1: Definition and classification of signals

Definition and Classification of Signals

A signal is a function that conveys information about the behavior or attributes of a physical phenomenon. Mathematically, it is represented as a dependent variable (e.g., voltage, pressure) varying with respect to an independent variable (e.g., time, space). Signals are fundamental in engineering for analyzing systems in communication, control, and signal processing.

Key Classifications:

1. Continuous-Time (CT) vs. Discrete-Time (DT) Signals:

   • CT signals: Defined for all values of a continuous independent variable (e.g., $x(t) = \sin(t)$).
   • DT signals: Defined only at specific, usually equally spaced, points (e.g., $x[n] = \sin(n)$).

2. Analog vs. Digital Signals:

   • Analog: Both time and amplitude are continuous (e.g., natural sound waves).
   • Digital: Time and amplitude are discrete and quantized (e.g., binary sequences).

3. Deterministic vs. Random Signals:

   • Deterministic: Follows a precise mathematical model; values are predictable (e.g., $x(t) = e^{-t}u(t)$).
   • Random: Unpredictable; characterized by probability distributions (e.g., thermal noise).

4. Periodic vs. Aperiodic Signals:

   • Periodic: Repeats every interval $T$ (for CT) or $N$ (for DT): $x(t + T) = x(t)$.
   • Aperiodic: Non-repeating (e.g., a decaying exponential).

5. Energy vs. Power Signals:

   • Energy signal: Finite total energy ($0 < E < \infty$) and zero average power. Example: A short-duration pulse.
   • Power signal: Infinite energy but finite average power ($0 < P < \infty$). Example: Periodic signals like sinusoids.

6. Even vs. Odd Signals:

   • Even: Symmetric about the vertical axis ($x(-t) = x(t)$).
   • Odd: Antisymmetric about the origin ($x(-t) = -x(t)$). Any signal can be decomposed into even and odd components.

Practical Significance:

Classifying signals helps select appropriate mathematical tools for analysis. For instance:

• CT signals use differential equations and Fourier transforms.
• DT signals leverage difference equations and Z-transforms.
• Energy/power distinctions determine convergence in transform domains.
• Periodic signals simplify analysis via Fourier series.

Understanding these categories provides a foundation for manipulating signals in subsequent topics like filtering, sampling, and spectral analysis.