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so in ESPRIT[1,2] we have some signal of the form:

$$y[n] = \sum_{k=1}^{K} a_i(k) e^{j\omega_k n} + N[n]$$

where N is noise, then we get the autocovariance matrix

$$R_{yy}=E[y[n]y[n]^*]$$

take the eigen-decomposition

$$\hat{R}_{yy} = U \Lambda U^*$$

and this is broken up into our signal subspace/noise subspace

$$U = [S \; N]$$

ESPRIT really loses me at the next step where it selects the first M and last M columns of the signal subspace:

$$S_1 = [I_{M-1} \; 0]S $$

$$S_2 = [0 \; I_{M-1}]S$$

then states: $$S_1 \phi = S_2 $$

not, sure what that statement means.

it then computes $$\phi = (S_2^* S_2)^{-1} S_2^* S_1$$

I'm having a hard time wrapping my head around how selection of the first and last M columns relate to the frequencies embedded within the signal calculated as the eigenvalues of $\phi$. The rotational invariance statement $S_1 \phi = S_2 $ is pretty non-intuitive.

[1] https://ieeexplore.ieee.org/document/32276

[2] http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=67D758D42882A6927A1947F292E3A27D?doi=10.1.1.13.1219&rep=rep1&type=pdf

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