Introduction to Signals and Systems

Why it matters

Signals and systems are fundamental to electrical engineering, as they form the basis for understanding how information is transmitted and processed in various applications, from telecommunications to control systems. Mastery of these concepts is crucial for designing and analyzing systems that are efficient and reliable.

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 only at discrete intervals.
• System: A system is an entity that processes signals. It can be characterized by its response to input signals, and it can be linear or nonlinear, time-invariant or time-variant.
• Continuous-Time Signals: These are signals defined for every value of time. Examples include idealized microphone voltage or temperature as a function of every time instant; periodically logged readings are discrete-time.
• 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.

Discrete time describes the time index; a digital signal also uses quantized amplitude values. A discrete-time signal need not be amplitude-quantized.

Formulas

• x(t) = A·sin(ωt + φ)
  • x(t): Signal as a function of time (unit: varies)
  • A: Amplitude (unit: varies)
  • ω: Angular frequency (unit: rad/s)
  • t: Time (unit: s)
  • φ: Phase angle (unit: rad)

Worked example

Problem: A continuous-time signal is given by x(t) = 5·sin(2π·50t + π/4). Find the amplitude, frequency, and phase of the signal.

1. Identify the amplitude: From the equation x(t) = A·sin(ωt + φ), the amplitude A is 5.
2. Determine the frequency: The angular frequency ω is 2π·50, so the frequency f is ω / (2π) = 50 Hz.
3. Find the phase: The phase φ is π/4 radians.

Answer: Amplitude = 5, Frequency = 50 Hz, Phase = π/4 rad.

Common mistakes

• Confusing continuous-time and discrete-time signals.
• Misinterpreting the phase angle in radians versus degrees.
• Forgetting to convert angular frequency to frequency.

For GATE EE

Questions often involve analyzing signal properties, system responses, and transformations like Fourier and Laplace. Practice problems on identifying signal characteristics and system behavior under different conditions.

Quick check

1. What is the difference between continuous-time and discrete-time signals?
2. Define an LTI system.
3. How do you convert angular frequency to frequency?

Answers: 1. Continuous-time signals are defined for every time value, while discrete-time signals are defined at discrete intervals. 2. An LTI system is linear and time-invariant. 3. Frequency f is ω / (2π).