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Signals and System Linear Time Invariance Introduction

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Continued from Signals & System Basic System Properties

Introducing Time Invariance

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Linear Time Invariance (LTI) and Linearity

Linear Time Invariance (LTI) plays a fundamental role in signal and system analysis for two major reasons:

  1. Many physical processes possess the property of LTI.
  2. LTI systems can be analysed using mathematical tools, which provide a structured approach to understanding system behaviour.

Mathematical Tools for LTI Analysis

LTI systems can be studied using various mathematical tools, including:

1. Convolution

  • When Needed: Used to compute the output of an LTI system given an input and its impulse response

2. Fourier Transform

  • When Needed: Essential for frequency domain analysis, stability, and system response characterization.

3. Laplace Transform

  • When Needed: Helps analyse continuous-time LTI systems and their stability. It is also useful for solving differential equations.

4. Eigenfunction Analysis

  • When Needed: Used to simplify LTI system response calculations by leveraging the fact that exponentials remain exponentials after system transformation.

We will explore these tools in detail in further sections.

Superposition Property in LTI Systems

  • If an input signal is expressed as a combination of simpler signals, the system’s response can be computed using the responses to those basic signals.
  • This enables modular analysis by breaking complex signals into simpler parts, making analysis easier.

References & Extras

LTI Systems in Real-World Applications

LTI systems are widely used in practical applications, such as:

1. Electrical Circuits (RL, RC, RLC Circuits in an Audio Processor)

  • Tools Used:
    • Differential Equations: To model circuit response.
    • Laplace Transform: To analyse frequency response and transient behavior.
    • Impulse Response & Convolution Integral: To determine output for any given input.

2. Audio Signal Processing (Equalizers, Noise Reduction)

  • Example: A graphic equalizer that boosts or attenuates certain frequencies in an audio signal.
  • Why LTI? The filter applies the same effect regardless of when the sound was played.
  • Tools Used:
    • Fourier Transform: To analyse how different frequencies are affected.
    • Convolution: To compute the output signal from impulse response.

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