Music, Time-Frequency Shifts, and Linear Independence
Chris Heil, Georgia Institute of Technology
Location: Stevenson 5211
Fourier series provide a way of writing almost any signal as a superposition of pure tones, or musical notes. Unfortunately, this representation is not local, and it does not reflect the way that music is actually generated by instruments playing individual notes at different times. We will discuss time-frequency representations, which are a type of local Fourier representation of signals. While such representations have limitations when it comes to music, they are powerful mathematical tools that appear widely throughout mathematics (e.g., partial differential equations and pseudodifferential operators), physics (e.g., quantum mechanics), and engineering (e.g., time-varying filtering). We ask one very basic question: are the notes in this representation linearly independent? This seemingly trivial question leads to surprising mathematical difficulties. This talk is intended to be introductory and accessible to beginning graduate students. Tea at 3:30 pm in SC 1425.