A high purity sinusoidal signal $($tone$)$ is defined as having a frequency of $6$ MHz$.$ Assuming that it has zero phase offset and an amplitude of $3$ volts: $($i$)$ Write a mathematical formula to represent this sinusoidal signal $($tone$).($ii$)$ Sketch the time domain diagram for this tone, labelling all significant parts. $($iii$)$ Sketch the frequency domain diagram for this tone.

Q $1($b$)[13$ Marks$]$ For the same sinusoidal signal with same frequency as in part $($a$)$ above, modulate it with a carrier signal such that the size of the transmitter antenna can be reduced to $0\mathrm{.}1$ meters. $($i$)$ Calculate the carrier signal frequency? $($ii$)$ Sketch the time domain plots of the modulating, carrier and modulated signals. $($iii$)$ Write the mathematical formula of the modulated signal and calculate the modulation index? Q$1($b$)$