Consider a $3$ x $3$ MIMO system. Let the channel matrix $H$ of this MIMO channel be:

$H=\left[\begin{array}{ccc}3& 0& 8\\ 0& 1& 0\\ 4& 0& 6\end{array}\right]$

Consider a transmit power of $P=-1\mathrm{.}75dB$ and receiver noise power ${}_n^2=3dB.$

$($a$)$ Derive the SVD of the channel matrix $H$ above $($hand analysis$)$

$($b$)$ Write a code that calculates the SVD of the channel matrix $H$

and computes the singular values

$($c$)$ Write a code for the water filling algorithm to find the

optimal transmit powers that maximize the MIMO channel

capacity.

$($d$)$ Compute the capacity of this MIMO channel in case of

$($i$)$ Multiplexing with and without channel knowledge.

$($ii$)$ Diversity with and without channel knowledge.

Note: Generalise the code so that it can work for any channel matrix $H$ which could have

different dimensions than the one provided and any given noise power and total transmit power. using MATLAB please.