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### Properties of LTI systems

Properties of system are with respect to input and output signal. Important properties of systems are as follows :
• Dynamic or static system
• Time variance
• Linearity
• Causality
• Stability
1. Static or dynamic system :

(I) Static system :

It is a system in which output at any instant of time depends on input sample at the same time.

Example :

y (n) : 7 x (n)

here 7 is constant which multiplies input x (n) but output at nth instant that means y (n) depends on input at the same (nth) time constant x (n). so this is a static system.

(II) Dynamic system :

It this system output at any instant of time depends on input sample at the same time as well as at other times.

Example :
Note that if x (n) represents input signal at present instant then,

past and future signal :

(a) x (n-k) - that means delayed input signal is called as past signal.
(b) x (n+k) - that means advanced input signal is called as future signal.

y (n) : x (n) + 6 x (n-1)
here x (n-1) is previous  sample. so system is dynamic.

y (n) = 5 x (n+2) + x (n)
here (n+2) indicates advanced version of input sample that means it is future sample; so tis is dynamic system.

2. Time variant or time invarient systems :

A system is time invarient if its input output characteristics do not chance with time.

A time invarient system means its input output characteristics are not changing with time shifting.

Let us consider we are applying input signal x (n) to the system and produces output y (n).

Now delay input by k samples. that means input becomes x (n-k). Apply this delayed input to the same system . Let us say the system now gives output y (n-k). then this system is called as time or shift invarient system.

Observe that initially the output is y (n) with input x (n). When input is delayed by "k" samples then the output is also delayed by the same "k" samples. Thus the input-output characteristics are not changed . so the system is invarient.

Example :
y (n) = x (n) - x (n-1)

y(n,k) = x (n-k) -x (n-k-1).............(1)

Replace n by n-k throughout the given equation :

y (n-k) = x (n-k) - x(n-k-1)............(2)

Compare equations (1) and (2)

Here y (n,k) = y (n-k).

Thus the system is time invarient.

3. Causal or anticausal system :

(I) Causal system :

A system is said to be a causal if output at any instant of time depends only an present and past inputs. But the output does not depend on future inputs.

Example :

y (n) = x (n) + x (n-1)
y (n)  = 6 x(n)

(II) Anticausal systems :

A system is said to be anticausal if its output depends not only on present and past inputs but also on future inputs.

y (n) = x (n) + x (n+1)
y(n) =  x(n) + n x(n+1)

4. Stable or Unstable system :

To define stability of a system we will use the term BIBO. It stands for bounded input bounded output. The meaning of word bounded is some finite of value. So bounded input means input signal is having some finite value.

(I) Stable system :

An initially relaxed system is BIBO stable if and only if every bounded input produces bounded output.

Here a relaxed system means when input to the system is zero then the output of system is also zero.

Example :

T[ x (n) ]  = a x(n) +5

(II) Unstable system :

An initially relaxed system is said to be unstable if bounded input produces unbounded output.

When unstable system is practically implemented then it causes overflow.

Unstable system shows erratic and extreme behavior.

Example :

5. Linear and non linear system :

When input x (n) is zero then output y (n) = 0. Thus the system is linear. Now first step is satisfied so we will check remaining step for linearity.

Example :

T x(n) = a x(n) + 6

step 1 : put x (n) = 0

y (n) = 0 + 6

step 2 :
x1 (n) = y1 (n) = a x1(n) + 6
x2(n) = y2(n) = a x2(n) + 6

y ‘(n)  = a1 y1 (n) + a2 y2 (n)

y '(n)  = a1 [a x1(n) + 6 ]  +  a2 y2 (n) + a2 [ a x2(n) + 6 ]........................(1)

step 3 :
a1  x1(n) + a x2(n)  = a [ a1  x1(n) + a x2(n) ] + 6

y'' (n) = a [ a1  x1(n) + a x2(n) ] + 6......................................................(2)

step 4 : compare  equation (1) and (2)

y '(n) = y'' (n) ; the system is non linear.
Properties of system are with respect to input and output signal. Important properties of systems are as follows :
• Dynamic or static system
• Time variance
• Linearity
• Causality
• Stability
1. Static or dynamic system :

(I) Static system :

It is a system in which output at any instant of time depends on input sample at the same time.

