Consider a linear measurement process with some noise: y = H x + vwith v ∼ N ( 0 , Σ )the covariance matrix Σ is not a diagonal matrix. As we know, using LS, the x ^ is x ^ = ( H T H ) − 1 H T yDoes it change when Σ is not a diagonal matrix?

Question

Consider a linear measurement process with some noise:  y  =  H  x  +  vwith  v  ∼      N    (  0  ,  Σ  )the covariance matrix   Σ is not a diagonal matrix. As we know, using LS, the             x      ^       is            x      ^        =  (      H    T    H      )          −      1            H    T    yDoes it change when   Σ is not a diagonal matrix?

sag2y8s · Accepted Answer

Sorry, I eventually realize it is a simple question. Here is the answer of mine.first, for a semi-definite matrix   Σ, we can find   Λ so that  Σ  =  Λ      Λ    T  Then, we have      Λ          −      1        v  ∼      N    (  0  ,  I  )So, we can apply the conventional LS to modified measurement process:      Λ          −      1        y  =      Λ          −      1        H  x  +      Λ          −      1        vthen            x      ^        =  (  (      Λ          −      1        H      )    T        Λ          −      1        H      )          −      1        (      Λ          −      1        H      )    T        Λ          −      1        y            x      ^        =  (      H    T        Σ          −      1        H      )          −      1            H    T        Σ          −      1        y

Consider a linear measurement process with some noise: y = H x + v with

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