# Solve derivative of least squares matrix equation

1. Mar 10, 2016

### beyondlight

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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
3. The attempt at a solution

$D= (s-s_{hat})^{H}(s-s_{hat})=0$
$D = (s-H^{-1}T^{-1}{THs})^{H}(s-H^{-1}T^{-1}THs)$
$D = ||s|| -s^{H}H^{-1}T^{-1}{THs}-s^{H}C-s^{H}H^{H}T^{H}(T^{-1})^{H}(H^{-1})^{H}s+(s^{H}H^{H}T^{H}(T^{-1})^{H}(H^{-1})^{H}s)(H^{-1}T^{-1}{THs})-sH^{H}T^{H}(T^{-1})^{H}(H^{-1})^{H}C-C^{H}s-C^{H}H^{-1}T^{-1}{THs}+C^{H}C$

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.

2. Mar 10, 2016

### Ray Vickson

The matrix $T$ disappears from your expression for $D$ as you have written it:
$H^{-1} T^{-1} T H = H^{-1} H = I$ (the unit matrix), because $T^{-1}T = I$ and $H^{-1}H = I$. So, what you have written is, basically, $D = (s-s)^H (s-s)$, which is just 0 for all $H, T$.

3. Mar 10, 2016

### beyondlight

But since s_hat is corrupted by noise then this will not be exactly true?

4. Mar 10, 2016

### Ray Vickson

I am just going by what you wrote. Perhaps what you wrote is not appropriate.