Unit
- Signal definition: A signal is a function that conveys information about a phenomenon. In engineering, signals represent physical quantities such as voltage, pressure, or light intensity that vary with time or space.
- Continuous-time signals: These are defined for every instant of time, often denoted as x(t). Real-world analog signals like microphone output or temperature readings are continuous in nature.
- Discrete-time signals: These are sequences of values at distinct time instants, written as x[n] where n is an integer. Digital systems process discrete-time signals obtained through sampling.
- Analog vs. digital: Analog signals have continuous amplitude and time; digital signals have both discrete time and quantized amplitude. Analog-to-digital conversion bridges the two domains.
- Periodic and aperiodic signals: Periodic signals repeat identically after a fixed period T. Many practical signals are aperiodic but can be analyzed using Fourier techniques.
- Energy and power signals: Energy signals have finite total energy (e.g., transient pulses). Power signals have infinite energy but finite average power (e.g., steady sine waves).
- Deterministic vs. random signals: Deterministic signals are exactly predictable (sinusoids). Random signals like noise require probabilistic treatment and statistical methods.
- Multidimensional signals: Images are 2D signals; video adds a time dimension. Medical MRI scans produce 3D volumes. Processing extends to higher dimensions.
- Sampling theorem: The Nyquist-Shannon sampling theorem states that a continuous-time signal must be sampled at a rate at least twice its highest frequency to avoid aliasing. This fundamental principle governs the conversion of analog signals to discrete-time signals for digital processing.
- Quantization and bit depth: Quantization maps continuous amplitudes to discrete levels, introducing quantization error. Higher bit depth reduces error but increases data size. For example, 16-bit audio provides 65,536 levels, balancing quality and storage.
- Signal processing applications: Signal processing is used in audio compression (MP3), image enhancement, radar, biomedical monitoring, and telecommunications. Techniques like filtering, Fourier analysis, and wavelet transforms are central to these applications.
- Convolution and linear systems: Convolution describes the output of a linear time-invariant system for a given input. It is fundamental in filtering operations, where a signal is convolved with a filter impulse response to modify its frequency content.
Code Example
Python
import numpy as np
import matplotlib.pyplot as plt
t = np.linspace(0, 1, 500)
analog_signal = np.sin(2 * np.pi * 5 * t)
n = np.arange(0, 50)
discrete_signal = np.sin(2 * np.pi * 0.1 * n)
plt.figure()
plt.plot(t, analog_signal, label='continuous')
plt.stem(n, discrete_signal, label='discrete')
plt.legend()
plt.show()The example generates a 5 Hz continuous-time sine wave and its sampled discrete-time counterpart using a sampling frequency of 50 Hz. Visual comparison highlights the difference in representation.
Quick Revision
- Signals are functions that carry information, classified as continuous or discrete in time.
- Analog signals have continuous amplitude and time; digital signals are quantized and sampled.
- Periodic signals repeat over a period; aperiodic signals do not.
- Energy signals have finite energy; power signals have finite average power.