Time Domain Representations of
Linear Time-Invariant Systems
Chapter 2
Introduction
• Several methods can be used to describe the
relationship between the input and output signals
of LTI system.
• Focus on system description that relate the output
signal to the input signal when both are
represented as function of time (time domain).
• Impulse response defined as output of LTI system
due to a unit impulse signal input applied at time
t=0 or n=0.
Analysis of Continuous-Time
Systems
Impulse Response
Let a system be described by
x(t) H
y(t)
and let the excitation be a unit impulse at time, t = 0. Then the
response y is the impulse response h.
H
d (t)
h(t)
Since the impulse occurs at time t = 0 and nothing has excited
the system before that time, the impulse response before time
t = 0 is zero (because this is a causal system). After time t = 0
the impulse has occurred and gone away. Therefore there is no
excitation and the impulse response is the homogeneous solution
of the differential equation.
Impulse Response
a2 h  t   a1 h  t   a0 h  t   d  t 
What happens at time, t = 0?
The left side of the equation must be a unit impulse.
The left side is zero before time t = 0 because the system has never
been excited.
The left side is zero after time t = 0 because it is the solution of the
homogeneous equation whose right side is zero. These two facts
are both consistent with an impulse. The impulse response might
have in it an impulse or derivatives of an impulse since all of
these occur only at time, t = 0. What the impulse response does
have in it depends on the form of the differential equation.
Impulse Response
Continuous-time LTI systems are described by differential
equations of the general form,
 t   an1 y n1  t    a1 y  t   a0 y  t 
m
m1
 bm x    t   bm1 x    t    b1 x   t   b0 x  t 
an y
n
For all times, t < 0:
If the excitation x(t) is an impulse, then for all time t < 0
it is zero. The response y(t) is zero before time t = 0
because there has never been an excitation before that time.
Impulse Response
For all time t > 0:
The excitation is zero. The response is the homogeneous
solution of the differential equation.
At time t = 0:
The excitation is an impulse. In general it would be possible
for the response to contain an impulse plus derivatives of an
impulse because these all occur at time t = 0 and are zero
before and after that time. Whether or not the response contains
an impulse or derivatives of an impulse at time t = 0 depends
on the form of the differential equation
 t   an1 y n1  t    a1 y  t   a0 y  t 
m
m1
 bm x    t   bm1 x    t    b1 x   t   b0 x  t 
an y
n
Convolution
• A systematic way to find systems respond.
• Convolution integral for CT systems.
The Convolution Integral
If a continuous-time LTI system is excited by an arbitrary
excitation, the response could be found approximately by
approximating the excitation as a sequence of contiguous
rectangular pulses of width Tp .
Exact
Excitation
Approximate
Excitation
The Convolution Integral
• All the pulses are rectangular and the same
width, the only differences between pulses
are when they occur and how tall they are.
• So the pulse responses all have the same
form except the delayed by some amount, to
account for time of occurrence, and
multiplied by a weighting constant, to
account for the height.
The Convolution Integral
Approximating the excitation as a pulse train can be expressed
mathematically by
 t  Tp
x  t    x  Tp  rect 
 T
 p
or

 t
  x  0  rect 

 Tp

 t  Tp
  x Tp  rect 

 Tp
 t  nTp 
x  t    x  nTp  rect 


n 
 Tp 
The excitation can be written in terms of pulses of width
Tp and unit area as

 t  nTp 
1
x  t    Tp x  nTp  rect 


Tp
n 
 Tp 

shifted unit area pulse

 

The Convolution Integral
Let the response to an unshifted pulse of unit area and width Tp
be the “unit pulse response” h p t .
Then, invoking linearity, the response to the overall excitation is
(approximately) a sum of shifted and scaled unit pulse responses
of the form
y t  

 T x  nT  h  t  nT 
n
p
p
p
p
As Tp approaches zero, the unit pulses become unit impulses,
the unit pulse response becomes the unit impulse response h(t)
and the excitation and response become exact.
A Graphical Illustration of the
Convolution Integral
The convolution integral is defined by
x t   h t  

 x   h  t    d

For illustration purposes let the excitation x(t) and the
impulse response h(t) be the two functions below.
A Graphical Illustration of the
Convolution Integral
In the convolution integral there is a factor h  t    .
We can begin to visualize this quantity in the graphs below.
A Graphical Illustration of the
Convolution Integral
The functional transformation in going from h() to h(t - ) is
 
  t
h   
 h    
 h     t    h  t   
A Graphical Illustration of the
Convolution Integral
The convolution value is the area under the product of x(t)
and h(t - ). This area depends on what t is. First, as an
example, let t = 5.
For this choice of t the area under the product is zero.
Therefore if
y(t )  x  t   h  t 
then y(5) = 0.
A Graphical Illustration of the
Convolution Integral
Now let t = 0.
Therefore y(0) = 2, the area under the product.
A Graphical Illustration of the
Convolution Integral
The process of convolving to find y(t) is illustrated below.
Convolution Integral Properties
x  t   Ad  t  t0   A x  t  t0 
If g  t   g 0  t   d  t  then g  t  t0   g 0  t  t0   d  t   g 0  t   d  t  t0 
If y  t   x  t   h  t  then y  t   x   t   h  t   x  t   h   t 
and y  at   a x  at   h  at 
Commutativity
x t   y t   y t   x t 
Associativity
 x  t   y  t   z  t   x  t    y  t   z  t 
Distributivity
 x  t   y  t   z  t   x  t   z  t   y  t   z  t 
System Interconnections
If the output signal from a system is the input signal to a second
system the systems are said to be cascade connected.
It follows from the associative property of convolution that the
impulse response of a cascade connection of LTI systems is the
convolution of the individual impulse responses of those systems.
Cascade
Connection
System Interconnections
If two systems are excited by the same signal and their responses
are added they are said to be parallel connected.
It follows from the distributive property of convolution that the
impulse response of a parallel connection of LTI systems is the
sum of the individual impulse responses.
Parallel
Connection
Stability and Impulse Response
A system is BIBO stable if its impulse response is
absolutely integrable. That is if

 h  t  dt is finite.

