1. 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)

2. 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
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3. Representation of DT Signals Using Unit Samples
Book Chapter#: Section#
Representation of DT Signals Using Unit Samples
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4. 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
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5. 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
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6. 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
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7. 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:
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8. 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
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9. 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
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10. 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
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11. 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
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12. 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
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13. 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:
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14. 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):
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15. 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

16. 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.
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17. 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.
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18. Representation of CT Signals
Book Chapter#2: Section#
Representation of CT Signals
Approximate any input x(t) as a sum of shifted, scaled pulses
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19. has unit area Book Chapter#: Section#
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20. The Shifting Property of the Unit Impulse
Book Chapter#: Section#
limit as
The Shifting Property of the Unit Impulse
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21. Response of CT LTI system
Book Chapter#: Section#
Response of CT LTI system
Impulse response :
Taking limits
Convolution Integral
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22. Operation of CT Convolution
Book Chapter#: Section#
Operation of CT Convolution
Flip
Slide
Multiply
Integrate
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23. PROPERTIES AND EXAMPLES
Book Chapter#: Section#
PROPERTIES AND EXAMPLES
Commutativity
Shifting property
Example:
An integrator
Step response:
So if input
output
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24. DISTRIBUTIVITY Book Chapter#: Section#
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25. ASSOCIATIVITY Book Chapter#: Section#
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26. Causality and Stability
Book Chapter#: Section#
Causality and Stability
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27. 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

28. 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.
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29. The Unit Doublet —Differentiator
Book Chapter#: Section#
The Unit Doublet —Differentiator
Impulse response = unit doublet
The operational definition of the unit doublet:
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30. 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
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31. 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 :
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32. Integrators (continued)
Book Chapter#: Section#
Integrators (continued)
the unit ramp
More generally, for n>0
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33. n and m can be positive or negative
Book Chapter#: Section#
Define
Then
n and m can be positive or negative
E.g.
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34. Sometimes Useful Tricks
Book Chapter#: Section#
Sometimes Useful Tricks
Differentiate first, then convolve, then integrate
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35. Example Book Chapter#: Section#
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36. Example (continued) Book Chapter#: Section#
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