Linear Time-Invariant (LTI) Systems

Why it matters

Linear Time-Invariant (LTI) Systems are fundamental in the analysis and design of signal processing and control systems. They simplify complex systems into manageable models, making it easier to predict system behavior and design appropriate responses.

Key ideas

• Linearity: An LTI system follows the principles of superposition, meaning T{ax1 + bx2} = aT{x1} + bT{x2}: both additivity and homogeneity are required.
• Time-Invariance: The system's behavior and characteristics do not change over time. If an input is delayed, the output is equally delayed.
• Impulse Response: The response of an LTI system to a unit impulse input, which characterizes the LTI zero-state response completely.
• Convolution: The output of an LTI system can be determined by convolving the input signal with the system's impulse response.
• Frequency Response: LTI systems can be analyzed in the frequency domain using tools like the Fourier Transform, which simplifies the convolution operation to multiplication.

Formulas

• y(t) = x(t) * h(t)

  • y(t): Output signal
  • x(t): Input signal
  • h(t): Impulse response
  • *: Convolution operator

• H(f) = ∫ from −∞ to ∞ h(t) e^(-j2πft) dt

  • H(f): Frequency response
  • h(t): Impulse response
  • f: Frequency (Hz)
  • j: Imaginary unit

Worked example

Given: An LTI system with impulse response h(t) = e^(-2t)u(t), where u(t) is the unit step function. Find the output y(t) for input x(t) = e^(-t)u(t).

1. Identify the impulse response and input:

   • h(t) = e^(-2t)u(t)
   • x(t) = e^(-t)u(t)

2. Apply convolution:

   • y(t) = ∫ x(τ)h(t-τ) dτ

3. Substitute the given functions:

   • y(t) = ∫ e^(-τ)u(τ) e^(-2(t-τ))u(t-τ) dτ

4. Evaluate the integral:

   • y(t) = ∫ e^(-τ) e^(-2t+2τ) dτ
   • y(t) = e^(-2t) ∫ e^(τ) dτ

5. Solve the integral:

   • y(t) = e^(-2t) [e^(τ)] from 0 to t
   • y(t) = e^(-2t) (e^(t) - 1)

6. Final output:

   • y(t) = (e^(-t) - e^(-2t))u(t)

Common mistakes

• Confusing the properties of linearity and time-invariance.
• Incorrectly applying the convolution integral limits.
• Forgetting to include the unit step function in the impulse response.

For GATE EC

Questions often involve finding the output of an LTI system given an input and impulse response, or analyzing the system's frequency response. Practice convolution and understanding system properties.

Quick check

1. What is the principle of superposition in LTI systems?
2. How does time-invariance affect the system's response to delayed inputs?
3. What is the significance of the impulse response in an LTI system?

Answers: 1. Responses obey weighted superposition, including additivity and homogeneity. 2. The output is equally delayed. 3. It characterizes the LTI zero-state response completely.