The document summarizes several key properties of systems: - Memory refers to whether a system's output depends on current and/or past input values. Causal systems are memoryless while anticausal systems require memory. - Linearity means a system's output is a linear combination of its inputs. A linear system satisfies superposition and homogeneity. - Time-invariance means a system's properties do not change over time such that a time shift of the input results in the same time shift of the output. - Invertibility refers to whether the input of a system can be recovered from its output. An invertible system has a corresponding inverse system.

1. Subject:- Signal And System
Topic:-System properties
Prepared By :
Dhruv Aradeshana (150110109004)
Tejas Dobariya (150110109010)
Dhruval Gondaliya (150110109011)

2. System properties
Memory
Invertibility
Causality
Stability
Time Invariance
Linearity

3. Causal System
A system is causal if it’s output is a function of only the current
and previous samples
Examples: Backward Difference
Counter Example: Forward Difference
]n[x]1n[x]n[y 
]1n[x]n[x]n[y 

4. Causal and noncausal system
Example: distinguish between causal and noncausal
systems in the following:
t
)(tu
1 2
(1) Case I )()( tuty 
t
)(ty
12
0)(
0)(1


tybut
tutwhen
Noncausal system

5. (2) Case II
causal system
)()(  tuty
t
)(ty
1 2
Delay system
(3) Case III )2()()(  tututy
causal system
At present past

6. (4) Case IV )2()()(  tututy
At present future
(5) Case V stepunitistuiftuty )()()( 2

t
)(ty
0)(
0)(0


tybut
tutwhen
Causal system
Causal system

7. Linear Time Invariant Systems
Definitions
• A linear system may be defined as one which obeys the
Principle of Superposition, which may be stated as
follows:
• A time-invariant system is one whose properties do not
vary with time. The only effect of a time-shift on an input
signal to the system is a corresponding time-shift in its
output.

8. A system is said to be time invariant if a time delay or
time advance of the input signal leads to an identical time
shift in the output signal.
)(tx
Time invariant
system
)(ty
)( 0ttx 
0t
)( 0tty 
0t

9. Example :
)(
)(
)(
tR
tx
ty 
0),()(
)(
)(
)(
)(
)(
)(
)(
)(
)()(
)(
)(
)(
0201
0
01
1
012
2
012
1
1








tfortytty
ttR
ttx
tybut
tR
ttx
tR
tx
ty
ttxtx
tR
tx
ty
Time varying system
)(tx )(ty

10. Invertibility
)(tx )(tx)(ty
A system is said to be Invertible if the input of the system
can be recovered from the output.
H Hinv
)}({)( txHty  )}({)( tyHtx inv

)}}({{)}({ txHHtyH invinv


11. )()( 0ttxty 
)(tx )(ty
IHH
ttxH
ttxH
inv
inv



)(
)(
0
0
Inverse system
)(tx )(ty
)()( 2
txty 
Example :
Example :

12. Linearity
)(1 tx
A system is said to be linear in terms of the system input
x(t) and the system output y(t) if it satisfies the following
two properties of superposition and homogeneity.
Superposition：
Homogeneity：
)(1 ty )(2 tx )(2 ty
)()( 21 txtx  )()( 21 tyty 
)(1 tx )(1 ty )(1 tax )(1 tay

13. • Linear System: A system is linear if and only if
Example: Ideal Delay System
]nn[x]n[y o
 
 
  ]nn[ax]n[xaT
]nn[ax]n[axT
]nn[x]nn[x]n[xT]}n[x{T
]nn[x]nn[x]}n[x]n[x{T
o1
o1
o2o112
o2o121




   
    (scaling)]n[xaT]n[axT
and
y)(additivit]n[xT]n[xT]}n[x]n[x{T 2121



14. )]()([])()([2])()([
)]()(2)([)]()(2)([
)](3)([)](3)([
)(3)(3)()()]()([3])()([
)(3)()()(2)(
)(3)()()(2)(
212121
222111
2211
21212121
22222
11111
tbytaytbytaytbytay
tytytybtytytya
txtxbtxtxa
txbtxatxbtxatbxtaxtbxtax
txtxtytyty
txtxtytyty






)(3)()()(2)( txtxtytyty 
Linear time
invariant system
)(tx )(ty
Example :
Linear system

15. Example :
)1()()(  txtxty
)()1()()1()()(
)()(
)1()()(
)()(
1
2
11
2
11
1
111
1
tyatxtxataxtaxty
taxtxlet
txtxty
txtxlet




)()( 1 tayty 
Non linear system
)(tx )(ty

16. Memory
• Memoryless
output for each value of independent variable at a
given time is dependent on the input at only that same
time
i.e. y[n] = 2*x[n] - x^2*[n] memoryless
All causal systems are memoryless systems.
All anticausal systems required memory.

17. Capacitor is a C-T system with memory
In physical systems memory is directly associated with the
storage of energy
y[n] = x[n-1] memory
D-T systems implemented with up, memory is associated
with storage registers
Memory typically suggests storing past values but
definition covers systems with outputs dependant upon
future values of input and output

18. Thank You…..