MUSIC algorithm terminology

Question

Looking for clarification on the notation described in the MUSIC algorithm in Ralph Schmidt's IEEE paper$^{[1]}$. The data model is:

$$X = AF + W$$

Schmidt defines the following:

X = received signal for each array element
A = steering vector
F = received signal
W = noise

M = number of array elements
D = number linear combination of incident wavefronts

X is dimension M x 1
A is dimension M x D
F is dimension D x 1
W is dimension M x 1

Questions

1. Where is the number of receivers accounted for? A practical system may have a $360^\circ$ element antenna, but only $K$ receivers, where $K \ll M$ (e.g.,$K=8$)
2. Where is the length of the data incorporated? Assuming $N$ samples for the received signal vector. Intuition leads me to believe that $F$ is of dimension $D\times N$
3. Does $F$ represent the received signal? If so, what does $X$ represent? Received signal at each element in the array?

$[1]\ $ R. Schmidt, "Multiple emitter location and signal parameter estimation," in IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp. 276-280, March 1986

Answer 1

Yeah, notation is not ideal.

1. It is not - he assumes that each of the $M$ antenna elements is connected to its own RF chain, i.e., there are also $M$ receivers available. If you have fewer receivers you need to modify your $A$, it needs to contain the response your $K$ receivers observe given a wave from a certain direction.

2. He doesn't put it but yeah, $F$ should have $N$ columns (and so should $X$ and $W$) that correspond to $N$ subsequent snapshots (i.e., samples in time) of the received RF data.

3. The received signal, i.e., the (basedband version of the) RF data at the ouput of the $M$ antennas observed in $N$ subsequent snapshots is the $M \times N$ matrix $X$. The matrix $F$ comprises only the waveforms itself, you could think of it as the signal "before" the antenna array. But that intuition is also slightly misleading. Point is that each antenna is observing a superposition of $D$ wavefronts corresponding to $D$ far-field sources. This can be written as $A F$.

Note that this data model is a gross oversimplification of the actual physics. It ignores near-field sources/reflections, multipath, diffuse components and many things more. It still works well enough in some applications. But the physical interpretation we can put into it has its limits.