Continuous-Time and Discrete-Time Systems

Why it matters

Continuous-time and discrete-time systems are fundamental in the analysis and design of signal processing applications, such as communication systems, control systems, and digital signal processing. Understanding these systems helps engineers to model real-world phenomena and design systems that can process signals effectively.

Key ideas

• Continuous-Time Systems: These systems process continuous-time signals, which are defined for every instant of time. Examples include analog electronic circuits and mechanical systems.
• Discrete-Time Systems: These systems process discrete-time signals, which are defined only at discrete intervals of time. Examples include digital filters and computer algorithms.
• System Properties: Systems can be characterized by properties such as linearity, time-invariance, causality, stability, and memory.
• Impulse Response: The response of a system to a unit impulse input, which determines the zero-state behavior for an LTI system, not an arbitrary nonlinear or time-varying system.
• Convolution: A mathematical operation used to determine the output of a linear time-invariant (LTI) system given its input and impulse response.

Formulas

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

  • y(t): Output signal
  • x(τ): Input signal
  • h(t - τ): Impulse response
  • t, τ: Time variables

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

  • y[n]: Output signal
  • x[k]: Input signal
  • h[n - k]: Impulse response
  • n, k: Discrete time indices

Worked example

Let x[0…2] = {1,2,3} and h[0…2] = {1,0,−1}, zero elsewhere. Compute their linear convolution. y[0] = 1 × 1 = 1. y[1] = 1 × 0 + 2 × 1 = 2. y[2] = 1 × (−1) + 2 × 0 + 3 × 1 = 2. y[3] = 2 × (−1) + 3 × 0 = −2. y[4] = 3 × (−1) = −3. Thus y[0…4] = {1,2,2,−2,−3}. The output length is 3 + 3 − 1 = 5; it is not obtained by multiplying arrays element-by-element or keeping only three entries.

Common mistakes

• Confusing continuous-time and discrete-time systems.
• Incorrectly applying the convolution formula, especially the limits of summation or integration.
• Assuming all systems are linear and time-invariant without verification.

For GATE EC

Questions often involve analyzing system properties, computing convolution, and determining system responses. Practice problems on convolution, system stability, and impulse response analysis are essential.

Quick check

1. What is the main difference between continuous-time and discrete-time systems?
2. How is the output of an LTI system determined?
3. What property does the impulse response of a system describe?

Answers: 1. Continuous-time systems process signals at every instant, while discrete-time systems process signals at discrete intervals. 2. By convolution of the input with the impulse response. 3. The system's behavior or response characteristics.