LTI System (Linear Time-Invariant System): Definition and Examples

LTI System (Linear Time-Invariant System)

Definition

An LTI (Linear Time-Invariant) system is a system that concurrently satisfies the linearity and time-invariance properties. These systems are fundamental in signal processing and control engineering because their behavior can be completely characterized by their impulse response.

Mathematical Representation

An LTI system is generally represented by the operator \( T \), which transforms an input signal \( x(t) \) into an output signal \( y(t) \):

\[ y(t) = T\{x(t)\} \]

Where:

• \( x(t) \) = Input signal
• \( y(t) \) = Output signal
• \( T\{\cdot\} \) = System operator

1. Linearity Property

A system is linear if it satisfies superposition and homogeneity. If:

\[ T\{x_1(t)\} = y_1(t) \quad \text{and} \quad T\{x_2(t)\} = y_2(t) \]

Then for any arbitrary constants \( a \) and \( b \):

\[ T\{a x_1(t) + b x_2(t)\} = a y_1(t) + b y_2(t) \]

2. Time-Invariance Property

A system is time-invariant if a delay in the input signal causes an identical delay in the output signal. If:

\[ T\{x(t)\} = y(t) \]

Then for any time delay \( t_0 \):

\[ T\{x(t - t_0)\} = y(t - t_0) \]

Step-by-Step Example

System Equation: \( y(t) = 3x(t) \)

Checking Linearity:

Testing the weighted sum of inputs:

\[ T\{ax_1(t) + bx_2(t)\} = 3(ax_1(t) + bx_2(t)) \]

\[ = a(3x_1(t)) + b(3x_2(t)) = ay_1(t) + by_2(t) \]

Result: The system is Linear.

Checking Time Invariance:

1. Delayed input output: \( y'(t) = 3x(t - t_0) \)

2. Delayed version of original output: \( y(t - t_0) = 3x(t - t_0) \)

Since \( y'(t) = y(t - t_0) \), the system is Time-Invariant.

Conclusion: \( y(t) = 3x(t) \) is a valid LTI system.

Key Characteristics of LTI Systems

Property	Technical Significance
Superposition	Simplifies complex signal analysis by breaking them into simpler components.
Time-Invariant	The system response is independent of the absolute time of application.
Impulse Response	Denoted as \( h(t) \), it allows for total system characterization.
Convolution	Output is determined by the formula: \( y(t) = x(t) * h(t) \).

Summary for Exams

• Definition: Must satisfy both Linearity and Time-Invariance.
• LTI Operator: \( T\{ax_1 + bx_2\} = aT\{x_1\} + bT\{x_2\} \).
• Time Shift: \( x(t-t_0) \to y(t-t_0) \).
• Common Examples: Scaling \( y(t) = k \cdot x(t) \), Summation, and Ideal Delays.
• Non-LTI Indicators: Squared terms \( x^2(t) \), absolute values \( |x(t)| \), or time-dependent coefficients like \( t \cdot x(t) \).

Return to Signal Processing Main Page →