Fourier Transform, Time–Frequency Duality, and Its Limitations
Fourier analysis provides a powerful way to represent signals via their frequency components, but it sacrifices all information about when particular frequencies occur—a serious limitation for many real-world signals.
Wavelet Functions as Time-Localized Little Waves
Wavelet transforms analyze signals using “little waves”—functions that oscillate like sinusoids but are localized in time, addressing Fourier’s inability to track when frequencies occur.
Continuous Wavelet Transform as a Time–Frequency Microscope
The continuous wavelet transform (CWT) uses scaled and shifted copies of a mother wavelet to produce a 2D representation of a 1D signal, capturing how different frequencies are expressed over time.
Convolution, Dot Product Geometry, and Complex Wavelets
The video reframes wavelet convolution using geometric intuition about dot products and extends real-valued wavelets to complex Morlet wavelets to cleanly extract time–frequency power.
Time–Frequency Uncertainty and Heisenberg Boxes in Wavelet Analysis
Wavelet analysis is sometimes misunderstood as “beating” the time–frequency uncertainty principle, but the video clarifies that it instead redistributes uncertainty in a smart, scale-dependent way illustrated by Heisenberg boxes.