Chapter 1

Linear Time-Invariant (LTI) Systems

A Linear Time-Invariant (LTI) system is a special class of systems widely used in signal processing, control systems, and communications. These systems are easy to analyze and have well-defined mathematical properties.

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1. Properties of LTI Systems

An LTI system satisfies two important properties:

1. Linearity:

   • The system follows the principles of superposition and scaling.
   • If input$x_1(t)$produces output$y_1(t)$and$x_2(t)$produces$y_2(t)$, then for any constants$a$and$b$:$\text{Input: } x(t) = a x_1(t) + b x_2(t)$$\text{Output: } y(t) = a y_1(t) + b y_2(t)$

2. Time Invariance:

   • The system does not change over time.
   • If input$x(t)$produces output$y(t)$, then a time-shifted input$x(t - T)$results in a time-shifted output$y(t - T)$.

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2. Representation of LTI Systems

LTI systems can be described in several ways:

A. Differential Equations (for Continuous-Time Systems)

An LTI system can be represented as:

$a_n \frac{d^n y(t)}{dt^n} + a_{n-1} \frac{d^{n-1} y(t)}{dt^{n-1}} + \dots + a_1 \frac{d y(t)}{dt} + a_0 y(t) = b_m \frac{d^m x(t)}{dt^m} + \dots + b_1 \frac{d x(t)}{dt} + b_0 x(t)$

This equation describes how the input

$x(t)$

and its derivatives affect the output

$y(t)$

.

B. Difference Equations (for Discrete-Time Systems)

For discrete-time systems, LTI systems are represented using:

$y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k]$

where

$x[n]$

is the input and

$y[n]$

is the output.

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3. Impulse Response of LTI Systems

• The impulse response$h(t)$(or$h[n]$in discrete time) is the system’s output when the input is a unit impulse function$\delta(t)$.
• Any arbitrary input$x(t)$can be expressed as a weighted sum of shifted impulses, leading to the convolution integral.

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4. Convolution in LTI Systems

The output of an LTI system is obtained by convolving the input signal with the system's impulse response:

Continuous-Time Convolution

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

Discrete-Time Convolution

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

Convolution allows us to compute the system's output for any given input.

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5. Frequency Domain Analysis of LTI Systems

LTI systems can also be analyzed using Fourier and Laplace Transforms:

• Fourier Transform: Describes the system’s behavior in the frequency domain.
  • The system’s frequency response is given by$H(f) = \mathcal{F} \{ h(t) \}$.
• Laplace Transform: Useful for solving differential equations and stability analysis.
  • The system function is$H(s) = \mathcal{L} \{ h(t) \}$.
• Z-Transform: Used for discrete-time LTI systems.

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6. Stability of LTI Systems

An LTI system is stable if it produces a bounded output for every bounded input. This is called Bounded Input, Bounded Output (BIBO) Stability.

Stability Condition (Continuous-Time)

An LTI system is stable if:

$\int_{-\infty}^{\infty} |h(t)| dt < \infty$

Stability Condition (Discrete-Time)

A discrete-time LTI system is stable if:

$\sum_{n=-\infty}^{\infty} |h[n]| < \infty$

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7. Applications of LTI Systems

LTI systems are widely used in:

• Communication Systems (e.g., filters, modulation, equalizers)
• Control Systems (e.g., feedback controllers, PID controllers)
• Signal Processing (e.g., audio and image filtering)
• Mechanical and Electrical Systems (e.g., circuit analysis, vibrations)

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