Skip to content

Signal Module

The Signal Module focuses on generating and managing synthetic waveforms. Central to its operation is the principle that any aperiodic signal can be approximated by a sum of sinusoidal components (Fourier Transform theory). In essence, complex real‑world signals, not strictly periodic, can be well‑represented or approximated by a set of sinusoidal basis functions.

Conceptual Overview

Constructing Base Signals

Signals are typically created by crafting a foundational sinusoidal waveform with possible sporadic interruptions. The base signals are formed by overlaying multiple sinusoidal waves. According to Fourier analysis, any periodic signal can be decomposed into pure sinusoidal components. Extending that concept, an aperiodic signal can be artificially treated as periodic by repetition over a long period \( T_0 \), enabling a similar decomposition. As \( T_0 \) approaches infinity, the discrete spectral components merge into the continuous Fourier Transform.

\[ x_P(t) = \sum_{n=-\infty}^{\infty} C_n e^{j 2\pi n f_0 t}, \]

where \( C_n \) are coefficients defined by:

\[ C_n = \frac{1}{T_0} \int_{-T_0/2}^{T_0/2} x_P(t) e^{-j 2\pi n f_0 t} \, dt. \]

In the limit as \( T_0 \to \infty \), these expressions become the standard Fourier Transform for aperiodic signals, justifying the construction of complex synthetic signals from purely sinusoidal forms.

References

  1. S. D. Apte (2016). Fourier Transform Representation of Aperiodic Signals. In Signals and Systems: Principles and Applications (pp. 409–512). Cambridge University Press.
  2. Fourier’s theorem and the spectrum of complex tones. (2023). Retrieved from https://mtsu.edu/faculty/wroberts/teaching/fourier_4.php.
  3. Haslwanter, T. (2021). Hands-on Signal Analysis with Python. Springer.