give the full answer---[Q1: 40 Marks] 1. A carrier signal $v_c(t) = 10 \cos(2\pi \times 300t)$ is modulated by a single-tone message signal $m(t) = 2\sin(2\pi \times 50t)$ to form the AM signal $s(t) = (10 + 2\sin(2\pi \times 50t))\cos(2\pi \times 300t)$. a. Obtain an expression for the sidebands of $s(t)$. [15 Marks] b. Sketch $m(t)$, $v_c(t)$, and $s(t)$ and their corresponding spectra. [15 Marks] c. What is the modulation index? [5 Marks] d. What is the bandwidth required to transmit this AM signal? [5 Marks]

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We have an amplitude modulation problem with a carrier signal at 300 hertz and a message signal at 50 hertz. The AM signal shows the characteristic envelope modulation pattern. We need to analyze the sidebands, sketch the signals and spectra, find the modulation index, and determine the required bandwidth. To find the sideband expression, we expand the AM signal and separate the carrier from the modulation terms. The second term contains the sidebands. Using the trigonometric identity for sine times cosine, we get the upper sideband at 350 hertz and lower sideband at 250 hertz. The spectrum shows the carrier at 300 hertz with sidebands on either side. The time domain shows the message signal as a sine wave with amplitude 2, and the AM signal with its characteristic envelope varying between 8 and 12. In the frequency domain, the spectrum contains the carrier at plus and minus 300 hertz, and sidebands at 250 and 350 hertz. This creates the complete AM spectrum with carrier and upper and lower sidebands.