Example :

y (n) : 7 x (n)

here 7 is constant which multiplies input x (n) but output at nth instant that means y (n) depends on input at the same (nth) time constant x (n). so this is a static system.

(II) Dynamic system :

It this system output at any instant of time depends on input sample at the same time as well as at other times.

Example :
Note that if x (n) represents input signal at present instant then,

past and future signal :

(a) x (n-k) - that means delayed input signal is called as past signal.
(b) x (n+k) - that means advanced input signal is called as future signal.

y (n) : x (n) + 6 x (n-1)
here x (n-1) is previous  sample. so system is dynamic.

y (n) = 5 x (n+2) + x (n)
here (n+2) indicates advanced version of input sample that means it is future sample; so tis is dynamic system.

2. Time variant or time invarient systems :

A system is time invarient if its input output characteristics do not chance with time.

A time invarient system means its input output characteristics are not changing with time shifting.

Let us consider we are applying input signal x (n) to the system and produces output y (n).

Now delay input by k samples. that means input becomes x (n-k). Apply this delayed input to the same system . Let us say the system now gives output y (n-k). then this system is called as time or shift invarient system.

Observe that initially the output is y (n) with input x (n). When input is delayed by "k" samples then the output is also delayed by the same "k" samples. Thus the input-output characteristics are not changed . so the system is invarient.

Example :
y (n) = x (n) - x (n-1)

y(n,k) = x (n-k) -x (n-k-1).............(1)

Replace n by n-k throughout the given equation :

y (n-k) = x (n-k) - x(n-k-1)............(2)

Compare equations (1) and (2)

Here y (n,k) = y (n-k).

Thus the system is time invarient.

3. Causal or anticausal system :

(I) Causal system :

A system is said to be a causal if output at any instant of time depends only an present and past inputs. But the output does not depend on future inputs.

Example :

y (n) = x (n) + x (n-1)
y (n)  = 6 x(n)

(II) Anticausal systems :

A system is said to be anticausal if its output depends not only on present and past inputs but also on future inputs.

y (n) = x (n) + x (n+1)
y(n) =  x(n) + n x(n+1)

4. Stable or Unstable system :

To define stability of a system we will use the term BIBO. It stands for bounded input bounded output. The meaning of word bounded is some finite of value. So bounded input means input signal is having some finite value.

(I) Stable system :

An initially relaxed system is BIBO stable if and only if every bounded input produces bounded output.

Here a relaxed system means when input to the system is zero then the output of system is also zero.

Example :

T[ x (n) ]  = a x(n) +5

(II) Unstable system :

An initially relaxed system is said to be unstable if bounded input produces unbounded output.

When unstable system is practically implemented then it causes overflow.

Unstable system shows erratic and extreme behavior.

Example :

5. Linear and non linear system :

When input x (n) is zero then output y (n) = 0. Thus the system is linear. Now first step is satisfied so we will check remaining step for linearity.

Example :

T x(n) = a x(n) + 6

step 1 : put x (n) = 0

y (n) = 0 + 6

step 2 :
x1 (n) = y1 (n) = a x1(n) + 6
x2(n) = y2(n) = a x2(n) + 6

y ‘(n)  = a1 y1 (n) + a2 y2 (n)

y '(n)  = a1 [a x1(n) + 6 ]  +  a2 y2 (n) + a2 [ a x2(n) + 6 ]........................(1)

step 3 :
a1  x1(n) + a x2(n)  = a [ a1  x1(n) + a x2(n) ] + 6

y'' (n) = a [ a1  x1(n) + a x2(n) ] + 6......................................................(2)

step 4 : compare  equation (1) and (2)

y '(n) = y'' (n) ; the system is non linear.