Signal and Systems
Prof. H. Sameti
Chapter #2:
1)
2)
3)
4)
5)
6)
7)
8)
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)
Book Chapter#: Section#
Exploiting Superposition and TimeInvariance
 𝑥[𝑛]
=
𝑘
𝑎𝑘 𝑥𝑘 [𝑛]
𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑦𝑠𝑡𝑒𝑚
𝑦[𝑛] =
𝑎 𝑦 [𝑛]
𝑘 𝑘 𝑘
 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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Book Chapter#: Section#
Representation of DT Signals Using
Unit Samples
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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[𝑘]𝛿[𝑛 − 𝑘]
=> 𝑥[𝑛] =
𝑘=−∞
Coefficients
Basic Signals
The Shifting Property of the Unit Sample
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Book Chapter#: Section#
 Suppose
the system is linear, and define ℎ𝑘 [𝑛] as the
response to 𝛿[𝑛 − 𝑘]:
𝛿[𝑛 − 𝑘] → ℎ𝑘 [𝑛]
 From superposition:
∞
𝑥[𝑛] =
∞
𝑥[𝑘]𝛿[𝑛 − 𝑘] → 𝑦[𝑛] =
𝑘→−∞
Computer Engineering Department, Signal and Systems
𝑥[𝑘]ℎ𝑘 [𝑛]
𝑘→−∞
5
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] =
∞
𝑥[𝑘]ℎ[𝑛 − 𝑘]
𝑥[𝑘]𝛿[𝑛 − 𝑘] → 𝑦[𝑛] =
𝑘→−∞
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𝑘→−∞
convolution sum
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Book Chapter#: Section#
Convolution Sum Representation of
Response of LTI Systems
∞
𝑥[𝑘]ℎ[𝑛 − 𝑘]
𝑦[𝑛] = 𝑥[𝑛] ∗ ℎ[𝑛] =
𝑘→−∞
Interpretation:
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Book Chapter#: Section#
Visualizing the calculation of
𝑦[𝑛] = 𝑥[𝑛] ∗ ℎ[𝑛]
 Choose value of n and consider it fixed
∞
𝑦[𝑛] =
𝑥[𝑘]ℎ[𝑛 − 𝑘]
𝑘→−∞
View as functions of k with n fixed
prod of
overlap for
prod of
overlap for
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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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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 − 𝑛0 ]
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 − 𝑛0 ] ⇒ 𝑥[𝑛] ∗ 𝛿[𝑛 − 𝑛0 ] = 𝑥[𝑛 − 𝑛0 ]
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Book Chapter#: Section#
Example 2:
𝑛
𝑦𝑛 =
- An Accumulator
𝑥[𝑘]
𝑘=−∞
Unit Sample response
h [n ] 
n
  [k ]  u [n ]
k 
x [n ]*u [n ] 
n
 x [k ]
k 
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Book Chapter#: Section#
The Commutative 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]
𝑛
ℎ[𝑘]
⇒ 𝑠[𝑛] =
𝑘→−∞
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Book Chapter#: Section#
The Distributive Property
𝑥[𝑛] ∗ (ℎ1 [𝑛] + ℎ2 [𝑛]) = 𝑥[𝑛] ∗ ℎ1 [𝑛] + 𝑥[𝑛] ∗ ℎ2 [𝑛]
Interpretation:
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Book Chapter#: Section#
The Associative Property
𝑥[𝑛] ∗ (ℎ1 [𝑛] ∗ ℎ2 [𝑛]) = (𝑥[𝑛] ∗ ℎ1 [𝑛]) ∗ ℎ2 [𝑛]
⇕ Commutative
𝑥[𝑛] ∗ (ℎ2 [𝑛] ∗ ℎ1 [𝑛]) = (𝑥[𝑛] ∗ ℎ2 [𝑛]) ∗ ℎ1 [𝑛]

Implication (Very special to LTI Systems):
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Properties of LTI Systems
Causality  h [ n ]  0
n0
a) Sufficient condition: Causality ⇒ ℎ 𝑛 = 0, 𝑛 < 0
b) Necessity: Proof
If h[n]=0 for n<0,
∞
𝑦𝑛 =
𝑥 𝑘 ℎ[𝑛 − 𝑘]
𝑘=−∞

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

 | h [k ] |  
Stability 
k 
a) sufficiency:
If 𝑥 𝑛 < 𝐵
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛
+∞
𝑦 𝑛 =|
ℎ 𝑘 𝑥 𝑛−𝑘 |
𝑘=−∞
+∞
𝑦𝑛 ≤
ℎ 𝑘 |𝑥 𝑛 − 𝑘 |
𝑘=−∞
+∞
𝑦 𝑛 ≤𝐵
ℎ𝑘
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛
𝑘=−∞
So we can conclude that if the impulse response is absolutely summable,
that is, if:
+∞
ℎ 𝑘 <∞
𝑘=−∞
Then, y[n] is bounded and hence, the system is stable.
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Book Chapter#: Section#
Properties of LTI Systems

