LTI Systems

What is a System?

Definition: A system is a process that transforms input signals into output signals.

Mathematical Representation:

\[\begin{aligned} y(t) &= T\{x(t)\} \quad \text{(Continuous-time)} \\ y[n] &= T\{x[n]\} \quad \text{(Discrete-time)} \end{aligned}\]

Examples:

Amplifier: \(y(t) = Kx(t)\)
Differentiator: \(y(t) = \frac{dx(t)}{dt}\)
Delay: \(y[n] = x[n-1]\)
Squarer: \(y(t) = [x(t)]^2\)

System Classification Overview

Major System Properties

Linearity: Superposition principle
Time Invariance: Shift invariance
Causality: Output depends only on present/past inputs
Stability: Bounded input gives bounded output
Memory: Depends on past/future inputs
Invertibility: Unique input-output mapping

GATE Focus

LTI Systems are the most important class - they combine Linearity and Time Invariance, making analysis mathematically tractable.

Linear Systems - Definition

A system is linear if it satisfies the superposition principle:

Superposition Principle

If \(T\{x_1(t)\} = y_1(t)\) and \(T\{x_2(t)\} = y_2(t)\), then:

\[T\{ax_1(t) + bx_2(t)\} = ay_1(t) + by_2(t)\]

for any constants \(a\) and \(b\).

Two Components:

Additivity: \(T\{x_1(t) + x_2(t)\} = T\{x_1(t)\} + T\{x_2(t)\}\)
Homogeneity: \(T\{ax(t)\} = aT\{x(t)\}\)

GATE Tip

A system is linear if and only if it satisfies BOTH additivity and homogeneity (scaling).

Linear vs. Non-Linear Systems

Linear Systems:

\(y(t) = 3x(t)\)
\(y(t) = \frac{dx(t)}{dt}\)
\(y[n] = x[n] + x[n-1]\)
\(y(t) = \int_{-\infty}^{t} x(\tau) d\tau\)

Non-Linear Systems:

\(y(t) = [x(t)]^2\)
\(y(t) = |x(t)|\)
\(y(t) = x(t) + 5\)
\(y(t) = \cos(x(t))\)

GATE Common Error

\(y(t) = x(t) + c\) is NOT linear if \(c \neq 0\)!

Test: \(T\{0\} = 0 + c = c \neq 0\) violates homogeneity.

Testing for Linearity

Method 1: Direct Test

Choose two arbitrary inputs \(x_1(t)\) and \(x_2(t)\)
Find \(y_1(t) = T\{x_1(t)\}\) and \(y_2(t) = T\{x_2(t)\}\)
Check if \(T\{ax_1(t) + bx_2(t)\} = ay_1(t) + by_2(t)\)

Method 2: Zero Input Test

If \(T\{0\} \neq 0\), system is non-linear
If \(T\{0\} = 0\), further testing required

Example: Test \(y(t) = 2x(t) + 3\)

\[\begin{aligned} T\{0\} &= 2(0) + 3 = 3 \neq 0 \end{aligned}\]

Therefore, system is non-linear.

Time-Invariant Systems

Definition: A system is time-invariant if a time shift in input causes the same time shift in output.

Time Invariance Condition

If \(T\{x(t)\} = y(t)\), then \(T\{x(t - t_0)\} = y(t - t_0)\) for all \(t_0\).

For discrete-time: If \(T\{x[n]\} = y[n]\), then \(T\{x[n - n_0]\} = y[n - n_0]\).

Physical Interpretation:

System characteristics don’t change with time
Same input produces same output regardless of when applied
Parameters are constant over time

GATE Insight

Time-invariant systems have constant coefficients in their differential/difference equations.

Time-Invariant vs. Time-Varying Systems

Time-Invariant:

\(y(t) = 3x(t)\)
\(y(t) = x(t-2)\)
\(y[n] = x[n] + x[n-1]\)
\(y(t) = \int_{-\infty}^{t} x(\tau) d\tau\)

Time-Varying:

\(y(t) = tx(t)\)
\(y(t) = x(2t)\)
\(y[n] = nx[n]\)
\(y(t) = \cos(t)x(t)\)

GATE Key Point

Time scaling operations like \(x(2t)\) or \(x(t/2)\) make a system time-varying!

