▫
Y
▫
Y
▫
N
▫
N
▫
Y
▫
Y
▫
Y
▫
Y
Time-Invariant
vs.
Time-Variant Systems

CO2035 –
Discrete-Time Signals and Systems
35
§
Linear System
▫
Obey superposition principle
▫
Definition
A system is linear if and only if:
T[a
1
x
1
(n) + a
2
x
2
(n)] = a
1
T[x
1
(n)] + a
2
T[x
2
(n)]
"
a
i
,
"
x
i
(n)
▫
Homogeneity
Let
a
2
= 0
®
T[a
1
x
1
(n)] = a
1
T[x
1
(n)]
▫
Additivity
Let
a
1
= a
2
= 1
®
T[x
1
(n) + x
2
(n)] = T[x
1
(n)] + T[x
2
(n)]
§
Non-Linear System
▫
The system does not obey superposition principle
Linear
vs.
Non-Linear Systems

CO2035 –
Discrete-Time Signals and Systems
36
§
Note:
Linearity
=
Homogeneity + Additivity
▫
If a system is
not homogeneous
, it is
not linear
.
▫
If a system is
not additive
, it is
not linear
.
Linear
vs.
Non-Linear Systems

CO2035 –
Discrete-Time Signals and Systems
37
§
Example 1
▫
The system is described by the input-output equation
y(n) = T[x(n)] = nx(n)
▫
For two input sequences x
1
(n) and x
2
(n), the corresponding output are:
y
1
(n) = nx
1
(n)
(I)
y
2
(n) = nx
2
(n)
(II)
▫
A linear combination of the two input sequences in the output
y
3
(n) =T[a
1
x
1
(n)+a
2
x
2
(n)] = n[a
1
x
1
(n)+a
2
x
2
(n)]=na
1
x
1
(n) + na
2
x
2
(n)
▫
On the other hand, a linear combination of the two outputs (I)&(II) results in the output.
a
1
y
1
(n)+a
2
y
2
(n)=a
1
nx
1
(n) + a
2
nx
2
(n)
▫
Obviously, the system
obeys
superposition principle. Therefore, the system is
Linear
.
Linear
vs.
Non-Linear Systems

CO2035 –
Discrete-Time Signals and Systems
38
§
Example 2
▫
The system is described by the input-output equation
y(n) = T[x(n)] = x
2
(n)
▫
For two input sequences x
1
(n) and x
2
(n), the corresponding output are:
y
1
(n) = x
1
2
(n)
(I)
y
2
(n) = x
2
2
(n)
(II)
▫
A linear combination of the two input sequences in the output
y
3
(n) =T[a
1
x
1
(n)+a
2
x
2
(n)] = [a
1
x
1
(n)+a
2
x
2
(n)]
2
(III)
▫
On the other hand, a linear combination of the two outputs (I)&(II) results in the output.
a
1
y
1
(n)+a
2
y
2
(n)=a
1
x
1
2
(n) + a
2
x
2
2
(n)
(IV)
▫
From
(III)
&
(IV)
, the system
does not
obey superposition principle. Therefore, the system
is
Non-Linear
.
Linear
vs.
Non-Linear Systems

CO2035 –
Discrete-Time Signals and Systems
39
§
Quiz:
Linear or not?
▫
Y
▫
N
▫
Y
▫
Y
▫
Y
▫
N
▫
N
▫
Y
Linear
vs.
Non-Linear Systems

CO2035 –
Discrete-Time Signals and Systems
40
§
Causal System
▫
Definition
A system T is said to be causal if the output of the system at any time n [i.e. y(n)]
depends only on present and past inputs
[i.e. x(n), x(n – 1), x(n – 2) …]. In
mathematical term, the output of a causal system satisfies an equation of the form
y(n) = F[x(n), x(n–1), x(n–2), …]
§
Noncausal System
▫
The system is said to be Noncausal if the output of the system does not abbey the
above definition.
Causal
vs.
Noncausal Systems

CO2035 –
Discrete-Time Signals and Systems
41
§
Quiz:
Causal or not?
▫
Y
▫
Y
▫
Y
▫
N
▫
N
▫
N
▫
Y
▫
Y
Causal
vs.
Noncausal Systems

CO2035 –
Discrete-Time Signals and Systems
42
§
Stable System
▫
BIBO:
Bounded Input-Bounded Output
▫
Definition
A relaxed system is said to be BIBO Stable if and only if very bounded input produces a
bounded output.

#### You've reached the end of your free preview.

Want to read all 80 pages?

- Fall '13
- Schumacher
- Digital Signal Processing, LTI system theory, discrete-time systems, 2n