{"id":361,"date":"2019-06-25T13:55:57","date_gmt":"2019-06-25T13:55:57","guid":{"rendered":"https:\/\/florian-roemer.de\/blog\/?p=361"},"modified":"2019-06-25T13:56:52","modified_gmt":"2019-06-25T13:56:52","slug":"erratum-for-deterministic-cramer-rao-bound-for-strictly-non-circular-sources-and-analytical-analysis-of-the-achievable-gains","status":"publish","type":"post","link":"https:\/\/florian-roemer.de\/blog\/erratum-for-deterministic-cramer-rao-bound-for-strictly-non-circular-sources-and-analytical-analysis-of-the-achievable-gains\/","title":{"rendered":"Erratum for “Deterministic Cramer-Rao Bound for Strictly Non-Circular Sources and Analytical Analysis of the Achievable Gains”"},"content":{"rendered":"\n

I am posting since since we were recently made aware of an error in our journal paper “Deterministic Cramer-Rao Bound for Strictly Non-Circular Sources and Analytical Analysis of the Achievable Gains” (T-SP vol. 64, no. 17)<\/a>. Unfortunately, T-SP does not allow (yet?) to publish errata alongside the original papers, nor to send updates. Therefore, right now the best we can do is to update the arxiv version<\/a> and inform the community via this blog post.<\/p>\n\n\n\n

In fact, the error is more or less a copy-paste error from the earlier conference version “Deterministic Cram\u00e9r-Rao Bounds for strict sense non-circular sources” (WSA 2007)<\/a> that contains the same error, although harder to spot. <\/p>\n\n\n\n

As the title says, both papers are concerned with Deterministic Cram\u00e9r-Rao Bounds (CRBs) for strictly non-circular sources. A closed-form expression of the CRB was derived and given by the expression (8) in WSA2007<\/a>, which reads as <\/p>\n\n\n\n

$$\\begin{align}
\\newcommand{ma}[1]{{\\mathbf {#1}}}
\\ma{C} = & \\frac{\\sigma^2}{2N} \\Big\\{ \\left(\\ma{G}_2-\\ma{G}_1 \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\ma{\\hat{R}}_{S,0}
\\\\ & + \\left[ \\left( \\ma{G}_1 \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right] \\left[ \\left( \\ma{G}_0-\\ma{H}_0^T \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1}
\\\\ & \\cdot \\left[ \\left(\\ma{H}_1^T- \\ma{H}_0^T \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]
\\\\ & + \\left[ \\ma{H}_1 \\odot \\ma{\\hat{R}}_{S,0} \\right] \\cdot \\left[ \\ma{G}_0 \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1} \\cdot \\left[ \\left( \\ma{H}_0^T \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]
\\\\ & + \\left[ \\ma{H}_1 \\odot \\ma{\\hat{R}}_{S,0} \\right] \\cdot \\left[ \\ma{G}_0 \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1}
\\cdot \\left[ \\left( \\ma{H}_0^T \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]
\\\\& \\cdot \\left[ \\left( \\ma{G}_0-\\ma{H}_0^T \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1}
\\cdot \\left[ \\left( \\ma{H}_0^T \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]
\\\\ &-\\left[ \\ma{H}_1 \\odot \\ma{\\hat{R}}_{S,0} \\right] \\cdot \\left[ \\left( \\ma{G}_0-\\ma{H}_0^T \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1}
\\cdot \\left[ \\ma{H}_1^T \\odot \\ma{\\hat{R}}_{S,0} \\right] \\Big\\}^{-1}, \\end{align}$$<\/p>\n\n\n\n

where the matrices $\\ma{G}_i, \\ma{H}_i$ for $i=0, 1, 2$ are all of size $d \\times d$ and given by<\/p>\n\n\n\n

$$\\begin{eqnarray}
\\ma{G}_0 & = & {\\rm Re}\\{\\ma{\\Psi}^* \\cdot \\ma{A}^H \\cdot \\ma{A} \\cdot \\ma{\\Psi}\\} \\\\
\\ma{H}_0 & = & {\\rm Im}\\{\\ma{\\Psi}^* \\cdot \\ma{A}^H \\cdot \\ma{A} \\cdot \\ma{\\Psi}\\} \\\\
\\ma{G}_1 & = & {\\rm Re}\\{\\ma{\\Psi}^* \\cdot \\ma{D}^H \\cdot \\ma{A} \\cdot \\ma{\\Psi}\\} \\\\
\\ma{H}_1 & = & {\\rm Im}\\{\\ma{\\Psi}^* \\cdot \\ma{D}^H \\cdot \\ma{A} \\cdot \\ma{\\Psi}\\} \\\\
\\ma{G}_2 & = & {\\rm Re}\\{\\ma{\\Psi}^* \\cdot \\ma{D}^H \\cdot \\ma{D} \\cdot \\ma{\\Psi}\\} \\\\
\\ma{H}_2 & = & {\\rm Im}\\{\\ma{\\Psi}^* \\cdot \\ma{D}^H \\cdot \\ma{D} \\cdot \\ma{\\Psi}\\}.
\\end{eqnarray}$$<\/p>\n\n\n\n

All nice and good, no problem so far. It was then though said in Section 4 how to generalize this to 2-D, where it was claimed that all we need to do is to replace $\\ma{D} \\in \\mathbb{C}^{M \\times d}$ by $\\ma{D}_{{\\rm 2D}} \\in \\mathbb{C}^{M \\times 2d}$ and $\\ma{\\hat{R}}_{S,0}$ by $\\ma{1}_{2 \\times 2} \\otimes \\ma{\\hat{R}}_{S,0}$. Well, the first statement is correct, the second one only partially. Unfortunately this went unnoticed into the TSP2016<\/a> paper, where the expression is given for $R$-D. <\/p>\n\n\n\n

