Linear Time-Invariant (LTI) Systems

Why it matters

Linear Time-Invariant (LTI) Systems are foundational in electrical engineering because they simplify the analysis and design of systems that process signals. They are used in various applications such as telecommunications, control systems, and signal processing, making them essential for engineers to understand and apply.

Key ideas

• Linearity: An LTI system is linear if it satisfies the principles of superposition and homogeneity. This means the response to a sum of inputs is the sum of the responses to each input, and scaling an input scales the response.
• Time-Invariance: A system is time-invariant if its behavior and characteristics do not change over time. This implies that a time shift in the input signal results in an identical time shift in the output signal.
• Impulse Response: The response of an LTI system to a unit impulse input is called the impulse response, denoted as h(t) for continuous-time systems or h[n] for discrete-time systems.
• Convolution: The output of an LTI system can be determined by convolving the input signal with the system's impulse response.
• Transfer Function: In the frequency domain, the behavior of an LTI system is described by its transfer function, which is the Laplace Transform of the impulse response for continuous-time systems or the Z-Transform for discrete-time systems.

Convolution describes the zero-state input–output response. Integrate over −∞ < τ < ∞ or sum over all integer k; support restrictions reduce these limits in a particular problem. Initial stored energy may add a zero-input response.

Formulas

• Convolution (Continuous-Time): y(t) = ∫ x(τ)h(t - τ)dτ

  • y(t): Output signal
  • x(τ): Input signal
  • h(t - τ): Impulse response
  • τ: Integration variable

• Convolution (Discrete-Time): y[n] = Σ x[k]h[n - k]

  • y[n]: Output signal
  • x[k]: Input signal
  • h[n - k]: Impulse response
  • k: Summation index

• Transfer Function (Continuous-Time): H(s) = L{h(t)}

  • H(s): Transfer function
  • L{}: Laplace Transform
  • h(t): Impulse response

• Transfer Function (Discrete-Time): H(z) = Z{h[n]}

  • H(z): Transfer function
  • Z{}: Z-Transform
  • h[n]: Impulse response

Worked example

Given: A continuous-time 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. Use the convolution integral: 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. Simplify and solve the integral: y(t) = ∫ e^(-τ) * e^(-2t + 2τ)u(τ)u(t-τ)dτ y(t) = e^(-2t) ∫ e^(τ)dτ from 0 to t y(t) = e^(-2t) [e^(τ)] from 0 to t y(t) = e^(-2t) [e^(t) - 1]

5. 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 or summation.
• Forgetting to include the unit step function when dealing with causal systems.

For GATE EE

Questions on LTI systems often involve finding the output of a system given its impulse response and input using convolution. Practice problems on convolution, both in continuous and discrete time, and understanding the properties of LTI systems are crucial.

Quick check

1. What is the principle of superposition in LTI systems?
2. How is the impulse response of an LTI system used?
3. What is the significance of the transfer function in LTI systems?

Answers: 1. The response to a sum of inputs is the sum of the responses to each input. 2. It is used to determine the output of the system for any input using convolution. 3. It describes the system's behavior in the frequency domain.