b) necessity:
 Assume we have a stable system.
 Suppose the input to the system is:
0,
𝑥 𝑛 = ℎ[−𝑛]
,
|ℎ −𝑛 |
This is a bounded input,
The output at n=0 is:
𝑖𝑓 ℎ −𝑛 = 0
𝑖𝑓 ℎ[−𝑛] ≠ 0
ℎ 𝑛 < 1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛
∞
𝑦0 =
𝑥 𝑘 ℎ[−𝑘]
𝑘=−∞
𝑦0 =
2
ℎ [𝑘]
∞
𝑘=−∞ |ℎ 𝑘 |
=
∞
𝑘=−∞ |ℎ
𝑘 | < ∞ since the system is assumed to be stable.
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Book Chapter#2: Section#
Representation of CT Signals
 Approximate any input x(t) as a sum of shifted, scaled
pulses
^
x(t )  x(k )
k   t  (k  1)
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Book Chapter#: Section#
  (t ) has unit area
 x(k)  (t  k)
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Book Chapter#: Section#
x(t ) 


k 
x(k )  (t  k )
limit as

x(t ) 

 0
x( ) (t   ) d

The Shifting Property of the Unit Impulse
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Book Chapter#: Section#
Response of CT LTI system
  (t )  h (t )

x (t ) 
 x(k)
k  

(t  k )   y (t ) 

 x(k)h
k  

(t  k ) 
Impulse response :
 (t )  h (t )
Taking limits   0

x (t ) 
 x( ) (t   )d


 y (t ) 
 x( )h(t   )d

Convolution Integral
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Book Chapter#: Section#
Operation of CT Convolution

y (t )  x(t ) * h(t ) 
 x( )h(t   )d

h( )
Flip
h( )
Integrate
Slide
h(t   ) Multiply
x( )h(t   )

 x( )h(t   )d

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Book Chapter#: Section#
PROPERTIES AND EXAMPLES
Commutativity
Shifting property
Example:
An integrator
x(t ) * h(t )  h(t ) * x(t )
x(t ) * (t  t0 )  x(t  t0 ),
t
y(t )   x( )d

t
So if
input
h(t )    ( )d  u(t )

Step response:
x(t ) * (t )  x(t )
x (t )   (t ) output
y (t )  h(t )
t
y(t )  x(t ) * h(t )  x(t ) * u(t )   x( )d

t
s(t )  u (t )* h(t )  h(t )* u(t )   h( )d

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Book Chapter#: Section#
DISTRIBUTIVITY
y(t )  x(t ) *[ h1 (t )  h2 (t )]
y(t )  x(t ) * h1 (t )  x(t ) * h2 (t )
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Book Chapter#: Section#
ASSOCIATIVITY
y(t )  [ x(t ) * h1 (t )] * h2 (t )
y(t )  x(t ) *[ h1 (t ) * h2 (t )]
y(t )  x(t ) *[ h2 (t ) * h1 (t )]
y(t )  [ x(t ) * h2 (t )] * h1 (t )
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Book Chapter#: Section#
Causality and Stability
 h(t )  0, t  0

  | h( ) | d  

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Book Chapter#: Section#
The impulse as an idealized “short”
pulse
dy (t )
1
1

y (t ) 
x(t )
dt
RC
RC
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.
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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,
 (t ) * h(t )  h(t )
for all h(t)
δ(t) is defined by what it does under convolution.
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Book Chapter#: Section#
The Unit Doublet —Differentiator
dx(t )
y (t ) 
dt
Impulse response = unit doublet
d (t )
u1 (t ) 
dt
The operational definition of the unit doublet:
dx(t )
x(t ) * u1 (t ) 
dt
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Book Chapter#: Section#
Triplets and beyond!
n0
un (t )  u1 (t ) * ... * u1 (t )
n times
n is number of
differentiations
Operational definitions
d n x(t )
x(t ) * u n (t ) 
dt n
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( n  0)
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Book Chapter#: Section#
Impulse response:
u1 (t )  u(t )
“-1 derivatives" = integral ⇒I.R.= unit step
Operational definition:
t
x(t ) * u1 (t )   x( )d
Cascade of n integrators :

un (t )  u1 (t ) * ... * u1 (t ), (n  0)
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Book Chapter#: Section#
Integrators (continued)
t
t


u2 (t )   u1 ( )d   u( )d
 t.u (t )
More generally, for n>0
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the unit ramp
t n 1
u n (t ) 
u (t )
(n  1)!
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Book Chapter#: Section#
Define
Then
u 0 (t )   (t )
u n (t ) *u m (t ) u n  m (t )
n and m can be positive or negative
E.g.
u1(t ) *u 1(t ) u 0 (t )
 du (t ) 

 
dt


Computer Engineering Department, Signal and Systems
 (t )
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Book Chapter#: Section#
Sometimes Useful Tricks
x(t ) * h(t )  x(t ) *  (t ) * h(t )
 x(t ) * u1 (t ) * u1 (t ) * h(t )
 {[ x(t ) * u1 (t )] * h(t )}* u1 (t )
Differentiate first, then convolve, then integrate
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Book Chapter#: Section#
Example
dx(t )
  (t  1)  2 (t  1)   (t  2)
dt
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Book Chapter#: Section#
Example (continued)
dx(t )
* h(t )  h(t  1)  2h(t  1)  h(t  2)
dt
 dx( )

x(t ) * h(t )   
* h( )d

 d

t
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