Introduction to Signals and Systems

Why it matters

Signals and systems are fundamental to understanding and designing modern communication, control, and electronic systems. They allow engineers to analyze how systems respond to different inputs, which is crucial for developing efficient and reliable technology.

Key ideas

• Signal: A signal is a function that conveys information about a phenomenon. It can be continuous-time or discrete-time, depending on whether it is defined for every instant of time or at discrete intervals.
• System: A system is an entity that processes input signals to produce output signals. Systems can be classified as linear or nonlinear, time-invariant or time-variant, and causal or non-causal.
• Continuous-Time Signals: These are signals defined for every value of time. Examples include analog audio signals and temperature readings.
• Discrete-Time Signals: These are signals defined only at discrete points in time, such as digital audio signals.
• Linear Time-Invariant (LTI) Systems: These systems have properties of linearity and time-invariance, making them easier to analyze using mathematical tools like the Fourier and Laplace transforms.

Formulas

• x(t) = A·sin(ωt + φ)

  • x(t): Continuous-time signal
  • A: Amplitude (same physical unit as the signal)
  • ω: Angular frequency (radians per second)
  • t: Time (seconds)
  • φ: Phase angle (radians)

• y[n] = Σ (h[k]·x[n-k])

  • y[n]: Output signal
  • h[k]: Impulse response of the system
  • x[n-k]: Input signal
  • Σ: Summation over all values of k

Worked example

Let x(t) = 5sin(200πt + π/4) for all real t and h(t) = exp(−50t)u(t). The stable LTI system has frequency response H(jω) = 1/(50 + jω). At ω = 200π rad/s, magnitude is 1/√(50² + (200π)²) and phase is −arctan(200π/50). Therefore y(t) = [5/√(50² + (200π)²)]sin[200πt + π/4 − arctan(200π/50)], approximately 0.007933sin(200πt − 0.7060). This is the response to a sinusoid existing for all time. If the input were multiplied by u(t), an additional startup transient would be present. The convolution limits for the original all-time sinusoid are −∞ to t, not 0 to t.

Common mistakes

• Confusing continuous-time and discrete-time signals.
• Incorrectly applying the properties of LTI systems.
• Errors in convolution calculations, especially with limits of integration.

For GATE EC

• Questions often involve analyzing LTI systems, calculating convolution, and transforming signals using Fourier or Laplace methods.
• Practice problems on signal classification, system properties, and basic transformations.

Quick check

1. What is the difference between continuous-time and discrete-time signals?
2. Define an LTI system.
3. What is the significance of the impulse response in a system?

Answers: 1. Continuous-time signals are defined for every instant of time, while discrete-time signals are defined at discrete intervals. 2. An LTI system is a system that is both linear and time-invariant. 3. The impulse response characterizes the output of a system in response to an impulse input.