2. Properties of System
• Stability
• Memory
• Causality
• Invertibility
• Time invariance
• Linearity
Prof: Sarun Soman, M.I.T, Manipal
3. Properties of Systems
Stabililty
A system is said to be bounded input, bounded output(BIBO)
stable iff every bounded input results in a bounded output.
Operator H is BIBO stable if the o/p signal satisfies the condition
)ݐ(ݕ ≤ ܯ௬ ≤ ∞ for all ‘t’
Whenever input signal x(t) satisfies the condition
)ݐ(ݔ ≤ ܯ௫ ≤ ∞ for all ‘t’
ܯ௬ and ܯ௫ finite positive numbers
Prof: Sarun Soman, M.I.T, Manipal
4. Properties of Systems
Example of unstable system
• Tacoma Narrows suspension bridge
• Collapsed on November 7, 1940.
• Collapsed due to wind induced vibrations
Prof: Sarun Soman, M.I.T, Manipal
6. Example1
Consider the moving average system with input output relation
ݕ ݊ =
ଵ
ଷ
ݔ ݊ + ݔ ݊ − 1 + ݔ ݊ − 2
Show that the system is BIBO stable.
Ans:
Assume that ]݊[ݔ < ܯ௫ < ∞ for all ‘n’
]݊[ݕ =
ଵ
ଷ
ݔ ݊ + ݔ ݊ − 1 + ݔ ݊ − 2
=
ଵ
ଷ
(ܯ௫ + ܯ௫ + ܯ௫)
= ܯ௫
Bounded input results in bounded o/p
Prof: Sarun Soman, M.I.T, Manipal
7. Example2
Consider a DT system whose i/p-o/p relation is defined by
ݕ ݊ = ݎ
]݊[ݔ
Where ݎ > 1. Show that the system is unstable.
Ans:
Assume that the input signal x[n] satisfies the condition
]݊[ݔ ≤ ܯ௫ ≤ ∞
]݊[ݕ = ݎ]݊[ݔ
= ݎ ]݊[ݔ
ݎ diverges as ‘n’ increases.
Prof: Sarun Soman, M.I.T, Manipal
8. Properties of Systems
Memory
A system is said to posses memory if its o/p signal depends on past or
future values of the input signal.
A system is said to be memoryless if its output signal depends only on
present value of input signal.
Eg.
Resistor
݅ ݐ =
ଵ
ோ
)ݐ(ݒ memoryless
Inductor
݅ ݐ =
ଵ
ݒ ߬ ݀߬
௧
ିஶ
memory
Prof: Sarun Soman, M.I.T, Manipal
10. Properties of Systems
Causality
A system is said to be causal if its present value of the o/p signal depends only
on the present or past values of the input signal.
A system is said to be non causal if its output signal depends on one or more
future values of the input signal.
Eg. Moving average system
ݕ ݊ =
ଵ
ଷ
(ݔ ݊ + ݔ ݊ − 1 + ݊[ݔ − 2])
Moving average system
ݕ ݊ =
ଵ
ଷ
(ݔ ݊ + ݔ ݊ + 1 + ݊[ݔ − 2])
ݕ 0 =
ଵ
ଷ
(ݔ 0 + ݔ 1 + )]2−[ݔ
causal
Non causal
Prof: Sarun Soman, M.I.T, Manipal
11. Properties of Systems
Invertibility
A system is said to be invertible if the input of the system can be
recovered from the o/p.
Note: necessary condition for invertibility distinct i/p must
produce distinct o/p
Prof: Sarun Soman, M.I.T, Manipal
12. Properties of Systems
Show that a square law system described by the input-output
relation ݕ ݐ = ݔଶ
()ݐ is not invertible.
Ans:
Distinct inputs x(t) and x(-t) produce the same output
Time invariance
A system is said to be time invariant if a delay or time advance
of the input signal lead to an identical time shift in the output
signal.
Prof: Sarun Soman, M.I.T, Manipal
13. Properties of Systems
Eg. Thermistor
Input output relation of thermistor
ݕଵ ݐ =
௫భ(௧)
ோ(௧)
Response ݕଶ ݐ of the thermistor to input ݐ(ݔ − ݐ)
ݕଶ ݐ =
௫భ(௧ି௧బ)
ோ(௧)
Prof: Sarun Soman, M.I.T, Manipal
14. Properties of Systems
Shifting the output ݕଵ ݐ
ݕଵ ݐ − ݐ =
௫భ(௧ି௧బ)
ோ(௧ି௧బ)
For a thermistor ܴ(ݐ − ݐ) ≠ ܴ()ݐ for ݐ ≠ 0
ݕଵ(ݐ − ݐ) ≠ ݕଶ()ݐ for ݐ ≠ 0
Thermistor is time variant
Linearity
A system is linear in terms of the system input x(t) and the
system output y(t) if it satisfies two properties.
Prof: Sarun Soman, M.I.T, Manipal
16. Properties of Systems
Consider a discrete time system described by the input-output
relation ݕ ݊ = ݊.]݊[ݔ Show that the system is linear.
Ans:
Let the input signal x[n] be expressed as the weighted sum
ݔ ݊ = ∑ ܽݔ[݊]ே
ୀଵ
Output of the system
= ݊ ∑ ܽ
ே
ୀଵ ݔ[݊]
= ∑ ܽ݊ே
ୀଵ ݔ[݊]
= ∑ ܽ
ே
ୀଵ ݕ[݊]
The system is linear
Prof: Sarun Soman, M.I.T, Manipal
21. Tutorial
Causality
ݕ −1 = ݔ −0.5
o/p depends on future value of input
Non-causal
Stability
Let )ݐ(ݔ ≤ ܤ௫ ≤ ∞
Then )ݐ(ݕ = ݔ
௧
ଶ
≤ ܤ௬ ≤ ∞
The system is stable.
