Consider a multiplex system in which four input sig$-$ nals m$($t$),$ m$($t$),$ m$3($t$),$ and m$(1)$ are respective$-$ ly multiplied by the carrier waves $[$cos$($omega$\_\{$a$\}*$t$)+$ cos$($omega$\_\{$b$\}*$t$)],[$cos$($omega$\_\{$a$\}*$t $+$ alpha$\_\{1\})+$ cos$($omega$\_\{$b$\}*$t $+$ beta$\_\{1\})][$cos$($omega$\_\{$a$\}*$t $+$ alpha$\_\{2\})+$ cos$($omega$\_\{$b$\}*$t $+$ beta$\_\{2\})][$cos$($omega$\_\{$a$\}*$t $+$ alpha$\_\{3\})+$ cos$($omega$\_\{$b$\}*$t $+$ beta$\_\{3\})]$ and and the resulting DSB$-$SC signals are summed and then transmitted over a common channel. In the re$-$ ceiver, demodulation is achieved by multiplying the sum of the DSB$-$SC signals by the four carrier waves separately and then using filtering to remove the un$-$ wanted components. Determine the conditions that the phase angles a$1,$ a$2,$ a$3$ and B$1,$ B$2,$ B$3$ must sat$-$ isfy in order that the output of the kth demodulator be m$($t$),$ where k $=1,2,3,4$