carrier wave E_{c} = 4sin(2pi * 500 * 10 ^ 3 * t) is amplitude-modulated by a message signal E infty =[0.2 sin 3(2 pi*500* t)+0.1 sin 5(2 pi*500* t)]. Determine the following:
D Upper and Lower sidebands
Sketch the complete spectrum of the modulated wave
overline 0 Total power in sidebands
To determine the upper and lower sidebands, we first need to find the modulated signal.
The modulated signal can be obtained by multiplying the carrier wave by the message signal:
E(t) = [1 + 0.2sin(3*2pi*500*t) + 0.1sin(5*2pi*500*t)] * 4sin(2pi*500*10^3*t)
Expanding this equation and simplifying, we get:
E(t) = 4sin(2pi*500*10^3*t) + 0.8sin(6pi*500*t) + 0.4sin(10pi*500*t) + 0.2sin(6pi*500*t) + 0.1sin(10pi*500*t)
The upper sideband frequencies are (500*10^3 + 3*500) Hz and (500*10^3 + 5*500) Hz.
The lower sideband frequencies are (500*10^3 - 3*500) Hz and (500*10^3 - 5*500) Hz.
Thus, the upper sidebands are at 500.15 kHz and 500.25 kHz, while the lower sidebands are at 499.85 kHz and 499.75 kHz.
To sketch the complete spectrum, we represent the carrier wave at 500 kHz with an amplitude of 4, the upper sidebands at 500.15 kHz and 500.25 kHz with amplitudes of 0.8 and 0.4, and the lower sidebands at 499.85 kHz and 499.75 kHz with amplitudes of 0.2 and 0.1 respectively.
The total power in the sidebands can be calculated by adding the power in each sideband.
Power in each sideband = (Amplitude of each sideband)^2 / 2
Total power in sidebands = (0.8^2 + 0.4^2 + 0.2^2 + 0.1^2) / 2 = 0.54
Therefore, the total power in the sidebands is 0.54.