- Signals and Systems Tutorial
- Signals & Systems Home
- Signals & Systems Overview
- Signals Basic Types
- Signals Classification
- Signals Basic Operations
- Systems Classification
- Signals Analysis
- Fourier Series
- Fourier Series Properties
- Fourier Series Types
- Fourier Transforms
- Fourier Transforms Properties
- Distortion Less Transmission
- Hilbert Transform
- Convolution and Correlation
- Signals Sampling Theorem
- Signals Sampling Techniques
- Laplace Transforms
- Laplace Transforms Properties
- Region of Convergence
- Z-Transforms (ZT)
- Z-Transforms Properties

- Signals and Systems Resources
- Signals and Systems - Resources
- Signals and Systems - Discussion

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

Systems are classified into the following categories:

- linear and Non-linear Systems
- Time Variant and Time Invariant Systems
- linear Time variant and linear Time invariant systems
- Static and Dynamic Systems
- Causal and Non-causal Systems
- Invertible and Non-Invertible Systems
- Stable and Unstable Systems

A system is said to be linear when it satisfies superposition and homogenate principles. Consider two systems with inputs as x_{1}(t), x_{2}(t), and outputs as y_{1}(t), y_{2}(t) respectively. Then, according to the superposition and homogenate principles,

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

$\therefore, $ T [a_{1} x_{1}(t) + a_{2} x_{2}(t)] = a_{1} y_{1}(t) + a_{2} y_{2}(t)

From the above expression, is clear that response of overall system is equal to response of individual system.

**Example:**

(t) = x^{2}(t)

Solution:

y_{1} (t) = T[x_{1}(t)] = x_{1}^{2}(t)

y_{2} (t) = T[x_{2}(t)] = x_{2}^{2}(t)

T [a_{1} x_{1}(t) + a_{2} x_{2}(t)] = [ a_{1} x_{1}(t) + a_{2} x_{2}(t)]^{2}

Which is not equal to a_{1} y_{1}(t) + a_{2} y_{2}(t). Hence the system is said to be non linear.

A system is said to be time variant if its input and output characteristics vary with time. Otherwise, the system is considered as time invariant.

The condition for time invariant system is:

y (n , t) = y(n-t)

The condition for time variant system is:

y (n , t) $\neq$ y(n-t)

Where y (n , t) = T[x(n-t)] = input change

y (n-t) = output change

**Example:**

y(n) = x(-n)

y(n, t) = T[x(n-t)] = x(-n-t)

y(n-t) = x(-(n-t)) = x(-n + t)

$\therefore$ y(n, t) ≠ y(n-t). Hence, the system is time variant.

If a system is both linear and time variant, then it is called linear time variant (LTV) system.

If a system is both linear and time Invariant then that system is called linear time invariant (LTI) system.

Static system is memory-less whereas dynamic system is a memory system.

**Example 1:** y(t) = 2 x(t)

For present value t=0, the system output is y(0) = 2x(0). Here, the output is only dependent upon present input. Hence the system is memory less or static.

**Example 2:** y(t) = 2 x(t) + 3 x(t-3)

For present value t=0, the system output is y(0) = 2x(0) + 3x(-3).

Here x(-3) is past value for the present input for which the system requires memory to get this output. Hence, the system is a dynamic system.

A system is said to be causal if its output depends upon present and past inputs, and does not depend upon future input.

For non causal system, the output depends upon future inputs also.

**Example 1:** y(n) = 2 x(t) + 3 x(t-3)

For present value t=1, the system output is y(1) = 2x(1) + 3x(-2).

Here, the system output only depends upon present and past inputs. Hence, the system is causal.

**Example 2:** y(n) = 2 x(t) + 3 x(t-3) + 6x(t + 3)

For present value t=1, the system output is y(1) = 2x(1) + 3x(-2) + 6x(4) Here, the system output depends upon future input. Hence the system is non-causal system.

A system is said to invertible if the input of the system appears at the output.

Y(S) = X(S) H1(S) H2(S)

= X(S) H1(S) · $1 \over ( H1(S) )$ *Since H2(S) = 1/( H1(S) )*

$\therefore, $ Y(S) = X(S)

$\to$ y(t) = x(t)

Hence, the system is invertible.

If y(t) $\neq$ x(t), then the system is said to be non-invertible.

The system is said to be stable only when the output is bounded for bounded input. For a bounded input, if the output is unbounded in the system then it is said to be unstable.

**Note:** For a bounded signal, amplitude is finite.

**Example 1:** y (t) = x^{2}(t)

Let the input is u(t) (unit step bounded input) then the output y(t) = u2(t) = u(t) = bounded output.

Hence, the system is stable.

**Example 2:** y (t) = $\int x(t)\, dt$

Let the input is u (t) (unit step bounded input) then the output y(t) = $\int u(t)\, dt$ = ramp signal (unbounded because amplitude of ramp is not finite it goes to infinite when t $\to$ infinite).

Hence, the system is unstable.

Advertisements