Q$1.[30$ marks$]$

Two signals, $m_1(t)$ and $m_2(t)($both band$-$limited to $10,000Hz)$ are to be transmitted simultaneously over a channel using a custom frequency division multiplexing scheme. The multiplexing system is shown as below,

If the spectrums of the two signals are given as below:

$($i$)$ Sketch the spectrum at point $($a$)$ and $($b$)$ of the receiver. $($Label all magnitudes and frequencies of the axes.$)$

$(10$ marks$)$

$($ii$)$ What must the bandwidth of the channel be required for the system?

$(5$ marks$)$

$($iii$)$ Design a transmitter to multiplex the signal $m_1(t)$ and $m_2(t)$ for the receiver. $($Show clearly all the details.$)$

$(15$ marks$)$