How to use MUltiple SIgnal Classification (MUSIC) algorithm to detect the wavenumber of a received signal at array of sensors?

Question

I am working on analyzing data obtained from Fiber Optic - Distributed Acoustic Sensor (DAS) to estimate the properties of acoustic waves that impinge FO cable, such as velocity of acoustic waves. FO-DAS provides real-time estimation of acoustic energy along the fiber (i.e., data over distance and time). Recently, I read about MUltiple SIgnal Classification (MUSIC) algorithm which can provide accurate estimation of the frequency of a received signal through detecting peaks in pseudospectrum. Through comparing MUSIC algorithm with other techniques such as Welch method, MUSIC algorithm provide more accurate estimation of the frequency as shown below.

I have the following two questions:

1. I used the following form of the steering vector to estimate the frequency of the signal (Eq. 6.53 in Mathematical Methods and Algorithms for Signal Processing). Suppose that, I have measurements of a received signal over FO cable length at specific time, how to modify the steering vector in MUSIC algorithm to estimate wavenumber of the received signal?

$s(f)=\left[1\ \ e^{j2\pi f}\ \ e^{j2\pi2f}\ \ \ \ \ \ \ \ ...\ \ \ \ \ e^{j2\pi\left(M-1\right)f}\right]^T$

2. Suppose that I have measurements obtained over the length of the cable at different time. How to modify the steering vector to estimate frequency and wavenumber of the received signal? I think, in this case, the steering vector should be 2D array of size (M by N) where M is the number of data samples (over time) and N is the number of distributed sensors (channels in case FO-DAS). I read some interesting questions and responses about how extend MUSIC algorithm for 2D (here and here), however they are relevant to estimating AOA rather than frequency and wavenumber. I started learning signal processing recently, so feel free to correct me if I am wrong.

Answer 1

A lossless cable only modifies the phase versus frequency given the Fourier relationship:

$$\mathscr{F}\{x(t-\tau)\} = e^{-j\omega \tau}X(\omega)$$

The dominant effect of a cable is time offset $(\tau)$ and the secondary effect is amplitude loss: the delay through a cable with a dielectric constant $\epsilon_r$ is $c/\sqrt{\epsilon_r}$ where $c$ is the speed of light, so in practical terms with Teflon dielectric where $\epsilon_r =2$, the delay is approximately 1.4 ns/foot or 4.7 ns/meter.

What we see here in plain words is that given any time domain function $x(t)$ that has a Fourier Transform $X(\omega)$, once delayed in time by a constant time offset $\tau$, the phase will be modified according to $e^{-j\omega \tau}$. It will help some to know that the general equation $Ke^{j\phi}$ with real magnitude $K$, and real phase $\phi$ is simply a phasor that has a magnitude of $K$ and phase $\phi$, equating this to geometric expression:$Ke^{j\phi} = K\angle{\phi}$. The Fourier Transform is generally a complex waveform with a magnitude and phase response versus frequency (the OP has only shown the magnitude response above). Thus if expressed with it's magnitude and phase components, $X(\omega)= K(\omega)e^{j\theta(\omega)}$, and we get:

$$ e^{-j\omega \tau}X(\omega) = e^{-j\omega \tau}K(\phi)e^{j\theta(\omega)} = K(\omega)e^{j(\theta(\omega) - \omega \tau)}$$

I don't believe that the result from MUSIC is sensitive to a change in phase for any given frequency (and the OP's plot shown is only displaying the magnitude), and assuming that is indeed true, no modification is necessary: We see from the math that the resulting magnitude before and after the delay, $|X(\omega)|$, and $|e^{-j\omega \tau}X(\omega)|$, is still $K(\omega)$ .

Answer 2

MUSIC algorithm can be used to estimate the wavenumbers of an incident signal through modifying the steering vector to be as follows:

$s(k)=\left[1\ \ e^{-jk}\ \ e^{-j2k}\ \ \ \ \ \ \ \ ...\ \ \ \ \ e^{-j\left(N-1\right)k}\right]^T$

where $N$ is the number of sensors or channels. I read the following paper which describes how to implement MUSIC algorithm to estimate frequency-wavenumber of multichannel data obtained from a seismic survey.

"Datta, Arjun. "On the application of the fk-MUSIC method to measurement of multimode surface wave phase velocity dispersion from receiver arrays." Journal of Seismology 23.2 (2019): 243-260."

The procedure is as follows:

1. Convert acquired data from spatiotemporal to spatiospectral domain. In this step, the data at each channel or sensor is transformed to frequency domain using Fast Fourier Transform.
2. For each frequency bin, autocorrelation matrix of data over channels is estimated. Then, MUSIC technique is used to estimate wavenumbers.

I have applied the technique described in the paper to synthetic example. The figure below shows the f-k spectrogram obtained using 2D FFT and MUSIC algorithm for the synthetic example.