Testing for Time Invariance

Step-by-Step Procedure:

Method 1: Shift input first
- Replace \(x(t)\) with \(x(t - t_0)\) in system equation
- Obtain \(y_1(t) = T\{x(t - t_0)\}\)
Method 2: Shift output
- Find \(y(t) = T\{x(t)\}\)
- Replace \(t\) with \(t - t_0\) to get \(y_2(t) = y(t - t_0)\)
Compare: If \(y_1(t) = y_2(t)\), system is time-invariant

Example: Test \(y(t) = tx(t)\)

\[\begin{aligned} y_1(t) &= T\{x(t - t_0)\} = tx(t - t_0) \\ y_2(t) &= y(t - t_0) = (t - t_0)x(t - t_0) \end{aligned}\]

Since \(y_1(t) \neq y_2(t)\), system is time-varying.

Linear Time-Invariant (LTI) Systems

Definition: A system that is both linear and time-invariant.

Why LTI Systems are Important

Superposition: Can analyze complex inputs as sum of simple components
Convolution: Complete characterization using impulse response
Frequency Analysis: Eigenfunction property of complex exponentials
Mathematical Tractability: Well-developed analysis tools

Representation:

Continuous: Differential equations with constant coefficients
Discrete: Difference equations with constant coefficients

GATE Gold Standard

Most communication systems, filters, and control systems are designed as LTI systems.

Impulse Response of LTI Systems

Definition: Impulse response \(h(t)\) or \(h[n]\) is the output when input is unit impulse.

\[\begin{aligned} h(t) &= T\{\delta(t)\} \quad \text{(Continuous)} \\ h[n] &= T\{\delta[n]\} \quad \text{(Discrete)} \end{aligned}\]

Key Properties:

Complete Characterization: \(h(t)\) or \(h[n]\) completely describes LTI system
Convolution: \(y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau\)
Discrete: \(y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k]h[n-k]\)

GATE Formula

For LTI systems: Output = Input \(\circledast\) Impulse Response

Convolution - The Heart of LTI Systems

Convolution Integral:

\[y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau = \int_{-\infty}^{\infty} x(t-\tau)h(\tau) d\tau\]

Convolution Sum:

\[y[n] = \sum_{k=-\infty}^{\infty} x[k]h[n-k] = \sum_{k=-\infty}^{\infty} x[n-k]h[k]\]

Properties of Convolution:

Commutative: \(x * h = h * x\)
Associative: \((x * h_1) * h_2 = x * (h_1 * h_2)\)
Distributive: \(x * (h_1 + h_2) = x * h_1 + x * h_2\)
Identity: \(x * \delta = x\)

GATE Memory Aid

Convolution is "flip, shift, multiply, integrate/sum"

Graphical Convolution Method

Steps for Computing \(x(t) * h(t)\):

Plot \(x(\tau)\) and \(h(\tau)\) vs. \(\tau\)
Flip: Create \(h(-\tau)\) (reflection about \(\tau = 0\))
Shift: Create \(h(t-\tau)\) (shift by \(t\))
Multiply: Find \(x(\tau) \cdot h(t-\tau)\)
Integrate: \(y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau\)
Repeat for different values of \(t\)

Key Insight: The area of overlap between \(x(\tau)\) and \(h(t-\tau)\) gives \(y(t)\).

GATE Shortcut

For simple functions (rectangular pulses, exponentials), use analytical formulas rather than graphical method in exams.

Causal Systems

Definition: A system is causal if the output at any time depends only on present and past inputs, not future inputs.

Causality Condition

For continuous-time: \(y(t_0)\) depends only on \(x(t)\) for \(t \leq t_0\)

For discrete-time: \(y[n_0]\) depends only on \(x[n]\) for \(n \leq n_0\)

Physical Interpretation:

Realizable: Can be implemented in real-time
Non-anticipatory: Cannot predict future
Physically meaningful: Respects causality principle

GATE Reality Check

All physically realizable systems must be causal. Non-causal systems exist only in mathematical analysis or offline processing.

