2
$\begingroup$

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

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.