Time-Frequency Analysis

Fourier Analysis · Axiom Academy

LESSON Joint Time-Frequency Representations Understanding how to analyze signals in both time and frequency simultaneously Instead of viewing a signal purely in time or purely in frequency, we can create a two-dimensional representation where: The horizontal axis represents time The vertical axis represents frequency The intensity or color at each point indicates the energy of that frequency at that time This creates a "map" showing how the frequency content of a signal evolves over time. 2. Why We Can't Have Perfect Resolution in Both Intuitively, we might want perfect resolution in both time and frequency: to know exactly what frequency occurs at exactly what instant. However, this is fundamentally impossible. Good time resolution requires analyzing a short window of the signal, but short windows cannot resolve low frequencies well Good frequency resolution requires analyzing a long window to distinguish close frequencies, but this averages over time and loses temporal precision Think of it like trying to measure the speed of a car: measuring over a short time gives precise timing but imprecise speed, while measuring over a long distance gives precise speed but loses moment-by-moment detail. 3. The Heisenberg Uncertainty Principle for Signals This trade-off is formalized by the uncertainty principle , borrowed from quantum mechanics but applicable to all signals and waves. t : the time uncertainty (duration of the analysis window)

This is the written version of the interactive lesson above. See the full Fourier Analysis course.