Causal vs. Non-Causal Systems

Causal Systems:

\(y(t) = x(t) + x(t-1)\)
\(y[n] = x[n] - x[n-2]\)
\(y(t) = \int_{-\infty}^{t} x(\tau) d\tau\)
\(y(t) = \frac{dx(t)}{dt}\)

Non-Causal Systems:

\(y(t) = x(t+1)\)
\(y[n] = x[n] + x[n+1]\)
\(y(t) = \int_{-\infty}^{\infty} x(\tau) d\tau\)
\(y(t) = x(-t)\)

GATE Key Point

Any system involving future inputs (\(x(t+T)\) where \(T > 0\)) is non-causal.

Causality in LTI Systems

For LTI Systems: Causality is determined by impulse response.

Causality Condition for LTI Systems

An LTI system is causal if and only if:

\[\begin{aligned} h(t) &= 0 \quad \text{for } t < 0 \quad \text{(Continuous)} \\ h[n] &= 0 \quad \text{for } n < 0 \quad \text{(Discrete)} \end{aligned}\]

Proof Concept:

\[y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau\]

For \(y(t)\) to depend only on \(x(\tau)\) with \(\tau \leq t\), we need \(h(t-\tau) = 0\) when \(\tau > t\), i.e., \(h(t-\tau) = 0\) when \(t-\tau < 0\).

GATE Quick Check

Plot \(h(t)\) or \(h[n]\) - if non-zero for negative time, system is non-causal.

Anti-Causal and Mixed Systems

Anti-Causal Systems:

Output depends only on future inputs
\(h(t) = 0\) for \(t > 0\) (LTI case)
Example: \(y(t) = x(t+1)\)

Mixed (Non-Causal) Systems:

Output depends on past, present, and future inputs
\(h(t) \neq 0\) for both \(t > 0\) and \(t < 0\)
Example: \(y(t) = x(t-1) + x(t+1)\)

Making Non-Causal Systems Causal:

Add sufficient delay: \(y(t) = x(t+1) \rightarrow y(t) = x(t+1-T)\) with \(T > 1\)
Trade-off: Introduces processing delay

BIBO Stability

Definition: A system is BIBO (Bounded-Input Bounded-Output) stable if every bounded input produces a bounded output.

BIBO Stability Condition

If \(|x(t)| \leq M_x < \infty\) for all \(t\), then \(|y(t)| \leq M_y < \infty\) for all \(t\).

Similarly for discrete-time: \(|x[n]| \leq M_x \Rightarrow |y[n]| \leq M_y\).

For LTI Systems: BIBO stability condition is:

\[\begin{aligned} \int_{-\infty}^{\infty} |h(t)| dt &< \infty \quad \text{(Continuous)} \\ \sum_{n=-\infty}^{\infty} |h[n]| &< \infty \quad \text{(Discrete)} \end{aligned}\]

GATE Must-Know

For LTI systems: Stability \(\Leftrightarrow\) Absolutely integrable/summable impulse response.

Testing for BIBO Stability

Method 1: Direct Test

Check if \(\int_{-\infty}^{\infty} |h(t)| dt < \infty\)
For discrete: \(\sum_{n=-\infty}^{\infty} |h[n]| < \infty\)

Method 2: Pole Analysis

Find system poles from transfer function
System is stable if all poles are in left half-plane (continuous) or inside unit circle (discrete)

Examples:

\[\begin{aligned} h(t) &= e^{-t}u(t) \quad \Rightarrow \quad \int_{0}^{\infty} e^{-t} dt = 1 < \infty \quad \text{(Stable)} \\ h(t) &= e^{t}u(t) \quad \Rightarrow \quad \int_{0}^{\infty} e^{t} dt = \infty \quad \text{(Unstable)} \\ h[n] &= \left(\frac{1}{2}\right)^n u[n] \quad \Rightarrow \quad \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n = 2 < \infty \quad \text{(Stable)} \end{aligned}\]

Marginal Stability

Definition: A system is marginally stable if some bounded inputs produce bounded outputs, but the system is on the verge of instability.

Characteristics:

Poles on imaginary axis (continuous) or unit circle (discrete)
Impulse response is not absolutely integrable/summable
Response to sinusoidal inputs remains bounded

Examples:

\[\begin{aligned} h(t) &= \cos(t) \quad \text{(Marginally stable)} \\ h[n] &= \cos\left(\frac{\pi n}{4}\right) \quad \text{(Marginally stable)} \\ h(t) &= u(t) \quad \text{(Marginally stable - integrator)} \end{aligned}\]

GATE Distinction

Marginally stable \(\neq\) BIBO stable. For practical systems, BIBO stability is required.