Unit Impulse Response and Unit
Step Response
In any LTI system let an excitation x(t) produce the response
y(t). Then the excitation
d
x t 

dt
will produce the response
d
y t 

dt
It follows then that the unit impulse response is the first
derivative of the unit step response and, conversely that the
unit step response is the integral of the unit impulse response.
Complex Exponential Response
Let an LTI system be excited by a complex exponential of
the form
x  t   est
The response is the convolution of the excitation with the
impulse response or
y  t   h  t   e st 


h

e





The quantity
s t  
d  e st

 s
h

e
d




complex constant
 s
h

e
d will later be designated the




Laplace transform of the impulse response and will be an
important transform method for system analysis.
Analysis of Discrete-Time
Systems
Impulse Response
Discrete-time LTI systems are described mathematically
by difference equations of the form
an y  n   an1 y  n  1   an D y  n  D 
 bn x  n   bn1 x  n  1 
 bn N x  n  N 
For any excitation x[n] the response y[n] can be found by
finding the response to x[n] as the only forcing function on the
right-hand side and then adding scaled and time-shifted
versions of that response to form y[n].
If x[n] is a unit impulse, the response to it as the only forcing
function is simply the homogeneous solution of the difference
equation with initial conditions applied. The impulse response
is conventionally designated by the symbol h[n].
Impulse Response
Since the impulse is applied to the system at time n = 0,
that is the only excitation of the system and the system is
causal. The impulse response is zero before time n = 0.
h  n  0 , n  0
After time n = 0, the impulse has come and gone and the
excitation is again zero. Therefore for n > 0, the solution of
the difference equation describing the system is the
homogeneous solution.
h  n  yh  n , n  0
System Response
• Once the response to a unit impulse is
known, the response of any LTI system to
any arbitrary excitation can be found
• Any arbitrary excitation is simply a
sequence of amplitude-scaled and timeshifted impulses
• Therefore the response is simply a
sequence of amplitude-scaled and timeshifted impulse responses
Simple System Response Example
More Complicated System
Response Example
System
Excitation
System
Impulse
Response
System
Response
Convolution
• A systematic way to find systems respond.
• Convolution sum for DT systems.
• Based on a simple idea, no matter how
complicated an excitation signal is, it is simply a
sequence of DT impulses.
• Therefore, with the assumption that the impulse
response to a unit impulse excitation occuring at
time n=0 has already found, we will use the
convolution technique to find system response.
The Convolution Sum
The response y[n] to an arbitrary excitation x[n] is of the
form
y  n 
x  1 h  n  1  x 0 h  n   x 1 h  n  1 
where h[n] is the impulse response. This can be written in
a more compact form
y  n 

 x  m h  n  m
m
called the convolution sum.
A Convolution Sum Example
A Convolution Sum Example
A Convolution Sum Example
A Convolution Sum Example
Convolution Sum Properties
Convolution is defined mathematically by
y n   x n  h n  

 x m h n  m 
m
The following properties can be proven from the definition.
Let
then
x n  Ad n  n0  Ax n  n0 
y n   x n  h n 
y n  n0  x n  h n  n0  x n  n0  h n 
y n  y n  1  x n  h n  h n  1 x n  x n  1 h n 
and the sum of the impulse strengths in y is the product of
the sum of the impulse strengths in x and the sum of the
impulse strengths in h.
Convolution Sum Properties
(continued)
Commutativity
x  n  y  n  y  n  x  n
Associativity
 x  n   y  n   z  n   x  n    y  n   z  n 
Distributivity
 x  n   y  n   z  n   x  n   z  n   y  n   z  n 
System Interconnections
The cascade connection of two systems can be viewed as
a single system whose impulse response is the convolution
of the two individual system impulse responses. This is a
direct consequence of the associativity property of
convolution.
System Interconnections
The parallel connection of two systems can be viewed as
a single system whose impulse response is the sum
of the two individual system impulse responses. This is a
direct consequence of the distributivity property of
convolution.
Stability and Impulse Response
It can be shown that a BIBO-stable system has an
impulse response that is absolutely summable. That is,

 h  n is finite.
n
Unit Impulse Response and Unit
Sequence Response
In any LTI system let an excitation x[n] produce the response
y[n]. Then the excitation x[n] - x[n - 1] will produce the
response y[n] - y[n - 1] .
Complex Exponential Response
Let an LTI system be excited by a complex exponential of
the form
x  n  z n
The response is the convolution of the excitation with the
impulse response or
y  n 


m
z m h  n  m 


m
z nm h  m 
which can be written as
y  n  z n


m
h  m z  m
complex
constant