Why is it wrong? Well, you can see that for $R$-D, the size of $\\ma{A}$ is unaffected while $\\ma{D}$ goes from having $d$ columns to having $R\\cdot d$ columns. Therefore, the size of $\\ma{G}_0$ and $\\ma{H}_0$ is unaffected ($d\\times d$) whereas $\\ma{G}_1$ and $\\ma{H}_1$ are now $R \\cdot d \\times d$ and $\\ma{G}_2$ and $\\ma{H}_2$ are now $R\\cdot d \\times R\\cdot d$. To make the CRB work, the augmentation of of $ \\ma{\\hat{R}}_{S,0} $ has to be done such that it is consistent with the dimensions of the $\\ma{G}_i$ and $\\ma{H}_i$. <\/p>\n\n\n\n

Concretely, this means that $ \\left(\\ma{G}_2-\\ma{G}_1 \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\ma{\\hat{R}}_{S,0} $ changes into $ \\left(\\ma{G}_2-\\ma{G}_1 \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\left(\\ma{1}_{R\\times R} \\otimes \\ma{\\hat{R}}_{S,0}\\right) $, which is the example treated in the paper. However, $ \\left( \\ma{G}_1 \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} $ becomes $ \\left( \\ma{G}_1 \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\left(\\ma{1}_{R \\times {\\color{red}1}} \\otimes \\ma{\\hat{R}}_{S,0} \\right)$ and $ \\left( \\ma{G}_0-\\ma{H}_0^T \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} $ remains unaffected.<\/p>\n\n\n\n

Long story short, here is the corrected version of the R-D CRB (equation (15) in TSP2016<\/a>):<\/p>\n\n\n\n

$$\\begin{align}
\\newcommand{ma}[1]{{\\mathbf {#1}}}
\\ma{C} = & \\frac{\\sigma^2}{2N} \\Big\\{ \\left(\\ma{G}_2-\\ma{G}_1 \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\left(\\ma{1}_{R\\times R} \\otimes \\ma{\\hat{R}}_{S,0} \\right)
\\\\ & + \\left[ \\left( \\ma{G}_1 \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\left(\\ma{1}_{R\\times 1} \\otimes \\ma{\\hat{R}}_{S,0} \\right)\\right] \\left[ \\left( \\ma{G}_0-\\ma{H}_0^T \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1}
\\\\ & \\cdot \\left[ \\left(\\ma{H}_1^T- \\ma{H}_0^T \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\left(\\ma{1}_{1\\times R} \\otimes \\ma{\\hat{R}}_{S,0}\\right) \\right]
\\\\ & + \\left[ \\ma{H}_1 \\odot \\left(\\ma{1}_{R\\times 1} \\otimes \\ma{\\hat{R}}_{S,0}\\right) \\right] \\cdot \\left[ \\ma{G}_0 \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1} \\cdot \\left[ \\left( \\ma{H}_0^T \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\left(\\ma{1}_{1\\times R} \\otimes \\ma{\\hat{R}}_{S,0} \\right) \\right]
\\\\ & + \\left[ \\ma{H}_1 \\odot \\left(\\ma{1}_{R\\times 1} \\otimes \\ma{\\hat{R}}_{S,0}\\right) \\right] \\cdot \\left[ \\ma{G}_0 \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1}
\\cdot \\left[ \\left( \\ma{H}_0^T \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]
\\\\& \\cdot \\left[ \\left( \\ma{G}_0-\\ma{H}_0^T \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1}
\\cdot \\left[ \\left( \\ma{H}_0^T \\ma{G}_0^{-1} \\ma{G}_1^T \\right) \\odot \\left(\\ma{1}_{1\\times R} \\otimes \\ma{\\hat{R}}_{S,0} \\right) \\right]
\\\\ &-\\left[ \\ma{H}_1 \\odot \\left(\\ma{1}_{R\\times 1} \\otimes \\ma{\\hat{R}}_{S,0}\\right) \\right] \\cdot \\left[ \\left( \\ma{G}_0-\\ma{H}_0^T \\ma{G}_0^{-1} \\ma{H}_0 \\right) \\odot \\ma{\\hat{R}}_{S,0} \\right]^{-1}
\\cdot \\left[ \\ma{H}_1^T \\odot \\left(\\ma{1}_{1\\times R} \\otimes \\ma{\\hat{R}}_{S,0}\\right) \\right] \\Big\\}^{-1}. \\end{align}$$ <\/p>\n\n\n\n

The pattern is clear: the “outer” terms get expanded, while the “inner” terms remain unaffected.<\/p>\n\n\n\n

We would like to thank Mr. Tanveer Ahmed for noticing the mistake! We sure hope we got it right this time! \ud83d\ude42 <\/p>\n","protected":false},"excerpt":{"rendered":"

I am posting since since we were recently made aware of an error in our journal paper “Deterministic Cramer-Rao Bound for Strictly Non-Circular Sources and Analytical Analysis of the Achievable Gains” (T-SP vol. 64, no. 17). Unfortunately, T-SP does not allow (yet?) to publish errata alongside the original papers, nor to send updates. Therefore, right […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[11],"tags":[12,13],"_links":{"self":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts\/361"}],"collection":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/comments?post=361"}],"version-history":[{"count":64,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts\/361\/revisions"}],"predecessor-version":[{"id":425,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts\/361\/revisions\/425"}],"wp:attachment":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/media?parent=361"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/categories?post=361"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/tags?post=361"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}