Signals and Systems

Systems and Their Properties

A system is any physical device, algorithm, circuit, or mathematical rule that accepts an input signal and produces an output signal. Its properties tell us how predictably, safely, and usefully it behaves.

Introduction

In Signals and Systems, a system is represented as a transformation from input to output. If the input is $$x(t)$$ and the output is $$y(t)$$, then the system describes the rule that converts $$x(t)$$ into $$y(t)$$.

This topic is important because before solving a circuit, filter, communication channel, control loop, or DSP algorithm, an engineer must know what kind of system it is.

Why This Topic Matters

Industry relevance

Audio systems must be stable and predictable.
Communication channels must be modeled to remove distortion and noise.
Control systems must avoid unstable output growth.
DSP filters are designed using linear time-invariant system theory.

Exam relevance

GATE asks direct property checks: linearity, time invariance, causality, and stability.
University exams ask definitions with examples and counterexamples.
Interviews often ask you to test a system using input-output logic instead of memorized statements.

Prerequisites

Meaning of input and output signals.
Basic function notation such as $$x(t)$$ and $$x[n]$$.
Time shifting and scaling ideas.
Basic algebra for testing superposition.
Concept of bounded and unbounded signal behavior.

Basic Intuition

Think of a system as a machine with behavior. A microphone preamplifier increases signal strength. A filter removes unwanted frequency components. A delay unit shifts a waveform in time. A sensor converts a physical quantity into an electrical signal.

A signal tells us what is changing. A system tells us how that change is modified.

Core Theory Explanation

System representation

$$y(t)=T\{x(t)\}$$

Here $$T$$ is the system operator. It may represent a circuit, a code algorithm, a mechanical device, a wireless channel, or a mathematical operation.

Major system properties

Property	Core idea	Engineering meaning
Linearity	Superposition holds	Signals can be analyzed separately and added
Time invariance	Behavior does not change with time	Same input today or later gives same shifted response
Causality	Output depends only on present and past inputs	Real-time systems cannot know the future
Stability	Bounded input gives bounded output	System does not blow up for normal signals
Memory	Output depends on values other than current input	Capacitors, inductors, delays, and filters store past behavior

Step-by-Step Mathematical Derivation

1. Linearity test

Assume two inputs produce two outputs:

$$x_1(t) \to y_1(t), \quad x_2(t) \to y_2(t)$$

The system is linear if scaling and addition pass through the system:

$$T\{a x_1(t)+b x_2(t)\}=aT\{x_1(t)\}+bT\{x_2(t)\}$$

Physically, this means the system does not create unexpected interaction between signals. If two tones enter a linear amplifier, the output is simply the sum of amplified tones.

2. Time invariance test

If input $$x(t)$$ gives output $$y(t)$$, delay the input by $$t_0$$:

$$x(t-t_0) \to y(t-t_0)$$

If the output is only delayed by the same amount and its shape is unchanged, the system is time invariant.

3. BIBO stability test

A bounded input satisfies:

$$|x(t)| \le M_x < \infty$$

The system is stable if the output also remains bounded:

$$|y(t)| \le M_y < \infty$$

Working Principle

An input signal enters the system.
The system applies an operation: amplification, attenuation, delay, differentiation, integration, filtering, or sampling.
Internal elements decide whether the output depends on present, past, or future values.
The output signal appears with changed amplitude, timing, frequency content, or shape.
System properties tell us whether that behavior is predictable, realizable, and safe.

Diagram Explanation

Input-System-Output Block Diagram Here

Linearity Superposition Diagram Here

Time Invariance Timing Diagram Here

BIBO Stability Waveform Here

Important Formulas

System operation

$$y(t)=T\{x(t)\}$$

Shows output as the result of applying a system rule to input.

Linearity

$$T\{a x_1+b x_2\}=aT\{x_1\}+bT\{x_2\}$$

Combines homogeneity and additivity.

Time invariance

$$x(t-t_0) \to y(t-t_0)$$

A delay in input causes the same delay in output.

Stability

$$|x(t)|\le M_x \Rightarrow |y(t)|\le M_y$$

Bounded input must produce bounded output.

Real-World Applications

Linear amplifiers in audio and RF circuits preserve waveform shape while increasing amplitude.
Time-invariant filters keep the same frequency response over time.
Causal systems are required for real-time control, audio processing, and communication receivers.
Stable systems are essential in power electronics, biomedical devices, and aircraft control.
Memory systems appear in digital filters, capacitive circuits, inductive circuits, and storage devices.

Solved Examples

Beginner example: memoryless or with memory

For $$y(t)=3x(t)$$, output depends only on current input. Therefore the system is memoryless.

Intermediate numerical: linearity

For $$y(t)=5x(t)$$:

$$T\{a x_1+b x_2\}=5(a x_1+b x_2)=a(5x_1)+b(5x_2)$$

So the system is linear.

Advanced problem: stability

For $$y(t)=tx(t)$$, even if $$x(t)=1$$ is bounded, output becomes $$y(t)=t$$, which grows without bound. Therefore the system is unstable.

Common Mistakes

Testing linearity with only one input instead of using superposition.
Assuming every circuit is time invariant; switched circuits may be time varying.
Calling a system causal just because it looks physically simple.
Confusing memoryless with causal. A memoryless system is causal, but a causal system can have memory.
Checking stability using one input only instead of bounded-input bounded-output logic.

Comparison Tables

System type	Condition	Example
Linear	Superposition holds	$$y(t)=2x(t)$$
Nonlinear	Superposition fails	$$y(t)=x^2(t)$$
Causal	No future input required	$$y(t)=x(t-2)$$
Non-causal	Needs future input	$$y(t)=x(t+2)$$

Interview Questions

What is a system in Signals and Systems?
How do you test linearity?
Is every memoryless system causal?
Why are real-time systems usually causal?
What does BIBO stability mean physically?
Give one practical example of a time-varying system.

Exam-Oriented Notes

For linearity, always test both additivity and homogeneity.
For time invariance, delay the input first and compare with delayed output.
For causality, check whether output uses future input like $$x(t+1)$$.
For stability, try bounded inputs that may expose unbounded output.
LTI systems are important because convolution and frequency response become powerful tools.

Revision Summary

A system maps input signals to output signals.
Linearity means superposition.
Time invariance means system behavior does not change with time.
Causality means no dependence on future input.
BIBO stability means bounded input gives bounded output.
Memory exists when output depends on past or future input values.

Practice Questions

Conceptual

Explain the difference between signal and system.
Why is causality important in real-time engineering?
Give one example of a system with memory.

Numerical

Test whether $$y(t)=4x(t)-2$$ is linear.
Check whether $$y(t)=x(t-3)$$ is causal.
Check stability of $$y(t)=e^t x(t)$$ for bounded input.

MCQs

A system satisfying superposition is: linear / nonlinear / causal / unstable.
A system depending on $$x(t+2)$$ is: causal / non-causal / memoryless / stable.
BIBO stability means bounded input gives: zero output / bounded output / infinite output / delayed output.

Introduction to Signals Next Mathematical Representation of Signals