A strange audio phenomenon, could there be a physical interpretation to it?

Question

https://mathoverflow.net/q/165038/14414

Motivation : Here is a motivation as to why this problem is so important.

Let $f(t)$ be an audio signal. We can safely asume it to be bandlimited to 0-20kHz as we cannot hear anything above that. Capture this signal in digital computer with appropriate sampling frequency and denote it as $f[n]$.

Now take Discrete Hilbert transform of $f[n]$ to get $f_h[n]$, (using the code $f_h$ = imag(hilbert(f)); in Matlab).

Compute the signal $f_{\theta}[n] = f[n]\cos\theta + f_h[n]\sin\theta$ for any value of $\theta$, then listen to the signal with different values for $\theta$.

They all sound exactly identical.

Similarly our $MI_{\omega_0,\omega_1}(t)$ is same for all $f_{\theta} = f\cos\theta + f_h\sin\theta$, for any value of $\theta$.

Question :

just try it. $<f,f_h> = 0$, they why do they produce same effect in the listner? Is it some quantum mechanical effect gone wrong?

Added :

Also see this metric space : metric space

I've recently filed a patent using this metric with a slight change, instead of arccos i used sqrt(2(1-cos(theta))), which makes it a Hilbertian metric. I had then embedded this metric space into an Hilbert space isometrically, to model using vectors.

MATLAB code :

[f,fs] = wavread('audio_file.wav');

fh = imag(hilbert(f));

theta = pi/4;

f_tht = fcos(theta) + fhsin(theta);

wavplay(f,fs);

wavplay(f_tht,fs);

Answer 1

To your Question:

1. There is no quantum mechanics involved. This is essentially a signal processing question, which is rooted in calculus.

2. Why does it sound the same?

The ear works essentially as a power spectrum analyzer, i.e. what you hear of a signal $f(t)$ is mainly determined by the powerspectrum $|{F(\omega)}|^2$, where ${F(\omega)}$ is the Fourier-transform of $f(t)$.

In your case: $F_{\theta}(\omega) = (\cos\theta) F(\omega) + (\sin\theta) F_h(\omega)$.

so that: $|F_{\theta}(\omega)|^2 = (\cos\theta)^2 |F(\omega)|^2 + (\sin\theta)^2 |F_h(\omega)|^2 + K$.

Where $K \propto F(\omega)^* F_h(\omega) + F(\omega) F_h(\omega)^*$, and $*$ denotes the complex conjugtate.

Using the relation between $F$ and $F_h$ given by the Hilbert-Transform we find that $K=0$ and $|F(\omega)|^2 = |F_h(\omega)|^2 $.

We conclude that

$|F_{\theta}(\omega)|^2 = ((\cos\theta)^2 + (\sin\theta)^2) |F(\omega)|^2 = |F(\omega)|^2 $.

(Where in the last step there is a trigonometric identity)

Summary: for all $\theta$ we find that $|F_{\theta}(\omega)|^2 = |F(\omega)|^2 $, so the ear (as a powerspectrum analyzer) hears the same.

Answer 2

You can express your signal as the series expansion:

$$f(t) = \sum_k a_k \cos(kt) + b_k \sin(kt)$$

The Hilbert transform is a linear operator, so:

$$f_h = \sum_k a_k H[\cos(kt)] + b_k + H[\sin(kt)] = \sum -a_k \sin(kt) + b_k \cos(kt)$$

So, $f_h$ has changed the phases f the different frequencies, but leaving the mangitudes unchanged. You hear the same thing because your ear is doing a Fourier Transform of the input (each frequency is detected by a different part of the cochlea), that remains invariant (modulo some phase) under Hilbert Transform.

What your transformation is doing is shifting each frequency by a a certain phase, but not affecting the magnitudes.