I have a question about audio analyzer, maybe unrelated to this class: The signal in real space is a function of intensity over time I(t), and I believe this image is taken at a perticular time t. At that time the signal should only be one value. Why is there a frequency spectra associated?

I think audio analyzers use a window around a point in time to determine an approximation of instantaneous frequency. As the window gets narrower, the frequency resolution decreases, but the time resolution increases. At the limit, when the window is narrowed to just a point in time, no frequencies will be resolvable. A more mathematical and detailed discussion can be found by looking at STFT.

Thanks a lot! This is real clear and helpful!