- #1

beyondlight

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## Homework Statement

I am designing a MIMO communication system, with input signal s, channel H and transform matrix T. The received signal is corrupted by noise.

## Homework Equations

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The received signal is r = Hs+n

And then it is transformed (compressed) by:

y = Tr

And then its estimate s_hat is computed:

s_hat = inv(TH)*y = inv(H)inv(T)THs + inv(H)inv(T)Tn

Set C = inv(H)inv(T)Tn

I want to find an optimum T based on the least squares solution:

D = norm(s-s_hat)^2

dD/dT = 0

## The Attempt at a Solution

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[itex]D= (s-s_{hat})^{H}(s-s_{hat})=0[/itex]

[itex]D = (s-H^{-1}T^{-1}{THs})^{H}(s-H^{-1}T^{-1}THs)[/itex]

[itex]D = ||s|| -s^{H}H^{-1}T^{-1}{THs}-s^{H}C-s^{H}H^{H}T^{H}(T^{-1})^{H}(H^{-1})^{H}s+(s^{H}H^{H}T^{H}(T^{-1})^{H}(H^{-1})^{H}s)(H^{-1}T^{-1}{THs})-sH^{H}T^{H}(T^{-1})^{H}(H^{-1})^{H}C-C^{H}s-C^{H}H^{-1}T^{-1}{THs}+C^{H}C[/itex]How do I find the derivative dD/dT? Suppose that I do find it, how then do I proceed to obtain T alone on one side of the equation?

I would also like to get some ideas on which book covers this kind of matrix algebra.