑧[1] = (−1) × 1 + 0 × 2 + 1 × (−1) = −2 𝑧[2] = (−1) × 2 + 0 × (−1) + 1 × (−1) = −3 𝑧[3] = (−1) × (−1) + 0 × (−1) = 1 𝑧[4] = (−1) × (−1) = 1 𝑧[𝑜] = 0 𝑜 > 4 Computer Engineering Department, Signal and Systems 9

Book Chapter#: Section# Properties of Convolution and DT LTI Systems  A DT LTI System is completely characterized by its unit sample response ] Ex. 1: ℎ[𝑜] = 𝜀[𝑜 − 𝑜 0 There are many systems with this response to ] 𝜀[𝑜 There is only one LTI System with this response to ] 𝜀[𝑜 𝑧[𝑜] = 𝑦[𝑜 − 𝑜 0 ] ⇒ 𝑦[𝑜] ∗ 𝜀[𝑜 − 𝑜 0 ] = 𝑦[𝑜 − 𝑜 0 ] Computer Engineering Department, Signal and Systems 10

Book Chapter#: Section# Example 2: 𝑜 - An Accumulator 𝑧 𝑜 = 𝑦[𝑙] 𝑙=−∞ Unit Sample response n     [ ] [ ] [ ] h n k u n  k   n [ ]* [ ] [ ] x n u n x k  k Computer Engineering Department, Signal and Systems 11

Book Chapter#: Section# The Commutativity Property ] 𝑧[𝑜] = 𝑦[𝑜] ∗ ℎ[𝑜] = ℎ[𝑜] ∗ 𝑦[𝑜 Ex: Step response 𝑡[𝑜 of an LTI system ] ] 𝑡[𝑜] = 𝑣[𝑜] ∗ ℎ[𝑜] = ℎ[𝑜] ∗ 𝑣[𝑜 𝑜 ] ⇒ 𝑡[𝑜] = ℎ[𝑙 𝑙→−∞ Computer Engineering Department, Signal and Systems 12

Book Chapter#: Section# The Distributivity Property ] 𝑦[𝑜] ∗ (ℎ 1 [𝑜] + ℎ 2 [𝑜]) = 𝑦[𝑜] ∗ ℎ 1 [𝑜] + 𝑦[𝑜] ∗ ℎ 2 [𝑜 Interpretation: Computer Engineering Department, Signal and Systems 13

Book Chapter#: Section# The Associativity Property ] 𝑦[𝑜] ∗ (ℎ 1 [𝑜] ∗ ℎ 2 [𝑜]) = (𝑦[𝑜] ∗ ℎ 1 [𝑜]) ∗ ℎ 2 [𝑜 ⇕ Commutativity ] 𝑦[𝑜] ∗ (ℎ 2 [𝑜] ∗ ℎ 1 [𝑜]) = (𝑦[𝑜] ∗ ℎ 2 [𝑜]) ∗ ℎ 1 [𝑜  Implication (Very special to LTI Systems): Computer Engineering Department, Signal and Systems 14

Properties of LTI Systems    0 n Causality [ ] 0 h n a) Sufficient condition: Causality ⇒ ℎ 𝑜 = 0, 𝑜 < 0 b) Necessity: Proof If h[n]=0 for n<0, ∞ 𝑧 𝑜 = 𝑦 𝑙 ℎ[𝑜 − 𝑙] 𝑙=−∞     Which is equivalent to: [ ] [ ] [ ] y n h k x n k  k 0 Meaning that the output at n depends only on previous inputs

Book Chapter#: Section# Properties of LTI Systems      Stability | [ ]| h k  a) sufficiency: k If 𝑦 𝑜 < 𝐶 𝑔𝑝𝑠 𝑏𝑚𝑚 𝑜 +∞ 𝑧 𝑜 = | ℎ 𝑙 𝑦 𝑜 − 𝑙 | 𝑙=−∞ +∞ 𝑧 𝑜 ≤ ℎ 𝑙 |𝑦 𝑜 − 𝑙 | 𝑙=−∞ +∞ 𝑧 𝑜 ≤ 𝐶 ℎ 𝑙 𝑔𝑝𝑠 𝑏𝑚𝑚 𝑜 𝑙=−∞ So we can conclude that if the impulse response is absolutely summable, that is, if: +∞ ℎ 𝑙 < ∞ 𝑙=−∞ Then, y[n] is bounded and hence, the system is stable. Computer Engineering Department, Signal and Systems 16

Book Chapter#: Section# Properties of LTI Systems  b) necessity:  Assume we have a stable system.  Suppose the input to the system is: 0, 𝑗𝑔 ℎ −𝑜 = 0 ℎ[−𝑜] 𝑦 𝑜 = 𝑗𝑔 ℎ[−𝑜] ≠ 0 |ℎ −𝑜 | This is a bounded input, 𝑦 𝑜 < 1 𝑔𝑝𝑠 𝑏𝑚𝑚 𝑜 The output at n=0 is: ∞ 𝑧 0 = 𝑦 𝑙 ℎ[−𝑙] 𝑙=−∞ ℎ 2 [−𝑙] ℎ 2 [𝑙] ∞ ∞ ∞ |ℎ 𝑙 | = 𝑙=−∞ |ℎ 𝑙 | < ∞ 𝑧 0 = 𝑙=−∞ |ℎ −𝑙 | = 𝑙=−∞ since the system is assumed to be stable. Computer Engineering Department, Signal and Systems 17

Book Chapter#2: Section# Representation of CT Signals  Approximate any input x(t) as a sum of shifted, scaled pulses ^   ( ) ( ) x t x k      ( 1) k t k Computer Engineering Department, Signal and Systems 18

Book Chapter#: Section#  has unit area ( t )        ( ) ( ) x k t k  Computer Engineering Department, Signal and Systems 19

Book Chapter#: Section#         ( ) ( ) ( ) x t x k t k   k   0 limit as         ( ) ( ) ( ) x t x t d  The Shifting Property of the Unit Impulse Computer Engineering Department, Signal and Systems 20

Book Chapter#: Section# Response of CT LTI system   ( ) ( ) t h t                   ( ) ( ) ( ) ( ) ( ) ( ) x t x k t k y t x k h t k       k k Impulse response :  t  ( ) ( ) h t   Taking limits 0                 ( ) ( ) ( ) ( ) ( ) ( ) x t x t d y t x h t d     Convolution Integral Computer Engineering Department, Signal and Systems 21

Book Chapter#: Section# Operation of CT Convolution         ( ) ( ) * ( ) ( ) ( ) y t x t h t x h t d          h  Flip Slide Multiply ( ) ( ) ( ) ( ) h h t x h t ( )  Integrate      ( ) ( ) x h t d  Computer Engineering Department, Signal and Systems 22

Book Chapter#: Section# PROPERTIES AND EXAMPLES Commutativity  ( )* ( ) ( )* ( ) x t h t h t x t Shifting property       ( ) * ( ) ( ), ( ) * ( ) ( ) x t t t x t t x t t x t 0 0 Example: t So if        )  ( ) ( ) ( ) ( ) x t t output ( ) ( y t h t y t x d An integrator input     t       t ( ) ( ) ( )     )  h t d u t ( ) ( ) * ( ) ( ) * ( ) ( y t x t h t x t u t x d     Step response: t     ( ) ( )* ( ) ( )* ( ) ( ) s t u t h t h t u t h d  23 Computer Engineering Department, Signal and Systems