Question

Amplitude modulation (AM, the same as AM radio) "modulates" a message signal m(t) – voice, speech,

music, etc. – by multiplying it by a relatively high-frequency sinusoidal wave: x() = m() cos(o).
(a) Let m(t) = 10 + 8 sin( − /3) and o = 13 rad/sec
Express x() in the form x() = 1 cos(1 + 1) + 2 cos( 2 + 2) + 3 cos( 3 + 3)
where 1 < 2 < 3, and find the , , and .

Answer 1

To express the AM signal x(t) as a sum of cosine functions, we use trigonometric identities to expand the product of m(t) and the carrier wave cos(ω0t). This results in three terms corresponding to the carrier frequency and upper and lower sidebands, giving us the amplitude A, frequency ω, and phase φ for each term in the form of Aicos(ωit + φi).

Step-by-step explanation:

To express the amplitude modulated signal x(t) with m(t) = 10 + 8 sin(ωt - π/3) and ω0 = 13 rad/sec in the form x(t) = A1cos(ω1t + φ1) + A2cos(ω2t + φ2) + A3cos(ω3t + φ3), we can use trigonometric identities to expand the product of m(t) and the carrier wave. Using the identity cos(A)sin(B) = 1/2[sin(A + B) + sin(A - B)], we can expand the modulation equation. Applying this:

• x(t) = (10 + 8 sin(ωt - π/3)) cos(ω0t) = 10 cos(ω0t) + 4 sin(ωt - π/3 + ω0t) - 4 sin(ω0t - ωt + π/3)
• This results in three terms where we can equate:

Therefore, the resulting AM radio waves consist of a carrier frequency and two sidebands.