If you like playing music or listening to music, you would probably know how different instruments playing their rthyum. You can hear how a jazz drum was druming |B S BB S|B S BB S| … (B- Bass drum, S– Snare drum). You may hear how the guitar (fill with chords) and the bass guitar ( making backup rthyum) playing their note. You may hear wonderful keyboard filling in their note. All are done instantaneously, and experienced musicians can even write them immediately.
Can we work it out by a computer? Can we plot an “instantaneous” spectrum of frequency. Let’s think mathematically. Let , its fourier-Plancherel transform defined by
We think as a incoming wave function as a musical signal, its fourier transform as the overall frequency spectrum. Our problem is asking whether at a particular time, we know the frequency spectrum or not. This turns out the unertainty principle makes all these instantaeous frequency impossible. Heuristically,
a function and its fourier transform cannot be supported on arbitrarily small sets together.
There are many variant forms.
Theorem 1. (Naive uncertainty principle) If is of compact support on , then cannot be of compact support.
Proof. If so, then we assume the support in on , then develop its fourier series on , by assumption, the series must be a trigonmetric polynomial. This is impossible, since the trigonmetric polynomial cannot vanish on a set of posifive measure. Q.E.D.
We work on more specfically, for two linear operators (may not be bounded) on a Hilbert space, we define the commutator
Theorem 2. (Hesienberg uncertainty principle) Let be selfadjoint operators on the a Hilbert space. Then for all
In particular on ,
Proof. (Sketch due to limited space) By selfadjointness
We then apply Cauchy-Schwarz inequality.
For the second one, we consider
defined on the Schwartz space of rapidly decreasing functions. Then apply the abstract inequality. Q.E.D.
More to come, we define a function is –concentrated on a measurable set if
Theorem 3. (Donoho and Stark uncertainty principle) Let , , is concentrated on , and is -concntrated. Then
Proof. Consider the Landau projection operators,
Then we have by concentratedness,
We have by triangle inequality,
But is a Hilbert-Schmidt operator, with (by direct computation), hence the result follows from this. Q.E.D.
We may say do a short time windowed fourier transform. Let be a bounded function of compact support, we define
Can we define some intstantaneous frequency conecpt, the answer is no. We state the following theorem without proof.
Theorem 4.(Liebs uncertainty principle) Suppose that . If and are such that
Uncertainty principle also appears in quantum mechanics where is the position and is the momentum.
Back to our time frequency analysis, we must overcome uncertainty principle in order to make precise analysis of musical signal. However, our ear is competent to it. Why? May be our brain never interpret music as fourier transform, it never does fourier transform, but understand the music in a mixture of experience, feeling and understanding ^^ We must appreciate our brain being far better than the computer.