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

Exploiting Superposition and Time-Invariance Book Chapter#: Section# Exploiting Superposition and Time-Invariance 𝑥[𝑛]= 𝑘 𝑎 𝑘 𝑥 𝑘 [𝑛] 𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑦𝑠𝑡𝑒𝑚 𝑦[𝑛]= 𝑘 𝑎 𝑘 𝑦 𝑘 [𝑛 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

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

The Shifting Property of the Unit Sample Book Chapter#: Section# That is … 𝑥[𝑛]=...+𝑥[−2]𝛿[𝑛+2]+𝑥[−1]𝛿[𝑛+1]+𝑥[0]𝛿[𝑛]+𝑥[1]𝛿[𝑛−1]+... =>𝑥[𝑛]= 𝑘=−∞ ∞ 𝑥[𝑘]𝛿[𝑛−𝑘 The Shifting Property of the Unit Sample Coefficients Basic Signals Computer Engineering Department, Signal and Systems

𝑥[𝑛]= 𝑘→−∞ ∞ 𝑥[𝑘]𝛿[𝑛−𝑘]→𝑦[𝑛]= 𝑘→−∞ ∞ 𝑥[𝑘] ℎ 𝑘 [𝑛 Book Chapter#: Section# Suppose the system is linear, and define ℎ 𝑘 [𝑛 as the response to 𝛿[𝑛−𝑘 : 𝛿[𝑛−𝑘]→ ℎ 𝑘 [𝑛 From superposition: 𝑥[𝑛]= 𝑘→−∞ ∞ 𝑥[𝑘]𝛿[𝑛−𝑘]→𝑦[𝑛]= 𝑘→−∞ ∞ 𝑥[𝑘] ℎ 𝑘 [𝑛 Computer Engineering Department, Signal and Systems

𝑥[𝑛]= 𝑘→−∞ ∞ 𝑥[𝑘]𝛿[𝑛−𝑘]→𝑦[𝑛]= 𝑘→−∞ ∞ 𝑥[𝑘]ℎ[𝑛−𝑘 Book Chapter#: Section# Now suppose the system is LTI, and define the unit sample response ℎ[𝑛 : 𝛿[𝑛]→ℎ[𝑛 From TI: 𝛿[𝑛−𝑘]→ℎ[𝑛−𝑘 From LTI: 𝑥[𝑛]= 𝑘→−∞ ∞ 𝑥[𝑘]𝛿[𝑛−𝑘]→𝑦[𝑛]= 𝑘→−∞ ∞ 𝑥[𝑘]ℎ[𝑛−𝑘 convolution sum Computer Engineering Department, Signal and Systems

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

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

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

Properties of Convolution and DT LTI Systems 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[𝑛]=𝛿[𝑛− 𝑛 0 There are many systems with this response to 𝛿[𝑛 There is only one LTI System with this response to 𝛿[𝑛 𝑦[𝑛]=𝑥[𝑛− 𝑛 0 ]⇒𝑥[𝑛]∗𝛿[𝑛− 𝑛 0 ]=𝑥[𝑛− 𝑛 0 Computer Engineering Department, Signal and Systems

𝑦 𝑛 = 𝑘=−∞ 𝑛 𝑥[𝑘] Example 2: - An Accumulator Unit Sample response Book Chapter#: Section# Example 2: 𝑦 𝑛 = 𝑘=−∞ 𝑛 𝑥[𝑘] - An Accumulator Unit Sample response Computer Engineering Department, Signal and Systems

The Commutativity Property Book Chapter#: Section# The Commutativity Property 𝑦[𝑛]=𝑥[𝑛]∗ℎ[𝑛]=ℎ[𝑛]∗𝑥[𝑛 Ex: Step response 𝑠[𝑛 of an LTI system 𝑠[𝑛]=𝑢[𝑛]∗ℎ[𝑛]=ℎ[𝑛]∗𝑢[𝑛 ⇒𝑠[𝑛]= 𝑘→−∞ 𝑛 ℎ[𝑘 Computer Engineering Department, Signal and Systems

