GATE EE

Linear Time-Invariant (LTI) Systems for Signals and Systems : GATE Exam Notes

Lecture Notes

SEC 01

System Fundamentals

1What 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: \(y(t) = [x(t)]^2\)

1System Classification Overview

1Major 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

1GATE Focus

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

SEC 02

Linearity

1Linear Systems - Definition

A system is linear if it satisfies the superposition principle:

1Superposition 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: \(T\{ax(t)\} = aT\{x(t)\}\)

1GATE Tip

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

1Linear 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:

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

1GATE Common Error

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

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

1Testing 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.

SEC 03

Time Invariance

1Time-Invariant Systems

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

1Time 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 donâ€™t change with time
Same input produces same output regardless of when applied
Parameters are constant over time

1GATE Insight

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

1Time-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:

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

1GATE Key Point

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

1Testing 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.

SEC 04

LTI Systems

1Linear Time-Invariant (LTI) Systems

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

1Why 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

1GATE Gold Standard

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

1Impulse 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: \(y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k]h[n-k]\)

1GATE Formula

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

1Convolution - 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: \(x * \delta = x\)

1GATE Memory Aid

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

1Graphical 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 \(y(t)\).

1GATE Shortcut

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

SEC 05

Causality

1Causal Systems

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

1Causality 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

1GATE Reality Check

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

1Causal 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:

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

1GATE Key Point

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

1Causality in LTI Systems

For LTI Systems: Causality is determined by impulse response.

1Causality 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 \(t-\tau < 0\).

1GATE Quick Check

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

1Anti-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

SEC 06

Stability

1BIBO Stability

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

1BIBO 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:

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

1GATE Must-Know

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

1Testing 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:

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

1Marginal 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:

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

1GATE Distinction

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

SEC 07

Memory

1Memory 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.

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

SEC 08

Invertibility

1Invertible 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:

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

1GATE 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: \(x[n] = y[n+1]\)

SEC 09

System Interconnections

1Series (Cascade) Connection

Overall Impulse Response:

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

Properties:

Order doesnâ€™t matter (commutative)
\(n\) systems: \(h(t) = h_1(t) * h_2(t) * \cdots * h_n(t)\)
Transfer functions multiply: \(H(s) = H_1(s) \cdot H_2(s)\)

1Parallel 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

1Feedback 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: \(h_2(t) = \delta(t)\)

SEC 10

Common LTI System Examples

1RC 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: \(\tau = RC\)

1Ideal 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

1Ideal 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

1Moving 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):

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

SEC 11

GATE Problem-Solving Strategies

1Common 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: \(H(s) = \frac{H_1(s)}{1 + H_1(s)H_2(s)}\)

1GATE Problem-Solving Tips

1Quick 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

1Common 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

1Time Management

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

1Summary and Key Takeaways

1Essential 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

1GATE 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!