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Basic System Properties.ppt
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