Array processing: Difference between revisions

Revision as of 05:04, 12 November 2018

Revision as of 20:52, 14 November 2018

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The same equation can be also expressed in the form of vectors:<br>

<math>\textstyle \boldmathbf x(t) = A(\theta)s(t) + n(t)</math>

If we assume now that M snapshots are taken at time instants t1, t2 … tM, the data can be expressed as:<br>

<math>\boldmathbf X = \boldmathbf A(\theta)\boldmathbf S + \boldmathbf N</math>

Where X and N are the r × M matrices and S is q × M:<br>

<math>\boldmathbf X = [x(t_{1}), ......, x(t_{M})]</math><br>

<math>\boldmathbf N = [n(t_{1}), ......, n(t_{M})]</math><br>

<math>\boldmathbf S = [s(t_{1}), ......, s(t_{M})]</math>

'''Problem definition'''<br>

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''' ''Approach overview'' '''<br>

<math>\textstyle 1.\ Subspace\ decomposition\ by\ performing\ eigenvalue\ decomposition:</math><br>

<math>\textstyle R_{x}=\boldmathbf A \boldmathbf R_{s} \boldmathbf A^{*} + \sigma^{2}I=\sum_{k=1}^M \lambda_{k}e_{k}r_{k}^{*}</math><br>

<math>\textstyle 2.\ span\{\boldmathbf A\}=spane\{e1,....,e_{d}\}=span\{\boldmathbf E_{s}\}.</math><br>

<math>\textstyle 3.\ Check\ which\ a(\theta)\ \epsilon span\{\boldmathbf E_{s}\}\ or\ \boldmathbf P_{A}a(\theta)\ or\ P_{\boldmathbf A}^{\perp}a(\theta),\ where\ \boldmathbf P_{A}\ is\ a\ projection\ matrix.</math><br>

<math>\textstyle 4.\ Search\ for\ all\ possible\ \theta\ such\ that: \left | P_{\boldmathbf A}^{\perp}a(\theta) \right |^{2} = 0\ or\ M(\theta)=\frac{1}{P_{A}a(\theta)} =\infty</math><br>

<math>\textstyle 5.\ After\ EVD\ of\ R_{x}:</math><br>

<math>\textstyle P_{A}^{\perp}=I-E_{s}E_{s}^{*}=E_{n}E_{n}^{*}</math><br>