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How to decide the most possible signal model between two model candidates besed on the received signal vector?

Assume the received signal vector is $y$, the possible signal model candidates could be:

(1) $y = Ax+n$, or

(2) $y = Bx+n$,

in which $x$ is the transmitted signal vector, and $A$ and $B$ are the system matrices for signal model candidate-1 and candidate-2 respectively, and $n$ is the Gaussian noise vector.

If $y$,$A$ and $B$ are all known, and the noise covariance matrix is $E[nn^H] = w^2I$, in which $w^2$ stands for noise power, and $I$ is the identity matrix, how to decide the most possible signal model candidate between the two. What's the optimal solution?

Thanks for any discussions.

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A good reference book could be

Hero, A. “Signal Detection and Classification” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999

in which different scenarios were discussed. Besides, some reference books are also helpful.

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