Task 8
In a MIMO system, a detection technique based on Minimum Mean Square Error (MMSE)
aims at identifying the input X from the output Y. Using such an MMSE detector, an estimate
of X, say $\bar{x}$, can be calculated by evaluating
$\frac{d\bar{x}(t)}{dt} = - (H^TH + \eta I) \bar{x}(t) + H^TY$
where $\eta$ is a constant parameter and I is the identity matrix. An initial condition of $\bar{x}(0) = H^TY$
can be assumed. An equilibrium point $x^*$ for the differential equation can be obtained as the
solution to the equation $\frac{d\bar{x}(t)}{dt} = 0$. Obtain $x^*$. Then, the residual error between $\bar{x}(t)$ and $x^*$
can be given as $e(t) := \bar{x}(t) - x^*$. Rewrite the differential equation above in terms of $e(t)$ in the
form $\frac{de(t)}{dt} = K e(t)$.
Task 9
Solve the resulting first order differential equation in Task 8 using a matrix exponential and
obtain an expression for $e(t)$.
