System properties for basics

1. Basic System Properties
Dr. S. N. Sharma
Electrical Engineering Department
Sardar Vallabhbhai National Institute Technology, Surat.
Subject : Network Systems

2. Basic System Properties
 Basic system properties P. 44-47, Signals and Systems, A V
Oppenheim, A S Willsky with Nawab
 Memoryless systems
 A system is said to be memoryless system if the output at the present
instant depends on the input on the present instant.
 That does not depend on past and present inputs.
 The example is the square law detector.
 Consider the system, the square law detector, using the input and the
output
)
(t
x
.
t
y
).
(
2
t
x
yt 

3. Basic System Properties
 Invertibility and inverse systemA system
 A system is said to be invertible if the distinct input produces the
distinct output. The system with the invertibility property are described
as the inverse system. Consider the system
 Where are the outputs and the inputs respectively. Thus can
be expressed as
 Note: This concept has found application in the Encoder design, since
the input of the encoder is recovered at the output of the the decoder.
 The decoder can be regarded as the inverse of the encoder. The encoder
is the inverse system.
),
(
2 t
x
yt 
)
(
and t
x
yt
.
2
1
t
t
t
t x
w
the
to
leading
y
w 


4. Basic System Properties
 Causal system
 A system is said to be the causal if the present and past values of the
input produce the present value of the output. The Causal system are
physically realizable.
 That is regarded as the non-anticipatve. That implies that one can not
anticipate the future value of the input from the present value of the
input. For the LTI system, the impulse response vanishes, i.e.
Note:
(i) Interpret the equation by guessing its system interpretation and physics.
(ii) Usefulness: advanced courses and competitive examinations
.
0
,
0 
 t
ht

5. Basic System Properties
 Time-varying system:
 A time-varying system is described
 The input argument indicates the system parameters are time-
varying. That is same as saying that the system with variable
coefficients. The linear version of the system is
 Another way of looking the time-time varying system: if the input is
delayed by certain time interval, then the output delayed by the same
time interval.
).
,
( t
t x
t
f
x 

t
)
,
( t
t x
t
f
x 

t
t
t x
A
x 


6. Basic System Properties
 Time-invariant system:
 A time-invariant system is described as
 The absence of the input argument indicates the system parameters
are not time-varying. That is same as saying that the system with
constant coefficients. The linear version of the above time-invariant
system is
 Another way of looking the time-time varying system: if the input is
delayed by certain time interval, then the output delayed by the same
time interval.
t
).
( t
t x
f
x 

)
( t
t x
f
x 
 .
t
t Ax
x 


7. Basic System Properties
 Time-invariant vs Time varying :
then the solution
 On the other hand, for the time-invariant case
 Stationarity :
 The stationarity is an important concept. That indicates that the signal
is stationary implies that signal and related property do not depend on
the time but the time interval.
 Note that time-invariant system has stationarity properties.
)
(t
u
B
x
A
x t
t
t
t 






d
u
B
ds
A
x
ds
A
x
t
t
t
s
t
t
t
s
t )
(
)
exp(
)
exp(
0
0
0
 
 


.
)
(
)).
(
exp(
))
(
exp(
0
0
0 

  d
u
B
t
A
x
t
t
A
x
t
t
t
t  





8. Basic System Properties
 Linearity:
(i) It has two parts additivity and homogeneity
(ii) The supervision theorem is a consequence of the linearity.
(iii) Linearity is tasted using the following: input and output relation in the
algebraic form or the linear homogeneous equation in state and input
setting in the time-domain.