Prof: Sarun Soman, M.I.T, Manipal
22. Tutorial
ݕ ݐ = )ݐ(ݔݐ stable or unstable?
Let )ݐ(ݔ be bounded.
ݕ ݐ will be unbounded , ‘t’ is unbounded
Unstable system
ݕ ݐ = ݁௫(௧)
Check for linearity, Stability, Time Invariance, Memory.
ݔଵ ݐ → ݕଵ ݐ = ݁௫భ(௧)
ݔଶ ݐ → ݕଶ ݐ = ݁௫మ(௧)
Let ݔଷ ݐ = ܽݔଵ ݐ + ܾݔଶ ݐ
Prof: Sarun Soman, M.I.T, Manipal
23. Tutorial
ݕଷ ݐ = ݁௫భ ௧ ା௫మ ௧
≠ ܽݕଵ ݐ + ܾݕଶ ݐ
The system is Non-linear
Time Invariance
delay i/p ݐ(ݔ − ݐ)
ݕଵ ݐ = ݁௫(௧ି௧బ) (1)
Delay o/p
ݕଶ ݐ − ݐ = ݁௫(௧ି௧బ) (2)
(1)=(2) system is time invariant
The system is memoryless.
Prof: Sarun Soman, M.I.T, Manipal
24. Tutorial
The i/p and o/p of the system
is shown. Determine if the
system could be memoryless &
causal.
Memory, Causal
Non-Causal
Memory
1 20 t
y(t)
1
10 t
x(t)
1
10 t
y(t)
1
-1
10 t
x(t)
1
Prof: Sarun Soman, M.I.T, Manipal
25. Tutorial
ݕ ݐ = [sin 6)ݐ(ݔ]ݐ Check for Memory, Time invariance,
Linearity, Stability and Causality.
Ans:
Present value of i/p depends on present value of o/p
Memoryless System
Delay in i/p ݐ(ݔ − ݐ)
ݕଵ ݐ = [sin 6ݐ(ݔ]ݐ − ݐ)
Delay o/p
ݕଶ ݐ − ݐ = [sin 6(ݐ − ݐ]ݐ(ݔ − ݐ)
ݕଵ ݐ ≠ ݕଶ ݐ
Time variant system
Prof: Sarun Soman, M.I.T, Manipal
26. Tutorial
ݔଵ ݐ → ݕଵ ݐ = sin 6ݐ ݔଵ ݐ
ݔଶ ݐ → ݕଶ ݐ = sin 6ݐ ݔଶ ݐ
Let ݔଷ ݐ = ܽݔଵ ݐ + ܾݔଶ ݐ
ݕଷ ݐ = [sin 6ݔ]ݐଷ()ݐ
= ܽ sin 6ݐ ݔଵ ݐ + ܾ sin 6ݐ ݔଶ ݐ
= ܽݕଵ ݐ + ܾݕଶ()ݐ
Linear System
Causal – o/p doesn't depend on future value of i/p.
Prof: Sarun Soman, M.I.T, Manipal
27. Tutorial
Let )ݐ(ݔ ≤ ܤ௫ ≤ ∞
)ݐ(ݕ = sin 6ݐ )ݐ(ݔ
sin 6ݐ ≤ 1
)ݐ(ݕ is bounded , system is stable.
Check for linearity, causality, time-invariance, stability
1.ݕ ݊ = ݊ଶ]݊[ݔ
Linear , timevariant, causal, unstable
2. ݕ ݊ = ]݊2[ݔ
Linear, time-variant, non-causal
Prof: Sarun Soman, M.I.T, Manipal
28. Sketch the waveform
ݕ ݐ = ݎ ݐ + 1 − ݎ ݐ + ݐ(ݎ + 2)
0-1 1 2 t
r(t+1)
0 1 2 t
-r(t)
0 1 2
t
r(t-2)
-1 0 1 2
y(t)
t
Prof: Sarun Soman, M.I.T, Manipal
29. LTI Systems
• Linear Time Invariant Systems
• Tools for analyzing and predicting the behavior of LTI systems.
• Characterizing an LTI system
– Impulse Response
– Linear constant-coefficient differential equation or difference equation
• Gives insight into system’s behavior.
• Useful in analyzing and implementing the system
i/p o/p
System
Prof: Sarun Soman, M.I.T, Manipal
30. Impulse Response
Impulse response is defined as the o/p of an LTI
system due to a unit impulse signal input applied at
time t=0 or n=0.
Symbol ℎ ݐ , ℎ[݊]
Impulse response completely characterizes the
behavior of any LTI system.
Given the impulse response, output can be
determined for any arbitrary input signal.
Convolution Sum – DT LTI systems
Convolution Integral – CT LTI systems
Prof: Sarun Soman, M.I.T, Manipal
31. Convolution Sum
Consider an LTI system
ߜ ݊ → ℎ ݊
Apply time invariance property
ߜ[݊ − ݇] → ℎ ݊ − ݇
Apply principle of Homogeneity
݊[ߜ]݇[ݔ − ݇] → ]݇[ݔℎ ݊ − ݇
Apply principle of superposition
∑ ݊[ߜ]݇[ݔ − ݇]ஶ
ୀିஶ →
Input is weighted superposition of time shifted impulse
Output is weighted super position of time shifted impulse
responses.
ݔ ݇ ℎ[݊ − ݇]
ஶ
ୀିஶ
Prof: Sarun Soman, M.I.T, Manipal