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

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

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

Properties of LTI Systems Book Chapter#: Section# Properties of LTI Systems sufficiency: If 𝑥 𝑛 <𝐵 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 𝑦 𝑛 =| 𝑘=−∞ +∞ ℎ 𝑘 𝑥 𝑛−𝑘 | 𝑦 𝑛 ≤ 𝑘=−∞ +∞ ℎ 𝑘 |𝑥 𝑛−𝑘 | 𝑦 𝑛 ≤𝐵 𝑘=−∞ +∞ ℎ 𝑘 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 So we can conclude that if the impulse response is absolutely summable, that is, if: 𝑘=−∞ +∞ ℎ 𝑘 <∞ Then, y[n] is bounded and hence, the system is stable. Computer Engineering Department, Signal and Systems

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

Representation of CT Signals Book Chapter#2: Section# Representation of CT Signals Approximate any input x(t) as a sum of shifted, scaled pulses Computer Engineering Department, Signal and Systems

has unit area Book Chapter#: Section# Computer Engineering Department, Signal and Systems

The Shifting Property of the Unit Impulse Book Chapter#: Section# limit as The Shifting Property of the Unit Impulse Computer Engineering Department, Signal and Systems

Response of CT LTI system Book Chapter#: Section# Response of CT LTI system Impulse response : Taking limits Convolution Integral Computer Engineering Department, Signal and Systems

Operation of CT Convolution Book Chapter#: Section# Operation of CT Convolution Flip Slide Multiply Integrate Computer Engineering Department, Signal and Systems

PROPERTIES AND EXAMPLES Book Chapter#: Section# PROPERTIES AND EXAMPLES Commutativity Shifting property Example: An integrator Step response: So if input output Computer Engineering Department, Signal and Systems

DISTRIBUTIVITY Book Chapter#: Section# Computer Engineering Department, Signal and Systems

ASSOCIATIVITY Book Chapter#: Section# Computer Engineering Department, Signal and Systems

Causality and Stability Book Chapter#: Section# Causality and Stability Computer Engineering Department, Signal and Systems

The impulse as an idealized “short” pulse Book Chapter#: Section# The impulse as an idealized “short” pulse Consider response from initial rest to pulses of different shapes and durations, but with unit area. As the duration decreases, the responses become similar for different pulse shapes. Computer Engineering Department, Signal and Systems

The Operational Definition of the Unit Impulse δ(t) Book Chapter#: Section# The Operational Definition of the Unit Impulse δ(t) δ(t) —idealization of a unit-area pulse that is so short that, for any physical systems of interest to us, the system responds only to the area of the pulse and is insensitive to its duration Operationally: The unit impulse is the signal which when applied to any LTI system results in an output equal to the impulse response of the system. That is, for all h(t) δ(t) is defined by what it does under convolution. Computer Engineering Department, Signal and Systems

The Unit Doublet —Differentiator Book Chapter#: Section# The Unit Doublet —Differentiator Impulse response = unit doublet The operational definition of the unit doublet: Computer Engineering Department, Signal and Systems

Triplets and beyond! Operational definitions n times n is number of Book Chapter#: Section# Triplets and beyond! n times n is number of differentiations Operational definitions Computer Engineering Department, Signal and Systems

Impulse response: “-1 derivatives" = integral ⇒I.R.= unit step Book Chapter#: Section# Impulse response: “-1 derivatives" = integral ⇒I.R.= unit step Operational definition: Cascade of n integrators : Computer Engineering Department, Signal and Systems

Integrators (continued) Book Chapter#: Section# Integrators (continued) the unit ramp More generally, for n>0 Computer Engineering Department, Signal and Systems

n and m can be positive or negative Book Chapter#: Section# Define Then n and m can be positive or negative E.g. Computer Engineering Department, Signal and Systems

Sometimes Useful Tricks Book Chapter#: Section# Sometimes Useful Tricks Differentiate first, then convolve, then integrate Computer Engineering Department, Signal and Systems

Example Book Chapter#: Section# Computer Engineering Department, Signal and Systems

Example (continued) Book Chapter#: Section# Computer Engineering Department, Signal and Systems