Signals & Systems
Spring 2009
Week 3
Instructor: Mariam Shafqat
UET Taxila
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Today's lecture
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Linear Time Invariant Systems
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Introduction
Discrete time LTI systems: Convolution Sum
Continuous time LTI systems: Convolution Integral
Properties of LTI systems
Quiz at the end of lecture
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Linear Time Invariant Systems
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A system satisfying both the linearity and the
time-invariance property
LTI systems are mathematically easy to
analyze and characterize, and consequently,
easy to design.
Highly useful signal processing algorithms
have been developed utilizing this class of
systems over the last several decades.
They possess superposition theorem.
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How superposition is applicable
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If we represent the input to an LTI system in
terms of linear combination of a set of basic
signals, we can then use superposition to
compute the output of the system in terms of
responses to these basic signals.
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Representation of LTI systems
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Any linear time-invariant system (LTI) system,
continuous-time or discrete-time, can be uniquely
characterized by its
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Impulse response: response of system to an impulse
Frequency response: response of system to a complex
exponential e j 2 p f for all possible frequencies f.
Transfer function: Laplace transform of impulse response
Given one of the three, we can find other two
provided that they exist
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Significance of unit impulse
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Every signal whether large or small can be
represented in terms of linear combination of
delayed impulses.
Here two properties apply:
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Linearity
Time Invariance
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Basic building Blocks
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For DT or CT case; there are two natural
choices for these two basic building blocks
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For DT: Shifted unit samples
For CT: Shifted unit impulses.
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Convolution Sum
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This fact together with the property of
superposition and time invariance, will allow
us to develop a complete characterization of
any LTI system in terms of responses to a
unit impulse.
This representation is called
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Convolution sum in discrete time case
Convolution integral in continuous time case.
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Impulse Response
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The response of a discrete-time system to a
unit sample sequence {d[n]} is called the unit
sample response or simply, the impulse
response, and is denoted by {h[n]}
The response of a discrete-time system to a
unit step sequence {m[n]} is called the unit
step response or simply, the step
response, and is denoted by {s[n]}
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Impulse Response Example
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The impulse response of the system
y[n]  1x[n]   2 x[n  1]  3 x[n  2]   4 x[n  3]
is obtained by setting x[n] = d[n] resulting in
h[n]  1d [n]   2d [n  1]  3d [n  2]   4d [n  3]
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The impulse response is thus a finite-length
sequence of length 4 given by
{h[n]}  {1,  2 , 3 ,  4}

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Impulse Response Example
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Example - The impulse response of the discretetime accumulator
y[n] 
n
 x[]
  
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is obtained by setting x[n] = d[n] resulting in
h[n] 
n
 d []  m [n]
  
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Time-Domain Characterization of LTI
Discrete-Time System
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Input-Output Relationship -A consequence
of the linear, time-invariance property is that
an LTI discrete-time system is completely
characterized by its impulse response
Knowing the impulse response one
can compute the output of the system for any
arbitrary input
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Time-Domain Characterization of LTI
Discrete-Time System
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Let h[n] denote the impulse response of a LTI
discrete-time system
We compute its output y[n] for the input:
x[n]  0.5d [n  2]  1.5d [n  1]  d [n  2]  0.75d [n  5]
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As the system is linear, we can compute its outputs
for each member of the input separately and add the
individual outputs to determine y[n]
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Time-Domain Characterization of LTI
Discrete-Time System
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Since the system is time-invariant
input
output
d [n  2]  h[n  2]
d [n  1]  h[n  1]
d [n  2]  h[n  2]
d [n  5]  h[n  5]
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Time-Domain Characterization of LTI
Discrete-Time System
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Likewise, as the system is linear
input
output
0.5d [n  2]  0.5h[n  2]
1.5d [n  1]  1.5h[n  1]
 d [n  2]  h[n  2]
0.75d [n  5]  0.75h[n  5]
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Hence because of the linearity property we
get
y[n]  0.5h[n  2]  1.5h[n  1]
 h[n  2]  0.75h[n  5]
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Time-Domain Characterization of LTI
Discrete-Time System
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Now, any arbitrary input sequence x[n] can be
expressed as a linear combination of delayed and
advanced unit sample sequences in the form
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x[n] 
 x[k ]d [n  k ]
k  
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The response of the LTI system to an input
will be
input
x[k ]d [n  k ]
output
x[k ] h[n  k ]
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Discrete Time LTI Systems:
THE CONVOLUTION SUM
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The representation of discrete time signal
in terms of impulses
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Simplest way is to visualize discrete time
signal in terms of individual impulses.
Here we use scaled unit impulse sequences.
Where the scaling on each impulse equals
the value of x[n] at the particular instant the
unit impulse occurs.
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Graphically
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Mathematically
This is called sifting property .
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Sifting property
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This property corresponds to the
representation of an arbitrary sequence as a
linear combination of shifted unit impulses ;
where the weights in the linear combination
are x[k] .
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The discrete time unit impulse response and
the convolution sum representation of LTI
systems
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Significance of sifting property
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Represent any input signal as a superposition
of scaled version of a very simple set of
elementary functions; namely; shifted unit
impulses:
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Each of which is non-zero at a single point in time
specified by the corresponding value of K.
Moreover property of time invariance states that
the response of a time invariant system to the
time shifted unit impulses are simply time shifted
version of one another.
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Contd….
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Suppose that the system is linear and define
hk[n] as a response of impulse[n-k]; then
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For superposition:
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Cntd…
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Now suppose that the system is LTI; and define
the unit sample response hk[n] as.
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For TI
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For LTI systems:
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Convolution sum/ Superposition sum
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The last equation is called superposition sum
or the convolution sum. Operation on the
right hand side is known as convolution of the
sequence x[n] and h[n].
We will represent the operation of the
convolution symbolically
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y[n]=x[n]*h[n]
LTI system is completely characterized by its
response to a single signal namely; its response
to the unit impulse.
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Convolution sum representation of LTI
system
Mathematically
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Graphically
Sum up all the responses for all K’s
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Contd….
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Develop the sequence y[n] generated by the
convolution of the sequences x[n] and h[n] shown
below
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y(0)=x(0)h(0)
h(1-k)=h[-(k-1)]
h(2-k)=h[-(k-2)]
y(1)=x(0)h(1)
+x(1)h(0)
y(2)=x(2)h(0)
+x(1)h(1)
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Continuous time systems: THE
CONVOLUTION INTEGRAL
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What we had for discrete time signals
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Convolution sum was the sifting property of
discrete time unit impulse – that is, the
mathematical representation of a signal as a
superposition of scaled and shifted unit
impulse functions.
For CT signals consider impulse as an
idealization of a pulse that is too short.
Rep CT signal as idealized pulses with
vanishingly small duration impulses.
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Rep of CT signal in terms of impulses
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App any signal x(t) as sum of shifted, scaled
impulses.
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Ideally
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Impulse has unit area:
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Sifting property of impulse
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Response of LTI system
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Convolution Integral
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Operation of convolution
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Example
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Properties of LTI systems
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Commutative
Distributive
Associative
With and without memory
Invertibility
Causality
Stability
The unit step response of an LTI system
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The end
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