9. Basic System Properties
 Stability of the System
 In the general framework, the bounded input produces the bounded
output. Then the system is stable.
 Suppose the system input is and the output is the impulse
response is Then input-output relation is
 The system is stable if and only if is absolutely integrable. The
absolute integrabilty implies
)
(t
u ),
(t
y
).
(t
h
.
)
(
)
(
)
(
)
(
)
(
0
0

 



t
t
d
u
t
h
d
t
u
h
t
y 





)
(t
h
.
)
(
0


 dt
t
h
t

10. Basic System Properties
 Useful Signal
 Impulse signal: In the continuous time case, the impulse function is the
impulse signal that is the Dirac delta function.
otherwise
 The unit step signal has the following properties:
otherwise
 The ramp signal
)
(t
u
,
)
( 

t
 0

t
,
0

0
,
1
)
( 
 t
t
u
,
0

.
0
),
(
)
( 
 t
t
tu
t
r

11. Basic System Properties
 Note:-
 Any signal multiplied with the unit step signal gives the right-sided
signal .For example where is the arbitrary signal , ,
is the unit step signal then is the right-sided signal.
 In Electrical Engineering, we study the right sided signal. For example,
is the double sided signal and the is the right-sided
signal.
 Relationship between the impulse signal and the unit step signal
, thus
 Alternatively,
)
(
)
( t
u
t
x )
(t
x )
(t
u
)
(
)
( t
u
t
x
t

sin )
(
sin t
u
t

)
(t

)
(t
u
)
(
)
( t
t
u
dt
d


 






t
d
t
d
t
u
0
.
)
(
)
(
)
( 






12. Basic System Properties
 Question: Why are the impulse, unit step and ramp signals are test
signals
 Answer:- The above input signals unfold the transient response and
steady state response of the system, conveniently.
 The qualitatively characteristics of the system, Peak overshoot, peak
time, rise time, peak time, settling time can be studied using the closed
form expression.
 Prove that the following:
 The input-output relation in the algebraic form
 Describes the system is not linear. Comment about
c
mx
y 

.
mx
y 

13. Basic System Properties
 Consider a system S whose input and the output are related by
 To determine whether system S is linear
 Answer: Yes. We prove it. The above input-output relation denotes the
time is the independent variable and and are the
dependendent variables. The first system is time-varying not time-
invariant.
 About the linearity
 Consider the input to the system is then the response
is
t
x t
y
t
t tx
y 
t t
x t
y
)
(
)
( 2
1 t
x
t
x t
t 
 
where
,
~
t
y
)
(
)
(
),
(
)
( 2
2
1
1 t
y
t
x
t
t
y
t
tx 


14. Basic System Properties
 The above is linear system. The linearity holds for the time-invariant
and time varying system. Thus, there is a concepts of linear time-varying
as well as time-invariant.
 Consider the input to the system is S whose input and output is
are related by
 To determine whether system S is linear.
 Answer: We prove it.
 The above input-output relation denotes the time is the independent
variable and and are the dependendent variables. The first
system is time-invariant not time-varying.
t
x t
y
2
t
t x
y 
t
t
x t
y
)
(
)
(
))
(
)
(
(
~
2
1
2
1 t
x
t
t
x
t
t
x
t
x
t
y t
t
t
t
t 


 



)
(
)
( 2
1 t
x
t
t
x
t t
t 
 

).
(
)
( 2
1 t
y
t
y t
t 
 


15. Basic System Properties
 About the linearity
 Consider the input to the system is then the
response is
(i)
 Then
 The first set of equation suggests that
 Exercise: Check the linearity of the following:
 Using the methods adopted in the previous two excercises.
)
(
)
( 2
1 t
x
t
x t
t 
 
where
,
~
t
y
)
(
)
(
),
(
)
( 2
2
2
1
2
1 t
y
t
x
t
y
t
x 

)
(
)
(
2
)
(
)
(
))
(
)
(
(
~
2
1
2
2
2
2
1
2
2
2
1 t
x
t
x
t
x
t
x
t
x
t
x
y t
t
t
t
t
t
t 




 




)
(
)
(
2
)
(
)
(
~
2
1
2
2
1
2
t
y
t
y
t
y
t
y
y t
t
t
t
t 


 


)
(
)
( 2
2
1
2
t
y
t
y t
t 
 

3
)
(
2
)
( 
 n
x
n
y