The systems are classified as,
· Static & dynamic system
· Time invariant and variant system
· Linear and non linear system
· Causal and non causal system
· Stable and unstable system

**Classification of system:**

The
systems are classified as,

·
Static & dynamic system

·
Time invariant and variant system

·
Linear and non linear system

·
Causal and non causal system

·
Stable and unstable system

**1. Static and dynamic system:**

·
Static system is said to be a memoryless system.

·
The output does not depend the past or future
input.

·
It only depends the present input for an output.

Eg, y(n) = x(n)

·
Dynamic system is said to be as system with memory.

·
Its output depend the past values of input for an
output.

Eg.Y(n) = x(n) + x(n - 1)

·
This static and dynamic systems are otherwise
called as memoryless and system with memory.

**2. Systems with and without
memory:**

A system
is called memory less if the output at any time t (or n) depends only on the
input at time t (or n); in other words, independent of the input at times
before of after t (or n). Examples of memory less systems:

**Examples of systems with memory:**

**3. Time invariant and time
variant system:**

If the
time shifts in the input signals results in corresponding time shift in the
output, then the system is called as time invariant.

The input
and output characteristics do not change with time.

f[x(t_{1}
– t_{2})] = y(t_{1} – t_{2})

For a
continuous time system,

f[x(t_{1}
– t_{2})] = y(t_{1} – t_{2})

For a
discrete time system,

F[x(n -
k)] = y(n - k)

If the
above relation does not satisfy, then the system is said to be a time variant
system.

A system
is called time-invariant if the way it responds to inputs does not change over
time:

**Examples of time-invariant systems:**

The RC
circuit considered earlier provided the values of R or C are constant.

**Examples of time-varying systems:**

The RC
circuit considered earlier if the values of R or C change over time.

Most
physical systems are slowly time-varying due to aging, etc. Hence, they can be
considered time-invariant for certain time periods in which its behavior does
not change significantly.

**4. Linear and non linear system:**

A system
is said to be linear if it satisfies the superposition principle.

Superposition
principle states that the response to a weighted sum of input signal be equal
to the weighted sum of the output corresponding to each of the individual input
signal

The
continuous system is linear if,

F[a_{1}x_{1}(t)
+ a_{2}x_{2}(t)] = a_{1}y_{1}(t) + a_{2}y_{2}(t)

The
discrete system is linear if,

F[a_{1}x_{1}(n)
+ a_{2}x_{2}(n)] = a_{1}y_{1}(n) + a_{2}y_{2}(n)

Otherwise
the system is non linear.

A system
is called linear if its I/O behavior satisfies the additivity and homogeneity
properties:

for any
complex constant a.

Equivalently,
a system is called linear if its I/O behavior satisfies the superposition
property:

where any
complex constants a and b.

**5. Causal and non causal system:**

A causal
system is one whose output depends upon the present and past input values.

If the
system depends the future input values, the system is said to be non causal.
Eg.for causal system.

Y(t) =
x(t) + x(t - 1)

Y(n) =
x(n) + x(n - 3)

Eg. For
non causal system,

Y(t) =
x(t+3) + x2(t)

Y(n) =
x(2n)

A system
is called causal or non-anticipative if the output at any time t (or n) depends
only on the input at times t or before t (or n or before n); in other words,
independent of the input at times after t (or n). All memory less systems are
causal. Physical systems where the time is the independent variable are causal.

Non-causal
systems may arise in applications where the independent variable is not the
time such as in the image processing applications.

**Examples of causal systems:**

**Examples of non-causal systems:**

**6. Stable and unstable system:**

When
every bounded input produces bounded output then the system is called as stable
system or bounded input bounded output (BIBO stable).

Otherwise
the system is unstable.

A system
is called stable if it produces bounded outputs for all bounded inputs

Stability
in a physical system generally results from the presence of mechanisms that
dissipate energy, such as the resistors in a circuit, friction in a mechanical
system, etc

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