Memory in Systems

Memoryless Systems:

Output depends only on present input
No storage of past information
Examples: \(y(t) = 2x(t)\), \(y[n] = [x[n]]^2\)

Memory Systems:

Output depends on past and/or future inputs
Requires storage elements
Examples: \(y(t) = x(t) + x(t-1)\), \(y[n] = \sum_{k=0}^{n} x[k]\)

For LTI Systems: Memory is characterized by impulse response duration.

\[\begin{aligned} \text{Memoryless:} \quad h(t) &= K\delta(t) \quad \text{or} \quad h[n] = K\delta[n] \\ \text{Finite Memory:} \quad h(t) &= 0 \text{ for } |t| > T \\ \text{Infinite Memory:} \quad h(t) &\neq 0 \text{ for some } |t| > T \text{ (any } T\text{)} \end{aligned}\]

Invertible Systems

Definition: A system is invertible if distinct inputs produce distinct outputs, and an inverse system exists.

Condition: \(T^{-1}\{T\{x(t)\}\} = x(t)\) for all \(x(t)\)

For LTI Systems: Invertible if and only if:

\[\int_{-\infty}^{\infty} h(t) dt \neq 0 \quad \text{or} \quad \sum_{n=-\infty}^{\infty} h[n] \neq 0\]

GATE Examples

Integrator: \(y(t) = \int_{-\infty}^{t} x(\tau) d\tau \Leftrightarrow\) Differentiator: \(x(t) = \frac{dy(t)}{dt}\)
Delay: \(y[n] = x[n-1] \Leftrightarrow\) Advance: \(x[n] = y[n+1]\)

Series (Cascade) Connection

Overall Impulse Response:

\[h(t) = h_1(t) * h_2(t) = h_2(t) * h_1(t)\]

Properties:

Order doesn’t matter (commutative)
\(n\) systems: \(h(t) = h_1(t) * h_2(t) * \cdots * h_n(t)\)
Transfer functions multiply: \(H(s) = H_1(s) \cdot H_2(s)\)

Parallel Connection

Overall Impulse Response:

\[h(t) = h_1(t) + h_2(t)\]

Properties:

Transfer functions add: \(H(s) = H_1(s) + H_2(s)\)
\(n\) systems: \(h(t) = h_1(t) + h_2(t) + \cdots + h_n(t)\)
Each system processes the same input independently

Feedback Connection

Overall Transfer Function:

\[H(s) = \frac{H_1(s)}{1 + H_1(s)H_2(s)}\]

Types:

Negative Feedback: Stabilizing effect
Positive Feedback: Can cause instability
Unity Feedback: \(h_2(t) = \delta(t)\)

RC Low-Pass Filter

Circuit: RC circuit with output across capacitor

Differential Equation:

\[RC\frac{dy(t)}{dt} + y(t) = x(t)\]

Impulse Response:

\[h(t) = \frac{1}{RC}e^{-t/RC}u(t)\]

Transfer Function:

\[H(s) = \frac{1}{1 + sRC}\]

Properties:

Stable: Pole at \(s = -1/RC\) (left half-plane)
Causal: \(h(t) = 0\) for \(t < 0\)
Low-pass: Attenuates high frequencies
Time constant: \(\tau = RC\)

Ideal Integrator

System Equation:

\[y(t) = \int_{-\infty}^{t} x(\tau) d\tau\]

Impulse Response:

\[h(t) = u(t)\]

Transfer Function:

\[H(s) = \frac{1}{s}\]

Properties:

Marginally Stable: Pole at \(s = 0\)
Causal: \(h(t) = 0\) for \(t < 0\)
Infinite Memory: \(h(t) \neq 0\) for all \(t > 0\)
Invertible: Inverse is differentiator

Ideal Differentiator

System Equation:

\[y(t) = \frac{dx(t)}{dt}\]

Impulse Response:

\[h(t) = \frac{d\delta(t)}{dt}\]

Transfer Function:

\[H(s) = s\]

Properties:

Unstable: No finite poles (unbounded response)
Causal: In generalized sense
Memoryless: In generalized sense
High-pass: Emphasizes high frequencies

Moving Average Filter

Continuous-Time:

\[y(t) = \frac{1}{T}\int_{t-T}^{t} x(\tau) d\tau\]

Discrete-Time:

\[y[n] = \frac{1}{M}\sum_{k=0}^{M-1} x[n-k]\]

Properties:

Stable: Finite impulse response
Causal: Uses only past inputs
Finite Memory: Memory length \(T\) or \(M\)
Smoothing: Reduces noise and fluctuations

Impulse Response (Discrete):

\[h[n] = \begin{cases} \frac{1}{M} & \text{for } 0 \leq n \leq M-1 \\ 0 & \text{otherwise} \end{cases}\]

Common GATE Question Types

Type 1: System Property Identification

Given system equation, determine if linear/nonlinear
Test for time-invariance using shift operations
Check causality from system description
Verify BIBO stability using impulse response

Type 2: Convolution Problems

Compute \(x(t) * h(t)\) analytically
Graphical convolution for piecewise functions
Properties of convolution for simplification

Type 3: System Interconnections

Series: \(H(s) = H_1(s) \cdot H_2(s)\)
Parallel: \(H(s) = H_1(s) + H_2(s)\)
Feedback: \(H(s) = \frac{H_1(s)}{1 + H_1(s)H_2(s)}\)

GATE Problem-Solving Tips

Quick Tests

Linearity: Test \(T\{0\} = 0\) first
Time-invariance: Look for time-dependent coefficients
Causality: Check for future inputs (\(t > 0\) terms)
Stability: For exponentials, check sign of exponent

Common Mistakes to Avoid

\(y(t) = x(t) + c\) is NOT linear if \(c \neq 0\)
\(y(t) = x(2t)\) is time-varying, not time-invariant
Marginal stability \(\neq\) BIBO stability
Remember to check both necessary and sufficient conditions

Time Management

Use properties and shortcuts when possible
Sketch impulse responses for quick analysis
Practice standard convolution integrals

Summary and Key Takeaways

Essential Concepts

LTI Systems: Foundation of linear signal processing
Convolution: Complete characterization via impulse response
System Properties: Linearity, time-invariance, causality, stability
Interconnections: Series, parallel, and feedback configurations

GATE Success Formula

Understand: Physical meaning of each property
Practice: Standard problem types repeatedly
Memorize: Key formulas and conditions
Apply: Systematic approach to problem-solving

Master LTI Systems = Master Signals and Systems!

System Fundamentals

What is a System?

System Classification Overview

Major System Properties

GATE Focus

Linearity

Linear Systems - Definition

Superposition Principle

GATE Tip

Linear vs. Non-Linear Systems

GATE Common Error

Testing for Linearity

Time Invariance

Time-Invariant Systems

Time Invariance Condition

GATE Insight

Time-Invariant vs. Time-Varying Systems

GATE Key Point

Testing for Time Invariance

Linear Time-Invariant (LTI) Systems

Why LTI Systems are Important

GATE Gold Standard

Impulse Response of LTI Systems

GATE Formula

Convolution - The Heart of LTI Systems

GATE Memory Aid

Graphical Convolution Method

GATE Shortcut

Causality

Causal Systems

Causality Condition

GATE Reality Check

Causal vs. Non-Causal Systems

GATE Key Point

Causality in LTI Systems

Causality Condition for LTI Systems

GATE Quick Check

Anti-Causal and Mixed Systems

Stability

BIBO Stability

BIBO Stability Condition

GATE Must-Know

Testing for BIBO Stability

Marginal Stability

GATE Distinction

Memory

Memory in Systems

Invertibility

Invertible Systems

GATE Examples

System Interconnections

Series (Cascade) Connection

Parallel Connection

Feedback Connection

Common LTI System Examples

RC Low-Pass Filter

Ideal Integrator

Ideal Differentiator

Moving Average Filter

GATE Problem-Solving Strategies

Common GATE Question Types

GATE Problem-Solving Tips

Quick Tests

Common Mistakes to Avoid

Time Management

Summary and Key Takeaways

Essential Concepts

GATE Success Formula